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3D LEVEL−I 1 The locus of the point, which moves such that its distance from (1, −2, 2) is unity, is (A) x2 + y2 + z2 − 2x + 4y − 4z + 8 = 0 (B) x2 + y2 + z2 − 2x − 4y − 4z + 8 = 0 (C) x2 + y2 + z2 + 2x + 4y − 4z + 8 = 0 (D) x2 + y2 + z2 − 2x + 4y + 4z + 8 = 0 *2 The angle between the lines whose direction ratios are 1, 1, 2; 3 − 1, − 3 − 1, 4 is
(A) cos−11
65 (B)
6
(C) 3 (D)
4
*3. The plane passing through the point (a, b, c) and parallel to the plane x + y + z = 0 is (A) x + y + z = a + b + c (B) x + y + z + (a + b + c) = 0 (C) x + y + z + abc = 0 (D) ax + by + cz = 0
4. The equation of line through the point (1, 2, 3) parallel to line x 4 y 1 z 102 3 8
are
(A) x 1 y 2 z 32 3 8
(B) x 1 y 2 z 31 2 3
(C) x 4 y 1 z 101 2 3
(D) none of these
5. The value of k, so that the lines x 1 y 2 z 33 2k 2
, x 1 y 5 z 63k 1 5
are perpendicular
to each other, is
(A) 107
(B) 87
(C) 67
(D) 1
*6. The angle between a line with direction ratios 2:2:1 and a line joining (3,1,4,) to (7,2,12)
(A) cos–123
(B) cos–1
32
(C) tan–123
(D) none of these
7. The equation of a plane which passes through (2, 3, 1) and is normal to the line joining the
points (3, 4, 1) and (2, 1, 5) is given by (A) x + 5y 6z + 19 = 0 (B) x 5y + 6z –19 = 0 (C) x + 5y + 6z +19 = 0 (D) x 5y 6z 19 = 0 8. Direction cosines of the line joining the points (0, 0, 0) and (a, a, a) are
(A) 1 1 1, ,2 2 2
(B) 1, 1, 1
(C) 1 1 1, ,3 3 3
(D) none of these
*9. The length of perpendicular from the point (–1, 2, –2)) on the line x 1 y 2 z 22 3 4
is
(A) 29 (B) 6 (C) 21 (D) none of these 10. Two lines not lying in the same plane are called (A) parallel (B) coincident (C) intersecting (D) skew 11. The distance of the point (x, y, z) from the x y plane is (A) x (B) y (C) 3 (D) z 12. A point (x, y, z) moves parallel to x axis. Which of three variables x, y, z remains fixed? (A) x and y (B) y and z (C) z and x (D) None of these *13. Let P (2, 3, 5), Q (1, 2, 3), R (7, 0, 1) then Q divides PR. (A) externally in the ratio 1 : 2 (B) internally in the ratio 1 : 2 (C) externally in the ratio 3 : 5 (D) internally in the ratio 1: 3 14. The xy plane divides the line segment joining (1, 2, 3) and (3, 4, 5) internally in the ratio (A) 3 : 5 (B) 3 : 4 (C) 4 : 3 (D) None of these 15. The direction cosines of the joining (1, 1, 1) and (1, 1, 1) are
(A) 1 1, ,02 2
(B) 2, 2,0
(C) 1 1, ,02 2
(D) 2, 2,0
16. Two lines with direction cosines 1 1 1 2 2 2l ,m ,n and l ,m ,n are at right angles iff (A) l1 l2 + m1 m2 + n1 n2 = 0 (B) l1 = l2, m1 = m2, n1 = n2 (C) l1 l2 = m1 m2 = n1 n2 (D) None of these 17. The foot of perpendicular from , , on x axis is (A) (, 0, 0) (B) (0, , 0) (C) (0, 0, ) (D) (0, 0, 0) 18. The direction cosines of a line equally inclined to the positive direction of axes are
(A) < 1, 1, 1> (B) 1 1 1, ,3 3 3
(C) 1 1 1, ,2 2 2
(D) None of these
19. A plane meets the coordinate axes at P, Q and R such that the centroid of the triangle is (1, 1, 1). The equation of plane is, (A) x + y + z = 3 (B) x + y + z = 9 (C) x + y + z = 1 (D) x + y + z = 1/3 *20. A plane meets the axes in P, Q and R such that centroid of the triangle PQR is (1, 2, 3). The
equation of the plane is
(A) 6x + 3y + 2z = 6 (B) 6x +3 y + 2z = 12 (C) 6x + 3y + 2z = 1 (D) 6x + 3y + 2z = 18 21. The direction cosines of a normal to the plane 2x 3y 6z + 14 = 0 are
(A) 2 3 6, ,7 7 7
(B) 2 3 6, ,7 7 7
(C) 2 3 6, ,7 7 7
(D) None of these
*22. The equation of the plane whose intercept on the axes are thrice as long as those made by
the plane 2x 3y + 6z 11 = 0 is (A) 6x 9y + 18z 11 = 0 (B) 2x 3y + 6z + 33 = 0 (C) 2x 3y + 6z = 33 (D) None of these 23. The angle between the planes 2x y + z = 6 and x + y + 2z = 7 is (A) /4 (B) /6 (C) /3 (D) /2 *24. The angle between the lines x = 1, y = 2 and y + 1 = 0 and z = 0 is (A) 00 (B) /4 (C) /3 (D) /2 LEVEL−II 1. The three lines drawn from O with direction ratios [1, −1, k], [2, −3, 0] and [1, 0, 3] are
coplanar. Then k = (A) 1 (B) 0 (C) no such k exists (D) none of these 2. A plane meets the coordinates axes at A, B, C such that the centroid of the triangle is (3, 3, 3). The equation of the plane is (A) x + y + z = 3 (B) x + y + z = 9 (C) 3x + 3y + 3z = 1 (D) 9x + 9y + 9z = 1 3. The equation of the plane through the intersection of the planes x − 2y + 3z − 4 = 0,
2x − 3y + 4z − 5 = 0 and perpendicular to the plane x + y + z − 1 = 0 is (A) x − y + 2 = 0 (B) x − z + 2 = 0 (C) y − z + 2 = 0 (D) z − x + 2 = 0
4. The coordinates of the point of intersection of the line x 1 y 3 z 21 3 2
with the plane
3x + 4y + 5z = 5 are (A) (5, 15, −14) (B) (3, 4, 5) (C) (1, 3, −2) (D) (3, 12, −10)
5. The angle between the line x 1 y 1 z 23 2 4
and the plane 2x + y − 3z + 4 = 0 is
(A) cos−14
406 (B) sin−1
4406
(C) 30 (D) none of these
*6. The angle between the lines whose direction cosines satisfy the equations l + m + n = 0, l2 + m2 n2 = 0 is given by
(A) 23 (B)
6
(C) 56 (D)
3
*7. The angle between the line x 2 y 1 z 32 1 2
and the plane 3x + 6y – 2z + 5 = 0 is
(A) cos–1421
(B) sin–1
421
(C) sin–1621
(D) sin–1 4
21
*8. Shortest distance between lines x 6 y 2 z 21 2 2
and x 4 y z 13 2 2
is
(A) 108 (B) 9 (C) 27 (D) None of these 9. The acute angle between the plane 5x – 4y + 7z – 13 = 0 and the y–axis is given by
(A) sin–1590
(B) sin–1
490
(C) sin–1790
(D) sin–1
490
10. The planes x + y – z = 0, y + z – x = 0, z + x – y = 0 meet (A) in a line (B) taken two at a time in parallel lines (C) in a unique point (D) none of these 11. The graph of the equation x2 + y2 = 0 in the three dimensional space is (A) z axis (B) (0, 0) point (C) y z plane (D) x y plane 12. A line making angles 450 and 600 with the positive directions of the x axis and y axis
respectively, makes with the positive direction of z axis an angle of (A) 600 (B) 1200 (C) both (A) and (B) (D) Neither (A) nor (B) 13. The angle between two diagonals of a cube is
(A) 1 1cos2
(B) 1 1cos3
(C) 1 1cos3
(D) 1 3cos2
14. If a line makes angles , , with the axes, then cos2 + cos2 + cos2 = (A) 1 (B) 1 (C) 2 (D) 2
15. The equation (x 1) . (x 2) = 0 in three dimensional space is represented by (A) a pair of straight line (B) a pair of parallel planes (C) a pair of intersecting planes (D) a sphere *16. The equation of the plane containing the line 2x + z 4 = 0 and 2y + z = 0 and passing
through the point (2, 1, 1) is (A) x + y z = 4 (B) x y z = 2 (C) x + y + z + 2 = 0 (D) x + y + z = 2 *17. The locus of xy + yz = 0 is, in 3 D ; (A) a pair of straight lines (B) a pair of parallel lines (C) a pair of parallel planes (D) a pair of intersecting planes
18. The lines 6x = 3y = 2z and x 1 y 2 z 32 4 6
are
(A) parallel (B) skew (D) intersecting (D) coincident
*19. The line 1 1 1x x y y z z0 1 2
is
(A) parallel to x axis (B) perpendicular to x axis (C) perpendicular to YOZ plane (D) None of these
20. For the line y 1x 1 z 3l :3 2 1
and plane P : x 2y z = 0 ; of the following assertions,
the one/s which is/are true : (A) l lies on P (B) l is parallel to P (C) l is perpendicular to P (D) None of these
21. The coordinates of the point of intersection of the line x 6 y 1 z 31 0 4
and the plane
x y z 3 are (A) (2, 1, 0) (B) (7, 1, 7) (C) (1, 2, 6) (D) (5, 1, 1)
*22. The Cartesian equation of the plane perpendicular to the line, x 1 y 3 z 42 1 2
and
passing through the origin is (A) 2x y + 2z 7 = 0 (B) 2x + y + 2z = 0 (C) 2x y + 2z = 0 (D) 2x y z = 0
Level – III *1. The length of projection of the segment joining (x1 , y1 , z1 ) and (x2 , y2 , z2 ) on the line
x y zl m n
is
(A) 2 1 2 1 2 1l x x m y y n z z (B) 2 1 2 1 2 1x x y y z z
(C) 2 1 2 1 2 1x x y y z zl m n
(D) None of these
2. The shortest distance between the lines x 1 y 2 z 3 x 2 y 3 z 5and2 3 4 3 4 5
is
(A) 16
(B) 16
(C) 13
(D) 13
3. The equation of the plane through the point (1, 2, 0) and parallel to the lines
x y 1 z 2 x 1 2y 1 z 1and3 0 1 1 2 1
is
(A) 2x + 3y + 6z 4 = 0 (B) x 2y + 3z + 5 = 0 (C) x + y 3z+ 1 = 0 (D) x + y + 3z = 1
*4. The distance of the plane through (1, 1, 1) and perpendicular to the line x 1 y 1 z 13 0 4
from the origin is
(A) 34
(B) 43
(C) 75
(D) 1
*5. The reflection of the point (2, 1, 3) in the plane 3x 2y z = 9 is
(A) 26 15 17, ,7 7 7
(B) 26 15 17, ,7 7 7
(C) 15 26 17, ,7 7 7
(D) 26 17 15, ,7 7 7
6. The coordinates of the foot of perpendicular from the point A (1, 1, 1) on the line joining the points B (1, 4, 6) and C (5, 4, 4) are
(A) (3, 4, 5) (B) (4, 5, 3) (C) (3, 4, 5) (D) (3, 4, 5) 7. The equation of the right bisecting plane of the segment joining the points (a, a, a) and (a, –a, a) ; a 0 is (A) x + y + z = a (B) x + y + z = 3a (C) x + y + z = 0 (D) x + y + z + a = 0 8. The angle between the plane 3x + 4y = 0 and the line x2 + y2 = 0 is (A) 00 (B) 300
(C) 600 (D) 900 9. If the points (0, 1, 2) ; (3, 4, 5) ; (6, 7, 8) and (x, x, x) are noncoplanar then x = (A) any real number (B) 1 (C) 1 (D) 0 *10. The equation of the plane through intersection of planes x + 2y + 3z = 4 and 2x + y z = 5
and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is (A) 7x 2y + 3z + 81 (B) 23y + 14x 9z + 48 = 0 (C) 23x + 14y 9z + 48 = 0 (D) 51x + 15y 50z + 173 = 0 11. The equation of the plane passing through the intersection of planes x + 2y + 3z + 4 = 0 and
4x + 3y + 2z + 1 = 0 and the origin is (A) 3x + 2y + z + 1 = 0 (B) 3x + 2y + z = 0 (C) 2x + 3y + z = 0 (D) x + y + z = 0 12. If the plane x + y z = 4 is rotated through 900 about the line of intersection with the plane x + y + 2z = 4 then equation of the plane in its new position is (A) 5x + y + 4z + 20 = 0 (B) 5x + y + 4z = 20 (C) x + 5y + 4z = 20 (D) None of these 13. The equation of the plane passing through the line of intersection of the planes 4x 5y 4z = 1 and 2x + y + 2z = 8 and the point (2, 1, 3) is (A) 32x 5y + 8z = 83 (B) 32x + 5y 8z = 83 (C) 32x 5y + 8z + 83 = 0 (D) None of these 14. The equation of the plane passing through the points (2, 1, 2) and (1, 3, 2) and parallel to x axis is (A) x + 2y = 4 (B) 2y + x + z = 4 (C) x + y + z = 4 (D) 2y + z = 4 15. The equation of the plane passing through the point (3, 3, 1) and is normal to the line
joining the points (2, 6, 1) and (1, 3, 0) is (A) x + 3y + z + 11 = 0 (B) x + y + 3z + 11 = 0 (C) 3x + y + z = 11 (D) None of these *16. If a point moves so that the sum of the squares of its distances from the six faces of a cube
having length of each edge 2 units is 46 units, then the distance of the point from (1,1, 1) is (A) a variable . (B) a constant equal to 7 units. (C) a constant equal to 4 units. (D) a constant equal to 49 units. 17. Planes are drawn parallel to the coordinate planes through the points (1, 2, 3) and (3, 4, 5). The length of the edges of the parallelepiped so found, are (A) 4, 6, 8 (B) 3, 4, 5 (C) 2, 4, 5 (D) 2, 6, 8 18. The length of a line segment whose projections on the coordinate axes are 6, 3, 2, is (A) 7 (B) 6 (C) 5 (D) 4
19. The direction cosines of a line segment whose projections on the coordinate axes are 6, 3, 2, are
(A) 6 3 2, ,7 7 7
(B) 6 3 2, ,7 7 7
(C) 6 3 2, ,7 7 7
(D) None of these
20. If P, Q, R, S are (3, 6, 4), (2, 5, 2), (6, 4, 4) , (0, 2, 1) respectively then the projection of PQ
on RS is (A) 2 units (B) 4 uints (C) 6 uints (D) 8 uints 21. Let f be a oneone function with domain (2, 1, 0) and range (1, 2, 3) such that exactly one
of the following statements is true. f (2) = 1, f (1) 1, f (0) 2 and the remaining two are false. The distance between points (2, 1, 0) and ( f (2), f (1), f (0) ) is
(A) 2 (B) 3 (C) 4 (D) 5
ANSWERS
LEVEL −I 1. A 2. C 3 A 4. A 5. A 6. A 7. A 8. C 9. D 10. (D) 11. (D) 12. (B) 13. (B) 14. (A) 15. A 16. (A) 17. (A) 18. (B) 19. (A) 20. (D) 21. (A) 22. (C) 23. (C) 24. (D) LEVEL −II 1. A 2. B 3. B 4. A 5. B 6. D 7. B 8. B 9. D 10. C 11. (D) 12. (C) 13. (B) 14. (A) 15. (B) 16. (D) 17. (D) 18. (D) 19. (B) 20. 21. (D) 22. (C)
Level – III 1. (A) 2. (B) 3. (D) 4. (C) 5. (B) 6. A 7. (C) 8. (A) 9. (A) 10. (D) 11. (B) 12. (B)
13. (A) 14. (D) 15. (A) 16. (B) 17. (D) 18. (A) 19. (A) 20. (A) 21. (D)
AD
LEVEL−I
1. Number of critical points of f (x) = 1x
|4x|2
2
are
(A) 1 (B) 2 (C) 3 (D) none of these 2. If the function f (x) = cos |x| – 2ax + b increases for all x R, then (A) a b (B) a = b/2 (C) a < –1/2 (D) a –3/2 3. Area of the triangle formed by the positive x-axis and the normal and the tangent to
x2 + y2 = 4 at (1, 3 ) is (A) 2 3 sq. units (B) 3 sq. units (C) 4 3 sq. units (D) none of these
4. A tangent to the curve y = 2x
2 which is parallel to the line y = x cuts off an intercept from the
y-axis is (A) 1 (B) –1/3 (C) 1/2 (D) –1/2 5. A particle moves on a co-ordinate line so that its velocity at time t is v (t) = t2 – 2t m/sec.
Then distance travelled by the particle during the time interval 0 t 4 is (A) 4/3 (B) 3/4 (C) 16/3 (D) 8/3 6. The derivative of f (x) = |x| at x = 0 is (A) 1 (B) 0 (C) –1 (D) does not exist 7. f (x) = [x2 + 3x4 + 5x6 + 5] have only ------------- value in (,) at x = ------------ 8. If y = a log x + bx2 + x has its extremum values at x = -1 and x =2 then a= ------- b = -------------- 9. The value of b for which the function f (x) = sin x –bx + c is decreasing in the interval (,)
is given by (A) b < 1 (B) b 1 (C) b > 1 (D) b 1
10. Equation of the tangent to the curve y = e–|x| at the point where it cuts the line x=1 (A) is ey + x =2 (B) is x + y = e
(C) is ex + y = 1 (D) does not exist 11. The greatest and least values of the function f(x) = ax + b x + c, when a > 0, b > 0, c > 0 in
the interval [0,1] are (A) a+b+c and c (B) a/2 b2+c, c
(C) 2
cba , c (D) None of these
12. The absolute minimum value of x4 – x2 – 2x+ 5
(A) is equal to 5 (B) is equal to 3 (C) is equal to 7 (D) does not exist
13. Through the point P (, ) where >0 the straight line 1by
ax
is drawn so as to form with
co-ordinates axes a triangle of area S. If ab >0, then the least value of S is (A) 2 (B) 1/2 (C) (D) None of these
14. If f(x) = A ln |x| + B x2 + x has its extreme values at x = 2 and x = 1 then
(A) A = 2, B = – 1/2 (B) A = – 2 , B = 1/2 (C) A = 2, B =1 (D) None of these
15. The function 2tan3x-3tan2x+12tanx + 3, x
2,0 is
(A) increasing (B) decreasing (C) increasing in (0, /4) and decreasing in (/4, /2) (D) none of these 16. The tangent to the curve y = 2x at the point whose ordinate is 1, meets the x – axis at the
point (A) (0, ln2) (B) (ln 2, 0) (C) (-ln2, 0) (D) (-1/ln2, 0) 17. The minimum value of ax + by, where xy = r2, is (r, ab >0) (A) 2r ab (B) 2ab r (C) –2r ab (D) None of these
18. The range of the function f(x) = sin-1
21x2 + cos-1
21x2 , where [.] is the greatest
integer function, is
(A)
ππ ,
2 (B)
2,0 π (C) {} (D)
2
,0 π
19. The domain of f(x) =
4
xx5log2
41 + 10Cx is
(A) (0, 1]U [4, 5) (B) (0, 5) (C) {1, 4} (D) None of these 20. A function whose graph is symmetrical about the origin is given by (A) f (x) = ex + e-x (B) f (x) = loge x (C) f (x + y) = f (x) + f (y) (D) none of these 21. Let f (x) be a function whose domain is [-5, 7]. Let g (x) = |2x + 5|, then the domain of fog (x)
is (A) [-5, 1] (B) [-4, 0] (C) [-6, 1] (D) none of these
22. 1xcosxcos1xsinxsinlim 24
24
x
is equal to,
(A) 0 (B) –1
(C) 1 (D) does not exist 23. Pick up the correct statement of the following where [ ] is the greatest integer function,
(A) If f (x) is continuous at x = a then [f (x)] is also continuous at x = a. (B) If f (x) is continuous at x = a then [f (x)] is differentiable at x = a. (C) If f (x) is continuous at x = a then f (x) is also continuous at x = a. (D) None of these
24. The greatest value of f (x) = cos (xe[x] + 7x2 –3x), x [-1, ) is
(A) –1 (B) 1 (C) 0 (D) none of these.
25. The equation of the tangent to the curve f (x) = 1 + e–2x where it cuts the line y = 2 is
(A) x + 2y = 2 (B) 2x + y = 2 (C) x – 2y = 1 (D) x – 2y + 2 = 0
26. The angle of intersection of curves y = 4 –x2 and y = x2 is……………………………………….
27. The greatest value of the function f (x) =
4
xsin
x2sin on the interval
2,0 is………………….
28. Let f(x) = x − sinx and g(x) = x − tanx, where x
2
,0 . Then for these value of x.
(A) f(x). g(x) > 0 (B) f(x) . g(x) < 0
(C) 0xgxf
(D) none of these
29. Suppose that f(x) 0 for all x [0, 1] and f is continuous in [0, 1] and 0dx)x(f1
0 , then
x [0, 1], f is (A) entirely increasing (B) entirely decreasing (C) constant (D) None of these
LEVEL−II 1. Let h (x) = f (x) + ln{f(x)} + {f (x)}2 for every real number x, then (A) h (x) is increasing whenever f (x) is increasing (B) h (x) is increasing whenever f (x) is decreasing (C) h (x) is decreasing whenever f (x) is increasing (D) nothing can be said in general 2. Let f (x) = a0 + a1x2 + a2x4 + …… + anx2n, where 0 < a0 < a1 < a2 < …… < an, then f (x) has (A) no minimum (B) only one minimum (C) no maximum (D) neither a maximum nor a minimum
3. The maximum value of xcosxsin
xcosxsin
in the interval
2,0 is
(A) 1/2 (B) 1/4
(C) 22
1 (D) 1/3
4. If y = .......xsinxsinxsin , then the value of dxdy
is
(A) 1yxsin
(B)
1yxsin
(C) 1y2xcos
(D) 1y2xcos
5. The curve y –exy + x = 0 has a vertical tangent at the point
(A) (1, 1) (B) at no point (C) (0, 1) (D) (1, 0)
6. A differentiable function f (x) has a relative minimum at x = 0 then the function
y = f(x) + ax + b has a relative minimum at x = 0 for (A) all a and all b (B) all b if a = 0 (C) all b > 0 (D) all a 0
7. Let f(x) =
Then.
0x,1xx
0x,xsin12
(A) f has a local maximum at x = 0 (B) f has a local minimum at x = 0 (C) f is increasing every where (D) f is decreasing everywhere 8. Let f(x) = xn+1 + a. xn, where ‘a’ is a positive real number, n I+ . Then x = 0 is a point of (A) local minimum for any integer n (B) local maximum for any integer n (C) local minimum if n is an even integer (D) local minimum if n is an odd integer 9. f(x) = max ( sinx, cosx) x R. Then number of critical points [ -2, 2] is /are ; (A) 5 (B) 7 (C) 9 (D) none of these 10. Let (x) = (f(x))3 –3(f(x))2 + 4f(x) + 5x + 3 sinx + 4 cos x x R, then (A) is increasing whenever f is increasing
(B) is increasing when ever f is decreasing (C) is decreasing whenever f is decreasing
(D) Nothing can be said
11. A function f(x) = 3x2x2x3x
2
2
is:
(A) Maximum at x = -3 (B) Minimum at x = -3 and maximum at x = 1 (C) No point of maxima or minima (D) Function is decreasing in it’s domain.
12. Let f(x) =
0xx6x5
0x)x3xsin(2
2. Then f(x) has
(A) local maxima at x = 0 (B) Local minima at x = 0 (C) Global maxima at x = 0 (D) Global minima at x = 0 13. If a, b, c, d are four positive real numbers such that abcd =1, then minimum value of (1+a)
(1+b) (1+c) (1+d) is (A) 8 (B) 12 (C) 16 (D) 20 14. If f(x) + 2f(1- x) = x2 + 2 xR, then f(x) is given as
(A) 32x 2 (B) x2 – 2
(C) 1 (D) None of these 15. xcosxsinlim
4/5x
, where [ . ] denotes the Integral part of x.
(A) is equal to –1 (B) is equal to –2 (C) is equal to –3 (D) Does not exist
16. If f (x) = x1
xx1ln2
x1
, then the value of f (0) so that f (x) is continuous at x = 0, is;
(A) 2 (B) 1 (C)1/2 (D) None of these
17. If f (x) = x1
x
, then
(A) f (x) is differentiable x R (B) f (x) is no where differentiable (C) f (x) is not differentiable at finite no. of point (D) None of these
18. If f1 (x) = sin x + tan x, f2 (x) = 2x then
(A) f1 (x) > f2 (x) x ( 0, /2) (B) f1 (x) < f2 (x) x ( 0, /2) (C) f1 (x) f2 (x) = 0 has exactly one root x ( 0, /2) (D) None of these
19. .Let f (x) =
1x1x
,3x2,a1x
. If f (x) has a local minima at x = 1. Then exhaustive set of
values of ‘a’ is; (A) a 4 (B) a 5 (C) a 6 (D) a 7
20. A differentiable function f (x) has a relative minimum at x = 0 then the function y = f (x) + ax +
b has a relative minimum at x = 0 for
(B) all a and all b (B) all b if a = 0 (D) all b > 0 (D) all a 0
21. The maximum value of f(x) = |x ln x| in x(0,1) is;
(A) 1/e (B) e (C) 1 (D) none of these
22. If f (x) = x
0
)1t( (et –1) (t – 2) (t + 4) dt then f (x) would assume the local minima at;
(A) x = –4 (B) x = 0 (C) x = 1 (D) x = 2.
23. f(x) = tan-1 (sinx + cosx) is an increasing function in
(A) (0,/4) (B) (0, /2) (C) (-/4, /4) (D) none of these.
24. Let f: RR, where f(x) = x3 - ax, aR. Then set of values of ‘a’ so that f(x) is increasing in its entire domain is; (A) (-, 0) (B) (0, ) (C) (-, ) (D) none of these
25. The curves y = 4x2 + 2x –8 and y = x3 –x + 10 touch each other at the point………………….. 26. Let f be differentiable for all x. if f (1) = -2 and f’ (x) 2 for all x [1, 6], then (A) f (6) < 8 (B) f (6) 8 (C) f (6) 5 (D) f (6) 5
27. The function f (x) = 4
2
x1x2 decreases in the interval……………………………………………..
28. The function f (x) = (x + 2) e –x increases in ------------------- and decreases in -------------------------------- 29. The function y = x –cot-1 x –log (x + 1x 2 ) is increasing on (A) (-, 0) (B) (-,) (C) (0, ) (D) R – {0}
30. Let f : (0, ) R defined by f(x) = x + xcosx
9 2
. Then minimum value of f(x) is
(A) 10 − 1 (B) 6 − 1 (C) 3 − 1 (D) none of these 31. Let a, n N such that a n3 then 33 a1a is always
(A) less than 2n31 (B) less than 3n2
1
(C) more than 3n1 (D) more than 2n4
1
32. The global minimum value of function f(x) = x3 + 3x2 + 10x + cosx in [-2,3] is (A) 0 (B) 3-2
(C) 16-2 (D) -15 33. The minimum value of the function defined by f(x) = Maximum {x, x+1, 2-x} is (A) 0 (B) 1/2 (C) 1 (D) 3/2 LEVEL−III 1. If the parabola y = ax2 + bx + c has vertex at (4, 2) and a [1, 3], then difference between
the extreme values of abc is equal to, (A) 3600 (B) 144 (C) 3456 (D) None of these 2. Let , and be the roots of f(x) = x3 + x2 –5x –1 = 0. Then [] +[] +[], where [.] denotes
the greatest integer function, is equal to (A) 1 (B) – 2 (C) 4 (D) – 3 3. The number of solutions of the equation x3 +2x2 +5x + 2cosx = 0 in [0, 2] is (A) 0 (B) 1 (C) 2 (D) 3 4. Let S be the set of real values of parameter for which the equation
f(x) = 2x3 – 3( 2+)x2 + 12x has exactly one local maximum and exactly one local minimum. Then S is a subset of
(A) (-4, ) (B) (-3, 3) (C) (3, ) (D) (-, 3) 5. Consider a function y = f (x) defined parametrecally as x = 2t + t , y = t t , t R. then function is
(A) Differentiable at x = 0 (B) non-differentiable at x = 0 (C) nothing can be said about differentiablity at x = 0 (D) None of these
6. If the line ax + by + c = 0 is normal to the curve x y + 5 = 0 then
(A) a > 0 , b > 0 (B) b > 0 , a < 0
(C) a < 0 , b < 0 (D) b < 0 , a > 0
7. The number of roots of x3-3x+1 = 0 in [1,2] is/are; (A) One (B) Two (C) Three (D)none of these
8. A cubic f(x) vanishes at x = -2 and has extrema at x = -1 and x = 31 such that
1
1 314dxxf
then f (x) = ………… 9. If g(x) = f(x) + f(1−x) and f(x) < 0, 0 x 1, then
(A) g(x) is decreasing in (0, 1) (B) g(x) is decreasing in
21,0
(C) g(x) is decreasing in
1,21 (D) g(x) is increasing in (0, 1)
10. Let g(x) > 0 and f(x) < 0 x R then (A) g(f(x + 1)) > g(f(x –1)) (B) f(g(x – 1)) < f(g(x + 1)) (C) g(f(x + 1) < g(f(x – 1)) (D) g(g(x + 1)) < g(g(x + 1))
11. The function
( )1 4ax bf x
x x
has a local maxima at (2, –1) then
(A) b = 1, a = 0 (B) a = 1, b = 0 (C) b = –1, a = 0 (D) None of these 12. 1 2( ) 2 , ( ) 3sin cosf x x f x x x x , then for x (0, /2):
(A) 1 2( ) ( )f x f x (B) 1 2f x f x
(C) 1 2( ) ( )f x f x (D) 1 2f x f x 13. y = f(x) is a parabola, having its axis parallel to y – axis. If the line y = x touches this parabola
at x = 1 then (A) (1) (0) 1f f (B) (0) (1) 1f f (C) (1) (0) 1f f (D) (0) (1) 1f f 14. If f(x) = 2 (2 1) 3x xe ae a x is increasing for all values of ‘x’ then (A) a (–, ) – {0} (B) a (–, 0] (C) a (0, ) (D) a [0, ) 15. If 2a + 3b + 6c = 0, then equation 2 0ax bx c has roots in the interval (A) (0, 1) (B) (2, 3) (C) (1, 2) (D) (0, 2) 16. The equation 23 4 0x ax b has at least one root in (0, 1) if (A) 4a + b + 3 = 0 (B) 2a + b + 1 = 0 (C) b = 0, a = -4/3 (D) None of these
17. If f(x) satisfies the conditions of Rolle’s theorem in [1, 2] then 2
1
( )f x dx is equal to
(A) 3 (B) 0 (C) 1 (D) –1 18. If f(x) =
2 22 /x ax e is a non-decreasing function then for a > 0; (A) x [a, 2a) (B) x (–, –a] [0, a] (C) x (–a, 0) (D) None of these
19. The function ( )1 tan
xf xx x
has
(A) One point of minimum in the interval (0, /2) (B) One point of maximum in the interval (0, /2) (C) No point of maximum, no point of minimum in (0, /2) (D) Two points of maximum in (0, /2) 20. The number of solutions of the equation ( ) ( ) 0,f xa g x where a > 0, g(x) 0 and has
minimum value of ½ is (A) 1 (B) 2 (C) 4 (D) 0
ANSWERS
LEVEL −I 1. A 2. C 3. A 4. D 5. C 6. D 7. 0 8. 2, −1/2 9. C 10. A 11. A 12. B 13. C 14. D 15. A 16. D 17. A 18. C 19. C 20. D 21. C 22. C 23. C 24. B 25. B 26. 2 2 27. 2 28. B 29. C LEVEL −II 1. A 2. B 3. C 4. D 5. D 6. B 7. A 8. C 9. B 10. A 11. C 12. B 13. C 14. A 15. B 16. C 17. C 18. A 19. B 20. B 21. A 22. D 23. C 24. A
25. 3, 34; − 1 74,3 9
26. B 27. 1 1, 0 ,2 2
28. (0, 1); R − (0, 1) 29. B 30. B 31. A 32. D 33. C LEVEL −III 1. C 2. 3. A 4. D 5. A 6. A, C 7. A 8. −x3 − x2 + x − 2
9. C 10. C 11. B 12. C 13. C 14. D 15. A 16. B 17. B 18. B 19. B 20. D
1
Area
LEVEL−I 1. Area common to the curves y = x3 and y = x is
(A) 125 (B)
65
(C) 85 (D) none of these
2. The area bounded by the parabola y2 = x, straight line y = 4 and y-axis is
(A) 3
64 (B) 3
16
(C) 7 2 (D) none of these 3. The area bounded by the curves y = |x| – 1 and y = – |x| + 1 is (A) 1 (B) 2 (C) 2 2 (D) 4 4. The area bounded by the curve y = sin x and the x-axis , for 0 x 2 is (A) 2 sq. units (B) 1 sq. units (C) 6 sq units (D) 4 sq. units 5. The area enclosed by y = ln x, its normal at (1, 0) and y-axis is (A) 1/2 (B) 3/2 (C) Not defined (D) none of these 6. The area bounded by y –1 = |x|, y =0 and |x| = 1/2 will be (A) 3/4 (B) 3/ 2 (C) 5/4 (D) none of these
7. The area bounded by the parabola y2 = 4 x and its latus rectum is
(A) 1 (B) ¾ (C) 8/3 (D) none of these
8. The area of the region bounded by y = |x-1| and y = 1 is
(A) 1/ 2 (B) 1 (B) 2 (D) none of these
9. The area of the region bounded by the parabola y = x2-3x with y 0 is (A) 3 (B) –3 (C) –9/2 (D) 9/2 10. The area of the smaller region bounded by the circle x2+y2 = 1 and |y| = x+1 is
(A) 21
4
(B) 12
(C) 2 (D) 1
2
11. The area bounded by the curves |x| + |y| 1 and x2 + y2 1 is (A) 2 sq. units (B) sq. units (C) - 2 sq. units (D) + 2 sq. units
2
12. Area bounded by f(x) = max.(sinx, cosx); 0 x /2 x = /2 and the coordinate axes is equal
to (A) 2 sq. units (B) 2 sq. units
(C) 2
1sq. units (D) None of these
13. If the area bounded by the curve , y =f(x), the lines x=1, x = b and the x-axis is (b-1)
cos (3b + 4), b > 1, then f(x) is (A) (x-5) sin (3x+4) (B) (x-1) sin (x+1)+ (x+1) cos (x-1) (C) cos (3x+4) –3(x-1) sin (3x+4) (D) (x-5) cos (3x+4)
14. The area of region that is completely bounded by the graph of f(x) = 2x – 1 and g(x) = 2 4x
is
(A) 3 (B) 203
(C) 323
(D) None of these
15. The area bounded by the curves 2 4y x and x + 2y = 4, is (A) 9 (B) 18 (C) 72 (D) 36 16. The area of the region bounded by the curve 2 2y x x and y x is
(A) 92
(B) 72
(C) 112
(D) None of these
17. The total area enclosed by , 1y x x and y = 0, is (A) 1 (B) 2 (C) 3 (D) 4 18. The area of the region bounded by the function 3( )f x x , the x-axis and the lines x = –1
and x = 1 is
(A) 14
(B) 13
(C) 18
(D) 12
19. The area of the region bounded by the curve y = x and 22 2y x is
(A) 13
(B) 16
(C) 19
(D) None of these
20. The area bounded by the axes and the curve 2y x is (A) 1 (B) 2 (C) 4 (D) None
3
LEVEL−II 1. Area bounded by the curves y = x2 + 2, y = –x, x = 0 and x = 1 is
(A) 2
17 (B) 6
17
(C) 6
19 (D) 6
13
2. The area bounded between the curves x = y2 and x = 3 – 2y2 is (A) 2 (B) 3 (C) 4 (D) 1 3. Area bounded by the curve ay = 3(a2 – x2) and the x-axis is (A) a2 (B) 2a2 (C) 3a2 (D) 4a2 4. Area bounded by the curves x2 = y and y = x + 2 and x-axis is
(A) 92
(B) 35
(C) 65 (D)
67
5. If Am represents the area bounded by the curve y = ln xm, the x-axis and the lines x= 1
and x= e, then Am+ m Am-1 is (A) m (B) m2 (C) m2/2 (D) m2-1 6. The area bounded by the curves y = ln x, y = | ln x| and the y-axis is (A) 3 (B) 2 (C) 4 (D) 8 7. If area bounded by y = f(x), the coordinate axes and the line x = a is given by aea, then
f(x) is (A) ex(x+1) (B) ex (C) x ex (D) xex+1 8. The area common to y2 = x and x2 = y is
(A) 1 (B) 2/3 (C) 1/3 (D) none of these
9. The area bounded by y = |x-1| and y = 3 -|x| is
(A) 2 (B) 3 (C) 4 (D) 1
10. The area cut off from the parabola 4y=3x2 by the straight line 2y=3x+12 is (A) 25 sq.units (B) 27 sq.units (C) 36 sq.units (D) 16 sq.units 11. The area bounded by the curve y = x2+ 2x+1, the tangent at (1, 4) and the y-axis is (A) 1 (B) 1/2 (C) 1/3 (D) 1/4
4
12. The area bounded by y = lnx, the x−axis and the ordinates x = 0 and x = 1 is (A) 1 (B) 3/2 (C) −1 (D) none of these 13. The area bounded by the straight lines y = 0, x + y – 2 = 0 and the straight line which equally
divides the common area included between the curves y = x2 and y = x is equal to (A) 1 sq. unit (B) 2sq, units (C) 3 sq. units (D) None of these 14. The area of the smaller region bounded by the circle 2 2 1x y and the lines 1y x is:
(A) 12 2 (B) 1
2
(C) 2
(D) 12
15. The area of the region bounded by 21 1y x and x y is
(A) 13
(B) 43
(C) 23
(D) 83
16. Area enclosed by the curve 2 1 1x y is
(A) 215
sq. units (B) 415
sq. units
(C) 2 sq. units (D) 4 sq. units 17. If the area bounded by a continuous function y = f(x), co-ordinate axes and the line x = a,
where a R+, is equal to a ea , then one such function can be (A) 1xe x (B) ( 1)xe x
(C) xe (D) None 18. Value of the parameter ‘a’ such that the area bounded by 2 2 1,y a x ax co-ordinate axes
and the line x = 1, attains the least value, is
(A) 14
(B) 34
(C) 12
(D) None of these
19. The area bounded by . xy x e and lines 1, 0x y is, (A) 4 (B) 6 (C) 1 (D) 2 20. The slope of the tangent to a curve y = f(x) at (x, f(x)) is 2x + 1. If the curve passes through
the point (1, 2), then the area of the region bounded by the curve, the x-axis and the line x = 1 is:
(A) 16
(B) 6
(C) 56
(D) 65
5
LEVEL−III
1. The area enclosed in the region 1by
ax
2
2
2
2
and 1by
ax
is
(A) ab21
4ab
(B)
4ab
(C) ab (D) none of these 2. The area of the loop of the curve x2 = y2(1-y) is (A) 2/15 (B) 15/14 (C) 4/15 (D) 8/15 3. The area common to the region determined by y x , and x2+y2 < 2 has the value (A) -2 (B) 2-1 (C) 3 - 3/2 (D) none of these 4. The area of the region for which 0 < y< 3 –2x-x2 and x> 0 is
(A) dxxx233
1
2 (B) dxxx223
0
2
(C) dxxx231
0
2 (D) dxxx223
1
2
5. The area enclosed between the curves y = sin2x and y = cos2 x in the interval 0 x is (A) 2 (B) ½ (C) 1 (D) None of these 6. The area between the curves y = xex and y = x e−x and the line x = 1 is (A) 2e (B) e (C) 2/e (D) 1/e 7. If An is the area bounded by y = (1-x2)n and coordinate axes, n N, then (A) An = An-1 (B) An < An-1 (B) An > An-1 (D) An = 2 An-1
8. Let ( ) min 1 , 1f x x x , then area bounded by f(x) and x-axis is:
(A) 16
(B) 56
(C) 76
(D) 116
9. Let 2 ; 0
( ); 0
x xf x
x x
Area bounded by the curve y = f(x), y = 0 and x = 3a is 92a
, then a =
(A) –1 or 12
(B) 1 or –12
6
(C) 1 or 12
(D) None
10. The interval [a, b] such that the value of 22b
a
x x dx is maximum, is
(A) [–2, 1] (B) [–2, –1] (C) [1, 2] (D) [–1, 2] 11. If A(n) represents the area bounded by the curve y = n lnx, where n N and n > 1, the x-axis
and the lines x = 1 and x= e, then the value of A(n) + n A(n – 1) is equal to
(A) 2
1n
e (B)
2
1n
e
(C) 2n (D) 2e x 12. Area of the region which consists of all the points satisfying the conditions 8x y x y
and xy 2, is equal to: (A) 2 (9 – ln8) sq. units (B) 4 (7 – ln2) sq. units (C) 4 (9 – ln8) sq. units (D) 4 (7 – ln8) sq. units 13. A point ‘P’ moves in xy – plane in such a way that 1x y , where [ ] denotes the
G.I .F. Area of the region representing all possible positions of the point ‘P’ is equal to (A) 8 sq. units (B) 4 sq. units (C) 16 sq. units (D) 2 2 sq. units 14. Area of the region bounded by 2 2 2 2x y and the axes is
(A) 38
sq. units (B) 32
sq. units
(C) 34
sq. units (D) None
15. The area of the smaller region in which the curve 3
100 50x xy
, where [ ] denotes G.I.F.,
divides the circle 2 22 1 4,x y is equal to
(A) 2 3 3
3
sq. units (B) 3 3
3
sq. units
(C) 5 3 3
3
sq. units (D) 4 3 3
3
sq. units
16. Area bounded by the curve
2xy e , x-axis and the lines x = 1, x = 2 is given to be equal to ‘a’
sq. units. Area bounded by the curve y = ln( )x , y-axis and the lines y = e and 4y e is equal to:
(A) 42e e a (B) 4e e a (C) 42 2e e a (D) 42 2e e a 17. Area bounded by the curves 2, 2xy e y x x and the line x = 0, x = 1 is equal to
7
(A)3 2
3e
sq. units (B) 4 5
4e
sq. units
(C) 4 74
e sq. units (D) 3 53
e sq. units
18. Value of the parameter ‘a’ such that area bounded by 2 3y x and the line y = ax + 2,
attain its minimum value is, (A) –1 (B) 0 (C) 1 (D) 1 19. Consider a triangle OAB formed by the points O (0, 0), A (2, 0), B 1, 3 . P(x, y) is an
arbitrary interior point of the triangle, moving in such a way that ( , ) ( , ) ( , ) 3d P OA d P AB d P OB , where d(P, OA), d(P, AB) and d(P, OB) represent the
distance of ‘P’ from the sides OA, AB and OB respectively. Area of the region representing all possible positions of the point ‘P’ is equal to
(A) 2 3 sq. units (B) 6 sq. units (C) 3 sq. units (D) None 20. Let f(x) = 2 ,ax bx c where a R and 2 4 0.b ac Area bounded by ( )y f x , x-axis
and the lines x = 0, x = 1 is equal to
(A) 1 3 (1) ( 1) 2 (0)6
f f f (B) 1 5 (1) ( 1) 8 (0)12
f f f
(C) 1 3 (1) ( 1) 2 (0)6
f f f (D) 1 5 (1) ( 1) 8 (0)12
f f f
ANSWERS
LEVEL −I 1. A 2. A 3. B 4. D 5. B 6. C 7. C 8. B 9. D 10. A 11. C 12. A 13. C 14. C 15. D 16. A 17. A 18. D 19. D 20. B LEVEL −II 1. B 2. A 3. D 4. 5. B 6. B 7. A 8. C 9. C 10. B 11. C 12. A 13. A 14. B 15. C 16. C 17. A 18. B 19. D 20. C LEVEL −III 1. A 2. C 3. D 4. C 5. B 6. C 7. B 8. C 9. A 10. D 11. C 12. D
8
13. A 14. C 15. D 16. A 17. D 18. B 19. C 20. D
1
BT LEVEL−I 1. The co-efficient of x in the expansion of (1-2x3+3x5)[1+(1/x)]8 is (A) 56 (B) 65 (C) 154 (D) 62 2. If the fourth term in the expansion of (px+1/x)n is 5/2 then the value of p is (A) 1 (B) 1/ 2 (C) 6 (D) 2 3. If x = 1/3, Then the greatest term in the expansion of (1+4x)8 is
(A) 564
43
(B) 56
5
34
(C) 565
43
(D) 56
4
52
4. The two consecutive terms in the expansion of (3+2x)74 whose coefficients are equal is (A) 30th and 31st term terms (B) 29th and 30th terms (C) 31st and 32nd terms (D) 28th and 29th terms
5. If z=55
2i
23
2i
23
, then
(A) Re(z) =0 (B) Im(Z) =0 (C) Re(z) >0, Im(z) >0 (D) Re(z) >0, Im(z) <0
6. The coefficient of xn in 2nn32
!nx1....
!3x
!2xx1
is
(A) !nn n (B)
!n2 n
(C) 2!n
1 (D) – 2!n
1
7. The sum of coefficients of even powers of x in the expansion of 11
x1x
is
(A) 11 11C5 (B) 211
11C6
(C) 11 611
511 CC (D) 0
8. The number of irrational terms in the expansion of 100
61
81
25
is equal to;
(A) 97 (B) 98 (C) 96 (D) 99 9. In the expansion of (1 + ax)n, n N, then the coefficient of x and x2 are 8 and 24
respectively. Then (A) a = 2, n = 4 (B) a = 4, n = 2
2
(C) a = 2, n = 6 (D) none of these 10. In the coefficients of the (m + 1)th term and the (m + 3) th term in the expansion of (1 +x)20
are equal then the value of m is (A) 10 (B) 8 (C) 9 (D) none of these 11. The number of distinct terms in the expansion of (2x + 3y –z + -7)n is (A) n + 1 (B) (n + 4)C4 (C) (n + 5)C5 (D) nC5 12. The coefficient of x5 in the expansion of (1 –x + 2x2)4 is………… 13. The two successive terms in the expansion of (1+x)24 whose coefficients are in the ratio 4 :1
are (A) 3rd and 4th (B) 4th and 5th (C) 5th and 6th (D) 6th and 7th
14. The expression nnnnnn CCCC 4..........4.4 2
210 , equals
(A) n22 (B) n32 (C) n5 (D) None of these
15. 602 when divided by 7 leaves the remainder (A) 1 (B) 6 (C) 5 (D) 2
16. The sum of the coefficients in the expansion of 21632 )31( xx is (A) 1 (B) –1 (C) 0 (D) None of these
17. The value of
0
11CC
n
n
1
21CC
n
n
……….
1
1n
nn
n
CC
is equal to
(A) !)1( 1
nn n
(B)!
)1(n
n n (C)
)!1(
1
nnn
(D) )!1(
)1( 1
nn n
18. The sum of the rational terms in the expansion of 10
51
32
is ………………
19. If in the expansion of (1 + x)m (1 –x)n, the co-efficient of x and x2 are 3 and –6 respectively,
then m is …………… 20. For 2 r n, nCr + 2 nCr–1 + nCr–2 is equal to ……………… 21. If (1 + x + 2x2)20 = a0 + a1x + a2x2 + ……………+ a40x40 then a1 + a3 + a5 + ……….+a37 equals
to ……………
22. The largest term in the expansion of (3 + 2x)50 where x = 51 is ……………
23. Let R = 1n21155
and f = R –[R] where [.] denotes the greatest integer function, then Rf
= ……………… 24. 23n –7n –1 is divisible by …………
3
25. If (1 –x + x2)n = a0 + a1x + a2x2 + ………………+a2nx2n, then a0 + a2 + a4 + …………+ a2n equals to …………
26. If the rth term in the expansion of 10
2x2
3x
contains x4, then r is equal to …………
27. 1.nC1 + 2.nC2 + 3.nC3 + ……..+ n.nCn is equal to
(A) n2.4
1nn (B) 2n+1 –3
(C) n.2n –1 (D) none of these 28. If the coefficient of (2r + 2)th and (r + 1)th terms of the expansion (1 +x)37 are equal then r = (A) 12 (B) 13 (C) 14 (D) 18
29. The value of 1n
C2...........
4C
23
C2
2C
2C2 n1n3423120
is equal to ………
30. If the co-efficient of rth, (r+1)th and (r+2)th terms in the expansion of (1+x)14 are in A.P., then the value of r is
(A) 5 (B) 6 (C) 7 (D) 9 31. If (1+ax)n = 1+8x +24x2+------- then (A) a= 3 (B) n= 5 (C) a= 2 (D) n =4 32. If ab 0 and the co-efficient of x7 in [ax2+(1/bx)]11 is equal to the co-efficient of x-7 in
11
2bx1ax
, then a and b are connected by the relation
(A) a = 1/b (B) a = 2/b (C) ab = 1 (D) ab = 2 LEVEL−II 1. Co-efficient of x5 in the expansion of (1+x2)5 (1+x)4 is (A) 40 (B) 50 (C) 30 (D) 60 2. The term independent of x in the expansion of (x+1/x)2n is
(A) !n
2).1n2(.5.3.1 n (B) !n!n
2).1n2(.5.3.1 n
(C) !n
)1n2(.5.3.1 (D) !n!n
)1n2(.5.3.1
3. If 6th term in the expansion of 8
102
3/8xlogx
x1
is 5600, then x is equal to
(A) 5 (B) 4 (C) 8 (D) 10
4
4. If coefficient of x2 y3 z4 in (x + y +z)n is A, then coefficient of x4y4z is
(A) 2A (B) 2
nA
(C) 2A (D) none of these
5. The coefficient of x6 in {(1 + x)6 + (1 + x)7 + ……….+ (1 + x)15} is (A) 16C9 (B) 16C5 – 6C5 (C) 16C6 –1 (D) none of these 6. If (1 +x)10 = a0 +a1x +a2x2 + ……+a10x10 then (a0 –a2 +a4 –a6 +a8 –a10)2 + (a1 –a3 +a5 –a7 +a9)2
is equal to (A) 310 (B) 210 (C) 29 (D) none of these 7. The remainder of 7103 when divided by 25 is……………
8. The term independent of x in the expansion of 3
x2x21
is…………
9. The number of irrational terms in the expansion of 55
101
21
32
is;
(A) 47 (B) 56 (C) 50 (D) 48 10. If ab 0 and the co-efficient of x7 in (ax2+(1/bx))11 is equal to the co-efficient of x-7 in
11
2bx1ax
, then a and b are connected by the relation
(A) a= 1/b (B) a =2/b (C) ab= 1 (D) ab=2
11. If (1 + 2x + 3x2)10 =
20
0r
rr xa then a2 is equal to;
(A) 210 (B) 620 (C) 220 (D) none of these
12. If Pn denotes the product of all the co-efficients in the expansion of (1+x)n, then n
1n
PP is equal
to
(A) !n1n n (B)
!1n1n 1n
(C) !n1n 1n (D)
!1n1n n
13. Value of
n
0r
2r
n
2rsinC , is equal to;
(A) 2n (B) 2n –1 (C) 2–n + 1 (D) 2n –1 –1
5
14. If 1 ba , then
n
0r
rnrr
n baC equals
(A) 1 (B) n (C) na (D) nb
15. If { x } denotes the fractional part of x , then
832n
, Nn is
(A) 3/8 (B) 7/8 (C) 1/8 (D) None of these. 16. The coefficient of mx in : ,)1.......()1()1( 1 nmm xxx nm is
(A) 11
mn C (B) 1
1
m
n C (C) mn C (D) 1m
n C
17. The expansion 5
21
35
21
3 1xx1xx
is a polynomial of degree ……
18. In the expansion of n
23
x1x
, n N, if the sum of the coefficients of x5 and x10 is 0 then n
is (A) 25 (B) 20 (C) 15 (D) none of these
19. The sum 21 10C0 – 10C1 + 2. 10C2 –22 . 10C3 + ……+ 29. 10C10 is equal to
(A) 21 (B) 0
(C) 103.21 (D) none of these
20. If the second, third and fourth terms in the expansion of (a+b) n are 135, 30 and 10/3 respectively, then (A) a = 3 (B) b = 1/3 (C) n = 5 (D) all the above LEVEL−III
1. The co-efficient of x53 in the expansion
100
0m
mm100m
100 2)3x(C is
(A) 100C53 (B) - 100C53 (C) 65C53 (D) 100C65
2. If n is an even natural number and coefficient of xr in the expansion of x1x1 n
is 2n, (|x| < 1),
then
(A) r n/2 (B) r 2
2n
(C) r 2
2n (D) r n
6
3. Let n be an odd natural number and A =
21n
1r rn C1 . Then value of
n
1r rn Cr is equal to
(A) n( A-1) (B) n( A+1)
(C) 2
nA (D) nA
4. ..........!5n!5
1!3n!3
1!1n!1
1
is equal to
(A) !n
2 1n for even values of n only (B)
!n12 1n
for odd values of n only
(C) !n
2 1n for all n N (D) none of these
5. The greater of two numbers 300! and 300300 is ……… 6. The co-efficient of x4 in the expansion of (1+x+x2+x3)11 is (A) 1001 (B) 990 (C) 900 (D) 895
7. Value of
n
1r
r
0mm
rr
n CC is equal to;
(A) 2n –1 (B) 3n -1 (C) 3n –2n (D) none of these
8. Value of
n
0r
2r
n Cr is equal to
(A) n . 2nCn (B) 2
Cn nn2
(C) n2 . 2nCn (C) 2
Cn nn22
9. If
n
1r rn C
r = , then value of
n
0r rn C1 is equal to;
(A) 2
n (B) n2
(C) 2
n (D) none of these
10. Value of
n
0rr
n xrnsinrxcosC is;
(A)2n –1 sin nx (B) 2n –1 cos nx (C) 2n cos nx (D) 2n sin nx 11. Value of
nji0j
nCi is;
(A) n.2n –3 (B) (n –1) . 2n –3 (C) n(n –1) . 2n –3 (D) none of these
7
12. The coefficient of xn in the polynomial ( x+ nC0) ( x+3 nC1) ( x+5 nC2) ……..( x+(2n + 1) nCn) is (A) n2n (B) n2n + 1 (C) (n +1)2n (D) n2n + 1
13. Value of 2
2
0
1.2
nn
rr
r Cr is equal to
(A)
1 22 2 1 22 1 2 2
n n nn n
(B)
2 1 22 2 1 22 1 2 2
n n nn n
(C)
2 1 22 2 2 12 1 2 2
n n nn n
(D) None of these
14. If R = 2 15 3 8
n and f = R – [R]; where [ ] denotes G. I. F., then R f is equal to
(A) 211 n (B) 2 111 n (C) 2 111 n (D) 11
15. Value of 2
0
n ni j
i j nC C
is
(A) 2 2. 2n nnn C (B) 2 21 2n n
nn C
(C) 2 21 2n nnn C (D) 2 21 2n n
nn C 16. The remainder when 1037 is divided by 25 is (A) 0 (B) 18 (C) 16 (D) 9 17. The number 100101 1 is divisible by (A) 10 (B) 210 (C) 310 (D) 410
18. Integral part of 2 15 5 11
n is
(A) Even (B) Odd (C) Neither (D) Can’t Say 19. Let 2( ) 10 3 4 5;n nf n n N . The greatest value of the integer which divides f(n) for all
‘n’ is (A) 27 (B) 9 (C) 3 (D) None
20. If 8
0
2 2 1,1 6
nn
rr
r Cr
then ‘n’ is
(A) 8 (B) 4 (C) 6 (D) 5
8
ANSWERS
LEVEL −I 1. C 2. B 3. B 4. A 5. B 6. B 7. D 8. A 9. A 10. C 11. B 12. −56 13. C 14. C 15. A 16. B 17. B 18. 41 19. 12 20. n+2Cr 21. 239 − 219 22. 50C6 344 (2x)6 23. 42n+1 24. 49
25. n3 12 26. 3 27. C 28. A
29. n 13 1n 1
30. D 31. C 32. C
LEVEL −II 1. D 2. A 3. D 4. C 5. A 6. B 7. −7 8. 3 3
0 12C C 9. B 10. C 11. A 12. A 13. B 14. A 15. C 16. A 17. 7 18. C 19. A 20. D LEVEL −III 1. B 2. D 3. B 4. C 5. 300! 6. B 7. B 8. B 9. B 10. A 11. C 12. C 13. A 14. C. 15. D 16. B 17. A, B, C, D 18. A 19. B 20. D
Quiz Bank-Circle-1
CIRCLE
LEVEL-I 1. The equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a circle, the condition will be (A) a = b and c = 0 (B) f = g and h = 0 (C) a = b and h = 0 (D) f = g and c = 0 2. The equation x2 + y2 + 2gx + 2fy + c = 0 represents a real circle if (A) g2 + f2 – c < 0 (B) g2 + f2 – c 0 (C) always (D) none of these
3. Equation of a circle with centre (4,3) touching the circle x2 + y2 = 1 is
(A) x2 +y2 – 8x – 6y – 9 = 0 (B) x2 + y2– 8x – 6y + 11 = 0 (C) x2 + y2 – 8x –6y – 11 = 0 (D) x2 + y2 – 8x – 6y + 9 = 0
4. A square is inscribed in the circle x2 + y2 –2x + 4y + 3 = 0. Its sides are parallel to the axes. Then
the one vertex of the square is (A) (1 + 2 , -2) (B) (1 - 2 , -2) (C) (1, -2 + 2 ) (D) none of these 5. The number of common real tangents that can be drawn to the circle x2 + y2 –2x –2y = 0 and x2 +
y2 –8x –8y + 14 = 0 is____________________________ 6. The lines 3x –4y + 4 = 0 and 6x –8y –7 = 0 are tangents to the same circle. The radius of the
circle is____________________________________________ 7. The straight line y = mx + c cuts the circle x2+y2 = a2 at real points if
(A) c)m1(a 22 (B) c)m1(a 22
(C) c)m1(a 22 (D) c)m1(a 22 8. A line is drawn through a fixed point P (,) to cut the circle x2+y2 = r2 at A and B. Then PA.PB is
equal to (A) (+)2-r2 (B) 2+2-r2 (C) (-)2+r2 (D) None of these 9. The locus of the centre of a circle of radius 2 units which rolls on the outside of the circle
x2 + y2 +3x –6y –9 = 0 is ………………………………………………………………………. 10. The values of a and b for which the two circles : x2 + y2 + 2(1 –a)x + 2(1 + b)y + (2 –c) = 0 and x2 + y2 + 2(1 + a)x + 2(1 –b)y + (2 + c) = 0 cut
orthogonally are …………….……………………………. 11. A circle of radius 2 lies in the first quadrant and touches both the axes of co-ordinates. Then the
equation of the circle with centre (6, 5) and touching the above circle externally is (A) (x –6)2 + (y –5)2 = 4 (B) (x –6)2 + (y –5)2 = 9 (C) (x –6)2 + (y –5)2 = 36 (D) none of these
Quiz Bank-Circle-2
12. Two circles x2 + y2 – 2x – 3 = 0 and x2 + y2 – 4x – 6y – 8 = 0 are such that (A) they touch each other (B) they intersect each other (C) one lies inside the other (D) each lies outside the other 13. The least distance of point (10, 7) from the circle x2 + y2 – 4x – 2y – 20 = 0 is (A) 10 (B) 15 (C) 5 (D) none of these 14. The number of common tangents to the circles x2 + y2 – x = 0 and x2 + y2 + x=0 is (A) 2 (B) 1 (C) 4 (D) 3 15. The radius of the circle passing through the point (2, 6) two of whose diameters are x + y = 6 and
x + 2y = 4 is (A) 10 (B) 2 5 (C) 6 (D) 4 16. The intercept on the line y = x by the circle x2 + y2 – 2x = 0 is AB. The equation of the circle with
AB as diameter is (A) x2
+ y2 + x + y = 0 (B) x2 + y2 = x + y (C) x2 + y2 – 3x + y = 0 (D) none of these 17. Equation of tangent to the circle x2 + y2 + 2x – 2y + 1 = 0 at (0, 1) (A) x = 0 (B) y = 0 (C) xy = 0 (D) none of these 18. The equation x2 + y2 – 2x + 4y + 5 = 0 represents (A) a point (B) a pair of straight lines (C) a circle (D) none of these 19. The equation of the chord of the circle x2 + y2– 4x = 0 which is bisected at the point (1, 1) is (A) x + y = 2 (B) 3x – y = 2 (C) x – 2y + 1 = 0 (D) x – y = 0 20. The line x + y = 1 is a normal to the circle 2x2 + 2y2 – 5x + 6y –1 = 0 if (A) 5 – 6 = 4 (B) 4 + 5 = 6 (C) 4 + 6 = 5 (D) none of these 21. The locus of the point (3h+2, k), where (h, k) lies on the circle x2+y2 = 1 is (A) a hyperbola (B) a circle (C) a parabola (D) an ellipse
Quiz Bank-Circle-3
LEVEL-II
1. The centre of the circle passing through the points (0, 0), (1, 0) and touching the circle x2+y2= 9 is
(A)
21,
23 (B)
23,
21
(C)
21,
21 (D)
2,21
2. The coordinates of mid point of the chord cut off by 2x – 5y + 18 = 0 by the circle
x2 + y2 – 6x + 2y – 54 = 0 are (A) (1, 4) (B) (2, 4) (C) (4, 1) (D) (1, 1) 3. Equation of tangent drawn from origin to the circle x2 + y2 – 2rx + 2hy + h2 = 0 are (A) x = 0 (B) y = 0 (C) (h2 – r2)x – 2rhy = 0 (D) (h2 – r2)x + 2rhy = 0 4. If 2 circles (x – 1)2 + (y – 3)2 = r2 and x2 + y2 – 8x + 2y + 8 = 0 intersect at 2 distinct points, then (A) 2 < r < 8 (B) r > 2 (C) r = 2 (D) r < 2 5. The equation of circle passing through (1, –3) and the points common to the two circles
x2 + y2 – 6x + 8y – 16 = 0, x2 + y2 + 4x – 2y – 8 = 0 is (A) x2 + y2 – 4x + 6y + 24 = 0 (B) 2x2 + 2y2 + 3x + y – 20 = 0 (C) 3x2 + 3y2 – 5x + 7y – 19 = 0 (D) none of these 6. The common chord of x2+ y2– 4x –4y = 0 and x2 + y2 = 16 subtends at the origin an angle equal to
(A) 6 (B)
4
(C) 3 (D)
2
7. The locus of the centre of the circle which touches externally the circle x2+y2–6x–6y+14=0 and
also touches the y-axis is given by the equations (A) x2 – 6x – 10y + 14 = 0 (B) x2 – 10x – 6y + 14 = 0 (C) y2 – 6x – 10y + 14 = 0 (D) y2 – 10x – 6y + 14 = 0 8. If the tangent at the P on the circle x2 + y2 + 2x + 2y = 7 meets the straight line 3x – 4y = 15 at a
point Q on the x-axis, then length of PQ is (A) 3 7 (B) 4 7 (C) 2 7 (D) 7 9. A straight line is drawn through the centre of the circle x2 + y2 – 2ax = 0, parallel to the straight
line x + 2y = 0 and intersecting the circle at A and B. Then the area of AOB is
(A) 5
a2
(B) 5
a3
(C) 3
a2
(D) 3
a3
Quiz Bank-Circle-4
10. The equation of the circle of radius 2 which touches the line x + y = 1 at (2, –1) is (A) x2 + y2 – 4x +2y+ 3= 0 (B) x2 + y2 + 6x +7= 0 (C) x2 + y2 – 2x +4y+ 3= 0 (D) none of these 11. If the co–ordinates of one end of a diameters of the circle x2 + y2 – 8x – 4y + c = 0 are (–3, 2),
then the co–ordinates of the other end are (A) (5, 3) (B) (6, 3) (C) (1, –8) (D) (11, 2) 12. The equation of the locus of the centre of circles touching the y–axis and circle x2 + y2 –2x= 0 is (A) x2 = 4y (B) x2 = – 4y (C) y2 = 4x (S) y2 = – 4x 13. The angle between a pair of tangents drawn from a point P to the circle x2 + y2 + 4x – 6y + 9 sin2 + 13 cos2 = 0 is 2. The equation of the locus of P is (A) x2 + y2 + 4x – 6y + 4 = 0 (B) x2 + y2 + 4x – 6y –9 = 0 (C) x2 + y2 + 4x – 6y – 4 = 0 (D) x2 + y2 + 4x – 6y + 9 = 0 14. The number of common tangents to the circles x2 + y2 – 6x – 14y + 48 = 0 and x2 + y2 – 6x = 0 is (A) 1 (B) 2 (C) 3 (D) 4 15. The equation of the smallest circle passing through the intersection of the line x + y = 1 and the
circle x2 + y2 = 9 is (A) x2 + y2 + x + y – 8 = 0 (B) x2 + y2 – x – y – 8 = 0 (C) x2 + y2 – x + y – 8 = 0 (D) none of these
16. A, B, C, D are the points of intersection with the co-ordinate axes of the lines ax + by = ab and bx + ay = ab then (A) A, B, C, D are concyclic (B) A,B,C,D forms a parallelogram (C) A, B, C, D forms a rhombus (D) None of these
17. If the lines 2x – 3y – 5 = 0 and 3x-4y = 7 are diameters of a circle of area 154 square units, then the equation of the circle is
(A) x2+y2+2x-2y-62 = 0 (B) x2+y2+2ax –2y – 47 = 0 (C) x2+y2-2x+2y-47 = 0 (D) x2+y2-2x+2y-62 = 0
18. The equation of the circle whose diameter is the common chord of the circle x2+y2+3x+2y+1= 0 and x2+y2+3x+4y+2 = 0 is
(A) x2+y2+8x+10y+2 = 0 (B) x2+y2-5x+4y+7 = 0 (C) 2x2+2y2+6x-2y-1 = 0 (D) None of these
19. The length of the tangent from any point on the circle 15x2 +15y2 – 48x + 64y = 0 to the two circles 5x2 + 5y2 – 24x + 32y + 75 = 0 and 5x2 + 5y2–48x + 64y + 300 = 0 are in the ratio of
(A) 1 : 2 (B) 2 : 3 (C) 3 : 4 (D) None of these
20. The tangents drawn from the origin to the circle x2+y2-2rx-2hy+h2 = 0 are perpendicular if (A) h = r (B) h = – r (C) r2+ h2 = 1 (D) r2 = h25.
Quiz Bank-Circle-5
21. If a variable circle of radius 4 cuts the circle x2 + y2 = 1 orthogonally then locus of its centre will be
(A) x2 + y2 = 16 (B) x2 + y2 =17 (C) x2 + y2 - 2x - 4y = 1 (D) 2x - 4y + 5 = 0
22. If four points
ii t
1,t ( i = 1, 2, 3, 4) are concyclic then t1t2 t3t4 =
(A) 1 (B) -1 (C) 4 (D) 1/4
23. The number of common tangents that can be drawn to the circle x2+y2–4x – 6y – 3 = 0 and x2 + y2 + 2x + 2y + 1 = 0 is
(A) 1 (B) 2 (C) 3 (D) 4
24. The circle x2 + y2 + 2ax + c = 0 and x2 + y2 + 2by + c = 0 touch if
(A) c1
b1
a1
22 (B) 222 c1
b1
a1
(C) 0c1
b1
a1
(D) none of these
25. The equation (x2 –a2)2 + (y2 –b2)2 = 0 represents points (A) which are collinear (B) which lie on a circle centred (0, 0) (C) which lie on a circle centre (a, b) (D) none of these 26. The equations of the circle which touch both the axes and the line x = a are
(A) x2+y2 ax ay+4a2
=0 (B) x2+y2 + ax ay+4a2
=0
(C) x2+y2 -ax ay+4a2
=0 (D) None of these
27. If the abscissae and ordinates of two points P and Q are the roots of the equation x2+2ax-b2 = 0
and x2+2px-q2 = 0 respectively, then the equation of the circle with PQ as diameter is (A) x2+y2+2ax+2py-b2-q2 = 0 (B) x2+y2-2ax-2py+b2+q2 = 0 (C) x2+y2-2ax-2py-b2-q2 = 0 (D) x2+y2+2ax+2py+b2+q2 = 0 28. If the distances from the origin of the centre of three circles x2+y2 +2ix –c2=0 (i= 1, 2, 3) are in
G.P. then the length of the tangent drawn to them from any point on the circle x2+y2 = c2 are in (A) A.P. (B) G.P. (C) H.P. (D) None of these 29. If the chord of contact of tangents drawn from a point on the circle x2 + y2 =a2 to the circle
x2 + y2 = b2 touches the circle x2 + y2 =c2, a, b, c> 0, then a, b, c are related as …… 30. The triangle PQR is inscribed in the circle x2 + y2 = 25. If Q and R have co-ordinates (3, 4) and
(–4, 3) respectively, then QPR is equal to (A) /2 (B) /3 (C) /4 (D) /6
Quiz Bank-Circle-6
31. If the circle x2 + y2 + 2x + 2ky + 6 = 0 and x2 + y2 + 2ky + k = 0 intersect ortohgonally, then k is (A) 2 or 3/2 (B) –2 or –3/2 (C) 2 or –3/2 (D) none of these 32. If the tangent to the circle x2 + y2 = 5 at the point (1, -2) also touches the circle x2 + y2 - 8x + 6y + 20 = 0, then its point of contact is (A) (3, 1) (B) (-3, 1) (C) (3, -1) (D) (-3, -1) 33. The equation of the circle having its centre on the line x + 2y –3 = 0 and passing through the point
of intersection of the circles x2 + y2 –2x –4y + 1= 0 and x2 + y2 –4x –2y + 1 = 0 is (A) x2 + y2 –6x + 1 = 0 (B) x2 + y2 –3x + 4 = 0 (C) x2 + y2 –2x – 2y + 1= 0 (D) x2 + y2 + 2x –4y + 4 = 0 34. Two circles x2 + y2 = 6 and x2 + y2 – 6x + 8 = 0 are given. Then the equation of the circle through
their point of intersection and the point (1, 1) is (A) x2 + y2 + x – y = 0 (B) x2 + y2 – 3x + 1 = 0 (C) x2 + y2 – 4y + 2 = 0 (D) none of these 35. Given that the circles x2 + y2 – 2x + 6y + 6 = 0 and x2 + y2 – 5x + 6y + 15 = 0 touch, the equation
to their common tangents is (A) x = 3 (B) y = 6 (C) 7x – 12y – 21 = 0 (D) 7x + 12y + 21 = 0 36. If an equilateral triangle is inscribed in the circle x2 + y2 = k2, the length of each side is equal to (A) k/3 (B) k3 (C) k (D) 2k 37. The equation of the circle through the origin and cutting intercepts of length 2 and 3 from the
positive sides of x and y is ____________________________ 38. If the circle x2+y2+4x+22y + c = 0 bisects the circumference of the circle x2 + y2 – 2x + 8y + d = 0
then c – d is equal to (A) 60 (B) 50 (C) 40 (D) 56 39. If an equilateral triangle is inscribed in the circle x2 + y2 = 25 then length of its each side is
(A) 5 2 (B) 235
(C) 5 3 (D) none of these 40. If the co–ordinates at one end of a diameter of the circle x2 + y2 – 8x – 4y + c = 0 are (11, 2) then
the co–ordinates at the other end are (A) (3, 2) (B) (–3, –2) (C) (–3, 2) (D) (3, –2) 41. S1 = x2 + y2 = 9, S2 = x2 + y2 – 8x – 6y + n2 = 0 , n Z. If the two circle have exactly two common
tangent then the number of possible value of n is (A) 7 (B) 8 (C) 9 (D) 10
Quiz Bank-Circle-7
42. If the common chord of x2 + (y – )2 = 16 and x2 + y2 = 16 subtends a right angle at the origin then is equal to
(A) 4 (B) 4 2 (C) 4 2 (D) 8 43. The locus of the middle point of chord of length 4 of the circle x2 + y2 = 16 is (A) a straight line (B) a circle of radius 2 (C) a circle of radius of radius 2 3 (D) an ellipse 44. The number of points with integral coordinates that are interior to the circle x2 + y2 = 16 is (A) 43 (B) 49 (C) 45 (D) 51 45. If equation of circle is ax2 + (2a – 3)y2 – 4x –1 = 0, then its centre is (A) (2, 0) (B) (2/3, 0) (C) (–2/3, 0) (D) none of these 46. The shortest distance between the circles x2 +y2 = 1 and x2 +y2 –10x –10y+ 41 = 0 is
(A) 41 -1 (B) 0 (C) 41 (D) 5 42
47. Two circles x2 + y2 + 2ax + c = 0 and x2 + y2 + 2by = 0 touch if
(A) a2 + b2 = c2 (B) c1 =
1 12 2a b
(C) 222 b1
a1
c1
(D) c2 = 4b2(a2 - c)
48. If y = 2x be the equation of a chord of the circle x2 + y2 = 2ax, then the equation of the circle, of
which this chord is a diameter, is (A) 2( x2+y2) – 5a( x + 2y ) = 0 (B) x2+y2 – 2a( x + 2y ) = 0 (C) 5(x2+y2) – 2a( x + 2y ) =0 (D) none of these.
49. PA is tangent to x2 + y2 = a2 and PB is tangent to x2 + y2 = b2 (b > a) . If APB = 2 , then locus
of point ‘P’ is (A) x2 − y2 = a2 + b2 (B) x2 + y2 = b2 − a2 (C) x2 + y2 = a2 + b2 (D) none of these 50. f(x, y) = x2 + y2 + 2ax + 2by + c = 0 represents a circle. If f(x, 0) = 0 has equal roots, each being 2
and f(0, y) = 0 has 2 and 3 as its roots, then centre of circle is
(A)
25,2 (B)
25,2
(C) data are not sufficient (D) data are inconsistent 51. Tangents PA and PB are drawn to x2+y2=4 from the point P(3, 0). Area of triangle PAB is equal to (A) 5
95 sq. units (B) 5
31 sq. units
(C) 59
10 sq. units (D) 53
20 sq. units
Quiz Bank-Circle-8
52. Radius of bigger circle touching the circle x2+y2 − 4x − 4y + 4 = 0 and both the co-ordinate axes is (A) 3 + 2 2 (B) 2 223
(C) 6 + 2 2 (D) 2 226 53. The lines 3x – 4y + = 0 and 6x – 8y + = 0 are tangents to the same circle. The radius of the
circle is
(A) 20
2 (B) 20
2
(C) 20
2 (D) none of these.
Quiz Bank-Circle-9
LEVEL-III 1. A circle of radius 5 units touches both the axes and lies in the first quadrant. If the circle makes
one complete roll on x-axis along the positive direction of x-axis, then its equation in the new position is
(A) x2 + y2 + 20x – 10y + 1002 = 0 (B) x2 + y2 + 20x + 10y + 1002 = 0 (C) x2 + y2 – 20x – 10y + 1002 = 0 (D) none of these 2. Let AB be a chord of circle x2 + y2 = 3 which subtends 450 angle at P where P is any moving point
on the circle. The locus of centroid of PAB is
(A) 31
31y
31x
22
(B) 31
31y
31x
22
(C) 31
31y
31x
22
(D) none of these
3. Two circles, each radius 5, have a common tangent at (1, 1) whose equation is 3x +4y – 7=0 then
their centre are (A) (4, –5), (–2,3) (B) (4, –3), (–2, 5) (C) (4, 5), (–2, –3) (D) none of these 4. The equation of the circle of radius 2 2 whose centre lies on the line x – y = 0 and which
touches the line x + y = 4 and whose centre’s co–ordinates satisfy the inequality x + y > 4 is (A) x2 + y2 – 8x – 8y + 24 = 0 (B) x2 + y2 = 8 (C) x2 + y2 – 8x + 8y + 24 = 0 (D) x2 + y2 + 8x + 8y + 24 = 0 5. The circle passing through distinct point (1, t), (t, 1) and ( t, t) for all values of t , passes
through the point (A) (-1, -1) (B) (1, 1) (C) ( 1, -1) (D) (-1, 1) 6. The equation of the locus of the midpoints of the chords of the circle 4x2 + 4y2 –12x + 4y + 1 = 0
that subtends an angle 32 at its centre is ______
7. The area of the triangle formed by the positive x-axis and the normal and tangent to the circle
x2 + y2 = 4 at the point (1, 3 ) is _____________________ 8. A circle is inscribed in an equilateral triangle of side a. the area of any square inscribed in this
circle is _______________________________________ 9. Tangents OP and OQ are drawn from the origin ‘O’ to the circle x2+y2+2gx+2fy+c=0. Then the
equation of the circumcircle of the triangle OPQ is (A) x2+y2+2gx+2fy = 0 (B) x2+y2+gx+fy = 0 (C) x2+y2-gx-fy=0 (D) x2+y2-2gx-2fy = 0 10. The locus of the mid points of the chords of the circle x2+y2+4x-6y-12 = 0 which subtends an
angle of 3 radians at its centre is
(A) (x+2)2+(y-3)2 = 6.25 (B) (x-2)2+(y+3)2 = 6.25 (C) (x+2)2+(y-3)2 = 18.75 (D) (x+2)2+(y+3)2 = 18.75
Quiz Bank-Circle-10
11. The locus of the mid-points of a chord of the circle x2 + y2 = 4, which subtends a right angle at the origin is
(A) x + y = 2 (B) x2 + y2 =1 (C) x2 + y2 = 2 (D) x + y = 1
12. If two distinct chords, drawn from the point (p, q) on the circle x2 +y2 = px + qy (where p, q 0 )
are bisected by the x-axis, then (A) p2 = q2 (B) p2 = 8q2 (C) p2 < 8q2 (D) p2 > 8q2 13. The locus of the centre of a circle which touches a given line and passes through a given
point, not lying on the given line, is (A) a parabola (B) a circle (C) a pair of straight line (D) none of these .
14. The tangents drawn from the origin to the circle x2+y2 + 2gx + 2fy + f2 =0 are perpendicular if (A) g = f (B) g = -f (C) g = 2f (D) 2g = f 15. Two circles with radii ‘r1’ and ‘r2’, r1 > r2 2 , touch each other externally. If ‘’ be the angle
between the direct common tangents, then
(A)
21
211
rrrrsin (B)
21
211
rrrrsin2
(C) = sin-1
21
21
rrrr (D) none of these.
16. Tangents are drawn to the circle x2 + y2 = 50 from a point ‘P’ lying on the x-axis. These tangents
meet the y-axis at points ‘P1’ and ‘P2’. Possible coordinates of ‘P’ so that area of triangle PP1P2 is minimum, are
(A) (10, 0) (B) (10 2 , 0) (C) (-10, 0) (D) (-10 2 , 0) 17. Two distinct chords of the circle x2 + y2 − 2x − 4y = 0 drawn from the point P(a, b) gets bisected
by the y-axis, then (A) (b + 2)2 > 4a (B) (b − 2)2 > 4a (C) (b − 2)2 > 2a (D) none of these 18. A circle S of radius ‘a’ is the director circle of another circle S1. S1 is the director circle of circle S2
and so on. If the sum of the radii of all these circles is 2, then the value of a is (A) 2 + 2 (B) 2 – 2
(C) 2 – 2
1 (D) 2 +2
1
19. Circles are drawn having the sides of triangle ABC as their diameters. Radical centre of these
circles is the (A) circumcentre of triangle (B) Incentre of triangle ABC (C) orthcentre of triangle ABC (D) centroid of ABC 20. The circle x2 + y2 + 2a1x + c = 0 lies completely inside the circle x2 + y2 + 2a2x + c =0, then (A) a1a2 > 0, c < 0 (B) a1a2 > 0, c > 0 (C) a1a2 < 0, c < 0 (D) a1a2 < 0, c > 0
Quiz Bank-Circle-11
ANSWERS
LEVEL −I 1. C 2. B 3. D 4. D 5. 3 6. 3/4 7. A 8. B
9. 2
23 169x y 32 4
10. a = b = 0 11. B
12. B 13. C 14. D 15. A 16. B 17. A 18. C 19. D 20. A 21. D LEVEL −II 1. D 2. A 3. A 4. A 5. D 6. D 7. D 8. C 9. A 10. C 11. D 12. C 13. D 14. D 15. B 16. A 17. C 18. C 19. A 20. A 21. B 22. A 23. C 24. A 25. B 26. C 27. A 28. B 29. G.P. 30. C 31. D 32. B 33. A 34. B 35. A 36. B
37. 2
2 3 13x 1 y2 4
38. B 39. C
40. C 41. C 42. C 43. C 44. C 45. B 46. D 47. D 48. C 49. C 50. D 51. C 52. B 53. A LEVEL −III 1. D 2. C 3. C 4. A
5. B 6.
2 23 1 9x y2 2 4
7. 2 3 units
8.
21 a6
9. B 10. c 11. C 12. D 13. A 14. A 15. B 16. A, C 17. B 18. B 19. C 20. B
COMPLEX NUMBER
LEVEL-I 1. If z1 , z2 are two complex numbers such that arg(z1+z2) = 0 and
Im(z1z2) = 0, then (A) z1 = - z2 (B) z1 = z2 (C) z 1= 2z (D) none of these 2. Roots of the equation xn –1 = 0, n I, (A) form a regular polygon of unit circum-radius . (B) lie on a circle. (C) are non-collinear. (D) A & B 3. Which of the following is correct (A) 6 + i > 8 – i (B) 6 + i > 4 - i (C) 6 + i > 4 + 2i (D) None of these 4. If (1+i3)1999 = a+ib, then (A) a = 21998, b = 219983 (B) a = 21999, b = 219993 (C) a=-21998, b = -219983 (D) None of these 5. If z = 1 + i 3 , then | arg ( z) | + | arg ( z ) | equals (A) /3 (B) 2/3 (C) 0 (D) /2
6. The equation 3i1zz3iizz
= 0 represents a circle with
(A) centre
23,
21 and radius 1 (B) centre
23,
21 and radius 1
(C) centre
23,
21 and radius 2 (D) centre
23,
21 and radius 2
7. Number of solutions to the equation (1 –i)x = 2x is (A) 1 (B) 2 (C) 3 (D) no solution 8. If ,0)arg( z then )arg()arg( zz
(A) (B)4 (C)
2
(D) 2
9. The number of solutions of the equation ,022 zz where Cz is
(A) one (B) two (C) three (D) infinitely many
10. If is an imaginary cube root of unity, then (1 + –2)7 equals (A) 128 (B) –128 (C) 128 2 (D) –128 2
11. If z1 and z2 be the nth roots of unity which subtend right angle at the origin. Then n must
be of the form (A) 4k + 1 (B) 4k + 2 (C) 4k + 3 (D) 4k 12. For any two complex numbers z1 and z2 | 7 z1 + 3z2|2 + |3z1 – 7 z2|2 is always equal to (A) 16(|z1|2 + |z2|2) (B) 4(|z1|2 + |z2|2) (C) 8(|z1|2 + |z2|2) (D) none of these
13. If is an nth root of unity other than unity itself, then the value of 1 + + 2 + ………+ n
–1 is ………………………………
14. Locus of ‘z’ in the Argand plane is 2,z then the locus of z + 1 is -
(A) a straight line (B) a circle with centre (1, 0)
(C) a circle with centre (0, 0) (D) a straight line passing through (0, 0) 15. Value of 1999 299 1 is (A) 1 (B) 2 (C) 0 (D) -1 16. Square root(s) of ‘–1’ is/ are -
(A) 1 12
i (B) 1 13
i
(C) 1 12
i (D) 1 12
i
17. The real value of ‘’ for which 3 2 sin1 2 sin
ii
is real is
(A) ,n n I (B) ,3
n n I
(C) ,2
n n I (D) ,2
n n I
18. Principal argument of 3z i is
(A) 56
(B) 6
(C) 56
(D) None
19. Which one is not a root of the fourth root of unity (A) i (B) 1
(C) 2i
(D) –i
20. If 3 22 4 8 0z z z then (A) 1z (B) 2z
(C) 3z (D) None
LEVEL-II
1. If a,b, c are three complex numbers such that c =(1– ) a + b, for some non-zero real number , then points corresponding to a,b, c are (A) vertices of a triangle (B) collinear (C) lying on a circle (D) none of these
2. If z be any complex number such that |3z –2| + |3z +2| = 4, then locus of z is (A) an ellipse (B) a circle (C) a line-segment (D) None of these 3. If arg 1z = arg(z2), then (A) z2 = k z1
-1 (k > 0) (B) z2 = kz1 (k > 0) (C) |z2| = | z 1| (D) None of these.
4. The value of the expression 2
21111 +3
21212 + 4
21313 + .
. . + (n+1)
21n1n , where is an imaginary cube root of unity, is
(A) 3
2nn 2 (B) 3
2nn 2 (C) 4
n41nn 22 (D) none of these
5. For a complex number z , | z-1| + |z +1| =2. Then z lies on a
(A) parabola (B) line segment (C) circle (D) none of these
6. If z1 and z2 are two complex numbers such that |z1| = |z2| + |z1 – z2|, then
(A) Im
2
1
zz = 0 (B) Re
2
1
zz = 0
(C)
2
1
2
1
zzIm
zzRe (D) none of these.
7. If 2
1
zz =1 and arg (z1 z2) = 0, then
(A) z1 = z2 (B) |z2|2 = z1z2 (C) z1z2 = 1 (D) none of these. 8. Number of non-zero integral solutions to (3+ 4i)n = 25n is (A) 1 (B) 2 (C) finitely many (D) none of these. 9. If |z| < 4, then | iz +3 – 4i| is less than (A) 4 (B) 5 (C) 6 (D) 9 10. If z is a complex number, then z2 + 2z = 2 represents
(A) a circle (B) a straight line (C) a hyperbola (D) an ellipse
11. If i1i1 = A + iB, then A2 +B2 equals to
(A) 1 (B) 2 (B) -1 (D) - 2 12. A,B and C are points represented by complex numbers z1, z2 and z3. If the circumcentre
of the triangle ABC is at the origin and the altitude AD of the triangle meets the circumcircle again at P, then P represents the complex number
(A) –3
21
zzz (B) –
1
32
zzz
(C) –2
13
zzz (D)
3
21
zzz
13. If |z1| = |z2| and arg(z1) +arg(z2) = /2 , then (A) arg(z1
-1) + arg(z2-1) = -/2 (B) z1z2 is purely imaginary
(C) (z1+z2)2 is purely imaginary (D) All the above.
14. If z1 and z2 are two complex numbers satisfying the equation 1izzizz
21
21 , then
2
1
zz
is a
(A) purely real (B) of unit modulus (C) purely imaginary (D) none of these 15. If the complex numbers z1, z2, z3, z4, taken in that order, represent the vertices of a
rhombus, then (A) z1 + z3 = z2 + z4 (B) |z1 – z2| = |z2 – z3|
(C) 42
31
zzzz
is purely imaginary (D) none of these
16. If 0z,z,kzzzzzz
2121
21 then
(A) for k = 1 locus of z is a straight line (B) for k {1, 0} z lies on a circle (C) for k = 0 z represents a point (D) for k = 1,z lies on the perpendicular bisector of the line segment joining
1
2
1
2zzand
zz
17. If the equation |z – z1|2 + | z – z2|2 = k represents the equation of a circle, where z1 2+
3i, z2 4 + 3i are the extremities of a diameter, then the value of k is
(A) 41 (B) 4
(C) 2 (D) None of these
18. If z be a complex number and ai , bi , ( i= 1,2,3) are real numbers, then the value of the
determinant
332211
332211
332211
azbazbazbzazbzazbzazbzbzazbzazbza
is equal to
(A) (a1 a2 a3 + b1 b2 b3 ) |z|2 (B) |z|2 (C) 0 (D) None of these
19. If z = x + iy satisfies the equation arg (z-2) = arg(2z+3i), then 3x-4y is equal to (A) 5 (B) -3
(C) 7 (D) 6
20. If a complex number x satisfies
1|z|2|z|26|z|2|z|log 2
2
2/1 <0 , then locus / region of the
point represented by z is (A) |z| = 5 (B) |z| <5 (C) |z|> 1 (D) 2<|z|<3
21. If for a complex number z= x + iy, sec–1
i
2z is an acute angle, then
(A) x = 2, y = 1 (B) x< 2, y < –1 (C) xy <0 (D) x = 2, y > 1
22. Number of solutions of Re (z2) = 0 and |Z| = a2, where z is a complex number and a >
0, is (A) 1 (B) 2 (C) 4 (D) 8
23. If the area of the triangle formed by the points represented by, Z, Z + iZ and iZ is 200,
then |Z| is ____________ 24. Let z is a variable complex number and a is a real constant. Then the solution set for z,
satisfying the equation, |z-a| + |z + a| = |a| is _____________
25. If Z1, Z2 be two non zero complex numbers satisfying the equation 1ZZZZ
21
21
then
2
1
2
1
ZZ
ZZ is _________.
26. If (x – iy) 1/3 = a – ib, then by
ax equals
(A) 2 (a2 + b2) (B) 4 (a + b) (C) 4 (a b) (D) 4 ab
27. If nn2i3 , where n is an integer, then
(A) n is a multiple of 5 (B) n is a multiple of 6 (C) n is a multiple of 10 (D) none of these 28. If points corresponding to the complex numbers z1, z2 and z3 in the Argand plane are A,B
and C respectively and if ABC is isosceles, and right angled at B then a possible value
of 23
21zzzz
is
(A) 1 (B) -1 (C) i (D) none of these 29. If z1 and z2 are two complex numbers satisfying the equation
1zzzz
21
21
, then 2
1zz
is a number which is
(A) Real (B) Imaginary
(C) Zero (D) None of these 30. If |z| = 1, then |z-1| is
(A) < |arg z| (B) >|arg z| (C) = |arg z| (D) None of these 31. If z1, z2 and z3, z4 are two pairs of conjugate complex numbers then
arg
4
1zz
+ arg
3
2zz
equals
(A) 2 (B)
(C) 2
3 (D) 0
32. If ||z + 2| |z 2|| = a2, z C is representing a hyperbola for a S, then S contains (A) [1, 0] (B) (, 0] (C) (0, ) (D) none of these
33. If |z| = 1 and z i, then iziz
is
(A) purely real (B) purely imaginary (C) a complex number with equal real and imaginary parts (D) none of these 34. The locus of z which satisfied the inequality log0.5|z –2| > log0.5|z – i| is given by (A) x+ 2y > 1 (B) x – y < 0 (C) 4x – 2y > 3 (D) none of these 35. Let Z1 and Z2 be the complex roots of ax2 + bx + c = 0, where a b c > 0. Then
(A) | Z1 + Z2 | 1 (B) |Z1 + Z2 | > 2 (C) |Z1 | = |Z2| = 1 (D) none of these 36. If the roots of z3 + az2 + bz + c = 0, a, b, c C(set of complex numbers) acts as the
vertices of a equilateral triangle in the argand plane, then (A) a2 + b = c (B) a2 = b (C) a2 + b = 0 (D) none of these 37. If |z1| = 4, |z2| = 4, then |z1 + z2 + 3 + 4i| is less than (A) 2 (B) 5 (C) 10 (D) 13 38. If z = x + iy satisfies Re{z -|z –1| + 2i} = 0, then locus of z is
(A) parabola with focus
21,
21 and directrix x + y =
21
(B) parabola with focus
21,
21 and directrix x + y =
21
(C) parabola with focus
21,0 and directrix y =
21
(D) parabola with focus
0,
21 and directrix x =
21
39. If |z +1| = z + 1 , where z is a complex number, then the locus of z is (A) a straight line (B) a ray (C) a circle (D) an arc of a circle 40. Length of the curved line traced by the point represented by z, when
arg41z
1z
, is
(A) 22 (B) 2
(C) 2 (D) none of these
41. If 02718128 23 izziz then
(A) 23z (B) 1z (C) 32z (D) 43z
42. If 2 iz and iz 351 then the maximum value of 1ziz is
(A) 312 (B) 231 (C) 231 (D) 7
43. ,)1z(i1sin 1
where z is not real, can be the angle of the triangle if
(A) 2)(,1)Re( zIz m (B) 1)(1,1)Re( zIz m (C) 0)()Re( zIz m (C) None of these
44. The value of )1ln( (A) does not exist (B) iln2 (C) i (D) 0
45. If 21 ,nn are positive integers then 2121 )1()1()1()1( 753 nnnn iiii is a real Number if and only if (A) 121 nn (B) 21 1 nn (C) 21 nn (D) 21 ,nn be +ve integers
46. Let 21 , zz be two nonreal complex cube roots of unity and 2
12
1 zzzz be the
equation of a circle with 21 , zz as ends of a diameter then the value of is
(A) 4 (B) 3 (C) 2 (D) 2
47. The center of the arc 4682
363arg
iziz
is
(A) (4,1) (B) (1,4) (C) (2,5) (D) (3,1)
48. The value of
6
1 72cos
72sin
k
kik
(A) i (B) i (C) 1 (D) –1
49. The complex numbers z1, z2 and z3 satisfying 2
3i1zzzz
32
31
are the vertices of a
triangle which is (A) of area zero (B) right angled isosceles (C) equilateral (D) obtuse angled isosceles
50. If |z| = 3 then the number 3z3z
is
(A) purely real (B) purely imaginary (C) a mixed number (D) none of these
51. If iz3 + z2 –z + i = 0, then |z| is equal to ………………………………………
52. If and are different complex numbers with || = 1, then
1 is equal to
53. If the complex numbers z1, z2, z3 are in A.P., then they lie on a
(A) circle (B) parabola
(C) line (D) ellipse
54. If z1 and z2 are two nth roots of unity, then arg
2
1
zz is a multiple of ………………….
55. The maximum value of |z| when z satisfies the condition z2z = 2 is ………………
56. All non-zero complex numbers z satisfying z = iz2 are…………………………………….
57. Common roots of the equation z3 + 2z2 + 2z +1 = 0 and z1985 + z100 + 1 = 0 is …………
LEVEL-III 1. If points corresponding to the complex numbers z1, z2, z3 and z4 are the vertices of a
rhombus, taken in order, then for a non-zero real number k (A) z1 – z3 = i k( z2 –z4) (B) z1 – z2 = i k( z3 –z4) (C) z1 + z3 = k( z2 +z4) (D) z1 + z2 = k( z3 +z4) 2. If z1 and z2 are two complex numbers such that | z1 – z2| = | |z1| - |z2| |, then
argz1 – argz2 is equal to (A) - /4 (B) - /2 (C) /2 (D) 0
3. If f(x) and g(x) are two polynomials such that the polynomial h(x) = x f(x3) + x2 g(x6) is divisible by x2 +x +1 , then
(A) f(1) = g(1) (B) f(1) - g( 1) (C) f(1) = g(1) 0 (D) f(1) = -g(1) 0 4. Consider a square OABC in the argand plane, where ’O’ is origin and A A(z0).
Then the equation of the circle that can be inscribed in this square is; ( vertices of square are given in anticlockwise order)
(A) | z – z0(1+ i)| =|z0| (B) 2
00 z
2i1z
z
(C)
00 z
2i1z
z
(D) none of these .
5. For a complex number z, the minimum value of |z| + | z - cos - isin| is (A) 0 (B) 1 (C) 2 (D) none of these 6. The roots of equation zn = (z +1)n (A) are vertices of regular polygon (B) lie on a circle (C) are collinear (D) none of these 7. The vertices of a triangle in the argand plane are 3 + 4i, 4+ 3i and 2 6 + i, then
distance between orthocentre and circumcentre of the triangle is equal to,
(A) 628137 (B) 628137
(C) 62813721
(D) 62813731
.
8. One vertex of the triangle of maximum area that can be inscribed in the curve
|z – 2 i| =2,is 2 +2i , remaining vertices is / are (A) -1+ i( 2 + 3 ) (B) –1– i( 2 + 3 ) (C) 1+ i( 2 – 3 ) (D) –1– i( 2 – 3 )
9. If
2
2
1
1
z3z2
z2z3 = k, then points A(z1) , B(z2), C(3, 0) and D(2, 0) (taken in clockwise
sense) will (A) lie on a circle only for k > 0 (B) lie on a circle only for k < 0 (C) lie on a circle k R (D) be vertices of a square k( 0, 1) 10. Let ‘z’ be a complex number and ‘a’ be a real parameter such that
z2 + az + a2 = 0, then (A) locus of z is a pair of straight lines
(B) arg(z) = 32
(C) |z| =|a| . (D) All
11. If z1, z2, z3 . . .. zn-1 are the roots of the equation zn-1 + zn-2 + zn-3 + . . .+z +1= 0,
where n N, n > 2, then (A) n, 2n are also the roots of the same equation. (B) 1/n, 2/n are also the roots of the same equation. (C) z1, z2, . . . , zn-1 form a geometric series. (D) none of these. Where is the complex cube root of unity. 12. The value of i log(x – i) + i2 +i3 log(x +i) + i4( 2 tan-1x), x> 0 ( where i = 1 ) is
(A) 0 (B) 1 (C) 2 (D) 3 13. If z = -2 + i32 , then z2n + 22n zn + 24n may be equal to (A) 22n (B) 0 (C) 3. 24n (D) none of these
14. The value of
135cos
1312sini 11
e169 is (A) 119 –120i (B) -i(120 +119i) (C) 119 + 120i (D) none of these 15. Let z1 and z2 be the complex roots of the equation 3z2 + 3z+ b = 0. If the origin, together
with the points represented by z1 and z2 form an equilateral triangle then the value of b is (A) 1 (B) 2 (C) 3 (D) None of these
16. If|z-2| = min {|z-1|,| z-3|}, where z is a complex number, then
(A) Re(z) = 23 (B) Re(z) =
25
(C) Re (z)
25,
23 (D) None of these
17. If x = 1 + i, then the value of the expression x4 – 4x3 + 7x2 – 6x + 3 is (A) -1 (B) 1 (C) 2 (D) None of these 18. If z lies on the circle centred at origin. If area of the triangle whose vertices are z, z and
z + z, where is the cube root of unity, is 4 3 sq. unit. Then radius of the circle is (A) 1 unit (B) 2 units (C) 3 units (D) 4 units 19. If i [0, /6], i = 1, 2, 3, 4, 5 and sin 1z4 + sin2 z3 + sin3 z2 + sin 4 z + sin5 = 2, then
z satisfies.
(A) 43|z| (B)
21|z|
(C) 43|z|
21
(D) None of these
20. If is the angle which each side of a regular polygon of n sides subtends at its centre,
then 1 + cos + cos2 + cos3 … + cos(n-1) is equal to (A) n (B) 0 (C)1 (D) None of these 21. Triangle ABC, A(z1), B(z2), C(z3) is inscribed in the circle |z| = 2. If internal bisector of the
angle A meets its circumcircle again at D(zd) then (A) 32
2d zzz (B) 31
2d zzz
(C) 122d zzz (D) none of these
ANSWERS LEVEL −I 1. C 2. D 3. D 4. A 5. B 6. B 7. A 8. A 9. D 10. D 11. D 12. A 13. 0 14. B 15. C 16. A 17. A 18. A 19. C 20. B LEVEL −II 1. B 2. C 3. A 4. C 5. B 6. A 7. B 8. D 9. D 10. C 11. A 12. B 13. D 14. A 15. A, B, C 16. A, B, C, D 17. B 18. C 19. D 20. B 21. D 22. A 23. 20 24. 25. 0 26. A 27. D 28. C 29. B 30. A 31. D 32. A 33. B 34. C 35. A 36. D 37. D 38. D 39. B 40. D 41. A 42. D 43. B 44. C 45. C 46. B 47. A 48. A 49. C 50. B 51. 1
52. 1 53. C 54. 2n
55. 1 + 3
56. 3 1,2 2
57. , 2
LEVEL −III 1. A 2. D 3. A 4. B 5. B 6. C 7. B 8. A 9. C 10. D 11. C 12. A 13. B, C 14. A, B 15. A 16. C 17. B 18. D 19. A 20. B 21. A
Determinants
LEVEL−I
1. Let f (x) = x(x – 1), then = )4(f)3(f)2(f)3(f)2(f)1(f)2(f)1(f)0(f
is equal to
(A) –2! (B) –3! – 2! (C) 0 (D) none of these
2. If f (x) = )1x(x)1x()2x)(1x(x)1x(x3
x)1x()1x(xx21xx1
, then f (100) is equal to
(A) 0 (B) 1 (C) 100 (D) –100
3. The determinant (x) = )x1(cbcac
bc)x1(babacab)x1(a
2
2
2
(abc 0) is divisible by
(A) 1 + x (B) (1 + x)2 (C) x2 (D) none of these
4. The value of the determinant pqrprqqrp
rqp111
222 is
(A) pqr (B) p + q + r (C) p + q + r – pqr (D) 0
5. If a, b, c > 0 and x, y, z R, then the determinant
1cccc
1bbbb
1aaaa
2zz2zz
2yy2yy
2xx2xx
is equal
to (A) ax + by + cz (B) a-x b–y c-z (C) a2x b2y c2z (D) 0 6. Given a system of equations in x, y, z: x + y + z = 6; x + 2y + 3z = 10 and x + 2y + az = b. If
this system has infinite number of solutions, then (A) a = 3, b = 10 (B) a = 3, b 10 (C) a 3, b = 10 (D) a 3, b 10 7. If each element of a determinant of 3rd order with value A is multiplied by 3, then the value of
the newly formed determinant is (A) 3A (B) 9A (C) 27A (D) none of these 8. If the value of 3rd order determinant is 11, then the value of the determinant formed by the
cofactors will be (A) 11 (B) 121 (C) 1331 (D) 14641
9. If a–1 + b–1 + c–1 = 0 such that c111
1b1111a1
= , then the value of is
(A) 0 (B) abc (C) –abc (D) none of these
10. If a, b, c are real numbers, then 1cc1c1bb1b1aa1a
is
(A) 0 (B) 6 (C) 9 (D) None of these 11. Let D be the determinant of order 3 3 with the entry Ii + k in lth row and kth column
(I = 1 . Then value of D is (A) imaginary (B) Zero (C) real and positive (D) real and negative
12. The value of the determinant abcccabbbcaa
2
2
2
111
is
(A) a3+b3+c3-3abc (B) a2+b2+c2-bc-ca-ab (C) a2b2+b2c2+c2a2 (D) None of these
13. Let =
1111
nxmlx
. Then, the roots of the equation are
(A) , , (B) l, m ,n (C) +, +, + (D) l+m, m+n, n+l
14. Let = bacacbcba
; a>0 , b>0, c >0. Then,
(A) 0 (B) a+b+c = 0 (C) >0 (D) R
15. The value of =
2
2
11
111 is
(A) 3 3 i (B) - 3 3 i (C) - 3 i (D) 3 i
16. If a, b, c are negative different real numbers, then = bacacbcba
is
(A) < 0 (B) 0 (C) > 0 (D) 0 17. The equation x + 2y + 3z = 1, x – y + 4z = 0, 2x + y + 7z = 1 have (A) one solution only (B) two solutions only (C) no solution (D) infinitely may solution
18. The value of and for which the system of equation x + y + z = 6, x + 2y + 3z = 10,
x + 2y + z = have unique solution are (A) = 3, R (B) = 3, = 10 (C) 3, = 10 (D) 3, 10
LEVEL−II
1. The value of 8m7m6m
5m4m5m
2m1mm
iiiiiiiii
, where i = 1 is
(A) 1 if m is multiple of 4 (B) 0 for all real m (C) –i if m is a multiple of 3 (D) none of these 2. If the equations a(y + z) = x, b(z + x) = y and c (x + y) = z, where a –1, b – 1, c –1
admit non-trivial solution, then (1 + a)–1 + (1 + b)–1 + (1 + c)–1 is (A) 2 (B) 1 (C) 1/2 (D) none of these 3. The number of values of t for which the system of equations (a – t)x + by + c = 0,
bx + (c – t)y + az = 0, cx + ay + (b – t)z = 0 has non-trivial solution is (A) 1 (B) 2 (C) 3 (D) 4
4. If , are non real numbers satisfying x3 – 1 = 0, then the value of
11
1 is
equal to (A) 0 (B) 3 (C) 3 + 1 (D) none of these 5. The system of equations ax + 4y + z = 0, bx + 3y + z = 0, cx + 2y + z = 0 has non trivial
solutions if a, b, c are in (A) A.P (B) G.P (C) H.P (D) none of these
6. The maximum value of x2sin41xcosxsin
x2sin4xcos1xsinx2cos4xcosxsin1
22
22
22
is
(A) 3 (B) 4 (C) 5 (D) 6 7. There are three points (a, x), (b, y) and (c, z) such that the straight lines joining any two of
them are not equally inclined to the coordinate axes where a, b, c, x, y, z R.
If
byaxcz
axczby
czbyax
= 0 and a + c = -b, then x , –2y , z are in
(A) A. P. (B) G.P. (C) H.P. (D) none of these 8. If x, y, z are the integers in A.P, lying between 1 and 9 and x51, y41 and z31 are
three digits numbers, then the value of zyx31z41y51x345
is
(A) x + y + z (B) x –y + z (C) 0 (D) None of these
9. If x y1 1 1 x y 1 x y 1
2 2
3 3
=a b1 1 1 a b 1 a b 1
2 2
3 3
, then the two triangles with vertices
(x1, y1), (x2, y2), (x3, y3), and (a1,b1), (a2,b2) (a3,b3) are (A) Congruent (B) Similar (C) Of equal area (D) Of equal altitude
10. Let a =nnna
nnana
333)1(242)1(
61
233
22
. Then
n
aa
1
is equal to
(A) 0 (B) (a-1) 2n (C) (a-1)n n (D) None of these
11. The determinant =
cossinsinsincossin
2cos)sin()cos(
is independent of
(A) (B) (C) and (D) None of these
12. Let =
a ab ac
ba b bc
ca cb c
2
2
2
1
1
1
. a, b, c R. Then,
(A) = 0 (B) <0
(C) >0 (D) None of these
13. If A +B +C = , then the value of 0tan)cos(
tan0sincossin)sin(
ABAABCBCBA
is
(A) sinA sinB sinC (B) sinA sinB+ sinC sinA +sinB sinC (C) 0 (D) sinA cosBsinC+sinA sinB cosC+cosA sinBsinC
14. Let 1= xaabxabbx
and 2 = xabx
. Then
(A) 1= 3(2)2 (B)
dxd
1 = 32
(C)
dxd
1 = 3(2)2 (D) 1 = 3(2)3/2
15. Let zypyxp
zyzypyxyxp
0= 0. Then
(A) x, y, z are in A.P (B) x,y,z are in G.P (C) x, y, z are in H.P (D) xy, yz , zx are in A.P
16. Let f(x) =32
3
016cossin
ppp
xxx where ‘p’ is a constant. Then )(3
3
xfdxd
at x =0 is
(A) p (B) p+p2 (C) p+p3 (D) independent of ‘p’
17. Let = 1sin1
sin1sin1sin1
, then lies in the interval
(A) [2, 3] (B) [3, 4] (C) [2, 4] (D) (2, 4)
18. If , , are roots of x3 + ax2 + b = 0, then the value of
is
(A) – a3 (B) a3 – 3b (C) a3 (D) a2 – 3b 19. Given ai
2 + bi2 + ci
2 = 1, (i = 1, 2, 3) and aiaj + bibj + cicj = 0 (i j, i, j = 1, 2, 3), then the value
of
2
321
321
321
cccbbbaaa
is
(A) 0 (B) 1/2 (C) 1 (D) 2
20. If (x) = 1xsinxsin
xcosxsin1xcosxsin1xcos1xcos1
, then
2/
0
dx)x( is equal to
(A) 1/4 (B) 1/2 (C) 0 (D) –1/2
21. If A + B + C = , then the value of determinant 1CcotCsin1BcotBsin1AcotAsin
2
2
2
is equal to
(A) 0 (B) 1 (C) -1 (D) None of these
LEVEL−III
1. If 3n2nn
3n2nn
3n2nn
zzz
yyy
xxx
= ( y-z )( z – x) ( x – y)
z1
y1
x1 , then
(A) n =2 (B) n = -2 (C) n = -1 (D) n = 1
2. Let m be a positive integer and r = )msin()m(sin)m(sin
1m21m1C1r2
2222
m2r
m
.
Then the value of rm
0r
is given by
(A) 0 (B) m2-1 (C) 2m (D) 2m sin2 (2m)
3. If (x) = xsinxtanxcos
xsinex1logxx1x
2
x2
32
then
(A) (x) is divisible by x (B) (x) = 0 (C) (x) = 0 (D) None of these 4. If fr(x), gr(x), hr(x), (r=1,2,3) are polynomials in x such that fr(A) = gr(A) = hr(A), r = 1,2,3 and
F(x) = f x f x f xg x g x g xh x h x h x
1 2 3
1 2 3
1 2 3
( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )
, then F (x) at x = a is
(A) 0 (B) 1 (C) fr(x)+ gr(x)+ hr(x) (D) None of these
5. Let f(x) = xxxecxx
xecxxxx
22
222
2
coscos1coscoscos
cotcosseccossec . Then dxxf
2/
0
)(
is equal to
(A)
4158
(B)
4158
(C) -
4158
(D) None of these
6. Let Dr = 151312
)5(4)3(22 111
nnn
rrr
. Then
n
rrD
1 is equal to
(A) ++ (B) (C) 2n 3n 5n (D) 0
7. If maximum and minimum values of the determinant
x2sin1xcosxsin
x2sinxcos1xsin
x2sinxcosxsin1
22
22
22
are and , then (A) +99 = 4
(B) 3 - 17 = 26 (C) (2n - 2n) is always an even integer for nN (D) a triangle can be constructed having it’s sides as , and - . 8. The parameter on which the value of the determinant
xdppxxdpxdppxxdp
aa
)sin(sin)sin()cos(coscos
1 2
does not depend upon is
(A) a (B) p (C) d (D) x L−I 1. B 2. A 3. C 4. D 5. D 6. 7. C 8. 9. B 10. A 11. B 12. D 13. A 14. D 15. A 16. C 17. 18. C L−II 1. D 2. 3. 4. A 5. 6. A 7. A 8. D 9. C 10. A 11. A 12. C 13. C 14. B 15. B 16. D 17. C 18. C 19. A 20. D L−III 1. C 2. A 3. 4. A 5. C 6. D 7. B 8. B
DE
LEVEL−I 1. If f(x), g(x) be twice differentiable function on [0, 2] satisfying
f (x) = g (x) , f (1) = 2, g(1) = 4 and f(2) = 3, g(2) = 9, then f(x) - g(x) at x = 4 equals (A) 0 (B) −10 (C) 8 (D) 2
2. baey x1
is a solution of 2xy
dxdy
when
(A) a = 1, b = 0 (B) a = 2, b = 0 (C) a = 1, b = 1 (D) a = 2, b = 2.
3. The solution of differential equation )x/y()x/y(
xy
dxdy
is
(A) x(y/x) = k (B) (y/x) = kx (C) y(y/x) = k (D) (y/x) = ky
4. Solution of differential equation of (x + 2y3) dy = ydx is (A) x = y3 + cy (B) y = x3 + cx (C) x2 + y2 = cxy (D) none of these
5. The curve, which satisfies the differential equation 2yydxxdyydxxdy
sin(xy) and passes
through (0, 1), is given by (A) y (1 - cos xy) + x = 0 (B) sinxy - x = 0 (C) siny + y = 0 (D) cosxy - 2y = 0
6. The solution of the differential equation y2
x2sin2y
dxdy
x is given by
(A) xy2 = cos2 x+ c (B) xy2 = sin2x+ c (C) yx2= cos2x+c (D) None of these 7. Differential equation whose general solution is y = c1x + c2/x for all values of c1 and c2 is
(A) 0dxdy
yx
dx
yd 2
2
2 (B) 0
dxdy
x
y
dx
yd22
2
(C) 0dxdy
x21
dx
yd2
2 (D) 0
x
ydxdy
x1
dx
yd22
2
8. A particle moves in a straight line with a velocity given by dtdx
= x + 1. The time taken by a
particle to travels a distance of 99 meters is (A) log10 e (B) 2 loge 10
(C) 2 log10 e (D) 21 log10 e
9. 2 32
2
d y dyx =0 is a differential equation ofdxdx
(A) degree 2, order 2 (B) degree 3, order 3 (C) order 2, degree 3 (D) None of these 10. The degree of a differential equation, written as a polynomial in differential coefficients, is
defined as (A) Highest of the orders of the differential coefficients occurring in it (B) Highest power of the highest order differential coefficients occurring in it (C) Any power of the highest order differential coefficients occurring in it.
(D) Highest power among the powers of the differential coefficients occurring in it
11. The order of the differential equation, whose general solution is y = C ex + C2 e2x + C3 e3x + C4
5cxe , Where C1, C2, C3, C4, C5 are arbitrary constants, is (A) 5 (B) 4 (C) 3 (D) none of these
12. I.F. for y ln is 0ylnxdydxy
(A) ln x (B) ln y (C) ln xy (D) none of these 13. Which one of the following is a differential equation of the family of curves y=Ae2x + Be-2x
y2dxdyxy4
3
dxdy Dy4
dxyd C
02xxydxdy2
dxydx B0y2
dxdy2
dxyd A
2
2
22
2
2
2
14. The differential equation of y = ax2 + bx + c is (A) y 0 (B) y = 0 (C) y + cx = 0 (D) y c 0
LEVEL−II 1. Which of the following transformations reduce the differential equation
xQ)x(Pdxd form the toinzlog
xzzlog
xz
dxdz 2
2
2
z
zlog Dzlog
1 Ce Bzlog A
2. The function f() =
0 xcoscos1dx
dd satisfies the differential equation
(A) d
df + 2f() cot = 0 (B) d
df - 2f() cot = 0
(C) d
df + 2f() = 0 (D) d
df - 2f() = 0
3. Solution of differential equation dxdy = sin(x+y) +cos(x+y) is
(A) 2
yxtan1log = x+c (B) x ylog 1 sec
2
= x+c
(C) )yxtan(1log = y+c (D) None of these
4. The degree of the differential equation 01dxdy
2dx
yd
dx
yd2
2
3
3 is
(A) 4 (B) 2 (C) 1 (D) None of these 5. The order of the differential equation of the family of circles with one diameter along the line
y – x is (A) 1 (B) 2 (C) 3 (D) none of these 6. If x-intercept of any tangent is 3 times the x-coordinate of the point of tangency, then the
equation of the curve, given that it passes through (1,1), is
(A) y = x1 (B) y =
2x
1 (C) y = x1 (D) none of these
7. The equation of the curve, passing through (2,5) and having the area of triangle formed by
the x-axis, the ordinate of a point on the curve and the tangent at the point 5 sq units, is
(A) xy = 10 (B) x2 = 10y (C) y2 = 10x (D) xy1/2 = 10
8. The family passing through (0, 0) and satisfying the differential equation 1yy
1
2 (where
yn = )dx
ydn
n is
(A) y = k (B) y = kx (C) y = k(ex+1) (D) y = k(ex-1)
9. If y = e4x + 2e–x satisfies the relation dxdyA
dxyd3
3
+ By = 0, then value of A and B respectively
are (A) –13, 14 (B) –13, –12 (C) –13, 12 (D) 12, –13
10. Solution of equation
)x(
ydx
)x(dy
dxdy
2
is
(A) y = x
c)x( (B) y = x
)x( +c (C) y = yx)x(
(D) y = )x( +x+c
11. The equation of curve through point (1, 0) and whose slope is xx1y
2 is
(A) (y-1) (x+1) +2x =0 (B) 2x(y-1) +x+1 =0
(C) y= x1x1
(D) None of these
12. If the slope of the tangent at (x,y) to a curve passing through (1, /4) is given by
y/x – cos2 (y/x) then the equation of the curve is (A) y = tan-1log(e/x) (B) y = x tan-1 log(e/x) (C) x = e1+cot(y/x) (D) x = e 1+ tan(y/x)
13. Differential equation of all parabolas whose axes are parallel to y-axis is
(A) 0dx
yd3
3
(B) cdy
xd2
2
(C) 0dy
xddx
yd2
2
3
3
(D) cdxdy2
dxyd2
2
14. The curve whose subnormal w.r.t any point is equal to the abscissa of that point is a (A) Circle (B) Parabola (B) Ellipse (D) Hyperbola 15. The family whose x and y intercepts of a tangent at any point are respectively double of the x
and y coordinates of that point is (A) x2 + y2 = c (B) x2 – y2= c (C) xy = c (D) None of these 16. Solution of differential equation (2x cosy + y2 cosx) dx + (2y sinx – x2
siny) dy = 0 is (A) y2 sinx + x2cosy = k (B) y2 cosy + x2sinx = k (C) y2 cosx + x2siny = k (D) None of these.
ANSWERS
LEVEL −I 1. B 2. A 3. B 4. A 5. A 6. A 7. D 8. B 9. A 10. B 11. B 12. B 13. C 14. A LEVEL −II 1. C 2. A 3. A 4. B 5. B 6. C 7. A 8. D 9. B 10. C 11. A 12. B 13. A 14. D 15. C 16. A
EL
LEVEL-I
1. If equation of ellipse is 16x2 + 25y2 = 400, then eccentricity of the ellipse (A) 2/5 (B) 4/5 (C) 3/5 (D) 1/5
2. If any tangent to the ellipse is 2 2
2 2
x y 1a b
intercepts lengths h and k on the axes,
then
(A)2 2
2 2
h k 1a b
(B)2 2
2 2
h k 2a b
(C)2 2
2 2
a b 1h k
(D) 2 2
2 2
a b 2h k
3. Two perpendicular tangents drawn to the ellipse 116y
25x 22
intersect on the curve
(A) x= 4 (B) y = 5 (C) x2 +y2 = 41 (D) x2 +y2 = 9 4. Equation to an ellipse whose centre is (-2, 3) and whose semi-axes are 3 and 2 and
major axis parallel to x-axis, is given by (A) 4x2 + 9y2 + 16 x – 54y – 61 = 0 (B) 4x2 + 9y2 - 16 x + 54y + 61 = 0 (C) 4x2 + 9y2 + 16 x – 54y + 61 = 0 (D) none of these 5. The angle between the tangents drawn from the point (1, 2) to the ellipse 3x2 + 2y2 = 5
is
(A) tan-1125
(B) tan-1
65
(C) tan-112
5 (D) tan-1
65
6. Eccentric angle of a point on the ellipse x2 + 3y2 = 6 at a distance 2 units from the
centre of the ellipse is
(A) 3or4 4 (B) 2or
3 3 (C) 5or
6 6 (D) none
of these 7. If latus rectum of the ellipse x2 tan2 + y2 sec2 = 1 is 1/2 then ( 0 < ) is
(A)12 (B)
6 (C) 8
12 (D) none of
these
8. Equation of the ellipse whose minor axis is equal to the distance between foci and whose latus rectum is 10, is given by
(A) 2x2 + 3y2 = 100 (B) 2x2 + 3y2 = 80 (C) x2 + 2y2 = 100 (D) none of these
9. If P is a point on the ellipse 2 2x y
16 20 =1 whose foci are S and S. Then PS + PS is
(A) 8 (B) 4 5 (C) 10 (D) 4
10. The distance between the directrices of the ellipse 2 2x y 1
4 9 is
(A) 95
(B) 245
(C) 185
(D) none of these
11. If F1 (0,0), F2 (3,4) and |PF1| +|PF2| =10, then the locus of is
(A) An ellipse (B) A straight line (C) A hyperbola (D) A line segment
12. The locus of a point represented by x = a t 12 t
, y = a t 12 t
is
(A) an ellipse (B) a circle (C) a pair of line (D) none of these 13. The eccentricity of the conic 7x2 + 16y2 = 112 is
(A) 237
(B) – 34
(C) 34
(D) none of these
14. The area of the ellipse 2 2x y 1
16 25 is
(A) 16 (B) 20 (C) 25 (D) 36 15. The locus of the point (3h+2, k), where (h, k) lies on the circle x2+y2 = 1 is (A) a hyperbola (B) a circle (C) a parabola (D) an ellipse 16. The equation of ellipse, whose focus is (1, 0), directrix is x = 4 and whose eccentricity is
a root of the quadratic equation 2x2 3x + 1 = 0, is-
(A)2 2x y 1
4 3 (B)
2 2x y 13 4
(C) 4x2 + 3y2 = 24 (D) None of these
17. Area of the quadrilateral formed by the tangents to the ellipse 2
2x y 14 at the end
points of its major and minor axes is (A) 8 (B) 4 (C) 16 (D) 2 18. The centre of the ellipse 3x2 + 6x + 4y2 8y 5 = 0, is (A) (1, 1) (B) (1, 1) (C) (1, 1) (D) None of these 19. Length of major axis of the ellipse, 3x2 6x + 4y2 8y 5 = 0, is (A) 4 (B) 1 (C) 3 (D) 2 20. Length of minor axis of the ellipse, 3x2 6x + 4y2 8y 5 = 0, is (A) 4 (B) 2 (C) 3 (D) 2 3
LEVEL-II
1. The equation 2 2x y 1 0
2 r r 5
represents an ellipse only if
(A) r > 2 (B) r < 5 (C) 2 < r < 5 (D) none of these
2. If any tangent to the ellipse 2 2x y 1
16 9 intercepts equal length l on the axes, then l
equals to (A) 25 (B) 7 (C) 12 (D) 5 3. An ellipse has its axes along co-ordinate axes. The distance between its foci is 2h and
the focal distance of an end of the minor axis is k. The equation of the ellipse is
(A) 2 2
2 2
x y 1h k
(B) 2 2
2 2 2
x y 1k k h
(C)2 2
2 2 2
x y 1k k h
(D) 2 2
2 2 2
x y 1k h h
4. Equation of the ellipse, referred to its axes as the x and y axes respectively, which
passes through the point (-3, 1) and the eccentricity 25
is
(A) 2x2 + 14y2 = 32 (B) 3x2 + 5y2 = 32 (C) 4x2 + 3y2 = 39 (D) none of these 5. Equation of tangents to the ellipse 9x2 + 10y2 = 144 from the point (2, 3) are (A) y = 3, x + y = 5 (B) x = 3, x – y = 5 (C) x + y = 3, x – y + 5 = 0 (D) none of these
6. If a tangent of slope ‘m’ at a point of the ellipse 2 2
2 2
x ya b
= 1 passes through (2a, 0) and
if ‘e’ denotes the eccentricity of the ellipse then (A) m2 + e2 = 1 (B) 2m2 + e2 = 1 (C) 3m2 + e2 = 1 (D) none of these 7. The tangent to the curve x = a( – sin ); y = a(1 + cos ) at the points = (2k + 1), k
Z are parallel to (A) y = x (B) y = –x (C) y = 0 (D) x = 0 8. The equation(s) of the tangent(s) to the ellipse 9(x - 1)2 + 4y2 = 36 parallel to the latus
rectum, is (are) (A) y = 3 (B) y = -3 (C) x = 3 (D) x = -3. 9. The area of the triangle formed by the points on the ellipse 25x2 + 16y2 = 400 whose
eccentric angles are /2, and 3/2 is
(A) 10 sq. units (B) 20 sq. units (C) 30 sq. units (D) 40 sq. units
10. If 3 bx+ay = 2ab touched the ellipse 2 2
2 2
x ya b
= 1 then eccentric angle is
(A) 6 (B)
4
(C) 3 (D)
2
11. The vaue of ‘c’ for which lie y = x + c is tangent to the ellipse 2x2 + 3y2 = 1 is
(A) 67
(B) 56
(C) 23
(D) 32
12. Foci of the ellipse; 25 (x + 1)2 +9(y +2)2 = 225, are (A) (1, 2) and (1, 6) (B) (2, 1) and (2, 6) (C) (1, 2) and (1, 6) (D) (1, 2) and (1, 6)
13. Let ‘E’ be the ellipse 2 2x y 1
9 4 and ‘C’ be the circle x2 + y2 = 9. Let P and Q be points
(1, 2) and (2, 1) respectively. Then (A) ‘Q’ lies in side ‘C’ but outside E (B) ‘Q’ lies outside both C and E (C) P lies inside both C and E (D) P lies inside ‘C’ but outside E 14. The equation 3 (x + y 5)2 + 2 (x y + 7)2 = 6 represents an ellipse, whose centre is (A) (1, 6) (B) (6, 1) (C) (1, 6) (D) (6, 1) 15. Eccentricity of the ellipse 3 (x + y 5)2 + 2 (x y + 7)2 = 6 is
(A) 12
(B) 23
(C) 13
(D) 12
16. One foot of normal of the ellipse 4 x2 + 9y2 = 36, that is parallel to the line 2 x + y = 3, is
(A) 9 8,5 5
(B) 9 8,5 5
(C) 9 8,5 5
(D) None of these
17. Equation of the ellipse whose axes are coordinate axes and whose length of latus
rectum, and eccentricity are equal and equal to ½ each is (A) 6 x2 + 12 y2 = 1 (B) 12 x2 + 6 y2 = 1 (C) 3 x2 + 12 y2 = 1 (D) 9 x2 + 12 y2 = 1
18. The line y = x 1 touches the ellipse 3 x2 + 4 y2 = 12, at (A) (1/2, 1/2) (B) (3, 2) (C) (1, 2) (D) None of these 19. The equation of common tangents to the curves x2 + 4 y2 = 8 and y2 = 4 x are (A) 2 y x 4 = 0 and 2 y + x + 4 = 0 (B) y 2 x 4 = 0 and y + 2 x + 4 = 0 (C) 2 y x 2 = 0 and 2 y + x + 2 = 0 (D) y 2 x 2 = 0 and y + 2 x + 2 = 0
20. If the line y = mx + c is a tangent to the ellipse 2 2
2 2
x y 1a b
then corresponding point of
contact is
(A)2 2a m b,c c
(B) 2 2a m b,
c c
(C)2 2a m b,c c
(D) 2 2a m b,
c c
LEVEL-III
1. The length of the major axis of the ellipse
2
2 2 (3x 4y 7)(5x 10) (5y 15)4
is
(A) 10 (B) 20/3 (C) 20/7 (D) 4 2. An ellipse has eccentricity 1/2 and one focus at the point P(1/2, 1). One of Its
directrix is the common tangent, nearer to the point P to the circle x2+y2 = 1 and the hyperbola x2-y2 = 1. Area of the ellipse is
(A) (B) 2 2
(C) 23 3 (D) none of these.
3. If the normal at the point P() to the ellipse 2 2x y 1
14 5 intersects it again at the point
Q(2), then cos = (A) 2/3 (B) -2/3 (C) 1/3 (D) -1/3 4. The equation of the ellipse centered at (1, 2) having the point (6, 2) as one of its focus
and passing through the point (4, 6) is
(A) 2 2x 1 3 y 2
136 64
(B) 2 2x 1 y 2
145 20
(C) 2 2x 1 y 2
118 32
(D) 2 2x 1 7 y 2
172 128
5. The tangent drawn to the ellipse 2x 11y 1
16 256
at the point P (); touches the circle
(x 1)2 + y2 = 16; then ‘’ equal to (A) /6 (B) /4 (C) /3 (D) None of these 6. Length of latus rectum of the ellipse, 3 (x + y 5)2 + 2 (x y + 7)2 = 6 is
(A) 243
(B) 223
(C) 13
(D) 23
7. Focii of the ellipse; 3 (x + y 5)2 + 2 (x - y + 7)2 = 6 are (A) (1/2, 13/2) and (3/2, 11/2) (B) (1/2, 11/2) and (3/2, 13/2) (C) (1/2, 11/2) and (-3/2, 11/2) (D) (1/2, 11/2) and (3/2, 13/2)
8. Locus of the midpoint of chords of the ellipse 2 2
2 2
x y 1a b
that are parallel to the line
y = 2 x + c, is (A) 2 b2 y a2 x = 0 (B) 2 a2 y b2 x = 0 (C) 2 b2 y + a2 x = 0 (D) 2 a2 y + b2 x = 0
9. Consider an ellipse2 2
2 2
x y 1a b
, centered at point ‘O’ and having AB and CD as its major
and minor axes respectively. If S 1 be one of the focus of the ellipse, radius of incircle of triangle OCS1 = 6 units, then area of the ellipse is equal to
(A) 16 sq. units (B) 654 sq. units
(C) 652
sq. uints (D) 65 sq. units
10. ‘P’ is any variable point on the ellipse 2 2
2 2
x y 1a b
having the points S1 and S2 as its foci .
maximum area of the triangle PS1S2 is (A) b2 c sq. units (B) a2 c sq. units (C) ab sq. units (D) abc sq. units 11. Consider an ellipse having its axes as coordinate axes and passing through the point (4, 1). If the line x +4y 10 = 0 is one of its tangent, then area of ellipse is (A) 10 (B) 20 (C) 25 (D) 15
12. S1 and S2 are foci of an ellipse ‘B’ be one of the extremity of its minor axes. If S1 S2 B is right angled then eccentricity of the ellipse is equal to
(A) 32
(B) 12
(C) 32
(D) None of these
13. If ‘L’ is the length of perpendicular drawn from the origin to any normal of the ellipse
2 2x y 1,25 16
then maximum value of ‘L’ is
(A) 5 (B) 4 (C) 1 (D) None of these
14. The maximum distance of the centre of the ellipse 2 2x y 1
9 4 from the chord of contact
of mutually perpendicular tangents of the ellipse is
(A) 913
(B) 313
(C) 613
(D) 3613
15. Tangents PA and PB are drawn to the ellipse 2 2x y 1
16 9 from the point P (0, 5). Area of
triangle PAB is
(A) 165
(B) 25625
(C) 352
(D) 102425
16. The straight line x 2y + 4 = 0 is one of the common tangents of the parabola y2 = 4x
and2 2
2
x y 14 b . The equation of the another common tangent to these curves is
(A) x + 2y + 4 = 0 (B) x + 2y 4 = 0 (C) x + 2y + 2 = 0 (D) x + 2y 2 = 0
17. A variable tangent of the ellipse 2 2x y 1
16 36 meets the tangents drawn at the extremities
of the major axis at point A1 and A2 Circle drawn on A1A2 as diameter will always pass through two fixed points whose coordinates are
(A) (0, 6) (B) (0, 5 2 ) (C) (0, 2 5 ) (D) (0, 4)
18. There are exactly two points on the ellipse 2 2
2 2
x y 1a b
whose distance from the center of
the ellipse are equal and equal to2 2a b
2 . Eccentricity of this ellipse is
(A) 32
(B) 23
(C) 13
(D) 23
19. For all admissible values of the parameter ‘a’ the straight line 2 ax + y 21 a 1 will
touch an ellipse whose eccentricity is equal to
(A) 32
(B) 23
(C) 32
(D) 23
20. The normal to the ellipse 4 x2 + 5 y2 = 20 at the point ‘P’ touches the parabola y2 = 4x,
the eccentric angle of ‘P’ is
(A) 1 1sin5
(B) 1 1tan2 5
(C) 1 1tan5
(D) 1 1cos
5
ANSWERS LEVEL −I 1. C 2. C 3. C 4. C 5. 6. C 7. A 8. C 9. B 10. C 11. A 12. D 13. C 14. B 15. D 16. A 17. A 18. C 19. A 20. D LEVEL −II 1. C 2. D 3. C 4. B 5. D 6. C 7. C 8. A 9. B 10. A 11. B 12. C 13. D 14. A 15. C 16. C 17. D 18. D 19. A 20. D LEVEL −III 1. B 2. D 3. B 4. B 5. C 6. B 7. A 8. D 9. B 10. A 11. A 12. B 13. C 14. A 15. B 16. A 17. C 18. D 19. A 20. D
FUNCTION
LEVEL−I
1. Let f(x) = ln(2x – x2) + sin2x , then
(A) Graph of f is symmetrical about the line x = 1 (B) Graph of f is symmetrical about the line x = 2 (C) maximum value of f is 1 (D) minimum value of f does not exist . 2. The domain of definition of f(x) = sec-1(cos2x) is (A) m, m I (B) /2 (C) /4 (D) none of these.
3. The period of f(x) = xcoscosxsincos21
is
(A) (B) /2 (C) /4 (D) 2
4. Domain of f (x) = 2xxlog 2
21
x
is , where [.] denotes the greatest integer function.
(A)
,23 (B) (2, )
(C)
2,23 (D) none of these
5. f(x) =
0x,K
0x,x if f(x) is having minimum value –10 then k =
(A) 2 (B) –10 (C) 9 (D) not possible
6 Domain of cos-1 [2x2 – 3] where [.] denotes greatest integer function, is
(A)
25,1 (B)
1,
25
(C)
25,11,
25 (D) None of these.
7. Which of the following function(s) from f : A A are invertible, where A = [-1,1]: (A) f(x) = x/2 (B) g(x) = sin (x/2) (C) h(x) = |x| (D) k(x) = x2 8 Solution of 0 < |x-3| 5 is (A) [-2,8] (B) [-2,3) U (3,8] (C) [-2,3) (D) none of these
9. Solution of
)6x(|4x|7x5x3x
0 is
(A) (-6,-5] U [3, 7) U (4, 7) (B) [3,7] (C) (-6,-5] (D) [3,4) (4,7]
10. f f(x) is a function that is odd and even simultaneously, then f(3) – f(2) is equal to (A) 1 (B) -1 (C) 0 (D) none of these 11 If f(x) and g(x) be two given function with all real numbers as their domain, then h(x) = (f(x) + f(-x)) (g(x) – g(-x)). is (A) always an odd function (B) an odd function when both the f and g are odd (C) an odd function when f is even and g is odd (D) none of these 12 If f(x) = sin{x}, f: R R , then f is (A) periodic (B) one-one (C) many-one (D)none of these
13 If f(x) = sin-1
2
2
x1x
then the range of f(x) is
(A) [-/2,/2] (B) [0,/2] (C) [0,/2) (D) [-/2,0)
14. If the period of )n/xtan(
)nxsin( , where nI, is 6, then
(A) n = 4 (B) n = -3 (C) n = 3 (D) none of these
15. If f(x) = {x} + sin ax (where { } denotes the fractional part function) is periodic, then (A) ‘a’ is a rational multiple of (B) ‘a’ is a natural number (C) ‘a’ is any real number (D) ‘a’ is any positive real number
16. If f(x) = sin ]a[ x, (where [.] denotes the greatest integer function), has as it’s fundamental period, then
(A) a = 1 (B) a [1,2) (C) a = 9 (D) a [4,5)
17. Range of the function f(x) = 2xx
1
is . . . . .
18 The function f (x) =
0x,x0x,x
is ({.} : fractional part}
(A) even (B) odd (C) neither (D) none of these 19 Period of |sin 2x| + |cos 8x| is: (A) /2 (B) /8 (C) /16 (D) None of these.
20 The domain of f(x) =
4
xx5log2
41 + 10Cx is
(A) (0, 1]U [4, 5) (B) (0, 5) (C) {1, 4} (D) None of these
21 The expression a1a is equal or …….…………. than ……………….… for
………….……..values of a. 22 The absolute value of an expression is always ……………………………….……………. 23 | x + y| = | x | + | y | holds good if and only if x and y are ………………………………….. 24 The solution of | x–3| = x is …………………………………………………….. 25 logba is meaningful only if a is ……………………….. and b is …………………..… or ………………………. 26 If Nlog ka = y logaN then y = …………………………………………………. 27 The expression ax2 + bx +c > 0 x R implies that a is …………………….… and …...………………...
28 The domain of f(x) = 1xx2
is
(A) (-1, 2) (B) R – (-1, 2] (C) R – [ -1, 2) (D) ( -1, 2] 29 The range of y = xsincoslog3 contain(s) (A) one element (B) infinitely many elements (C) the function is undefined (D) none of these
30 The domain and range of f(x) = x3cos2
1
are respectively
(A) R - 3
1n2 , R (B) R, R – [ 1/3, 1]
(C) R, [1/3, 1] (D) none of these 31 The equation x > [x] holds true for, where [ - ] denotes GIF (A) all integral values of x (B) all x R (C) all positive integers (D) R – I 32 The function and its inverse (A) are symmetric about y = x line (B) meet each other along the line y = x (C) are symmetric about y + x = 0 line (D) never intersect each other . 33 Let f (-x) = f (x). Then f (x) must be (A) an even function (B) an odd function (C) a periodic function (D) neither odd nor even
34. If f(x) = irrationalisxwhenx1
rationalisxwhenx, then fof (x) is given as
(A) 1 (B) x (C) 1 + x (D) None of these 35. If x -{x} = 2 then x belongs to…………………………………………………………………….
36. Domain of the function f (x) = xsincoslog3 is ……………………………………………… 37. If f (x) = cos []x + cos [x], where[.] stands for greatest integer function, then f (/2) equals to…………………………………………………………………………………………………….. 38 Solution set of inequation cos x -1/2 is
(A)
32n2,
32n2 (B)
32n2,
32n2
(C)
3
2n,32n (D) none of these
39 Solution set of inequation tan x > 3 is
(A) 2
x3
2n
(B)
2nx
3n
(C) 3
n2x3
n2
(D) none of these
40 Range of f(x) = sin– 1 1xx2 is
(A)
2
,3
(B)
4
,3
(C)
2
,3
(D) none of these
41 Let f(x) = sinx + cos
2a4 x . Then the integral values of ‘a’ for which f(x) is a periodic
function are given by (A) {2, – 2} (B) [– 2, 2) (C) (–2, 2) (D) none of these 42. The function f(x) = (1 – x)1/3 is (A) one– one & onto (B) many– one & onto (C) one– one & into (D) many– one & into 43 Let f: R R be any function. Define g : R R by g(x) = |f(x)| for all x, then g is (A) onto if f is onto (B) one– one if f is one– one (C) continuous if f is continuous (D) differentiable if f is differentiable 44. The domain of definition of f(x) = sec-1(cos2x) is (A) m , m I (B) /2 (C) /4 (D) none of these. 45. Which of the following functions is /are periodic (A) Sgn (e–x) (B) sinx + |sinx|
(C) min (sinx, |x|) (D) ]x[221x
21x
Where [x] denotes the greatest integer function 46 The function defined as f : [0, ] [–1, 1], f (x) = cos x is (A) one-one onto (B) many-one onto (C) one-one into (D) many-one into 47 Find the period of the function f (x) = cos [2]x + cos [–2]x
(A) (B) 2
(C) 2
(D) 4
3
48 y = log|x| |x|, then find the domain (A) R (B) R – {–1, 1} (C) R – {0} (D) R – {0, –1, 1}
49. The range of the function f (x) = 1x
x4
2
is
(A)
21,0 (B) 10,
2
(C) (0, ) (D) (0, 2]
50. [sin x] = [cos x] for all x 2k
, k is an integer
(A) true (B) false 51. If f (x) is an invertible function then (f o f–1) (x) = x for all x R (A) true (B) false 52. The range of the function ln (x2-2x+6) is (A) (n6, ) (B) [ln 5, ) (C) (0, ) (D) R (set of real numbers) 53. Domain of log1/2log4log3[( x - 4)2] is, [.] denotes the integer function . (A) (- , 2] [ 6, ) (B) (- , 2] [ 6, 8) (C) ( 2, 6) (D) [ 2, 6]
54. The graph of y = x + x1 is symmetrical
(A) about x – axis (B) about y - axis (C) in opposite quadrants (D) None of these 55. Period of the function |cos 2x| is (A) 2 (B)
(C) 2 (B)
4
56. The domain of f (x) = sin–1 (|x – 1| – 2) is (A) [–2, 0] [2, 4] (B) (–2, 0) (2, 4) (C) [–2, 0] [1, 3] (D) (–2, 0) (1, 3) 57. If f (x) = x2, g (x) = x , then what is g o f (x) is (A) |x| (B) x (C) –x (D) –|x| 58. Minimum of ]27)3x[( 32
2 is (A) 227 (B) 1 (C) 2 (D) 2–27
59. The function defined as f :
2,
2 [–1, 1], f (x) = sin x is
(A) one-one onto (B) many-one onto
(C) one-one into (D) many-one into
60. The range of the function f (x) = |3x|
3x is
(A) {–1, 1} (B) R (C) R – {3} (D) R – {–1} 61. The solution set of log {x} = 0 is (A) {} (B) [1, –1] (C) (0, –1) (D) [0, 1]
62. The domain of the function f (x) = |x|]x[
1
is
(A) [0, ) (B) R (C) (–, 0] (D) {}
63. If f (x) = x1
1
, then f [f {f (x)}] is
(A) x – 1 (B) 1 – x (C) x (D) –x
64. The value of x for log1/3
x2x < – 1 lies in
(A) (0, 1) (1, ) (B) (0, 1) (2, ) (C) (0, 1) [2, ) (D) (0, 1] [2, ) 65. The range of the function f (x) = 11 – 3 sin x is (A) [6, 14] (B) [8, 14] (C) [8, 12] (D) [8, 11]
66. The period of the function f (x) = {x} + sin 3 x + tan 2x
(A) 1 (B) 2 (C) 3 (D) not periodic
67. The domain of the function f (x) = ]x[xsin 1
is
(A) [–1, 0) {1} (B) (–, 0) {1} (C) (–1, 0) {1} (D) not defined
68. If f (x) = [x] and g (x) = |x|, then g o f
35 – f o g
35 is
(A) 0 (B) –1 (C) 1 (D) none of these 69. Which of the following is not periodic? (A) f (x) = cos x (B) f (x) = |cos x| (C) f (x) = cos x2 (D) f (x) = cos2 x 70. The solution set of log [x] = 0 is (A) [1, 2) (B) [1, 2] (C) (1, 2] (D) (1, 2)
71. The domain of the function f (x) = 2]x[2]x[
is
(A) R (B) R – {2} (C) R – [2, 3) (D) not defined
72. Domain of function f(x) = 2xx1log
1
10
.
(A) (−3, −2) −
25 (B) [0, 1] −
21
(C) [−2, 1) − {0} (D) none of these
73. If f(x) = xx3x2
1x223
, interval when f(x) 0
(A) R (B) R − [−1, 0] (C) R+ (D) none of these
74. Domain of function f(x) =
21x3sin3x21 1
(A)
21,1 (B)
21,
31
(C)
1,31 (D)
1,21
75. Which of the following function is non-periodic (A) f(x) = {x} (B) f(x) = cot(x + 7)
(C) f(x) = 1 − xcot1
xsin2
−
xtan1xcos2
(D) f(x) = x + sinx
76. Let f(x) = x2 and g(x) = x then (A) gof(−2) = −2 (B) gof(4) = 4 (C) gof(3) = 6 (D) gof(2) = 4 77. The domain of f(x) = 2xx2log is, x = (A) 1 (B) 2 (B) 3 (D) none of these
78. The range of f(x) = x33x
, x 3 is
(A) R (B) R − {−1} (C) R − {1} (D) none of these
79. The range of f(x) = x1
x
is
(A) R − {−1, 1} (B) R (C) R − {1} (D) none of these
80. Let f(x) = 1x1x
, x −1 then f−1(x) is
(A) x1x1
(B)
x1x1
(C)
x11
(D) none of these
81. If f(x) = 1 + x, 0 is the inverse of itself then the value of is (A) −1 (B) 1 (D) 2 (D) none of these
82. The value of n I for which the function f(x) =
nxsin
nxsin has 4 as its period is equal to
(A) 2 (B) 2 (C) 1 (D) none of these
LEVEL−II 1. Which of the following is correct? (A) sin1 > sin2 (B) sin1 < sin2 (C) sin2 > sin3 (D) sin2 < sin3. 2. The range of the function sin2x – 5 sinx – 6 is (A) [– 10, 0] (B) [– 1, 1] (C) [0, ] (D) [– 49/4, 0]
3. If f(x) = (1 – xn)1/n, 0 < x < 1, n being an odd positive integer and h(x) = f(f(x)), then h
21 is
equal to (A) 2n (B) 2 (C) n. 2n-1 (D) none of these 4. If f : I I be defined by f(x) = [x + 1], where [.] denotes the greatest integer function, then
f-1(x) is equal to (A) x – 1 (B) [x + 1]
(C) 1x1
(D) 1x
1
5. Which pair of functions is identical? (A) sin-1(sinx), sin(sin-1x) (B) lnex, elnx (C) lnx2, 2 lnx (D) none of these. 6. If g is the inverse function of f and f(x) = sinx, then g(x) is equal to (A) sin(g(x)) (B) cosec(g(x)) (C) tan(g(x)) (D) none of these. 7. Value(s) of x for which tangent drawn to the curve f(x) = |1– 2 e-|x|| would be lying entirely
below the curve, is given by (A) x (ln2, ) (B) x ( – ln2 , 0) (C) x ( – , – ln2) (D) x ( 0, ln2) 8. Solution set of [sin-1x] > [cos-1x], where [.] denotes greatest integer function
(A) [sin1, 1] (B)
1,2
1
(C) (cos1, sin1) (D) None of these 9. If P(x) be a polynomial satisfying the identity P(x2) +2x2 +10x = 2x P( x+1) +3 , then
P(x) is (A) 2x +3 (B) 3 x- 4 (C) 3x + 2 (D) 2 x –3
10. Let f(x) =
2x,3x
2x,1x2
3
. Then
(A) f-1(x) =
2x,3x
2x,1x2/1
3/1
(B) f-1(x) =
7x,3x
7x,1x2/1
3/1
(C) f-1(x) =
7x,3x
1x,1x2/1
3/1
(D) f-1(x) does not exist
11. Which of the following is/are true, (you may use f(x) = xlnxlnln )
(A) (ln 2.1)ln2.2 > (ln 2.2)ln2.1 (B) (ln 4)ln5 < (ln5)ln4 (C) (ln30)ln31 > (ln31)ln30 (D) (ln28)ln30 < (ln30)ln28 12. sin ax + cosax and |sinx| + |cosx| are periodic functions of same fundamental period if a
equals (A) 0 (B) 1 (C) 2 (D) 4
13. If {x} denotes the fractional part of x, then
154 n2
, nN, is
(A) 151 (B)
1514
(C) 87 (D) None of these
14. If f (x) = minimum {sin x, cos x} x R. then range of g (x) = [f (x)] is, [ ] denotes the
greatest integer function (A) {–1, 0, 1} (B) {0, 1} (C) {–1, 0} (D) none of these 15. If f(x-1/x) = x2 + 1/x2, x 0, then f(x) is (A) is an even function (B) always greater or equal to2 x R (C) onto if f : R [3, ) (D) none of these
16 .If f(x) =
002
xforxxforx , then fof(x) is given by
(A) x2 for x 0, x for x < 0 (B) x4 for x 0, x2 for x < 0 (C) x4 for x 0, -x2 for x < 0 (D) x4 for x 0, x for x < 0
17. The range of the function f(x) = sin-1
21x2 + cos-1
21x 2 , where [.] is the greatest
integer function, is
(A)
,2
(B)
2,0
(C) (D)
2
,0
18. If |x| + [x] = 2x (where [.] denotes the greatest integer function), then number of solutions of the equation in [-1,1) is/are
(A) one only (B) infinitely many (C) two only (D) none of these
19. If f(x) = cos
2xsinx , (where [.] denotes the greatest integer function), then
(A) f(x) is periodic (B) f(x) is odd (C) f(x) is even (D) f(x) is non-periodic
20. Let f : (2,4) (1,3) where f(x) = x – [x/2] (where [.] denotes the greatest integer function), then f -1(x) is
(A) not defined (B) x – 1 (C) x + 1 (D) none of these
21. The fundamental period of cos (cos 2x) + cos (sin3x) is (A) (B) 2 (C) /4 (D) /2
22. Let f : R R, where f(x) = 2|x| – 2-x, then (A) Range of f(x) is all non-negative R (B) f(x) is many-one (C) f(x) is into (D) f(x) is non-periodic8
23. If f(x) = xsinxsin
1x4 2
, then the domain of f(x) is
(A) [-2,0] (B) (0,2] (C) [-2,2} (D) [-2,0)
24. Number of real roots of 3x + 4x + 5x – 6x = 0 is/are (A) two (B) more then two (C) one (D) equation will not have any real root 25. Range of function [ | sinx| + | cosx| ] , where [.] denotes the greatest integer function is .
. . . 26. f: { x, y, z } { a, b, c} be a one one function. It is known that only one of the
following statements is true (i) f(x) b (ii) f(y) = b (iii) f(z) a then f-1(a) = (A) x (B) y (C) z (D) none of these
27. The function f(x) = 12x
1ex
x
is
(A) even (B) odd (C) neither even nor odd (D) none of these 28. Let f : [-10,10] R, where f(x) = sin x + [x2/a] be an odd function. Then set of values of
parameter ‘a’ is/are: (A) (-10,10) ~ {0} (B) (0,10) (C) [100,) (D) (100,) 29. If fog = |sin x| and gof = sin2 x then f(x) and g(x) are: (A) f(x) = xsin , g(x) = x2 (B) f(x) = |x|, g(x) = sin x
(C) f(x) = x , g(x) = sin2x (D) f(x) = sin x , g(x) = x2 30. If f(x) + 2f(1- x) = x2 + 2 xR, then f(x) is given as
(A) 32x 2 (B) x2 – 2
(C) 1 (D) None of these 31. Let f (x) be a function whose domain is [-5, 7]. Let g (x) = |2x + 5|, then the domain of fog (x) is
(A) [-5, 1] (B) [-4, 0] (C) [-6, 1] (D) none of these 32. Let f:[-/3, 2/3] [0,4] be a function defined as f(x) = 3 sin x – cos x + 2. Then f -1(x) is
given by
(A) sin-1 62
2x π
(B) sin-1 62
2x π
(C) 3
2 – cos-1
2
2x (D) None of these.
33. The function f : R R, f(x) = xx
xx
eeee
is
(A) one-one and onto (B) one-one and into (C) many-one and onto (D) many-one and into
34. The function f(x) =
1x|,x|x1x1,x1x1
1x|,x|x is (where [ - ] denotes GIF)
(A) even (B) odd (C) neither even nor odd (D) symmetric with y-axis
35. Let f(x) =
x2forxeccosxtan2xforb
2x0forxcosxsin
2
, Then its odd extension is
(A) -tan2 x – cosec x, – <x<- 2 (B) - tan2 x + cosec x, – < x <- 2
-b for x = - 2 -b for x = - 2
-sin x + cos x for - 2 < x < 0 sin x – cos x for - 2
< x < 0
(C) -tan2 x + cosec x, - < x < - 2 (D) None of these
b for x = - 2
sin x – cos x , - 2 < x< 0
36. Period of the function f (x) = cos (cos x) + cos (sin x) is…………………………………………. 37. Let f : (-, 1] (-, 1] such that f (x) = x (2 –x). then f-1 (x) is (A) 1 + x1 (B) 1- x1 (C) x1 (D) none of these 38. Number of solutions of the equation cos x = |x|, x [-/2, /2] is (A) 1 (B) 2 (C) 3 (D) 4
39. The number solutions of equation tan x = x in interval
23,0
(A) 1 (B) 2 (C) 3 (D) 4
40. Let f be a function satisfying f(x +y) =f(x).f(y) for all x,y R. If f(1) =3 then
n
1r)r(f is equal to
(A) )13(23 n (B) )1n(n
23
(C) 3 n+1 – 3 (D) None of these 41. Let f: R R be a function such that f(x) = x3 +x2 + 3x + sinx. Then (A) f is one– one and into (B) f is one– one and onto (C) f is many– one and into (D) f is many– one and onto 42. Let f(x) = sec–1[1 +cos2x] where [.] denotes the greatest integer function. Then (A) the domain of f is R (B) the domain of f is [1,2] (C) The range of f is [1,2] (D) the range of f is [sec– 11, sec– 12] 43. Range of the function f(x) = bxxa , where a > b > 0
(A) (– , ba ] (B) [ ba , ba2 ]
(C) [ ba , ) (D) none of these 44. If |f(x) + 6 – x2| = |f(x)|+ |4 – x2| + 2, then f(x) is necessarily non– negative in (A) [– 2, 2] (B) (– , – 2) (2, ) (C) [– 6, 6] (D) none of these
45. The period of f(x) = xcoscosxsincos21
is
(A) (B) /2 (C) /4 (D) 2 46. Total number of roots of the equation 3cosx = |sinx|, belonging to [ - 2, 2], are; (A) 6 (B) 8 (C) 10 (D) 12 47. If f(x) = [x2] – [x]2, where[.] denotes the greatest integer function, and x [0,2], then the set
of values of f(x) is (A) {-1, 0} (B) {-1,0,1} (C) {0} (D) {0,1,2}
48. Range of f (x) = 2 cos 22
x9
is
(A) [–1, 2] (B) [1, 0] (C) (0, 1) (D) [1, 2] 49. If [x]2 – 5[x] + 6 = 0, then x belongs to (A) [2, 4) (B) [2, 4) – {3} (C) {3} (D) {2}
50. Range of y = cos–1 xsin2
2
is
(A)
2
,0 (B)
32cos,0 1
(C) [0, cos–1 2] (D)
,32cos 1
51. Number of solution of sin x + cos x = 2 are (A) 1 (B) 2 (C) 0 (D) infinite 52. The period of the function f (x) = 2 + (–1)[x] is (A) 1 (B) 0 (C) 2 (D) 0.5 53. The number of solutions of |ln |x|| = 2x5 is (A) 1 (B) 2 (C) 3 (D) 4
54. The function f (x) = 1xx7x4x
2
2
, where f : R R is
(A) one-one into (B) many-one into (C) one-one onto (D) many-one onto 55. Total number of solutions of 2|cos x| = 3|sin x|, belonging to the interval [–10, 10] are; (A) 20 (B) 40 (C) 80 (D) none of these
56. If f: [1,) [2,) is given by f(x) = x + x1 then f−1(x) equals
(A) 2
4xx 2 (B) 2
4xx 2 (C) 2
4xx 2 (D) none of these
57. The solution of the inequality log1/2sin-1x > log1/2 cos-1x is
(A) x
21,0 (B) x
1,
21
(C) x
21,0 (D) None of these
58. Total number of roots of the equation 7|x| (|5 − |x||) = 1, are; (A) 6 (B) 8 (C) 4 (D) 12 59. The range of the function f(x) = 4x + 2x + 4-x + 2-x + 3 is (A) [3/4, ) (B) (3/4, ) (C) (7, ) (D) [7, ) 60. Let reflection of function f(x) = (4 − (x − 7)3)1/5 about a line y = x is g(x) then (A) g(x) = 7 − (4 − x3)1/5 (B) g(x) = x (C) g(x) = − x2 + 1 (D) g(x) = 7 + (4 − x5)1/3 61. The period of the function f(x) = sin4x + cos4x (A) (B) /4 (C) /2 (D) 2 62. The function f: R R given by f(x) = x3 + ax2 + bx + c is one−one if (A) a < b (B) a2 < 3b (C) a2 > 3b2 (D) a2 = c2
63. Let f(x) = 3x2x
is an invertible function then domain f−1(x) is
(A) R+ (B) R − {3}
(C) R − {1} (D) none of these
64. Let g(x) = 1 + x − [x] and f(x) =
0x,10x,00x,1
. Then for all x, fog(x) is equal to
(A) x (B) 1 (C) f(x) (D) g(x)
LEVEL−III
1. If the derivative of f(x) w.r. t. x is xf
xsin21 2
, then f(x) is a periodic function with period
(A) (B) 2 (C) /2 (D) none of these.
2. If tan-1(x + h) = tan-1(x) + (h siny) (siny) – (h siny)2.2
y2sin + (h siny)3.3
y3sin + ….,
where x ( 0, 1), y (/4, /2) , then (A) y = tan-1x (B) y = sin-1x (C) y = cot-1x (D) y = cos-1x
3. The domain of the function f(x) = xlncosxlnsin
x 2/1
is
(A) In
21n3
n2 e,e
(B)
In
45n2
41n2
e,e
(C) In
43
n341
n2e,e
(D)
In
43
n343
n2e,e
4. If f(x) = log[x-1][ x|x| , where [.] denotes greatest integer function, then
(A) domain of f = ( 2, ) (B) range of f = { 0, 1} (C) domain of f = [ 3, ) (D) range of f = {0} 5. Let f(x) = sinx + ax + b. Then f(x) = 0 has (A) only one real root which is positive if a > 1, b < 0 (B) only one real root which is negative if a > 1, b > 0 (C) only one real root which is negative if a < -1, b < 0 (D) none of these. 6. If f(x) = [x2] – [x]2 , where [.] denotes the greatest integer function, and x [ 0, n], n N,
then the number of elements in the range of f(x) is (A) 2n + 1 (B) 4n – 3 (C) 3n – 3 (D) 2n – 1 7. Total number of solutions of x2 –2x –[x] = 0 is equal to (A) 2 (B) 4 (C) 6 (D) none of these
8. If f(x) = |xcos||xsin|1
(where [.] denotes the greatest integer function), then
(A) f(x) is an even function (B) f(x) is an odd function (C) range of f(x) contains only one element (D) none of these
9. Let f and g be functions from the interval [0, ) to the interval [0, ), f being an increasing
function and g being a non-increasing function. If f{g(0)} =0 then (A) f{g(x)} f{g(0)} (B) g{f(x)} g{f(0)} (C) f{g(2)} =0 (D) None of these
10. If P(x) be a polynomial satisfying the identity P(x2) +2x2 +10x = 2x P(x+1) +3 , then P(x) is
(A) 2x +3 (B) 3 x- 4 (C) 3x + 2 (D) 2 x –3
11. If ksin2x + k1 cosec2x = 2, x (0, /2), then cos2x +5 sinx cosx + 6 sin2x is equal to
(A) 2
2
k6k5k (B)
2
2
k6k5k
(C) 6 (D) none of these 12. The number of distinct values of f (x) = [x3] – [x]3 for x [0, 2] (A) 4 (B) 5 (C) 7 (D) 8 13. If f(x) is an odd function also periodic function with period 2 then f(4) equal to (A) 1 (B) 2 (C) 0 (D) none of these 14. Domain of f(x) satisfying ( )2 2 2x f x is (A) (, –1) (B) [0, 1] (C) (–1, 1) (D) (–, 1) 15. If f : R R, where f(x) = ax + cosx is an invertible function then (A) a (–2, –1] [1, 2) (B) a [–2, 2] (C) a (–, –1] [1, ) (D) a [–1, 1] 16. Total number of solutions of 2 4 [ ] 0x x (where [ ] denotes G. I. F.) is (A) 0 (B) 1 (C) 2 (D) 3 17. The fundamental period of f(x) = [x] + [2x] + [3x] + …. +[nx]; where x N and [ ] G. I. F.;
is (A) 1 (B) n (C) 1/n (D) Non-periodic
18. If 2 ; 0
( )2 ; 0
x xf x
x x
; then f(f(x)) is given by
(A) 2 , 02 , 0
x xx x
(B) 4 , 04 , 0
x xx x
(C) 4 , 04 , 0
x xx x
(D) 2 , 02 , 0
x xx x
19. Period of f(x) = x – [x + a] + b + a sin(2x); where a, b R and [ ] denotes G. I. F.; os (A) (B) 1 (C) a + b (D) Don’t exist 20. : R (0, /2] where (x) = 1 2cot ( )x x a complete set of values of ‘a’ such that (x) is
onto is:
(A) [3/4, ) (B) [1/2, ) (C) [1, ) (D) [1/4, )
ANSWERS LEVEL −I 1. A, C, D 2. A 3. B 4. B 5. D 6. D 7. A, B 8. A 9. A 10. C 11. A 12. A 13. C 14. C 15. A 16. D 17. [2, ) 18. A 19. /2 20. C 21. Greater, 2, all 22. positive
23. both positive & both negative 24. x = 32
25. positive, positive & 1 26. 1k
27. positive, D < 0 28. C 29. A 30. C 31. D 32. A 33. B 34. B 35. [2, 3] 36. x n 37. cos4 38. A 39. B 40. A 41. A 42. A 43. C 44. A 45. C 46. B 47. B 48. D 49. B 50. B 51. B 52. B 53. A 54. C 55. C 56. A 57. A 58. B 59. A 60. A 61. A 62. D 63. C 64. B 65. B 66. D 67. A 68. A 69. C 70. A 71. C 72. C 73. B 74. B 75. D 76. D 77. A 78. D 79. D 80. A 81. A 82. A LEVEL −II 1. C 2. A 3. D 4. A 5. D 6. B 7. B, D 8. A 9. A 10. B 11. C 12. D 13. A 14. A 15. A 16. D 17. A 18. C 19. A, C 20. C 21. A 22. A, C 23. D 24. C 25. 1 26. B 27. B 28. D 29. C 30. A 31. C 32. B
33. B 34. A 35. B 36. 2
37. B 38. B 39. B 40. A 41. B 42. A 43. B 44. A 45. B 46. B 47. D 48. D 49. A 50. B 51. C 52. C 53. D 54. B 55. B 56. A 57. C 58. C 59. D 60. D 61. C 62. B 63. C 64. B LEVEL −III 1. A 2. C 3. B 4. C 5. A, B, C 6. D 7. A 8. A, C 9. B 10. A 11. D 12. C 13. C 14. D 15. C 16. C 17. A 18. B 19. B 20. D.
FUNCTION
2. The domain of definition of f(x) = sec-1(cos2x) is (A) m, m I (B) /2 (C) /4 (D) none of these. 7. Which of the following function(s) from f : A A are invertible, where A = [-1,1]: (A) f(x) = x/2 (B) g(x) = sin (x/2) (C) h(x) = |x| (D) k(x) = x2 8 Solution of 0 < |x-3| 5 is (A) [-2,8] (B) [-2,3) U (3,8] (C) [-2,3) (D) none of these
9. Solution of )6x(|4x|7x5x3x
0 is
(A) (-6,-5] U [3, 7) U (4, 7) (B) [3,7] (C) (-6,-5] (D) [3,4) (4,7]
13 If f(x) = sin-1
2
2
x1x
then the range of f(x) is
(A) [-/2,/2] (B) [0,/2] (C) [0,/2) (D) [-/2,0)
14. If the period of )n/xtan(
)nxsin( , where nI, is 6, then
(A) n = 4 (B) n = -3 (C) n = 3 (D) none of these 19 Period of |sin 2x| + |cos 8x| is: (A) /2 (B) /8 (C) /16 (D) None of these. 40 Range of f(x) = sin– 1 1xx2 is
(A)
2
,3
(B)
4
,3
(C)
2
,3
(D) none of these
46 The function defined as f : [0, ] [–1, 1], f (x) = cos x is (A) one-one onto (B) many-one onto (C) one-one into (D) many-one into 55. Period of the function |cos 2x| is (A) 2 (B)
(C) 2 (B)
4
57. If f (x) = x2, g (x) = x , then what is g o f (x) is (A) |x| (B) x (C) –x (D) –|x|
63. If f (x) = x1
1
, then f [f {f (x)}] is
(A) x – 1 (B) 1 – x (C) x (D) –x
1. 20x x2cos1
xtanx2x2tanxlim
is
(A) 2 (B) –2 (C) 1/2 (D) –1/2
4. f(x) =
1x,1x
1x,cbxax2
. If f(x) is continuous for all values of x, then;
(A) b = 1, a + c = 0 (B) b = 0, a + c = 2 (C) b = 1, a + c = 1 (D) none of these 5. The equation of the tangent to the curve f (x) = 1 + e–2x where it cuts the line y = 2 is (A) x + 2y = 2 (B) 2x + y = 2 (C) x – 2y = 1 (D) x – 2y + 2 = 0
10. 20x xxcos1lim
is equal to
(A) (B) 1/4 (C) 1/2 (D) 1
11 x
lim 1x21x2
is equal to
(A) 1 (B) 0 (C) -1 (D) 1/2
12. x2tan
xlim 10x is equal to
(A) 0 (B) 1/2 (C) 1 (D)
13. If f(x) = (1 – xn)1/n , 0 < x < 1, n being an odd positive integer and h(x) = f(f(x)), then h
21 is
equal to (A) 2n (B) 2 (C) n. 2n-1 (D) 1 17 The number of points of non differentiability for the function f (x) = |log |x|| are (A) 2 (B) 4 (C) 5 (D) 3
18 x
|x|lim0x
=
(A) 0 (B) 1 (C) –1 (D) doesn’t exist 22 Function f (x) = tan x is continuous in the interval
(A) R –
In:
2)1n2( (B) R – {n : n I}
(C) R+ (D) R – {0}
27 The value of
x4
sinx4
cosxlimx
is
(A) 2 (B)
4 (C) 1 (D)
31 The value of x
balimxx
0x
is
(A) loge
ba (B) loge
ab (C) loge (ab) (D) none of these
32 If f (x) =
2x,nxsin
2x,1mx
is continuous at x = 2 , then
(A) m = 1, n = 0 (B) m = 2
n + 1 (C) n = 2
m (D) m = n = 2
34. The value of 2
24
x x)x1(x1lim
is equal to
(A) 0 (B) –1 (C) 2 (D) 1 3. Area of the triangle formed by the positive x-axis and the normal and the tangent to
x2 + y2 = 4 at (1, 3 ) is
(A) 2 3 sq. units (B) 3 sq. units
(C) 4 3 sq. units (D) none of these
4. A tangent to the curve y = 2x
2 which is parallel to the line y = x cuts off an intercept from the y-axis is
(A) 1 (B) –1/3 (C) 1/2 (D) –1/2 5. A particle moves on a co-ordinate line so that its velocity at time t is v (t) = t2 – 2t m/sec. Then
distance travelled by the particle during the time interval 0 t 4 is (A) 4/3 (B) 3/4 (C) 16/3 (D) 8/3 11. The greatest and least values of the function f(x) = ax + b x + c, when a > 0, b > 0, c > 0 in the
interval [0,1] are (A) a+b+c and c (B) a/2 b2+c, c
(C) 2
cba , c (D) None of these
12. The absolute minimum value of x4 – x2 – 2x+ 5
(A) is equal to 5 (B) is equal to 3 (C) is equal to 7 (D) does not exist
13. Through the point P (, ) where >0 the straight line 1by
ax
is drawn so as to form with co-
ordinates axes a triangle of area S. If ab >0, then the least value of S is (A) 2 (B) 1/2 (C) (D) None of these
HYPERBOLA
1. If e, e are the eccentricities of hyperbolas 2 2 2 2
2 2 2
x y y x1 and 1,a b b a
then
(A) e = e (B) e = –e
(C) e e = 1 (D) 2 2
1 1 1e e
2. Centre of the hyperbola x2 + 4 y2 +6xy +8 x- 2y + 7 = 0 is , (A) (1, 1) (B) ( 0, 2) (C) (2, 0) (D) none of these . 3. The eccentricity of the hyperbola 2x2 –y2 = 6 is (A) 2 (B) 2 (C) 3 (D) 3
4. The radius of the director circle of the hyperbola 2 2
2 2
x y 1a b
is
(A) a –b (B) a b (C) 2 2a b (D) 2 2a b 5. The tangent to the curve x = a( – sin ); y = a(1 + cos ) at the points = (2k +
1), k Z are parallel to (A) y = x (B) y = –x (C) y = 0 (D) x = 0
6. The legth of latus rectum for hyperbola 2 2x y 1
16 9 is
(A) 323
(B) 92
(C) 83
(D) none of these
7. The straight line y = 3x+ c will be tangent to hyperbola 2 2x y 1
25 16 if c2 is
equal to (A) 119 (B) 225 (C) 209 (D) 144
8. Coordinates of the foci of the hyperbola 2 2x 1 y 2
19 16
are
(A) (1, 7) and (1, 3) (B) (6, 2) and (4, 2) (C) (1, 3) and (1, 7) (D) None of these
9. The eccentricity of the hyperbola passing through (3, 0) and (3 2 , 2) and having its axes along the coordinate axes is
(A) 136
(B) 132
(C) 133
(D) 134
10. The centre of the hyperbola 4 x2 8 x 5 y2 + 10 y = 21, is (A) (1, 1) (B) (1, 1) (C) (1, 2) (D) (2, 1) 11. Which of the following expressions ( t being the parameter) can’t represent a
hyperbola?
(A) tx y x tyt 0; 1 0a b a b
(B) a 1 b 1x t ; y t2 2 2 t
(C) t t t tx e e ; y e e
(D) 2 2 2 tx 2 cos t 3 ; y 2 2cos 12
12. Centre of the hyperbola 2
2x 1y 1
4
, is
(A) (0, 1) (B) (1, 0) (C) (2, 0) (D) (0, 2)
13. Centre of the hyperbola 2
2x 1 y 14 16
is
(A) (0, 1) (B) (1, 0) (C) (2, 0) (D) (0, 2) 14. Length of the latus rectum of the hyperbola xy = c2 is (A) 2 c (B) 4 c (C) 2 2 c (D) 2 c
15. Coordinates of the foci of the hyperbola: 2 2x 1 y 1
116 9
(A) (1, 7) and (1, 3) (B) (1, 3) and (1, 7) (C) (6, 2) and (4, 2) (D) (4, 2) and (6, 2) 16. Eccentricity of the hyperbola: 4 x2 8 x 5 y2 + 10 y = 21 is
(A) 53
(B) 43
(C) 35
(D) 34
17. Length of latus rectum of the hyperbola: 4 x2 8 x 5 y2 + 10 y = 21 is
(A) 58
(B) 12
(C) 2 (D) 85
18. Eccentricity of the hyperbola 2 2x 1 y 1
19 16
; is
(A) 54
(B) 53
(C) 43
(D) 32
19. Length of latus rectum of the hyperbola; 2 2x 1 y 2
19 16
; is
(A) 92
(B) 94
(C) 74
(D) 322
20. Centre of the hyperbola 2 2x y x y
14 9
; is
(A) (0, 0) (B) (1, 1) (C) (1, 1) (D) (1, 1)
ANSWERS
1. D 2. D 3 D 4. C 5. C 6. B 7. C 8. B 9. C 10. B 11. D 12. B 13. B 14. C 15. D 16. C 17. D 18. B 19. D 20. A
INDEFINITE INTEGRAL
1. If
xtanxcot1x4cos dx = k cos 4x + c, then
(A) k = –21 (B) k = –
81 (C) k = –
41 (D) none of these
2.
xcos1)xsin1(ex
dx is equal to
(A) log |tan x| + c (B) ex tan
2x + c (C) sin ex cot x + c (D) ex cot x + c
3. dxxcos is equal to
(A) 2[ x sin x + cos x ] + c (B) sin x + c (C) 2[ x cos x – sin x ] + c (D) none of these
4. xsinbxcosadx
2222 is equal to
(A) cxtanbatan 1
(B) cxcotabtan
ab1 1
(C) cxtanabtan
ab1 1
(D) cxtanabtan 1
5. dx)xtan1(xsecex is equal to
(A) ex sec x + c (B) ex sec x tan x + c (C) ex tan x + c (D) none of these
6.
dx
x1x
x1x 2
22/3
is equal to
(A) cx1x
25 2/5
(B) cx1x
52 2/5
(C) cx1x2
2/1
(D) none of these
7. bxaxdx =
(A) cbxaxba
1.32 2/32/3
(B) cbxax
ba1.
21 2/12/1
(C) cbxaxba
1.23 2/32/3
(D) none of these
8.
dx6x3x
1x3 3
2
=
(A) c6x3x21 2/13
(B) - c6x3x
21 2/13
(C) c6x3x21 3/23 (D) none of these
9. dxxsec4 =
(A) tanx + 3
xtan2
+ c (B) tanx + 3
xtan3
+ c
(C) tanx + 3
xtan4
+ c (D) 4
xtan4
+ c
10.
2/
0
36 dcossin =
(A) 652 (B)
632
(C) 631 (D)
1303
11. If dxxcosxsin
xcot = A xcot + B, then A =
(A) 1 (B) 2 (C) −1 (D) −2
12. If
x1xtan1x3x
1x2
124
2
dx = k log cx
1xtan2
1 , then k is equal to
(A) 1 (B) 2 (C) 3 (D) 5
13. xcosx2cos dx is equal to
(A) 2sinx + log|(secx tanx)| + c (B) 2sinx – log |(secx – tanx)| + c (C) 2sinx + log |(secx + tanx)| + c (D) 2sinx – log|(secx + tanx)| + c
14. xcos1xsin1e x
dx is
(A) xcos1
e x
+ c (B) excot
2x + c (C) ex tan
2x + c (D) None of these
15. dxx1x 2/52/13 is equal to
(A)
cx131x1
52x1
71
54 2/32/52/52/52/72/5
(B)
cx1x151x1
71
54 2/32/52/52/52/72/5
(C)
cx1x152x1
54 2/32/52/52/52/72/5
(D) none of these
16. If cxf21xdxcosxf 2 , then f(x) can be
(A) x (B) 1 (C) cosx (D) sinx 17. The value of the integral xcosxcose 3xsin2
sinx dx is
(A) cxsin3e21 2xsin2
(B) cxcos211e 2xsin2
(C) cxsin2xcos3e 22xsin2 (D) cxsin3xcos2e 22xsin2
18 2xx2
xd is equal to
(A) sin–1 (1 - x) + c (B) – cos–1 (1 – x) + P (C) sin-1 (x – 1) + c (D) cos-1 (x – 1) + P
19. I = dx
e1dx
x is equal to
(A) loge
x
x
ee1 + c (B) loge
x
x
e1e + c
(C) loge (ex) (ex +1) + c (D) loge 1e x2 + c
20. I = dxx1
xx1e 2
2xtan 1
is equal to
(A) cex xtan 1
(B) x e cx tan
(C) x1 ce xtan 1
(D) none of these
ANSWERS
1. B 2. B 3. A 4. C 5. A 6. B 7. A 8. C 9. B 10. A 11. D 12. A 13. D 14. C 15. A 16. D 17. 18. C 19. B 20. A
LCD
1. 20x x2cos1
xtanx2x2tanxlim
is
(A) 2 (B) –2 (C) 1/2 (D) –1/2
2. f(x) =
1x,1x
1x,cbxax2
. If f(x) is continuous for all values of x, then;
(A) b = 1, a + c = 0 (B) b = 0, a + c = 2 (C) b = 1, a + c = 1 (D) none of these 3. The equation of the tangent to the curve f (x) = 1 + e–2x where it cuts the line y = 2 is (A) x + 2y = 2 (B) 2x + y = 2 (C) x – 2y = 1 (D) x – 2y + 2 = 0
4. xsinxxsinxlim
x
= …………………………………………………….
5. 3x
x x13xlim
= ……………………………………………………..
6. 1x
xsinbxcosa1xlim 30x
, then a = …………………………..b = ………………………
7. 20x xxcos1lim
is equal to
(A) (B) 1/4 (C) 1/2 (D) 1
8 x
lim 1x21x2
is equal to
(A) 1 (B) 0 (C) -1 (D) 1/2
9. If f(x) = (1 – xn)1/n , 0 < x < 1, n being an odd positive integer and h(x) = f(f(x)), then h
21 is
equal to (A) 2n (B) 2 (C) n. 2n-1 (D) 1
10 Among
xsinxseclim 1
0x …. (1)
and
xxsinseclim 1
0x …. (2)
(A) (1) exists, (2) does not exist (B) (1) does not exist, (2) exists (C) both (1) and (2) exist (D) neither (1) nor (2) exists
11 A function f (x) is defined as f (x) =
1x,3bx1x,21x,ax3x2
What are the values of a and b respectively such that f (x) is continuous at x = 1. (A) 1, –2 (B) 0, –5 (C) –1, 0 (D) 2, –3
12 Given a function f(x) continuous x R such that
xfloge
11logxflim xf0x = 0,
then f(0) is (A) 0 (B) 1 (C) 2 (D) 3
13 The value of
x4
sinx4
cosxlimx
is
(A) 2 (B)
4 (C) 1 (D)
14 The value of x
balimxx
0x
is
(A) loge
ba (B) loge
ab (C) loge (ab) (D) none of these
15 If f (x) =
2x,nxsin
2x,1mx
is continuous at x = 2 , then
(A) m = 1, n = 0 (B) m = 2
n + 1 (C) n = 2
m (D) m = n = 2
16. The value of 2
24
x x)x1(x1lim
is equal to
(A) 0 (B) –1 (C) 2 (D) 1
17. xxtan
eelimxxtan
0x
is equal to
(A) 1 (B) e (C) –1 (D) 0
18. The function f (x) = 2]x[1])x[tan(
, where [.] denotes greatest integer function, is
(A) discontinuous at some x (B) continuous at all x, but f (x) does not exist for some x (C) f (x) exists for all x (D) none of these
19 If the function f(x) =
2x,ABx
2x1,x31x,BAx
2
be continuous at x = 1 and discontinuous at
x = 2, then (a) A = 3 + B, B 3 (b) A = 3 + B, B = 3 (c) A = 3 + B (d) none of these
41. If (x) =
1x,caxbx
1x,bax2
2, b 0. Then f(x) is continuous and differentiable at x = 1 if
(a) c = 0, a = 2b (b) a = b, c R (c) a = b, c = 0 (d) a = b, c 0. 42. If f(x) = x3 sgn x, then (a) f is derivable at x = 0 (b) f is continuous, but not derivable at x = 0 (c) LHD at x = 0 is 1 (d) RHD at x = 0 is 0. 43. If f(x) = (x – x0) (x) and (x) is continuous at x = 0, then 0xf is equal to (a) (x0) (b) (x0) (c) x0 (x0) (d) none of these.
44 If f (x) =
0xfor0
0xforx
xsin where [x] denotes greatest integer function, then xflim
0x =
(A) 1 (B) 0 (C) -1 (D) doesn’t exist
45. If the function
0x,ex
)x2sin(
0x,k
x2
2
)x(f is continuous, then k is
(A) 2 (B) 3 (C) 4 (D) 5.
46. For a function y = f(x), 2x1xdxdy
. Find the point of local maximum and minimum for
the function y = f(x). ............................................................ 47. Find the function y = f(x) for the above function if it is given that y = 2 at x = 0. ............................................................ 48. The value of derivative of f (x) = |x –1| + |x –3| at x = 2 is (A) –2 (B) 0 (C) 2 (D) not defined 49. The function f (x) = |sin x| –1 is
(A) continuous everywhere (B) not differentiable at x = 3
(C) differentiable at x = 0 (D) differentiable everywhere
50. Let f (x) =
3x2,x22x0,4x3
, if f (x) is continuous at x = 2, then is
(A) –1 (B) –2 (C) 2 (D) none of these
51. The number of points at which the function f (x) = |x|log
x is discontinuous is
(A) 1 (B) 2
(C) 3 (D) 4 52. The number of values of x x [0, 2] at which the real function f (x) = |x –1/2| + |x –1| + tan x
is not differentiable is (A) 2 (B) 3 (C) 1 (D) 0
LEVEL−II 1. The function (x2 – 1) xcos2x3x2 is not differentiable at (A) –1 (B) 0 (C) 1 (D) 2
2. For x R, x
x 2x3xlim
is
(A) e (B) e– 1 (C) e– 5 (D) e5
3.
x2
xcos6Lim2
x, where [.] denotes the greatest integer function, is equal to;
(A) - 3 (B) - 4 (C) -2 (D) none of these
4. Let f(x) = 4x
1
xtan
x (0, /2) ~ {/4}, then the value of f(/4} such that f(x) becomes
continuous at x = 4 is equal to;
(A) e (B) e (C) e1 (D) e2
5. Let f(x)=[5+3 sinx] x R. Then total number of points of discontinuity of f(x) in [0, ] is
equal to; (A) 5 (B) 6 (C) 7 (D) 4 6. f(x) = sin-1(sinx), x [-2, 2]. Total number of critical points of f(x) is ; (A) 3 (B) 4 (C) 5 (D) 2 7. If the line ax + by + c = 0 is normal to the curve x y + 5 = 0 then
(A) a > 0 , b > 0 (B) b > 0 , a < 0 (C) a < 0 , b < 0 (D) b < 0 , a > 0 8. The maximum value of f(x) = |x ln x| in x(0,1) is; (A) 1/e (B) e (C) 1 (D) none of these 9. f(x) = 3x3 +4ex – k is always increasing then value of k = (A) 2 (B) –4/9 (C) 4/9 (D) all of these 10. x]2x[]x2[lim
2x
is
(A) 0 (B) 3 (C) –3 (D) does not exist
11. Let f (x) be a twice differentiable function and f (0) = 2 then 22x x
x4fx2f3xf2lim
is
(A) 6 (B) 1 (C) 12 (D) 3 12 Let h (x) = f (x) –{f (x)}2 + {f (x)}3 for all real values of x then (A) h is whenever f (x) is (B) h is whenever f(x) is 0
(C) h is whenever f is (D) nothing can be said in general 13. Let f (x) > 0, g (x) < 0 for all x R, then (A) f {g (x)} > f {g (x + 1)} (B) f {g (x)} > f {g (x –1)} (C) g {f (x)} >< g {f (x + 1)} (D) g {f (x)} > g {f (x –1)}
14. xlxlnlim
x = ………………………………………………………….. [.] G. I. F
15. n1
nnn
n753lim
= ……………………………………………
16. If , are the roots of ax2 + bx + c = 0 then 2
2
x xcbxaxcos1lim
= ………………………
17. x11xx1lim
1x
= ………………………………………………………………
18. f (x) = sin-1(cos x) then points of nondifferentiability between [0, 2] = …………………….. 19. Let f (x + y) = f (x) . f (y) for all x & y, if f (5) = 2 and f’(0) = 3, then f’ (5) = …………………….
20. f(x) =
2x,2x]x[x
2x,b
2x,xx2
|2xx|a2
2
( where [.] denotes the greatest integer function ). If f(x)
is continuous at x = 2, then (A) a = 1, b = 2 (B) a = 1, b = 1 (C) a = 0, b = 1 (D) a = 2, b = 1
21. Let f(x) =
0x,10x,00x,1
and g(x) sinx + cosx, then points of discontinuity of f{g(x)} in (0,
2) is
(A)
43,
2 (B)
47,
43
(C)
35,
32 (D)
37,
45
22. If and are the roots at ax2 + bx + c = 0 then
x/12
xcbxax1lim is
(A) a ( – ) (B) ln|a( – )| (C) ea( – ) (D) ea| – |
23. 4x
lim
1xcot
1xcos2
is equal to
(A) 2/1 (B) 1/2
(C) 22
1 (D) 1
24. The function f(x) = [x]2 – [x2] where [y] denotes the greatest integer less than or equal to y),
is discontinuous at (A) all integers (B) all integers except 0 and 1 (C) all integers except 0 (D) all integers except 1
25. If the derivative of f(x) w.r. t x is xf
xsin21 2
, then f(x) is a periodic function with period
(A) (B) 2 (C) /2 (D) none of these.
26.
x
xsin))7y2y(min(lim 2
0x = ? (where [.] denotes greatest integer function)
(A) 4 (B) 5 (C) 6 (D) none of these
27.
x
xtan100lim0x
= ? (where [.] denotes greatest integer function)
(A) 100 (B) 99 (C) 101 (D) 0
28. If f (x) = |cos 2x|, then f
04
is equal to
(A) 2 (B) 0 (C) –2 (D) doesn’t exist 29. xcos/1
2/x)x(sinlim
=
(A) 0 (B) e (C) 1 (D) doesn’t exist
30. 40x x)xcos1cos(1lim
equals to
(A) 21 (B)
81 (C)
41 (D)
161
31. xtanIn1
xtan2lim4/x
equals to
(A) e (B) 1 (C)0 (D) e–1
32. 2
2
0x xxcossinlim
equals to
(A) 0 (B) (C) – (D) not exist 33. f (x) = max {x, x3},then the number of points where f (x) is not differentiable, are (A) 1 (B) 2 (C) 3 (D) 4
34. 1xln
1etanlim2x
2x
(A) 2 (B) -2 (C) 1 (D) –1
35. The function defined by f (x) =
0x,0
0x,xxsin 2
is
(A) continuos and derivable at x = 0 (B) neither continuous nor derivable at x = 0 (C) continuous but not derivable at x = 0 (D) none of these 36. x2/12
0x)xtan1(lim
is equal to
(A) 1 (B) 0 (C) e1/2 (D) e–1/2 37. The left hand derivative of f (x) = [x] sin (x) at x = k, k is an integer is (A) (–1)k(k – 1) (B) (–1)k – 1(k – 1) (C) (–1)kk (D) (–1)k–1k
38. yx
xy
yx yxyxlim
is
(A) ylog1
eylog
(B) eylog
ylog1 (C) ylog1ylog1
(D) (1 – log y) log ey
39. x
0x)x(sinlim
is
(A) 1 (B) 0 (C) (D) does not exist
40 If f(x) is a continuous function x R and the range of f(x)=(2, 26 ) and g(x) =
axf is
continuous x R ([.] denotes the greatest integer function), then the least positive integral value of a is
(A) 2 (B) 3 (C) 6 (D) 5
41 Let
nlim)x(f
5x2tan31
n21
. then the set of values of x for which f(x) = 0, is
(A) |2x| 3 (B) |(2x)| 3 (C)|2x| 3 (D)| 2x | 3
42 If f (x) = )1xlog(
)1esin( 2x
, then )x(flim2x
is equal to
(A) –2 (B) –1 (C) 0 (D) 1
43 If f (x) =
0x,k
0x,x
)bx1log()ax1log( and f (x) is continuous at x = 0, the value of k
is (A) a – b (B) a + b (C) log a + log b (D) none of these
44 The expression of dxdy
of the function y = .........xaxa is
(A) )xlogy1(x
y2
(B)
)xlogy1(xylogy2
(C)
)ylogxlogy1(xylogy2
(D)
)ylogxlogy1(xylogy2
45 The value of 2
23
x x31x2x/1sinxlim
is
(A) 0 (B) – 1/3 (C) –1 (D) – 2/3
46 1xln
1etanlim2x
2x
(A) 2 (B) -2 (C) 1 (D) –1
47 Let f(x) =
0x,1isxsin.)x(fThen.0x,0
0x,1
(A) differentiable at x = 0 (B) continuous at x = 0 (C) not continuous at x =0 (D) none of these
48 The function f(x) = x4sin
xsinxcos is not defined at x = /4. The value which should be
assigned to f at x = /4, so that it is continuous there, is (A) 0 (B) 1 (C) -1 (D) none of these
49 ,]x[
]x[xlnLimx
([.] denotes the greatest integer function)
(A) has value –1 (B) has value 0 (B) has value 1 (D) does not exist
50. The function 1e
1extan
xtan
is discontinuous
(A) at n, n I (B) at (2n+1) 2 , n I (C) No where (D) Every where
51 If a, b, c, d are positive, then dxc
x bxa11lim
=
(A) ed/b (B) ec/a (C) e(c+d)/(a+b) (D) e 52 The length of the largest interval in which the function 3 sin x – 4 sin3 x is increasing, is
(A) 2 (B)
3
(C) 2
3 (D)
53. The interval in which f (x) = |8x6x| 2
e increases, is (A) (–, 2) (3, 4) (B) R (C) (2, 3) (4, ) (D) (2, 4) 54. If x + |y| = 2y, then y as a function of x is (A) continuous but not differentiable at x = 0 (B) continuous and differentiable at x = 0 (C) differentiable for all x (D) none of these 55. If y= a log |x| + bx2 + x has its extremum values at x = –1 and x = 2, then (A) a = 2, b = –1 (B) a = 2, b = –1/2 (C) a = –1/2, b = 1/2 (D) none of these
56. The points of extremum of the function (x) = x
1
22/t dt)t1(e2
, is/are
(A) x = 0 (B) x = 1/2 (C) x = –2 (D) x = 1 57. Let f(x) = xn+1 + a. xn, where ‘a’ is a positive real number. Then x = 0 is a point of (A) local minimum for any integer n (B) local maximum for any integer n (C) local minimum if n is an even integer (D) local minimum if n is an odd integer 58. Least natural number ‘a’ for which x+ ax-2 > 2 x ( 0, ) is (A) 1 (B) 2 (C) 5 (D) none of these 59. Let f(x) =
nlim (sin x)2n ,then f is
(a) continuous at x = /2 , (b) discontinuous at x = /2 (c) discontinuous at x = – /2 (d) discontinuous at infinite number of points.
60. Let f(x) =
0x,0
0x,x1sinxn
, then f(x) is continuous, but not differentiable at x = 0, if
(a) n (0, 1] (b) n [1, ) (c) n (-, 0) (d) n= 0 61. If f(x) = 21xx , then f (x) equals
(a) 0 for all x (b) 2 1xx
(c)
1x0for)1x2(4
1xforand0xfor0 (d)
0xfor)1x2(40xfor0
62. If the function f(x) =
,6
x0,e
0x,b
0x6
,)xsin1(
x3tanx2tan
xsina
is continuous at x = 0, then
(a) a = 32a,log b
e (b) 32a,log a
e
(c) a = 2b,log be (d) none of these
63. The function f(x) =
1x,]x[3x2
1x,2xsin
(a) is continuous at x = 1 (b) is differentiable at x = 1 (c) is continuous but not differentiable at x = 1 (d) none of these
64. The value of p for which the function f(x) =
0x,4log12
0x,
3x1log
pxsin
14
3
2
3x
is continuous at
x = 0 is (a) 1 (b) 2 (c) 3 (d) 4
65.
222x n1n.......
n12
n11lim is equal to
(A) 0 (B) –1/2 (C) 1/2 (D) none of these
66 x
2x1lim
, where [x] is the greatest integer function, is equal to
(A) 1 (B) –1 (C) 1 (D) doesn’t exist
67 x2
x x2xlim
=
(A) e-4 (B) e-6 (C) e-2 (D) none of these 68 If f (x) = [x sin x] { where [x] denotes greatest integer function}, then f (x) is (A) continuous at x = 0 (B) continuous in (-1, 0) (C) differentiable at x = 1 (D) differentiable in (-1, 1) 69 In order that function f (x) = (x + 1)cot x is continuous at x = 0, f (0) must be defined as (A) 0 (B) e (C) 1/e (D) none of these
70
x
n
x exlim = 0, (n is integer), for
(A) no value of n (B) all value of n (C) only negative value of n (D) only positive value of n
71 n1
nn
n54lim
is equal to
(A) 4 (B) 5 (C) e (D) none of these
72 xsinsinlim 1
2x
equals, where [.] denotes the greatest integer function
(A) 2 (B) 0
(C) 1 (D) does not exist
73 The value of derivative of f (x) = |x –1| + |x –3| at x = 2 is (A) –2 (B) 0 (C) 2 (D) not defined 74 The number of points where the function f (x) = x2 –1 + |ln |x|| is not differentiable is (A) 1 (B) 2 (C) 3 (D) none of these 75 f (x) is a continuous function and takes only rational values. If f (0) = 3, then f (2) equals (A) 5 (B) 0 (C) 1 (D) none of these
76 x
21x1xlim
0x
is equal to
(A) 1 (B) −1 (C) 2 (D) 0
77. xcos1x2.xlim
x
0x
is equal to
(A) log2 (B) 21 log2 (C) 2 log2 (D) none of these
78. If f(x) = x
2
2
3xx3x5x
. Then xflim
x is
(A) e4 (B) e3 (C) e2 (D) None of these
79. 3
11
0x xxtanxsinlim
is equal to
(A) 1/2 (B) 2 (C) −1/2 (D) None of these
80. x
asinaxasinxalim22
0x
is equal to
(A) a2 cosa + a sina (B) a2 cosa + 2a sina (C) 2a2 cosa + a cosa (D) None of these
81. Let f : R R is a differentiable function and f(1) = 4. Then the value of
dt1x
t2limxf
41x
is
(A) 8 f(1) (B) 4f(1) (D) 2f(1) (D) None of these
82. If f(x) =
0x,2x3x0x3x
2 and g(x) = f(|x|) + |f(x)| , then g(x) is continuous at
(A) R − {0} (B) R+ (C) R − {1, 2} (D) R − {0, 1, 2}
83. The value the limit x/ax/a
x/ax/a
0x eeeelim
, a > 0 is
(A) 0 (B) 1 (C) infinity (D) does not exist
84. The number of points where g(f(x)) is discontinuous given that g(x) = 1xx
12
and
f(x) = 3x
1
is
(A) 1 (B) 2 (C) 3 (D) 4
85. The value of 2x/1
2
2
0x x31x51lim
is
(A) e2 (B) e3 (C) e5 (D) none of these 86. The number of points at which the function f(x) = |x − 0.5| + |x − 1| + tanx does not have a
derivative in the interval (0, 2) is (A) 1 (B) 2 (C) 3 (D) 4 87. Let f(x + y) = f(x) f(y) x, y R. Suppose that f(3) = 3 and f(0) = 11 then f(3) is given by (A) 22 (B) 44 (C) 28 (D) 33
88. The function f(x) =
1x,4
132x3
4x
1x,3x2 then which of the following is not true
(A) continuous at x = 1 (B) continuous at x = 3 (C) differentiable at x = 1 (D) differentiable at x = 3 89. The function f(x) = max{1 − x, 1 + x, 2}, x (−, ) is
(A) differentiable at all points (B) differentiable at all points except at x = 1 and x = −1 (C) continuous at all points except at x = 1 and x = −1, where it is discontinuous (D) None of these 90. Let f(x) = [tan2x] where [.] is greatest integer function then (A) xflim
0x does not exist (B) f(x) is continuous at x = 0
(C) f(x) is not differentiable x = 0 (D) f(0) = 1
LEVEL−III 1. The number of critical points of f (x) = max (sin x , cos x) for x (0 , 2 ) (A) 2 (B) 5 (C) 3 (D) non
2. If f (x) = x
0
)1t( (et –1) (t – 2) (t + 4) dt then f (x) would assume the local
minima at; (A) x = - 4 (B) x = 0 (C) x = -1 (D) x = 2. 3. Let f (x) = [cos x + sin x], 0 < x < 2 where [x] denotes the greatest integer less than or
equal to x. The number of points of discontinuity of f (x) is (A) 6 (B) 5 (C) 4 (D) 3
4. 1x
xcoslim1
1x
…………………………………………………………………….
5. f (x) = |x|log
1 is discontinuous at x = ………………………………………………
6. The value of the limit
xsinxsin/1xsin/1xsin/1
0x
2222
n.....21lim
(A) (B) 0
(C) 2
1nn (D) n
7. 40x x
xcosxsincoslim
is equal to
(A) 1/5 (B) 1/6 (C) 1/4 (D) ½
8. If tan-1 (x +h) = tan-1(x) + (h siny)(siny) – (h siny)2 . 2
y2sin + (h siny)3.3
y3sin + . .. .,
where x ( 0, 1), y (/4, /2) , then (A) y = tan-1x (B) y = sin-1x (C) y = cot-1x (D) y = cos-1x 9. The value of ]))x(sin(tantancos[lim 11
x
is equal to
(A) -1 (B) 2 (C) 21
(D) 2
1
10. If
x
0
2
0x1
taxsinxdttlim , then the value of a is
(A) 4 (B) 2 (C) 1 (D) none of these
12 For some g, let f(x) = x(x+3) eg(x) be a continuous function. If there exists only one point x = d such that f(d) = 0, then
(A) d < -3 (B) d > 0 (C) -3 d 0 (D) -3 <d < 0
13
1n
n n11ln1lim is equal to
(A) 0 (B) 1 (C) e (D) none of these
14 The value of ]x[
1nn
x e1nxxlim
, n I is
(A) 1 (B) 0 (C) n (D) n(n –1)
15 Given a function f(x) continuous x R such that
xflog
e11logxflim xf0x
= 0,
then f(0) is (A) 0 (B) 1 (C) 2 (D) 3 16 Let R be the set of real numbers and f : R R be such that for all x and y in R | f (x) – f (y) | | x –y |7. Then f (x) is. (A) linear (B) constant (C)quadratic (D) none of these.
17. Find the value of
xcot
x1lim 220x
(A) 2/5 (B) 2/3 (C) 1/4 (D) 1/5.
18
x
x2cos121
lim0x
is
(A) 1 (B) –1 (C) 0 (D) doesn’t exist 19 Given that f (x) is a non-zero differentiable function such that f (x + y) = f (x). f (y), x, y R,
and f (0) = 1 then ln f (1) is equal to (A) 0 (B) 1 (C) e (D) none of these
20 The largest interval where the function f (x) = |x|1
x
is differentiable
(A) (–, ) (B) (0, ) (C) (–, 0) (0, ) (D) none of these
21
2
x/1
0x x2exex1
lim
is equal to
(A) 24
e11 (B) − 24
e11 (C) 24e (D) None of these
22 The value of the limit
37xcos2
12431limxxx
0x is
(A) 0 (B) − 6(log3) (log4) (C) 1 (D) none of these
23 Let 2
yfxf2
yxf
, for all x, y R and if f(x) is differentiable, and f(0) = −1, f(0) = 1
then the function f(x) is (A) −x + 1 (B) x + 1 (C) x2 − 1 (D) x −1
24 The points of discontinuity of the function fog where g(x) = 1x
1
and f(x) = 2xx
12
are
(A) 21 , 2, 1 (B) 2, 1 (B) 2, 1
2 (D) none of these
ANSWERS
LEVEL −I 1. C 2. B 3. A 4. C
5. B 6. C 7. 1 8. e2 9. −2, −1 10. C 11. D 12. B 13. D 14. D 15. C 16. A 17. A 18. D 19. A 20. B 21. B 22. B 23. D 24. A 25. A 26. B 27. B 28. C 29. A 30. B 31. A 32. C 33. C 34. A 35. C 36. B 37. A 38. D 39. A 40. A 41. A 42. B 43. B 44. D 45. D 46. −2, 1 47. f(x) = x3/3 + x2/2 – 2x + 2 48. B 49. A 50. B 51. C 52. A LEVEL −II 1. D 2. C 3. A 4. D 5. A 6. B 7. A, C 8. A 9. D 10. C 11. A 12. C 13. B, D 14. 0 15. 7 16. (2a + b)2 / 2 17. −1 18. 0, , 2 19. 6 20. B 21. B 22. C 23. B 24. B 25. A 26. B 27. B 28. A 29. C 30. B 31. D 32. C 33. C 34. C 35. A 36. C 37. A 38. C 39. D 40. C 41. A 42. D 43. B 44. A 45. B 46. C 47. B 48. D 49. A 50. B 51. A 52. B 53. C 54. A 55. B 56. D 57. C 58. B 59. D 60. B 61. C 62. A 63. C 64. D 65. B 66. D 67. A 68. A 69. B 70. B 71. B 72. C 73. B 74. B 75. D 76. D 77. C 78. A 79. A 80. B 81. A 82. A 83. D 84. C 85. A 86. C 87. D 88. D 89. B 90. B LEVEL −III 1. C 2. C 3. B 4. 1 / 2 5. 0, 1 6. D 7. D 8. C 9. D 10. A 12. D 13. A 14. B 15. A 16. B 17. B 18. D 19. B 20. A 21. A 22. B 23. A 24. A
1
PARABOLA LEVEL-I
*1. The parametric equation of the parabola is x = t2 + 1, y = 2t + 1. The equation of
its directrix is
(A) x = 0 (B) x + 1 = 0 (C) y = 0 (D) none of these *2. The tangents to the parabola y2 = 4x at the points (1, 2) and (4, 4) meets on
the line (A) x = 3 (B) x + y = 4 (C) y = 3 (D) none of these 3. Normal at point to the parabola y2 = 8x, where abscissa is equal to ordinate,
will meet the parabola again at a point (A) (12, –18) (B) (–12, 18) (C) (–18, 12) (D) (18, –12) 4. If the tangents to the parabola y2 = 4ax at the points (x1, y1) and (x2, y2) meet at
the point (x3, y3) then (A) y3 = 21yy (B) 2y3 = y1 + y2
(C) 213 y
1y1
y2
(D) none of these
5. If tangents at A and B on the parabola y2 = 4ax intersect at the point C, then
ordinates of A, C and B are (A) always in A.P. (B) always in G.P. (C) always in H.P. (D) none of these 6. The point P on the parabola y2 = 4ax for which |PR – PQ| is maximum, where R
(– a, 0), Q (0, a), is (A) (a, 2a) (B) ( a, -2a) (C) (4a, 4a) (D) (4a, -4a) *7. The point (1, 2) is one extremity of focal chord of parabola y2 = 4x. The length
of this focal chord is (A) 2 (B) 4 (C) 6 (D) none of these 8. If normals at two points of a parabola y2 = 4ax intersect on the curve, then the
product of ordinates is (A) 2a2 (B) 4 a2 (C) 6a2 (D) 8a2 9. If AFB is a focal chord of the parabola y2 = 4ax and AF = 4, FB = 5, then the
latus-rectum of the parabola is equal to
(A) 9
80 (B) 809
(C) 9 (D) 80
2
10. The length of the chord of the parabola x2 = 4y passing through the vertex and having slope cot is
(A) 4 cos . cosec2 (B) 4 tan sec (C) 4 sin. sec2 (D) none of these 11. The straight line y = mx + c touches the parabola y2 = 4a(x + a) if
(A) c = am – ma (B) c = m –
ma
(C) c = am + ma (D) none of these
*12. The equation of the tangent to the parabola y2 = 16x inclined at an angle of 600 to
x-axis is (A) 3x – 3 y + 4 = 0 (B) 3x + 3 y + 4 = 0 (C) 3x –y + 4 = 0 (D) none of these *13. For all parabolas x2 + 4x + 4y + 16 = 0, the equations of the axis and the directrix
are given by (A) x + 2 = 0, y – 2 = 0 (B) x – 2 = 0, y + 2 = 0 (C) x + 2 = 0, y + 2 = 0 (D) none of these *14. If (4, 0) is the vertex and y-axis the directrix of a parabola, then its focus is (A) (8, 0) (B) (4, 0) (C) (0, 8) (D) (0, 4) 15. The slope of the normal at the point (at2, 2at) of the parabola y2 = 4ax is
(A) t1 (B) t
(C) –t (D) –t1
*16. If ASB is a focal chord of a parabola such that AS = 2 and SB = 4, then the latus
rectum of the parabola is
(A) 38 (B)
316
(C) 3
25 (D) none of these
17. The normal to the parabola y2 = 8x at (2, 4) meets the parabola again at (A) (18, 12) (B) (18, –12) (C) (–18, 12) (D) none of these *18. The value of k for which the line x + y + 1 = 0 touches the parabola y2 = kx is (A) –4 (B) 4 (C) 2 (D) –2 *20. The equation of directrix of the parabola x2 + 4x + 4y + 8 = 0 is (A) y = –1 (B) y = 1
3
(C) y = 0 (D) y = 23
21. The area of the triangle formed by the tangent and the normal to the parabola y2
= 4ax both drawn at the same end of the latus rectum and the axis of the parabola is
(A) 2 2 a2 (B) 2a2 (C) 4a2 (D) none of these 22. If two normals at P and Q of a parabola y2 = 4ax intersect at a third point R on
the curve, then the product of ordinates of P and Q is (A) 8a2 (B) 4a2 (C) 2a2 (D) none of these 23. The length of the subnormal to the parabola y2 = 4ax at any point is equal to (A) a 2 (B) 2 2 a (C) 2/a (D) 2a *24. The number of tangents to the parabola y2 = 8x through (2, 1) is (A) 0 (B) 1 (C) 2 (D) none of these *25. If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the
values of k is
(A) 81 (B) 8
(C) 4 (D) 41
*26. If the point P (4, – 2) is one end of the focal chord PQ of the parabola y2 = x, then
the slope of the tangent at Q is
(A) – 41 (B)
41
(C) 4 (D) – 4 *27. The equation of the parabola whose vertex and focus lie on the x– axis at
distances a and a1 from the origin respectively, is (A) y2 = 4(a1 – a)x (B) y2 = 4(a1 – a) (x – a) (C) y2 = 4(a1 – a) (x – a1) (D) none of these *28. If (2, 0) is the vertex and y– axis the directrix of the parabola, then the focus is (A) (2, 0) (B) (– 2, 0) (C) (4, 0) (D) (– 4, 0) 29. If the normals at t1 and t2 meets on the parabola then
(A) t2 = – t1 – 1t2 (B) t1t2 = 2
(C) t1 t2 = – 1 (D) none of these
4
*30. The graph represented by the equations x = sin2t, y = 2 cost is (A) parabola (B) circle (C) hyperbola (D) none of these 31. If y = –4 is the directrix and (–2, –1) the vertex of a parabola then its focus is at
…………………………………………………
32. The condition that the line by
ax = 1 be a normal to the parabola y2 = 4px is
……………………………………………………… 33. If k = ………………, the line y = 2x + k is normal to the parabola y2= 4x at
……………. 34. The value of k for which the equation x2 + y2 + 2kxy + 2x + 4y + 3 = 0 represents
a parabola are ………………………………………………… 35. The point of intersection of the tangents of the parabola y2 = 4x at the points,
where the parameter t has the value 1 and 2 are (A) (3, 8) (B) (4, 5) (C) (2, 3) (D) (4, 6) 36. If the line y = x + k is a normal to the parabola y2 = 4x then k can have the value (A) 22 (B) 4 (C) –3 (D) 3 37. The tangents from the origin to the parabola y2 + 4 = 4x inclined of (A) /6 (B) /4 (C) /3 (D) /2 38. Normal at point to the parabola y2 = 4ax where abscissa is equal to ordinate,
will meet the parabola again at a point (A) (6a, – 9a) (B) (–6a, 9a) (C) (–9a, 6a) (D) (9a, – 6a) *39. If the focus of the parabola is (–2, 1) and the directrix has the equation x + y = 3
then the vertex is (A) (0, 3) (B) (–1, 1/2) (C) (–1, 2) (D) (2, –1) 40. The locus of the point from which tangents to a parabola are at right angles is a (A) straight line (B) pair of straight lines
(C) circle (D) none 41. Given the two ends of the latus rectum, the maximum number of parabolas that
can be drawn is (A) 1 (B) 2
5
(C) 0 (D) infinite *42. The Cartesian equation of the curve whose parametric equations are x = t2 + 2t +
3 and y = t + 1 is (A) y = ( x– 1) 2 + 2( y–1) + 3 (B) x = ( y – 1)2 + 2( y–1) +5 (C) x = y2 +2 (D) None of these
*43. If line y = 2x + 41 is tangent to y2 = 4ax, then a is equal to
(A) 1/ 2 (B) 1 (C) 2 (D) None of these 44. The shortest distance between the parabola y2 = 4x and the circle x2 + y2 + 6x –
12y + 20 = 0 is (A) 4 52 (B) 0 (C) 3 2 +5 (D) 1 45. The equation (13x – 1)2 + ( 13y – 1)2 = k (5x – 12y + 1)2 will represent a parabola
if (A) k = 2 (B) k = 81 (C) k = 169 (D) k =1 *46. If l, m be the lengths of segments of any focal chord of a parabola y2 = 4ax then
length of semi–latus rectum is
(A) 2ml (B)
mllm
(C) mlml2
(D) ml
47. The normal chord of a parabola y2 = 4ax at a point whose ordinate is equal to
abscissa subtends a right angle at the (A) focus (B) vertex (C) end of the latus rectum (D) none of these 48. If a tangent to the parabola y2 = ax makes an angle of 45°with x – axis, its point
of contact will be (A) (a/2, a/4) (B) (-a/2, a/4) (C) (a/4, a/2) (D) (-a/4, a/2) 49. The triangle formed by the tangents to a parabola y2 = 4ax at the ends of the
latus rectum and the double ordinate through the focus is (A) equilateral (B) isosceles (C) right angled isosceles (D) depends on a 50. The equation x2 + 4xy + y2 + x + 3y + 2 = 0 represents a parabola if is (A) –4 (B) 4 (C) 0 (D) none of these
6
LEVEL-II 1. From point P two tangents are drawn from it to the parabola y2 = 4x such that the
slope of one tangent is three times the slope of the other. The locus of P is (A) straight line (B) circle (C) parabola (D) none of these *2. The chord AB of the parabola y2 = 4ax cuts the axis of the parabola at C. If
A = (at12, 2at1), B = (at22, 2at2) and AC : AB = 1: 3 , then (A) t2 = 2t1 (B) t2 + 2t1 = 0 (C) t1 + 2t2 = 0 (D) none of these 3. If the normals drawn at the end points of a variable chord PQ of the
parabola y2 =4ax intersect at parabola, then the locus of the point of intersection of the tangent drawn at the points P and Q is
(A) x +a = 0 (B) x – 2a = 0 (C) y2- 4x +6 = 0 (D) none of these 4. If the normals at the end points of a variable chord PQ of the parabola y2 – 4y –
2x = 0 are perpendicular, then the tangents at P and Q will intersect at (A) x + y = 3 (B) 3x – 7 = 0 (C) y+3 = 0 (D) 2x + 5 = 0 *5. The number of focal chord(s) of length 4/7 in the parabola 7y2 = 8x is (A) 1 (B) zero (C) infinite (D) none of these . 6. The equation of common tangent touching the circle x2 – 4x + y2 = 0 and the
parabola y2 = 4x is (A) 2 y = 2x + 1 (B) 2 y = –(x + 2) (C) 2 y = x + 2 (D) none of these 7. Three normals to the parabola y2 = x are drawn through a point (c, 0) then
(A) c = 41 (B) c =
21
(C) c > 21 (D) none of these
8. Tangents are drawn from ( -2, 0) to y2 = 8x, radius of circle(s) that would touch
these tangents and the corresponding chord of contact, can be equal to, (A) 4 12 (B) 4 12
(C) 8 2 (D) None of these. 9. The coordinates of the point on the parabola y = x2 + 7x +2, which is nearest to
the straight line y = 3x – 3 are (A) ( -2, -8) (B) ( 1, 10) (C) ( 2, 20) (D) ( -1, -4)
7
10. The equation of the common tangent to the parabola y2 = 32x and x2 = 108y is (A) x = 0 (B) 2x + 3y + 36 = 0 (C) 2x – 3y – 36 = 0 (D) 2x – 3y + 36 = 0 11. The locus of the middle points of the chords of the parabola y2 = 4ax which
subtend a right angle at the vertex is ………………………. 12. Three normals are drawn from a point (c, 0) to the parabola y2 = x. One normal is
always the x-axis. the value of c for which the other two normals are perpendicular to each other is ……………………………..
13. Three distinct normals are drawn from a point to a parabola. The ordinates of the
foot of two normals are –1 and 3 on the parabola. The ordinate of the foot of third normal is…………………………………..
14. If two of the three feet of normals drawn from a point to the parabola y2 = 4x be
(1, 2) and (1, –2) then the third foot is (A) (2, 22) (B) (2, –22)
(C) (0, 0) (D) none 15. Let y2 = 4ax be a parabola and x2 + y2 + 2bx = 0 be a circle. Then condition on a
and b so that parabola and circle touch each other externally is (A) ab > 0 (B) ab < 0 (C) ab < –1 (D) none of these
*16. The parametric coordinates of any point on the parabola y2 = x can be (A) (sin2, sin) (B) (cos2, cos) (C) (sec2, sec) (D) (tan2, tan) *17. Slope of tangent to x2 = 4y from (-1, -1) can be
(A) 2
51 (B) 2
51
(C) 2
51 (D) 2
51
18. A line passing through the focus of the parabola y2 = 4ax, intersects the parabola in two distinct points. Slope of the line is
(A) any real number (B) greater than 1 and less than 1 (C) less than 1 or greater than 1 (D) none of these 19. The length of the common chord of the curves y2 – 4x–4 = 0 and 4x2 + 9y2 – 36 = 0 is
(A) 2 3 units (B) 3 2 units (C) 4 units (D) 6 units
20. x+ y = a represents (A) a part of parabola (B) ellipse (C) Hyperbola (D) Line segment
8
21. A line through the focus of parabola y2 = 4(x –2) having slope ‘m’ meets the
curve in distinct real points, then exhaustive set of values of ‘m’ is; (A) m (-1, 1) (B) m (-2, 2) (C) m (-, ) (D) none of these 22. If (y + b) = m1 (x + a) and (y + b) = m2 (x + a) be tangents of y2 = 4ax then; (A) m1 + m2 = 0 (B) m1 m2 = 0
(C) m1 m2 = -1 (D) m1 = -m2 2m
2
*23. A tangent to the parabola x2 = 4ay is inclined at an angle 6 with the x-axis, then
coordinates of point of contact is;
(A) 3a2,a3 (B)
3a2,
3a
(C)
3a2,
3a (D)
3a,
3a2
24. The length of focal chord of the parabola y2 = 4ax at a distance b from the vertex
is c then (A) 2a2 = bc (B) a3 = b2c (C) ac = b2 (D) b2c = 4a3
9
LEVEL-III 1. The circle drawn with variable chord x + ay – 5 = 0 (a being a parameter) of the
parabola y2 =20x as diameter will always touch the line (A) x + 5 = 0 (B) y + 5 = 0 (C) x + y + 5 = 0 (D) x – y + 5 = 0 2. The set of points on the axis of the parabola 2((x –1)2 + ( y –1)2) = (x + y)2 ,
from which 3 distinct normals can be drawn to the parabola, is the set of points (h, k) lying on the axis of the parabola such that
(A) h > 3 (B) h > 3/2 (C) k > 3/2 (D) k > 3 3. Radius of the circle passing through the origin and touching the parabola y2 = 4x
at (1, 2) (A) 5/6 (B) 5 2 /6 (C) 5/ 2 (D) none of these 4. If the parabola y = f (x), having axis parallel to y-axis, touches the line y = x at
(1, 1) then; (A) 2f (0) + f (0) = 1 (B) 2f (0) + f (0) = 1 (C) 2f (0) - f (0) = 1 (D) 2f (0) - f (0) = 1 *5. The length of latus rectum of the parabola whose focus is (a sin2, a cos2) and
directrix is the line y = a, is (A) 24 cosa (B) 24 sina
(C) 4 cos 2a (D) 4 sin 2a 6. Chord AB of the parabola 2 4y ax subtends a right angle at the origin. Point of
intersection of tangents drawn to parabola at ‘A’ and ‘B’ lie on the line - (A) x + 2a = 0 (B) y + 2a = 0 (C) x + 4a = 0 (D) y + 4a = 0 7. A circle is drawn to pass through the extremities of the latus rectum of the
parabola 2 8y x . It is given that this circle also touches the directrix of the parabola. Radius of this circle is equal to
(A) 2 (B) 21 (C) 8 (D) 4 8. The circle 2 2 2 2 0x y gx fy c cuts the parabola 2 4x ay at points
( , )i i iP x y , i = 1, 2, 3, 4; then
(A) 0iy (B) 0ix
(C) 4( 2 )iy f a (D) 2( 2 )ix g a
10
9. Maximum number of common normals of 2 4y ax and 2 4x by can be equal to
(A) 3 (B) 4 (C) 6 (D) 5 10. Maximum distance between the curves 2 1y x and 2 1x y is equal to
(A) 3 2
4 (B)
5 24
(C) 7 2
4 (D)
24
11. Sides of an equilateral triangle ABC touch the parabola 2 4y x , then points A, B
and C lie on
(A) 22 3 4y x a ax (B) 2
2 43
x ay ax
(C) 2
2 43
y ax ay
(D) 22 3 4x y a ay
12. Length of the latus rectum of the parabola whose parametric equation is :
2 1x t t ; 2 1y t t , where t R, is equal to (A) 8 (B) 4 (C) 2 (D) 2 13. A circle having its centre at (2, 3) is cut orthogonally by the parabola 2 4y x .
The possible intersection point of these curves can be (A) (1, 2) or (3, 2 3 ) (B) (1, 2) or (4, 4) (C) (9, 6) or (3, 2 3 ) (D) None 14. The vertex of the parabola 21 2( 2)x y x y is
(A) (2, –1) (B) 13 17,4 4
(C) 1 3,2 2
(D) 19 35,8 8
15. The axis of the parabola 21 2 2x y x y is (A) y = x + 2 (B) x – y = 1 (C) x + y = 2 (D) x + y = 1 16. The line x + y = a touches the parabola 2y x x and
2 2 5( ) sin sin cos cos , 1, ( ( ))3 3 4
f x x x x x g b g f x
, then `
11
(A) a = b (B) a = 2b (C) a + b = 0 (D) a + 2b = 0 17. The co-ordinates of the point on the parabola 2 8y x , which is at minimum
distance from the circle 22 6 1x y are (A) (2, 4) (B) (–2, 4) (C) (–2, –4) (D) (2, –4) 18. If three normals can be drawn to the parabola 2y x from the point (C, 0), then
the two normals other than the axis of the parabola are perpendicular to each other if C =
(A) 34
(B) 43
(C) 34
(D) 43
19. If 1( )
1f x
x
and , ( ) be the values of x, where f(f(x)) is not defined,
then a ray of light parallel to the axis of the parabola 2 4y x after reflection from the internal surface of the parabola will necessarily pass through the point
(A) , (B) ,
(C) , (D) None *20. If 1t and 2t be the ends of a focal chord of the parabola 2 4y ax , then the
equation 21 2 0t x ax t has
(A) imaginary roots, (B) both roots positive (C) one positive and one negative roots (D) both roots negative
12
ANSWERS LEVEL −I 1. A 2. C 3. D 4. B 5. A 6. A 7. B 8. D
9. A 10. A 11. C 12. A 13. C 14. A 15. C 16. B 17. B 18. B 20. C 21. C 22. A 23. D 24. A 25. C 26. C 27. B 28. C 29. B 30. A 31. (-2, 2) 32. a3b = 2pba2 + pb3 33. -12, (4, -4) 34. 1 35. C 36. C 37. B 38. D 39. C 40. A 41. B 42. C 43. A 44. A 45. D 46. C 47. A 48. C 49. C 50. B LEVEL −II 1. C 2. B 3. B 4. D 5. B 6. D 7. C 8. B 9. A 10. B 11. y2 – 2ax + 8a2 = 0
12. c = 34
13. -2 14. C 15. B 16. D 17. A, B 18. D 19. C 20. A 21. D 22. C 23. D 24. D LEVEL −III 1. A 2. A, B, C, D 3. C 4. B 5. B 6. C 7. D 8. B 9. D 10. A 11. A 12. D 13. B 14. C 15. D 16. A 17. D 18. A 19. B 20. C
Permutation and Combination
1. The number of ways of selecting two numbers from the set {1, 2, …,12} whose sum is divisible by 3 is
(A) 66 (B) 16 (C) 6 (D) 22 1. (D) Any natural number is either of the form 3 k or 3k – 1 or 3k +1. Sum of two numbers will be
divisible by 3 if and only if either both are of the form 3 k or one is of the form 3k-1 and other is of the form 3k+1. This can be done in 4C2+4C1 4C1 = 6+16 = 22
2. The number of flags with three strips in order that can be formed using 2 identical red, 2
identical blue and 2 identical white strips is (A) 24 (B) 20 (C) 90 (D) 8 2. (A)
No. required flags = 3! coefficient of x3 in 32
!2xx1
= 6 4 = 24
3. If nPr = nPr+1 and nCr = nCr-1, then (n, r) are (A) (2,3) (B) (3,2) (C) (4,2) (D) (4,3) 3. (B) npr = npr+1 n-r = 1 ... (1)
ncr = ncr-1 2r – 1 = n ... (2) Solving (1) & (2) we vet n = 3, r = 2 4. The number of 9 digit numbers that can be formed by using the digits 1,2,3,4 and 5 is (A) 9C1 8C2 (B) 59
(C) 9C5 (D) 9! 4. (B) 5. The number of diagonals that can be drawn by joining the vertices of an octagon is (A) 28 (B) 48 (C) 20 (D) None of these 5. (C) 8C2-8 = 20 6. Number of ways in which 5 identical objects can be distributed in 8 persons such that no person
gets more than one object is (A) 8 (B) 8C5 (C) 8P5 (D) None of these 6. (B) No. of ways = Coefficient of x5 in (x0 + x)8 = 8C5
7. Number of ways in which 7 girls & 7 boys can be arranged such that no two boys and no two
girls are together is (A) 12!(2!)2 (B) 7! 8! (C) 2(7!)2 (D) None of these 7. (C)
Corresponding to one arrangement of the boys, there are two ways in which the girls can be arranged; position (1) remaining vacant is position (2) remaining vacant
(1) B – B – B – B – B – B (2) 2(7!) (7!) ways. 8. The number of ordered triplets (a, b, c), a, b, c N, such that a + b + c 20 is (A) less than 100 (B) less than 1000 (C) equal to 1000 (D) more than 1000 8. (D) a + b + c 20 a + b + c + d = 20, a, b, c 1, d 0 a1 + b1 + c1 + d = 17, a1, b1, c1, d 0 No. of solutions = 17 + 4 1C41 = 20C3 = 1140 9. In a hockey tournament, a total of 153 matches were played. If each team played one match
with every other team, the total number of teams that participated in the tournament were (A) 20 (B) 18 (C) 16 (D) 14 9. (B) Given nC2 = 153 n2 – n – 306 = 0 n = 18. 10. In how many ways can we distribute 5 different balls in 4 different boxes when order is not
consider inside the boxes and empty boxes are not allowed (A) 120 (B) 150 (C) 240 (D) None of these
10. (C)
5C2 (4!) = 240 11. The number of rectangles that you can find on a chessboard is (A) 144 (B) 1296 (C) 256 (D) None of these. 11. B 12. The number of even divisors of 1008 is (A) 23 (B) 21 (C) 20 (D) None of these. 12. A
13. If n n n
r 1 r 1 r 1P P Pa b c , then
(A) ab, b, ac are in A.P. (B) ab, b, ac are in G.P. (C) 2b a b c (D) None of these. 13. C 14. In a college of 300 students, every student reads 5 newspapers and every newspaper is read by
60 students. The number of newspapers is (A) at least 30 (B) at least 20 (C) exactly 25 (D) None of these.
14. C 15. The number of arrangements of the letters of the word BANANA in which two N’s do not appear
adjacently. (A) 40 (B) 60 (C) 80 (D) 100 15. A 16. The number of triangles which can be formed from 12 points out of which 7 are collinear is (A) 105 (B) 210 (C) 175 (D) 185
16. B 17. The number of ways in which 5 male and 2 female members of a committee can be seated
around a round table so that the two females are not seated together is (A) 480 (B) 600 (C) 720 (D) 840 17. B 18. A set contains (2n + 1) elements. The member of subsets of the set which contain at most n
elements is (A) 2n (B) 2n + 1 (C) 2n – 1 (D) 22n
18. D 19. A polygon has 44 diagonals. The number of its sides is (A) 9 (B) 10 (C) 11 (D) 12 19. C 20. Everybody in a room shakes hand with everybody else. The total number of hand shakes is
153. The total number of persons in the room is (A) 16 (B) 17 (C) 18 (D) 19
20. C 21. Eight chairs are numbered from 1 to 8. Two woman and three men wish to occupy one chair
each. First the women chose the chairs from amongst chairs marked 1 to 4; then the men select the chairs from amongst the remaining. The number of possible arrangement is
(A) 6C3 4C4 (B) 4P2 4P3 (C) 4C3 4P3 (D) 4P2 6P3 22. In an examination there are 3 multiple choice questions and each question has 4 choices.
Number of sequences in which a student can fail to get all answers correct is (A) 11 (B) 15 (C) 80 (D) 63 23. A box contains two white balls, three black balls and four red balls. The number of ways in
which three balls can be drawn from the box so that atlest one of the balls is black is (A) 74 (B) 84 (C) 64 (D) 20 24. Number of subsets of a set containing n distinct objects is (A) nC1 + nC2 + …+ nCn (B) nC0 + nC1 + nC2…+ nCn (C) 2n 1 (D) 2n + 1 25. In a group of boys, two boys are brothers and in this group 6 more boys are there. In how many
ways can they sit if the brothers are not to sit along with each other? (A) 2 6! (B) 7P2 6! (C) 7C2 6! (D) none of these 26. In a 12 storey building 3 persons enter a lift cabin, It is known that they will leave the lift at
different storeys. In how many ways can do so if the lift does not stop at the second storey. (A) 720 (B) 240 (C) 120 (D) 36 27. The number of five digits telephone numbers having atleast one of their digits repeated is (A) 90000 (B) 100000 (C) 30240 (D) 69760 28. The number of arrangement of the letters of the word ‘BANANA’ in which two N’s donot appear
adjacent is (A) 40 (B) 60 (D) 80 (D) 100 29. The number of straight lines that can be formed by joining 20 points of which 4 points are
collinear is (A) 183 (B) 186 (C) 197 (D) 190 30. Number of numbers greater than 1000 but less than 4000 that can be formed by using the digit
0, 1, 2, 3, 4 when repetition is allowed is (A) 125 (B) 105 (C) 375 (D) 625
31. There are ‘n’ seats round a table marked 1, 2, 3, ……, n. The number of ways in which m (n) persons can take seats is;
(A) npm (B) nCm (m –1)! (C) n –1Cm (m)! (D) n –1pm –1 32. Number of divisors of the form 4n + 2, n 0 of the integer 240 is; (A) 4 (B) 8 (C) 10 (D) none of these 33. Six identical coins are arranged in a row. The total number of ways in which the number of
heads is equal to the number of tails is; (A) 40 (B) 20 (C) 9 (D) 18 34. How many different nine digit numbers can be formed from the number 227788558 by
rearranging it’s digits so that odd digits occupy the even positions? (A) 16 (B) 36 (C) 60 (D) none of these 35. The number of proper divisors of 1800 which are also divisible by 10 is; (A) 16 (B) 18 (C) 17 (D) none of these 36. Let A = {x : x is a prime and x 31}. The number of different rational numbers whose numerator
and denomirator belong to A is; (A) 110 (B) 109 (C) 111 (D) none of these 37. Let n1 and n2 be two, four digit numbers. How many such pairs can be there so that n2 can be
substracted from n1 without borrowing? (A) 453 . 36 (B) 454 (C) 553 . 45 (D) none of these 38. Consider a rectangle ABCD. Three, four, five and six points are marked respectively on the
sides AB, BC, CD and DA (none of them being the vertex of the rectangle). Number of triangles that can be formed with these points as vertices, so that there is atmost one angular point of the triangle on any side of rectangle ABCD is;
(A) 232 (B) 342 (C) 282 (D) none of these 39. Brijesh has 10 friends among who two are married to each other. She wishes to invite 5 of them
for a party. If the married couple don’t accept to attend the party, if invited together, then the number of different ways in which she can invite 5 friends is;
(A) 8C5 + 2. 9C5 (B) 8C5 + 9C5 + 8C4 (C) 8C5 + 2. 8C4 (D) none of these 40. Let kjia and r be any vector such that j.r,i.r and k.r are positive integers. If 3
a.r 10, then number of all such vectors r is; (A) 12C7 (B) 10C7 (C) 11C7 (D) none of these
41. The number of distinct rational numbers x such that 0 < x < 1 and x = qp , where
p, q {1, 2, 3, 4, 5, 6} is (A) 15 (B) 13 (C) 12 (D) 11 42. The total number of 5-digit numbers of different digits in which the digit in the middle is the
largest is 9
(A)
9
4n4
nP (B) 33 (3!)
(C) 30 (3!) (D) none of these 43. The number of 6-digit numbers in which the sum of digits is divisible by 5 is (A) 180000 (B) 540000 (C) 5 105 (D) none of these 44. The number of divisors of the form (4n+2) (n 0) of the integer 240 is (A) 4 (B) 8 (C) 10 (D) 3 45. The number of non-negative integral solutions of a + b+ c = n, n N, n 3, is (A) (n–1)C2 (B) (n–1)P2 (C) n(n – 1) (D) none of these 46. The number of ways to give 20 apples to 3 boys, each receiving at least 4 apples, is (A) 10C8 (B) 90 (C) 20C20 (D) none of these 47. The position vector of a point P is kzjyixr
, where x, y, z N and kjia
. If
10ar
, the number of possible positions of P is (A) 36 (B) 72 (C) 66 (D) none of these 48. In a plane three are two families of lines y = x + r, y = – x + r, where r {0, 1, 2, 3, 4} . The
number of squares of diagonals of the length 2 formed by the lines is (A) 9 (B) 16 (C) 25 (D) none of these 49. There are n seats round a table numbered 1, 2, 3,…,n. The number of ways in which m (n)
person can take seats is (A) nPm (B) nCm (m – 1)! (C) (n–1)P(m–1) (D) nCm+1 m! 50. The rank of the word RACE if the words formed by letters of word RACE are arranged in
the dictionary order is _____________ 51. The number of n-digit numbers, no two consecutive digits being the same, is (A) n! (B) 9! (C) 9n (D) n9 51. (C)
The first digit can be chosen in 9 ways( other than zero), the second can be chosen in 9 ways ( any digit other then the first digit), the third digit can be chosen in 9 ways( any digit other then the second digit ) and so on. Hence required number of numbers is 9 9 . . . . 9 ( n times) = 9n .
52. The number of divisors of 3630, which have a remainder of 1 when divided by 4, is (A) 12 (B) 6 (C) 4 (D) none of these. 52. (B) 3630 = 2 3 5 112. Now a divisor will be of the form (4n+1) if divisor is form the help of (4n+1) type number or
by (4n+3) types number taken even times. Hence divisors are 1, 5, 3 11, 112, 5 112, 5 3 11, i.e., 6. 53. The number of solutions of the inequation 10Cx-1 > 3 . 10Cx is (A) 0 (B) 1 (C) 2 (D) 9 53. (C)
10Cx –1 > 3 . 10Cx x11
1
> x3 4x > 33 x 9 , but x 10.
So x = 9, 10. Hence there are two solutions 54. Triplet (x, y , z) is chosen from the set { 1, 2, 3, . . . . n }, such that x y < z. The number of
such triplets is (A) n3 (B) nC3 (C) nC2 (D) none of these 54. (D) Any three numbers x, y, z from {1, 2, 3, . . . .} can be chosen in nC3 ways and we get
unique triplet ( x, y, z) , x< y < z . Again any two numbers x, z can be chosen from {1, 2, 3, . . . , n } in nC2 ways and we get the triplet ( x, x, z) , x< z . Hence total number of required triplets is nC2+ nC3 .
55. If m and n are positive integers more than or equal to 2, m > n, then (mn)! is divisible by (A) (m!)n (B) (n!)m (C) ( m+n)! (D) (m - n) ! 55. (A) , (B), (C) , (D)
n!m
!mn is the number of ways of distributing mn distinct objects in n persons equally.
Hence n!m
!mn is an integer ( m!)n | ( mn)! . Similarly (n!)m |(mn)!.Further m+n < 2 m
mn ( m+n)! | (mn)! and m –n < m < mn ( m -n)! | (mn)! 56. Let S be the set of 6-digit numbers a1a2a3a4a5a6 (all digits distinct)
where a1 > a2 > a3 > a4 < a5 < a6 . Then n(S) is equal to (A) 210 (B) 2100 (C) 4200 (D) 420
56. (B) First, 6 distinct digits can be selected in 10C6 ways. Now the position of smallest digit in
them is fixed i.e. position 4. Of the remaining 5 digits, two digits can be selected in 5C2
ways. These two digits can be placed to the right of 4th position in one way only. The remaining three digits to the left of 4th position are in the required order automatically.
So n(S) = 10C6 5C2 = 210 10 = 2100 . 57. The number of positive integral solutions of the equation x1 x2 x3 = 60 is (A) 54 (B) 27 (C) 81 (D) None of these. 57. (A) Here x1x2 x3 = 22 3 5.Let number of two’s given to each of x1 , x2 , x3 be a, b, c . Then
a+b+c = 2, a, b, c 0 The number of integral solutions of this equations is equal to coefficient of x2 in (1-x)-3 i.e.
4C2 i.e. the available 2 two’s can be distributed among x1, x2 and x3 in 4C2 = 6 ways.Similarly, the available 1 three can be distributed among x1 , x2 , x3 in 3C2 = 3 ways( = coefficient of x in (1 – x)-3 )
Total number of ways = 4C2 3C2 3C2 = 6 3 3 = 54 ways. 58. For the series 21, 22, 23, . . . . , k –1, k ; the A.M. and G.M. of the first and last number
exist in the given series. If ‘k’ is a three digit number, then ‘k’ can attain (A) 5 values (B) 6 values (C) 2 values (D) 4 values 58. (C) 21, 22, 23, . . . . k –1, k
A.M. = k.21M.G,2
k21
k = 21. 2 , I also 100 k 999 and k should be odd
21
99921
100 2 4. 76 2 47. 57 = 3, 4, 5, 6 but should be odd odd
= 3,5 ‘k’ can assume 2 different values . 59. Consider a set {1, 2, 3, . . . ., 100 } . The number of ways in which a number can be
selected from the set so that it is of the form xy , where x, y, N and 2 , is (A) 12 (B) 16 (C) 5 (D) 11 59. (A) Perfect square = 100 – 1 = 9( excluding one ) Perfect cubes = 31100 3/1 Perfect 4th powers = 31100 4/1 Perfect 5th powers = 11100 5/1 Perfect 6th powers = 11100 6/1
Now, perfect 4th powers have already been counted in perfect squares and perfect 6th powers have been counted with perfect squares as well as with perfect cubes. Hence the total ways = 9+ 3+ 1 – 1 = 12 .
60. Number of natural numbers < 2 .104 which can be formed with the digits 1, 2, 3 only is
equal to
(A) 2
33.23 46 (B) 2
33.23 46
(C) 2
137 (D) none of these
60. (A) Total number of numbers will be equal to the sum of numbers of all possible 1–digit, 2-
digit, 3-digit, 4-diigit and 5-digit numbers. Total number of numbers =3 + 32 + 33 +34 + 34
= 2
33.2332
133 464
5
.
61. The sum of the factors of 7!, which are odd and are of the form 3t + 1 where t is a whole
number, is (A) 10 (B) 8 (C) 9 (D) 15 61. (B) 7! = 24 32 5 7 Since the factor should be odd as well as of the form 3t + 1, the factor cannot be a multiple
of either 2 or 3. So the factors may be 1, 5, 7and 35 of which only 1 and 7 are of the from 3t +1, whose sum is 8.
62. Number of positive integers n less than 15, for which n! + (n+1)! + (n+2)! is an integral
multiple of 49, is (A) 3 (B) 4 (C) 5 (D) 6 62. (A) n! + ( n+1)! + (n+2)! = n! { 1+n +1 + ( n + 2)(n+1) } = n!( n+2)2 Either 7 divides n + 2 or 49 divides n! n = 5, 12, 14 . 63. Let n be a positive integer with f(n) = 1! + 2! + 3! + . . . + n! and P(x), Q(x) be polynomials
in x such that f(n+2) = P(n)f(n+1) + Q(n)f(n) for all n 1. Then (A) P(x) = x + 3 (B) Q(x) = -x –2 (C) P(x) = -x –2 (D) Q(x) = x + 3 63. (A), (B) f(n) = 1! + 2! + 3! + . . . . . + n! f(n+1) = 1! + 2! + 3! + . . . . . + (n+1)! f(n+2) = 1! + 2! + 3! + . . . . . + (n+2)! f(n+2) – f(n+1) = ( n +2)! = ( n+2 )( n + 1)! = (n +2)[ f( n+1) – f(n) ] f( n+2) = (n+3)f( n+1) – (n+2)f(n) P(x) = x +3, Q(x) = – x –2 64. The number of ordered pairs (m, n) ( m, n { 1, 2, . . ., 20} ) such that 3m +7n is a multiple of 10, is
(A) 100 (B) 200 (C) 4! 4! (D) none of these 64. (A) The last digit of powers of 3 will be 3, 9, 7, 1 and it repeats in the same order. The last digit
of powers of 7 will be 7, 9, 3,1 and it repeats in same order. Now 3m + 7n will be a multiple of 10 as 3+7, 9+1, 7+3, 1+9.
(m, n) will be of the form(4t+1, 4k+1), ( 4t +2, 4k), ( 4t+3, 4k+3) and (4t , 4k+2). So total number of ways = 5 5 + 5 5 + 5 5 + 5 5 =100 65. The number of four-digit natural numbers in which odd digits occur at even places and
even digits occur at odd places and digits are in increasing order from left to right, (A) is less than 36 (B) is greater than 100 (C) lies between 60 and 100 (D) none of these. 65. (A)
I II III IV
Two distinct odd digits for the second and fourth places can be selected in 4C2 = 6 ways
(since we cannot take 1, as first digit will be at least 2). Now these can be arranged in increasing order in one way only. Similarly two distinct even digits for the first and third places can be selected in 4C2 = 6 ways (since we cannot take 0). Now these can be arranged in increasing order in one way only.
Now total number of ways of filling the four places is 6 6 = 36. But this contains the numbers of the type 6385 which are not needed. So number of such
numbers will be less than 36. 66. The number of permutations of the letters of the word HINDUSTAN such that neither
the pattern ‘HIN’ nor ‘DUS’ nor ‘TAN’ appears, are (A) 166674 (B) 169194 (C) 166680 (D) 181434 66. (B)
Total number of permutations = !2!9
Number of those containing ‘HIN’ = 7!
Number of those containing ‘DUS’ = !2!7
Number of those containing ‘TAN’ = 7! Number of those containing ‘HIN’ and ‘DUS’ = 5! Number of those containing ‘HIN’ and ‘TAN’ = 5! Number of those containing ‘TAN’ and ‘DUS’ = 5! Number of those containing ‘HIN’, ‘DUS’ and ‘TAN’ = 3!
Required number =
2!7!7!7
!2!9 3 5! – 3! = 169194.
67. Nine hundred distinct N-digit numbers are to be formed by using 6, 8 and 9 only. The
smallest value of N for which this is possible, is (A) 6 (B) 7 (C) 8 (D) 9
67. (B) (3)6 = 729 < 900 and (3)7 = 2187 > 900 68. y = x + r and y = - x + r where r takes all decimal digits. Then the number of squares in xy
plane formed by these lines with diagonals of 2 units length are (A) 81 (B) 100 (C) 64 (D) 49 68. (C) Draw all ten lines y = x + r and other ten lines y = -x + r. We can observe that required
squares are 82 = 64 69. Let y be an element of the set A = {1, 2, 3, 5, 6, 10, 15, 30} and x1, x2, x3 be positive integers
such that x1x2x3 = y, then the number of positive integral solutions of x1x2x3 = y is (A) 64 (B) 27 (C) 81 (D) None of these 69. (A) The number of solutions of the given equation is the same as the number of solution of the
equation x1x2 x3 x4 = 30 = 2 3 5 ( here x4 is dummy variable ) Hence number of solutions is 43 = 64. 70. The number of ways in which we can choose 2 distinct integers from 1 to 100 such that
difference between them is at most 10 is (A) 100C2 – 90C2 (B) 100C98 – 90C88 (C) 100C2 – 90C88 (D) None of these 70. (A), (B), (C) Let the chosen integers be x1 and x2 . Let there be a integer before x1, b integer between x1 and x2 and c integer after x2 a+b +c = 98. Where a 0 , b 10 , c 0 Now if we consider the choices where difference is at least 11, then the number of solution
is 88 + 3 –1C3 –1 = 90C2 Number of ways in which b is less than 10 is 100C2 – 90C2 which is equal to (A), (B)
and (C) option. 71. How many words can be formed by taking four different letters of the word MATHEMATICS? (A) 796 (B) 1680 (C) 2454 (D) 18 72. In an examination there are 3 multiple choice questions and each question has 4 choices.
Number of sequences in which a student can fail to get all answers correct is (A) 11 (B) 15 (C) 80 (D) 63 73. Number of ways in which 6 persons can be seated in a row so that two particular persons are
never seated together is equal to (A) 480 (B) 72 (C) 120 (D) 240 74. The number of ways in which N positive signs and n negative sign (Nn) may be placed in a row
so that no two negative signs are together is (A) NCn (B) N+1Cn (C) N! (D) N+1Pn
75. The number of diagonals of hexagon is (A) 3 (B) 6 (C) 9 (D) 12 76. The number of 10 digits that can be written by using the digits 1 and 2 is (A) 1010 (B) 10P2 (C) 210 (D) 10! 77. The number of all the odd divisors of 3600 is (A) 45 (B) 4 (C) 18 (D) 9 78. Number of all four digit numbers having different digits formed of the digits 1, 2, 3, 4 and 5 and
divisible by 4 is (A) 24 (B) 30 (C) 125 (D) 100 79. Let A be the set of 4-digit numbers a1a2a3a4 where a1> a2> a3> a4, then n(A) is equal to (A) 126 (B) 84 (C) 210 (D) none of these 80. A polygon has 44 diagonals, then n is equal to (A) 10 (B) 11 (C) 12 (D) 13 81. nCr + 2.nCr + 1 + nCr+2 is equal to (2 r n) (A) 2.nCr + 2 (B) n +1Cr +1 (C) n + 2Cr + 2 (D) none of these 82. Number of ways in which 6 persons can be seated around a table so that two particular persons
are never seated together is equal to (A) 480 (B) 72 (C) 120 (D) 240 83. How many words can be made from the letters of the word INSURANCE, if all vowels come
together (A) 18270 (B) 17280 (C) 12780 (D) none of these 84. If a, b, c, d, e are prime integers, then the number of divisors of ab2c2de excluding 1 as a factor,
is (A) 94 (B) 72 (C) 36 (D) 71 85. The number of 5-digit numbers in which no two consecutive digits are identical is (A) 9283 (B) 984 (C) 95 (D) None of these
PROBABILITY LEVEL-I
1. From a group of 10 persons consisting of 5 lawyers, 3 doctors and 2 engineers, four persons
are selected at random. The probability that the selection contains at least one of each category is
(A) 1/2 (B) 2/3 (C) 2/3 (D) none of these 2. If one ball is drawn at random from each of the three boxes containing 3 white and 1 black, 2
white and 2 black, 1 white and 3 black balls then the probability that 2 white and 1 black balls will be drawn is
(A) 13/32 (B) 1/4 (C) 1/32 (D) 3/16 3. The probability of occurrence of a multiple of 2 on a dice and a multiple of 3 on the other dice
of both are thrown together is (A) 7/26 (B) 1/32 (C) 11/36 (D) 1/4 4. A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of
the head appearing on the fifth toss equals (A) 31/32 (B) 1/32 (C) 1/2 (D) 1/5 5. Let A and B be two independent events such that their probabilities are 3/10 and 2/5. The
probability of exactly one of the events happening is (A) 23/50 (B) 1/2 (C) 31/50 (D) none of these 6. A second-order determinant is written down using the numbers 1, –1 as elements. Then the
probability for which determinant is non-zero is (A) 3/8 (B) 5/8 (C) 1/8 (D) 1/2 7. There are 7 seats in a row. Three persons take seats at random. The probability that the
middle seat is always occupieace and no two persons are consecutive is (A) 9/70 (B) 9/35 (C) 4/35 (D) none of these 8. A, B, C are three events for which P (A) = 0.6, P (B) = 0.4, P (C) = 0.5, P (A B) = 0.8,
P (A C) = 0.3 and P (A B C) = 0.2. If P (A B C) 0.85, then the interval of values of P (B C) is
(A) [0.2, 0.35] (B) [0.55, 0.7] (C) [0.2, 0.55] (D) none of these 9. The probability that at least one of the events A and B occurs is 0.6. If A and B occur
simultaneously with probability 0.2, then P( A ) + P( B ) is (A) 0.4 (B) 0.8 (C) 1.2 (D) 1.4 10. A fair die is thrown until a score of less than 5 points is obtained. The probability of obtaining
not less than 2 points on the last thrown is (A) 3/4 (B) 5/6 (C) 4/5 (D) 1/3 11. Let 'E' and 'F' be two independent events. The probability that both 'E’ and 'F’ happen is
1/12 and the probability that neither 'E' nor 'F' happens is 1/2, then , (A) P(E) = 1/3, P(F) = 1/4 (B) P(E) = 1/2, P(F) = 1/6
(C) P(E) = 1/6, P(F) = 1/2 (D) P(E) = 1/4, P(F) = 1/3 12. A die is thrown three times and the sum of three numbers obtained is 15. The probability of
first throw being 4 is
(A) 181 (B)
51 (C)
54 (D)
1817
13. The probability that a shooter will hit a target is give as 51 . Then the probability of atleast
one hit in 10 shots is
(A) 1051 (B)
10
541
(C) 1 – 1051 (D)
10
54
14. There are 4 envelopes with addresses and 4 concerning letters. The probability that letter
does not go into concerning proper envelope, is or There are four letters and four addressed envelopes. The chance that all letters are not
dispatched in the right envelope is
(A) 2419 (B)
2321 (C)
2423 (D)
241
15. Three identical dice are rolled. The probability of that the same number will appear on each
of them is
(A) 61 (B)
361 (C)
181 (D)
283
16. In a box containing 100 bulbs, 10 are defective. What is the probability that out of a sample
of 5 bulbs, none is defective.
(A) 10–5 (B) 5
21
(C) 5
109
(D) 109
17. A pair of dice is thrown and the numbers appearing have sum greater than or equal to 10.
The probability of getting sum 10 is
(A) 61 (B)
41 (C)
31 (D)
21
18. If P(A) = 32 , P(B) =
21 and P(AB) =
65 then the events A and B are
(A) mutually exclusive (B) independent (C) independent and mutually exclusive (D) none of these 19. In a given race the odds in favour of four horses A, B, C,D are 1:3, 1:4, 1:5,1:6 respectively.
Assuming that a dead heat is impossible, find the chance that one of them wins the race.
(A) 420319 (B)
420219 (D)
400319 (D) none of these
20. A number is chosen at random from the numbers 10 to 99. By seeing the number a man will
laugh if product of the digits is 12. If he choose three numbers with replacement then the probability that he will laugh at least once is
(A) 1 – 3
53
(B) 3
4543
(C) 1 – 3
254
(D) 1 – 3
4543
21. If 6
p21and4
p1,3
p31 are the probabilities of three mutually exclusive events, then the
set of all value of p is
(A)
21,
31 (B)
21,
31 (C)
65,
31 (D) none of these
22. One hundred identical coins, each with probability p, of showing up heads are tossed. If
0 < p < 1 and the probability of heads showing on 50 coins is equal to that of the heads showing in 51 coins, then value of p is
(A) 21 (B)
10149 (B)
10150 (D)
10152
23. A fair dice is tossed until a number greater than 4 appears. The probability that an even
number of tosses shall be required is
(A) 21 (B)
53 (C)
51 (D)
32
24. There are four machines and it is known that exactly two of them are faulty. There are tested
one by one, in a random order till both the faulty machine’s are identified. Then the probability that only two tests are needed is
(A) 31 (B)
61 (C)
21 (D)
41
25. If the integers m and n are chosen at random between 1 and 100. Then the probability that a
number of form 7m + 7n is divisible by 5 equals
(A) 41 (B)
71 (C)
81 (D)
491
LEVEL-II 1. All the spades are taken out from a pack of cards. From these cards, cards are drawn one
by one with out replacement till the ace of spades comes. The probability that the ace comes in the 4th draw is
(A) 1/13 (B) 12/13 (C) 4/13 (D) none of these 2. 8 coins are tossed simultaneously. The chance that head appears at least five of them is
(A) 8C5 (B) 8
58
21C
(C) 25693 (D) none of these
3. A number of six digits is written down at random. Probability that sum of digits of the number
is even is (A)1/2 (B) 3/8 (C) 3/7 (D) none of these 4. Fifteen coupons are numbered 1, 2, 3, - - - 15. Seven coupons are selected at random one
at a time with replacement. The probability that the largest number appearing on the selected coupon is 9, is
(A) 6
169
(B)7
158
(C) 7
53
(D) none of these
5. A bag contains three white, two black and four red balls. If four balls are drawn at random
with replacement, the probability that the sample contains just one white ball is; (A) (2/3)4 (B) 32/81 (C) (1/3)4 (D) none of these. 6. A purse contains 4 copper coins, 3 silver coins, the second purse contains 6 copper coins
and 2 silver coins. A coin is taken out of any purse, the probability that it is a copper coin is (A) 4/7 (B) 3/4 (C) 3/7 (D) 37/56 7. Three numbers are chosen at random without replacement from the set A = {x| 1 10, xN}.
The probability that the minimum of the chosen numbers is 3 and maximum is 7, is
(A) 121 (B)
151 (C)
401 (D) None of these
8. Two distinct numbers are selects from the numbers 1, 2, 3, .. . , 9. Then probability that
their product is a perfect square is (A) 2/9 (B) 4/9 (C) 1/9 (D) none of these 9. A student appears for test I, II and III. The student is successful if he passes either in test I,
II or I, III. The probability of the student passing in test I, II and III are respectively p. q and 1/2. If the probability of the student to be successful is 1/2 then
(A) p = q = 1 (B) p = q = 1/2 (C) p = 1, q = 0 (D) p = 1, q = 1/2 10. Two small squares on a chess board are chosen at random. Probability that they have a
common side is, (A) 1/3 (B) 1/9 (C) 1/18 (D) none of these
11. A fair coin is tossed a fixed number of times . If the probability of getting 7 heads is equal to
getting 9 heads, then the probability of getting 2 heads is, (A) 15/28 (B) 2/15 (C) 15/213 (D) none of these
12. A fair die is tossed eight times. Probability that on the eighth throw a third six is observed is,
(A) 8C3 8
5
65 (B) 8
52
7
6.5C (C) 7
52
7
6.5C (D) none of these
13. There are n persons (n 3), among whom are A and B, who are made to stand in a row in
random order. Probability that there is exactly one person between A and B is
(A) 1)n(n
2n (B)
1)n(n2)2(n
(C) 2/n (D) none of these
14. If the papers of 4 students can be checked by any one of the 7 teachers, then the
probability that all the 4papers are checked by exactly 2 teachers is; (A) 2/ 7 (B) 32/ 343 (C) 6/49 (D) None of these 15. If ‘head’ means one and ‘tail’ means two , then coefficient of quadratic equation ax2 + bx + c
= 0 are chosen by tossing three fair coins. The probability that roots of the equations are imaginary is
(A) 5/8 (B) 3/8 (C) 7/8 (D) 1/8 16. In a bag there are 15 red and 5 white balls. Two balls are chosen at random and one is found
to be red. The probability that the second one is also red is (A) 12/19 (B) 13/19 (C) 14/19 (D) 15/19
17. Pair of dice is rolled together till a sum of either 5 or 7 is obtained. Then the probability that 5
comes before 7 is
(A) 91 (B)
61 (C)
52 (D) none of these
18. A determinant is chosen at random the set of all determinants of order 2 with elements 0 or
1 only. Then the probability that the value of the determinant chosen is positive is
(A) 161 (B)
163 (C)
165 (D)
167
LEVEL-III 1. Three of the six vertices of a regular hexagon are chosen at random. The probability that the
triangle with three vertices is equilateral equals to
(A) 21 (B)
51 (C)
101 (D)
201
2. A and B play a game of tennis. The situation of the game is as follows; if one scores two
consecutive points after a deuce he wins; if loss of a point is followed by win of a point, it is deuce. The chance of a server to win a point is 2/3. The game is at deuce and A is serving. Probability that A will win the match is, (serves are changed after each game)
(A) 3/5 (B) 2/5 (C) 1/2 (D) 4/5 3. Six different balls are put in three different boxes, no box being empty. The probability of
putting balls in the boxes in equal numbers is, (A) 3/10 (B) 1/6 (C) 1/5 (D) none of these
4. Three persons A1, A2 and A3 are to speak at a function along with 5 other persons. If the
person speak in random order, the probability that A1 speaks before A2 and A2 speaks before A3 is’
(A) 1/6 (B) 3/5 (C) 3/8 (D) none of these
ANSWERS LEVEL −I 1. A 2. A 3. C 4. C 5. A 6. 7. D 8. A 9. C 10. A 11. A 12. B 13. B 14. A 15. B 16. C 17. D 18. B 19. A 20. D 21. 22. 23. 24. B 25. LEVEL −II 1. A 2. C 3. A 4. D 5. B 6. D 7. C 8. C 9. C 10. C 11. C 12. B 13. B 14. C 15. C 16. C
17. C 18. B LEVEL −III 1. C 2. C 3. B 4. A
1
PS
LEVEL-I 1. nth term of 5, 3, 1, –1, –3, –5, ……… is (A) 2n – 7 (B) 7 – 2n (C) 2n + 3 (D) 2n + 5
2. nth term of 1, 21 ,
31 , …. is
(A) 1n
1
(B) 1n
1
(C) n1 (D)
1nn
3. Sum of the series 21 +
221 +
41 + ……… is
(A) 1 + 21 (B) 1 (C)
121
(D) 12
2
4. Number of integers between 100 and 200, that are divisible by 5 are (A) 10 (B) 20 (C) 9 (D) 19
5. H.M of 3 and 31 is
(A) 35 (B) 1 (C)
320 (D)
53
6. The nth terms of the two series 3 + 10 + 17 + …. and 63 + 65 + 67 + …… are equal, then the
value of n is (A) 9 (B) 13 (C) 19 (D) none of these 7. If n A.M’s are inserted between two quantities a and b, then their sum is equal to
(A) n(a + b) (B) 2n (a + b)
(C) 2n(a + b) (D) 2n (a – b)
8. If a, b, c are in H.P, then the value of cbcb
abab
is
(A) 1 (B) 2 (C) 3 (D) none of these 9. If a, b, c are in A.P., a, x, b are in G.P. and b, y, c are in G.P., then x2, b2, y2 are in (A) H.P (B) G.P (C) A.P (D) none of these 10. If a, b, c, d, e are in A.P, then (e – a) is equal to (A) 2(b + d) (B) 2(b – d) (C) 2(d – b) (D) none of these 11. If (2x – 1), (4x – 1), (7 + 2x) ……. are in G.P, then next term of the sequence is (A) 625/3 (B) 125/3 (C) 81 (D) 9 12. In any triangle ABC the angles A, B, C are in A.P, then the value of sin 2B is given by (A) 1/2 (B) 3 /2 (C) 1/ 2 (D) none of these
2
13. If 1 + 2 + 3 + ……. + 49 = x, then 13 + 23 + 33 + …… + 493 is given by (A) x3 (B) x2 (C) x2 + x (D) none of these 14. If a, b, c are in A.P and a, b, d are in G.P, then a, a – b, d – c will be in (A) A.P (B) G.P (C) H.P (D) none of these
15. rth term of sequence .......975
1753
1531
1
is given by
(A) )4r)(2r(r
1
(B) )5r2)(3r2)(1r2(
1
(C) )3r2)(1r2)(1r2(
1
(D) none of these
16. If vr = r)1r(1
1
, then vr –1 is equal to
(A) r)1r(1
1
(B) r)1r(1
1
(C) )2r)(1r(1
1
(D) none of these
17. The value of
)x1n(11log........
x211log
x111log
x11logxlog
(A) nxlog (B) nxlog (C) )xnlog( (D) x)1nlog(
18. If a, b, c, d are in H.P., then ab + bc + c d is equal to……… 19. If the first term of a G.P is 1 and the sum of the third and fifth terms is 90. Then the common
ratio if G.P is (A) 1 (B) 2 (C) 3 (D) 4
20. If a, b, c are in A.P., then ab1,
ca1,
bc1 will be in
(A) A.P. (B) G.P. (C) H.P. (D) None of these 21. The numbers 1, 4, 16 can be three terms (not necessarily consecutive) of (A) no A.P. (B) only 1 or 2 G.Ps (C) infinite number of A.Ps (D) infinite number of G.Ps
22. If Sn =
n
1rr
2
2termsr.....221 , then Sn is equal to
(A) 2n – (n + 1) (B) n × (n + 1)/2 (C) (n2 + 3n + 2)/6 (D) n – 1 + (1/2n)
23. If Sn = nP + Q2
1nn , where Sn denotes the sum of the first ‘n’ terms of an A.P. then the
common difference is (A) P + Q (B) 2P + 3Q (C) 2Q (d) Q 24. a, b, c R+ and from an A.P. if abc = 4, then the minimum value of b is (A) (2)2/3 (B) (2)1/3 (C) (4)2/3 (D) none of these
3
25. If b + c, c + a, a + b are in H.P., then a2, b2, c2 will be in
(A) G.P. (B) H.P. (C) A.P. (D) none of these 26. Every term of a G.P. is positive and every term is the sum of two preceding terms. Then the
common ratio of the G.P. is
(A) 2
51 (B) 2
51
(C) 2
15 (D) 1
27. If the roots of the equation a(b − c)x2
+ b(c − a)x + c(a − b) = 0 are equal, then a, b, c are in (A) A.P. (B).G.P. (C) H.P. (D) none of these
28. If a, b, c R+, then ba
abca
accb
bc
is always
(A) cba21
(B) abc31
(C) cba31
(D) abc21
29. If a, b, c are in A.P., then a3 + c3 − 8b3 is equal to (A) 2abc (B) 6abc (C) 4abc (D) none of these
30. If bc
1c1
ba1
a1
= 0 and a + c −b 0, then a, b, c are in
(A) A.P. (B) G.P. (C) H.P. (D) none of these 31. Three non-zero numbers a, b and c are in A.P.. Increasing a by 1 or increasing c by 2 the
number become in G.P., then ‘b’ equals to (A) 10 (B) 12 (C) 14 (D) 16 32. Let the positive numbers a, b, c, d be in A.P. then abc, abd, acd, bcd are (A) not in A.P./G.P./H.P. (B) in A.P. (C) in G.P. (D) in H.P. 33. Consider an infinite series with first term a and common ratio ‘r’. If its sum is 4 and the
second term is 43 , then
(A) a = 47 , t =
73 (B) a = 2, r =
83
(C) a = r,23
21 (D) a = 3, r =
41
34. The value of
n
1r1r
r
balog is
(A)
n
n
balog
2n (B)
n
1n
balog
2n
4
(C)
1n
1n
balog
2n (D)
1n
1n
balog
2n
5
LEVEL-II 1. If a, b, c are in H.P. and a > c > 0 , then
ba1
cb1
(A) is positive (B) is zero (C) is negative (D) has no fixed sign. 2. If the sum Sn of n terms of a progression is a cubic polynomial in n, then the progression
whose sum of n terms is Sn – Sn-1 is (A) an A. P. (B) a G. P. (C) a H.P. (D) an A. G. P.
3. Let p, q, r R+ and 27pqr ( p + q + r)3 and 3p + 4q + 5r = 12 then p3 + q4 + r5 is equal to (A) 3 (B) 6 (C) 2 (D) none of these 4. Let a, b and c be positive real numbers such that a + b + c = 6. Then range of ab2c3 is (A) (0, ) (B) (0, 1) (C) (0, 108] (D) (6, 108] 5. log45 , log20 5, log1005 are in (A) A.P. (B) G.P. (C) H.P. (D) none of these 6. If the product of three positive real numbers say a, b, c be 27, then the minimum value of
ab + bc + ca is equal to (A) 274 (B) 273 (C) 272 (D) 27 7. If three distinct real numbers a, b, c are in G.P and a + b + c = ax , then
(A)
,43x –{1, 3} (B) x R+ (C) x (-1, ) (D) none of these
8. If ,cb
1
ac
1
, ba
1
are in A.P. then 9ax + 1, 9bx+1, 9cx+1, x 0 are in
(A) G.P. (B) G.P. only if x < 0 (C) G.P. only if x > 0 (D) none of these 9. The sum of an infinitely decreasing G.P. is equal to 4 and the sum of the cubes of its terms
is equal to 64/7. Then 5th term of the progression is
(A) 41 (B)
81 (C)
161 (D)
321
10. Number of increasing geometrical progression(s) with first term unity, such that any three
consecutive terms, on doubling the middle become in A.P. is (A) 0 (B) 1 (C) 2 (D) infinity 11. Sum of n terms of a sequence be n2 + 2n, then it is (A) AP (B) GP (C) HP (D) none of these
12. Sum of
33
22
x1x
x1x
x1x + ……… is
(A) –1 (B) x11x
(C) 0 (D) none of these
13. The third term of a G.P is 4. The product of first five terms is (A) 43 (B) 45 (C) 44 (D) none of these 14. The sum of n terms of the series 12 – 22 + 32 – 42 + …… is, where n is even number
6
(A) –2
)1n(n (B) 2
)1n(n
(C) –n(n + 1) (D) none of these 15. After inserting n A.M’s between 2 and 38, the sum of the resulting progression is 200. The
value of n is (A) 10 (B) 8 (C) 9 (D) none of these 16. If the numbers a, b, c, d, e form an A.P., then the value of a – 4b + 6c – 4d + e is (A) 1 (B) 2 (C) 0 (D) none of these 17. If S1 = {1}, S2 = {2, 3}, S3 = {4, 5, 6}, S4 = {7, 8, 9, 10}, then first term of S20 is given by (A) 20 (B) 190 (C) 191 (D) none of these 18. The polygon has 25 sides, the length of which starting from the smallest sides are in A.P. If
perimeter is 2100 cm and length of largest side is 20 times that of the smallest side then the length of smallest side and common difference of A.P is
(A) 6, 631 (B) 8, 6
31 (C) 8, 5
31 (D) none of these
19. The fourth term of a G.P is 8, the product of the first seven terms is (A) 219 (B) 220 (C) 221 (D) 224 20. If 3x+7y + 4z = 21, where x, y, z are positive real numbers, then maximum value of x4y5z3 is
equal to
(A) 12
457 1057 (B) 12
457 1057 (C) 34
5711
76
(D)
3457
10
65
21. If A, G and H be the A.M, G.M and H.M respectively of two distinct positive integers, then the
equation Ax2 –|G|x –H = 0 has (A) both roots as fractions (B) at least one root as a negative fraction (C) exactly one positive root (D) at least one root as integer
22. If a1, a2, a3, ………an are in H.P, then 1n21
n
n31
2
n32
1
a...aaa,...
a...aaa,
a...aaa
are in (A) A.P (B) G.P (C) H.P (D) A.G.P 23. The tenth common term between the series 3 + 7 + 11 + ….. and 1 + 6 + 11 + …. is (A) 191 (B) 193 (C) 211 (D) none of these
24. ......321
721
513
332322
to is
(A) 3 (B) 4 (B) 5 (D) 6 25. The number of divisors of 1029, 1859 and 122 are in (A) A.P (B) G.P (C) H.P (D) none of these
26. If the first two terms of a H.P. are 53 and
109 respectively then the largest term of
H.P. is
7
(A) 2nd term (B) 3rd term (C) 4th term (D) none of these 27. If log10x + log10 y 2 then the smallest possible value of x2 + y2 is (A) 200 (B) 2000 (C) 100 (D) none of these 28. If ab = 4a + 9b, a> 0, b> 0 then minimum value of ab is (A) 13 (B) 14 (C) 12 (D) none of these 29. If ax3 + bx2 + cx + d is divisible by ax2 + c, then d is equal to
(A) 2ab (B)
abc
(C) bac (D) none of these
30. The sum of the products of the nine numbers 1, 2, 3, 4, 5 taking two at a time is (A) 155 (B) 30 (C) –30 (D) none of these
31. If in a series tn = !2n1n
then
10
0nnt is equal to
(A) 1– !10
1 (B) 1– !11
1
(C) 1– !12
1 (D) none of these
32. The value of
n
2r
32nr is equal to
(A) 94
1nn 22
(B) 9
61n1n2n2
(C) 94
1nn1n 2
(D) none of these
33. The harmonic means of the roots of equation 0528x54x25 2 is (A) 2 (B) 4 (C) 6 (D) 8 34. If x2 + 9y2 + 25z2 = 15yz + 5xz + 3xy then x, y, z are in (A) A.P. (B) G.P. (C) H.P. (D) none of these
35. If x12 + x2
2 + x32 + ….+ x50
2 = 50 and Ax.....xxx
12
502
32
22
1
then
(A) Aminimum = 1 (B) Amaximum = 1 (C) Aminimum = 50 (D) Amaximum = 50 36. If n is an odd integer greater than or equal to 1 then the value of
31n333 1)1(............)2n()1n(n is
8
(A) 4
)12()1( 2 nn (B)
4)12()1( 2 nn
(C)4
)12()1( 2 nn (D) None of these
37. A monkey while trying to reach the top of a pole of height 12 meters takes every time a jump
of 2 meters but slips 1metre while holding the pole. The number of jumps required to reach the top of the pole is . (A) 6 (B) 10 (C) 11 (D) 12
38. The sum of n terms of the series 2 2 2 2 2 21 2.2 3 2.4 5 2.6 ...... is 2
)1n(n 2
when n is even. When n is odd, the sum is
(A) 2
)1n(n2 (B)2
)1n(n 2
(C) )1n2()1n(n 2 (D) None of these.
39. If 4
b3)1n2(3.n.........3.33.23.1a
n32 then (a,b) is :
(A) (n –2, 3) (B) )3,1n( (C) )3,n( (D) )3,1n(
40. The sum of infinite series .......10.71
7.41
4.11 is
(A) 31
(B) 3 (C) 41
(D)
41. If a,b,c,d are positive real numbers such that 2 dcba , then
M= )dc)(ba( satisfies the relation (A) 1M0 (B) 2M1 (C) 3M2 (D) 4M3 42. If A.M. and G.M. between two numbers be A and G respectively, then the numbers are
(A) 22 GAA (B) 22 GAG
(C) 22 AGA (D) None of these
43. The H.M. of two numbers is 4 and their A.M. and G.M. satisfy the relation 2A + G2 = 27, then the numbers are : (a) –3,1 (b) 5, –25 (c) 5, 4 (d) 3, 6
44. If n = 55 then 2n is equal to (a) 385 (b) 506 (c) 1185 (d) 3025
45. If an is an A.P. and a1 + a4 + a7 + …..+ a16 = 147, then a1 + a6 + a11 + a16 =
(a) 96 (b) 98 (c) 100 (d) none of these
46. The interval for which the series ......)1()1(1 2xx may be summed, is (a) 1,0 (b) 2,0 (c) 1,1 (d) 2,2
47. The interior angles of a polygon are in A.P. the smallest angle is 120 and The common difference is 5. Then, the number of sides of polygon is : (a) 5 (b) 7 (c) 9 (d) 15
9
48. 36log..........logloglog 1664 3333 xxxx is
(a) 3x (b) 34x (c) 9x (d) 3x
49. If 1n1n
nn
baba
be the geometric mean between two distinct positive reals a and b, then the
value of n is (A) 0 (B) 1/2 (C) –1/2 (D) 1 50. If log 2, log (2x –1) and log (2x + 3) are in A.P then x is equal to (A) 5/2 (B) log2 5 (C) log3 2 (D) 3/2
51. The values of x for which x1
1,x1
1,x1
1
are in A.P lies in
(A) (0, 2) (B) (1, ) (C) (0, ) (D) none of these 52. If three positive real numbers a, b, c (c > a) are in H.P. then log [(a + c) (a + c –2b)] is equal
to (A) 2 log (c –b) (B) 2 log (a + c) (C) 2 log (c –a) (D) log (abc)
53. The value of the expression 1.(2 –) (2 – 2) + 2.(3 –) (3 – 2) + ……….+ (n –1).(n –) (n – 2), where is an imaginary cube root of unity is………………………………………….
54. Co-efficient of x99 in the polynomial (x –1) (x –2) (x –3)………… (x –100) is
……………………………..
55. The sum of first n terms of the series 1615
87
43
21
+ ……….is equal to
……………………………. 56. log3 2, log6 2, log12 2, are in ………………………………………………………… 57. If an A.P, the pth term is q and the (p + q)th term is 0. the qth term is (A) –p (B) p (C) p + q (D) p –q
58. If the sum of the series 1 + ............x8
x4
x2
32 to is a finite number then
(A) x < 2 (B) x > 21
(C) x > –2 (D) x < –2 or x > 2 59. If a > 1, b > 1 then the minimum value of logb a + loga b is (A) 0 (B) 1 (C) 2 (D) none of these 60. The product of n positive numbers is 1. Their sum is
10
(A) a positive integer (B) divisible by n
(C) equal to n + n1 (D) greater than or equal to n
61. If (1 + x) (1 + x2) (1 + x4) …………(1 + x128) =
n
0r
rx then n is
(A) 255 (B) 127 (C) 63 (D) none of these 62. If tn denotes the nth term of the series 2 + 3 + 6 + 11 + 18 + …..then t50 is (A) 492 –1 (B) 492 (C) 502 + 1 (D) 492 + 2
63. Let tn = n (n!). Then
15
1nnt is equal to
(A) 15! –1 (B) 15! +1 (C) 16! –1 (D) none of these 64. The sum of 19 terms of an A.P, whose nth terms is 2n + 1 is (A) 390 (B) 399 (C) 499 (D) none of these 65. Three numbers whose sum is 15 are in A.P, if 8, 6 and 4 be added to then respectively then
these are in G.P, then the numbers are (A) 4, 6, 8 (B) 1, 5, 9 (C) 2, 5, 8 (D) 3, 5, 7
66. If x + y + z = 3, then z1
y1
x1
is , x , y, z > 0
(A) 3 (B) 3 (C) 4 (D) none of these 67. If x = log5
3 + log75 + log9
7 then
(A) x ≥ 3/2 (B) x ≥ 3 21
(C) x > 3
32
(D) none of these
68. If tr = 2r/2 + 2-r/2 then
10
1r
2rt is equal to
(A) 202
1210
21
(B) 192
1210
21
(C) 12
1220
21
(D) 202
123 10
10
69. If (a, b), (c, d), (e, f) are the vertices of a triangle such that a, c, e are in G.P. with common
ratio r and b, d, f are in G.P. with common ratio s then the area of the triangle is
(A) rs2s1r2
ab (B) rs1s1r
2ab
(C) rs1s1r2
ab (D) rs1s1r
2ab
11
70. a, b, c R+, then the minimum value of a(b2 + c2) + b(c2 + a2) + c(a2 + b2) is equal to (A) abc (B) 2abc (C) 3abc (D) none of these 71. a, b, c R+ ~ {1} and loga100, 2logb10, 2logc5 + logc4 are in H.P., then (A) 2b = a + c (B) b2 = ac (C) b(a + c) = 2ac (D) none of these 72. If (m + 1)th , (n + 1)th and (r + 1)th terms of an A.P. are in G.P. and m, n, r in H.P., then
ratio of the first term of the A.P. to its common difference in terms of n is
(A) 2n (B) −
2n
(C) 3n (D) −
3n
73. Suppose a, b, c are in A.P. and a2, b2, c2 are in G.P.. If a < b< c and a + b + c = 23 , then the
value of a is
(A) 22
1 (B) 32
1
(C) ,31
21 (D)
21
21
74. The value of 21/4.41/8.81/16……. is (A) 1 (B) 2
(C) 23 (D)
25
75. Coefficient of x9 in the polynomial (x – 5)(x – 8)(x – 11)…..(x – 32) is given by (A) 185 (B) 153 (C) –185 (D) –153
12
LEVEL-III
1.
1n4 1n4n equals to
(A) 0 (B) 1 (C) (D) none of these . 2. If 3x2 – 2(a – d) x + (a2 + 2(b2 +c2) + d2) = 2(ab + bc + cd), then (A) a, b, c, d are in G .P. (B) a, b, c, d are in H .P. (C) a, b, c, d are in A .P. (D) None of these 3. The sum of numbers in the nth group of the following
(1, 3), ( 5, 7, 9, 11), ( 13, 15, 17, 19, 21, 23) , . . . . is
(A) 3
2n1nn (B) 2n3 (C) n2(n +1)2 (D) 4n3
4. If S denote the sum to infinity and Sn the sum of n terms of the series ...
271
91
311
such that S – Sn < 300
1 , then the least value of n is
(A) 4 (B) 5 (C) 6 (D) 7 5. If a, b, c are three positive real numbers, then the minimum value of the expression
cba
bac
acb
is
(A) 1 (B) 2 (C) 3 (D) None of these 6. If xi > 0, i = 1, 2, …., 50 and x1 + x2 + … + x50 = 50, then the minimum value of
5021 x1...
x1
x1
equals to
(A) 50 (B) (50)2 (C) (50)3 (D) (50)4
7. The value of ....18.14
114.10
110.61
equals to
(A) 224
1 (B) 61 (C)
241 (D)
3241
8. Let rth term of a series be given by Tr = 42 rr31r
. Then
n
1rrn
Tlim is
(A) 3/2 (B) 1/2 (C) –1/2 (D) –3/2 9. A sequence a1, a2 …. an of real numbers is such that a1 = 0, |a2| = |a1 – 2|, |a3| = |a2 – 2|,
….. |an| = |an-1 – 2|. Then the maximum value of the arithmetic mean of these numbers is (A) 1 (B) 4n (C) n (D) none of these
10. If x1, x2, … x20 are in H.P. then x1 x2 + x2x3 + … + x19x20 = (A) x1 x20 (B) 19 x1x20 (C) 20 x1x20 (D) none of these
11. The first two terms of an H.P. are 52 and
2312 . The value of the largest term of the H.P. is
(A) 7372 (B) 6 (C)
61 (D) none of these
13
12. ....7.5
35.3
23.11
222222 up to n terms equals to
(A) 1n21n (B)
21n221nn
(C) 1n2
n
(D) None of these
13. If abc = 8 and a, b, c > 0, then the minimum value of (2 + a) (2 + b) (2 + c) is (A) 32 (B) 64 (C) 8 (D) 10
14. Coefficient of x49 in the polynomial
101......31
50x.......531
2x31
1x is
(A) 101......31
121
(B) –
101......31
1121
(C) 101......31
49
(D) 101......31
50
15. Let nfrn
1r
4
, then
n
1r
41r2
(A) f (2n) –16 f (n); n N (B) f (n) –16 f
2
1n , when n is odd
(C) f (n) –16 f
2n , when n is even (D) none of these
16. The co-efficient of 2nx in )nx.().........3x)(2x)(1x( is
(A) 24
)1n3)(1n(n 2 (B) 24
)2n3)(1n(n 2
(C)24
)4n3)(1n(n 2 (D) None of these
17. If a,b,c, are digits, then the rational number represented by 0.cababab….is (a) cab/990 (b) (99c + ab) / 990 (c) (99c + 10a + b) / 99 (d) (99c + 10a + b ) / 990
18. If 90
........31
21
11 4
444
then ........
51
31
11
444 is equal to
(a) 96
4 (b)
45
4 (c)
90
4 (d)
46
4
19.
n
i
i
j
j
k1 1 1.......1
(a) 6
)2)(1( nnn (b) 2n (c)
6)2)(1( nnn
(d) none of these
20 If In = 4/
0
n dxxtan , then 645342 II
1,II
1,II
1
are in
(A) A.P (B) G.P (C) H.P (D) none of these
14
21 If x > 1, y > 1, z > 1 are in G.P, then zln1
1,yln1
1,xln1
1
are in…………………..
22. If ax = by = cz = du and a, b, c, d are in G.P., then x, y, z, u are in …………………….. 23. Let a1, a2, a3, …….., a10 be in AP and h1, h2, h3, …….., h10 be in H.P. If a1 = h1 = 2 and a10 =
h10 = 3 then a4h7 is (A) 2 (B) 3 (C) 5 (D) 6 24. In the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4 ....... , where n consecutive terms have the value n,
the 150th term is (A) 17 (B) 16 (C) 18 (D) none of these 25. If a, a1, a2…….a2n-1, b are in A.P, a, b1, b2…….b2n-1, b are in G.P. and a c1, c2…….c2n-1, b are
in H.P. where a, b are positive then the equation anx2 – bnx + cn = 0 has its roots (A) real and unequal (B) real and equal (C) imaginary (D) do not exist
26. If
n
1k
k
1m
m = an4 + bn3 + cn2 + dn + e, then
(A) a = 121 , e =
121 (B) a = 0, e = 0
(C) a = 0, e = 121 (D) a =
121 , e = 0
27. In the above question find the values of b, c and d ? ………………………………………………………. 29. If mth, nth and pth terms of an A.P. and G.P. are equal and are respectively x, y, z then (A) xyyzzx = xzyxzy (B) (x − y)x (y − z)y = (z − x)z (C) (x − y)z ( y − z)x = (z − x)y (D) none of these 30. Coefficient of x8 in (x –1) (x – 2) (x –3) …. (x – 10) is
(A) 980 (B) 1395 (C) 1320 (D) none of these . 31. If the sum to n terms of an A.P. is cn(n –1), where c 0. The sum of the squares of these
terms is
(A) c2n2(n + 1)2 (B) 32 c2n (n –1) (2n –1)
(C) 32 c2n (n + 1) (2n + 1) (D) none of these
ANSWERS LEVEL −I 1. B 2. C 3. A 4. D 5. D 6. B 7. A 8. B 9. C 10. C 11. B 12. B 13. B 14. B 15. C 16. C 17. C 18. 3ad 19. C 20. A 21. C 22. D 23. D 24. A 25. C 26. B 27. C 28. A
15
29. D 30. C 31. B 32. D 33. D 34. D LEVEL −II 1. A 2. A 3. A 4. C 5. A 6. D 7. A 8. A 9. B 10. B 11. A 12. A 13. B 14. A 15. B 16. C 17. C 18. B 19. C 20. A 21. C 22. C 23. A 24. B 25. A 26. C 27. A 28. C 29. B 30. D 31. C 32. D 33. B 34. C 35. A 36. A 37. C 38. A 39. D 40. A 41. A 42. A 43. D 44. A 45. B 46. B 47. A 48. D 49. B 50. B 51. B
52. C 53. 22n n 11
4
54. -5555 55. n – 1 + 2-n
56. H.P. 57. B 58. D 59. C 60. D 61. A 62. D 63. C 64. B 65. D 66. B 67. C 68. B 69. C 70. D 71. D 72. B 73. D 74. B 75. C LEVEL −III 1. D 2. C 3. D 4. C 5. D 6. A 7. C 8. C 9. A 10. B 11. B 12. B 13. B 14. B 15. A 16. B 17. D 18. A 19. A 20. C 21. H.P. 22. H.P. 23. B 24. A 25. B 26. B
27. 1 1 1, ,6 2 3
29. A 30. C 31. B
QEE 1. The equation whose roots are opposite in sign to those of the equation x2 3x 4 = 0 is
given by (A) 4x2 3x + 1 = 0 (B) x2 + 3x 4 = 0 (C) x2 + 3x + 4 = 0 (D) none of these 2. Sum of the roots of the equation x5 5x3 + x + 1 = 0 is given by (A) 0 (B) 5 (C) 1 (D) none of these 3. If the roots of quadratic equation ax2 + bx + c = 0 are equal in magnitude and opposite in
sign then (A) a = 0 (B) c = 0 (C) a = c (D) none of these 4. One of the roots of the quadratic equation (sin2 ) x2 x + cos2 = 0 is given by (A) 1 (B) 2 (C) 1 (D) none of these
5. If and are the roots of ax2 + bx + c = 0, then the equation whose roots are 1 1 and
is
given by (A) ax2 + cx + b = 0 (B) cx2+ bx + a = 0 (C) (ac b2) x2 + bx + c = 0 (D) none of these
6. If 1 1x 2 3
; then x belongs to
(A) ( , 5] (B) [2, 5] (C) (2, 5] (D) none of these 7. The number of real roots of the equation 12 5x7x2 2
is (A) 0 (B) 1 (C) 2 (D) 4 8. The real roots of the equation 5x4xlog 2
77 = (x − 1) are (A) 1 and 2 (B) 2 and 3 (C) 3 and 4 (D) 4 and 5 9. If roots of quadratic equation ax2 + 2bx +c = 0 are not real, then ax2+ 2bxy+ cy2+ dx+ ey+f=0
represent (A) Ellipse (B) Circle (C) Parabola (D) Hyperbola 10. 3x10 − 5x2 + 7 = 0 is an (A) equation (B) expression (C) identity (D) none of these 11. Expression x2 + px + q will be a perfect square of linear expression if (A) p2 − 4q = 0 (B) p2 + 4q = 0 (C) q2 = p2 (D) none of these
12. If a, b, c are the roots of the equation x3 − px2 + qx − r = 0 then the value of 222 c1
b1
a1
is
(A) r
pr2q2 (B) r
pr2q2
(C) 2
2
rpr2q (D) 2
2
rpr2q
13. If a, b, c R, the roots of a equation (x − a)(x − b) + (x − b)(x − c) + (x − c)(x − a) = 0 are (A) rational (B) irrational (C) imaginary (D) real 14. Root of equation 3x−1 + 31−x = 2 is (A) 2 (B) 3 (C) 4 (D) none of these 15. If (1 + m)x2 − 2(1 + 3m)x + (1 + 8m) = 0 has equal roots, then m is equal to (A) 0, 1 (B) 0, 2 (C) 0, 3 (D) none of these 16. If the roots of the equation (a2 + b2) x2 + 2x (ac + bd) + c2 + d2 = 0 are real, then (A) ad = bc (B) ab = cd (C) ac = bd (D) none of these
17. If r be the ratio of the roots of the equation ax2 + bx + c = 0, then
r1r 2
is equal to
(A) bca2
(B) acb2
(C) abc2
(D) none of these
18. If the roots of the equation x2 + px + q = 0 differ from the roots of the equation x2 + qx + p = 0
by the same quantity, then p + q is equal to (A) −1 (B) −2 (C) −3 (D) −4
19. The quadratic equation whose one of the roots is 52
1
is
(A) x2 + 4x − 1 = 0 (B) x2 + 3x − 1 = 0 (C) x2 + 4x + 1 = 0 (D) none of these 20. Let , be the roots of x2 − x + p = 0 and , be the roots of x2 − 4x + q = 0. If , , , are
in G.P. , then the integral value of p and q respectively are (A) −2, −32 (B) −2, 3 (C) −6, 3 (D) −6, −32 21. If , are roots of x2 − p(x + 1) − c = 0 then ( + 1) ( + 1) is equal to (A) c (B) c − 1 (C) 1 − c (D) none of these 22. For a b, if the equations x2 + ax + b = 0 and x2 + bx + a = 0 have a common root, then the
value of (a + b) is (A) −1 (B) 0 (B) 1 (D) 2
23. If the roots of the equation 11
caxbxx2
are equal and opposite then the value of is
(A) baba
(B) c
(C) c1 (D)
baba
24. The equation 1x41x1x has (A) no solution (B) one solution (C) two solution (D) none of these
25. If , are the roots of the equation ax2 + 3x + 2 = 0, then the sign of expression
22
is (A) positive (B) negative (C) can’t say (D) none of these
26. If and be the roots of the equation ax2 + bx + c = 0, then
ba
22
is
equal to (A) a (B) b (C) c (D) none of these 27. If (a2 − 1)x2 + (a − 1)x + a2 − 4a + 3 = 0 be an identity in x. Then value of a is (A) 1 (B) 2 (C) 3 (D) none of these 28. If 3x+1 = 3log26 , then x is (A) 3 (B) 2 (C) log32 (D) log23 29. If and are the roots of 2x2 – 5x + 7 = 0, then equation whose roots are 2 + 3, 3 + 2
is (A) x2 – 25 x + 82 = 0 (B) 2x2 – 25 x + 82 = 0 (C) x2 – 20 x + 64 = 0 (D) none of these 30. The set of all the possible values of a, so that 6 lies between the roots of the equation
x2 + 2(a – 3)x + 9 = 0 is (A) (– , 0) (6, 0) (B) (– , – 3/4) (C) (0, ) (D) none of these 31. The number of values of a for which (a2 – 3a + 2)x2 + (a2 – 5a + 6)x + a2 – 4 = 0 is an identity
in x is (A) 0 (B) 2 (C) 1 (D) 3 32. x Z, the number of values of x for which x2 – 5x + 6 0 and x2 – 2x > 0 is (A) 1 (B) 2 (C) 3 (D) 4 33. If , are the roots of the equation x2 – px + q = 0 then product of the roots of the quadratic
equation whose roots are 2 - 2, 3 - 3 is (A) p(p2 – q)2 (B) p(p2 – q) (p2 – 4q)
(C) p(p2 – 4q) (p2 + q) (D) none of these 34. If x [2, 4] then for the expression x2 – 6x + 5 (A) the least value = -4 (B) the greatest value = 4 (C) the least value = 3 (D) the greatest value = -3
35. The value of x for which
03x1x
2x1x23
4
is
(A) [−1, 1] (B) (−1, 1] (C) (−1, 1) (D) none of these 36. If a and b are non–zero roots of the equation x2 + ax + b = 0 then the least value of
x2 + ax + b = 0 is (A) 0 (B) – 9/4 (C) 9/4 (D) none of these 37. (x – 3)2(x + 2) 0 for all values of x belonging to interval (A) [–2, ) (B) (–, –2] (C) [–2, 3) (D) none of these 38. The roots of quadratic equation are always rational if and only if (A) D is a perfect square (B) D is a perfect square and coefficients are rational (C) D is not a perfect square (D) D is not a perfect square and coefficients irrational 39. The graph of quadratic equation expression f (x) = ax2 + bx + c with a > 0 is always above x-
axis iff (A) D = 0 (B) D > 0 (C) D < 0 (D) none of these 40. Quadratic equations (a – b)x2 + (b – c)x + (c – a) = 0 and (2a – b – c)x2 + (2b – c – a)x + (2c – a – b) = 0 have a common root, given by (A) a (B) c (C) b (D) 1 41. If one of the root of a quadratic equation with rational coefficients is rational, then other root
must be (A) imaginary (B) irrational (C) rational (D) none of these 42. If two roots of quadratic equation ax2 + bx + c = 0 are , , then the roots of the quadratic
equation ax2 – bx + c = 0 are given by
(A) 1,1 (B) –, –
(C) 221,1
(D) none of these
43. In the quadratic equation (2a – 3)x2 + ax + a – 5 = 0, the value of a can never be (A) 3/2 (B) 0 (C) 5 (D) none of these 44. The quadratic equation whose roots are –2 and 4 is given by (A) x2 – 2x – 8 = 0 (B) x2 – 2x + 8 = 0 (C) x2 + 2x + 8 = 0 (D) none of these
45. If p, q be two positive numbers, then the number of real roots of quadratic equation
px2 + q|x| + 5 = 0 is (A) 1 (B) 0 (C) 2 (D) 4 46. If p and q are roots of the quadratic equation x2 + mx + m2 + a = 0, then the value of
p2 + q2 + pq is (A) 0 (B) a (C) –a (D) m2 47. The number of real roots of the equation |x|2 – 3|x| + 2 = 0 is (A) 4 (B) 3 (C) 2 (D) 1 48. The diagram shows the graph of
y = ax2 + bx + c, then (A) a > 0 (B) b < 0 (C) c > 0 (D) b2 – 4ac = 0
(x1, 0) (x2, 0)
y
x
49. The equation whose roots are 1 and 0, is (A) x2 – 2x + 1 = 0 (B) x2 – 1= 0 (C) x2– x = 0 (D) none of these 50. One root of px2 – 14x + 8 = 0 is six times the other then p is (A) 0 (B) 3 (C) 1/3 (D) 1 51. Roots of the equation (x – a)(x – b) = h2 are (A) real and equal (B) real and unequal (C) imaginary (D) none of these 52. If x1/2 + x1/4 = 12, then x is (A) 16 or 81 (B) 81 or 256 (C) 81 (D) 16 or 256 53. One root of a quadratic equation is 2 + 3 , then product of roots will be (A) 7 (B) 4 (C) 0 (D) 1 54. The expression –x2 + 3x + 9 is always (A) positive (B) negative (C) 0 (D) none of these 55. If 3x2 − 2mx − 4 = 0 and x2 − 4m + 2 = 0 have a common root, then m is
(A) 12
(B) 13
(C) 13
(D) 12
56. Set of values of x which satisfies )6x)(1x()2x)(4x( 2
0, is
(A) (–2, 1) (6, ) (B) [–2, 1) (6, ) (C) (–, –2] (6, ) (D) [–2, 1) (1, 6) 57. If and are the roots of the equation ax2 + bx + c = 0, then (a 0) (A) a( + ) + c = 0 (B) a( + ) + b = 0 (C) a + + = 0 (D) b( + ) + a = 0 58. If the product of the roots of the equation x2 – 5x + 2log4 = 0 is 8, then is (A) 22 (B) 22 (C) 3 (D) none of these 59. The set of values of ‘a’ for which 1 lies between the roots of the equation x2 + ax + 4 = 0, is (A) (–, –5) (B) (4, ) (C) (5, ) (D) (–5, 4) 60. If ax2 + bx + c < 0, x R then ax2 + bx + c = 0 has (a 0) (A) two real roots (B) one real root (C) complex roots (D) none of these 61. If x2 + ax + b = 0 and x2 + bx + a = 0 have a common root , (a b) then (A) a + b + 1 = 0 (B) a + b = 1 (C) + 1 = 0 (D) none of these 62. The set of values of p for which the roots of the equation 3x2 + 2x + p(p –1) = 0 are of the
opposite sign is (A) (–, 0) (B) (0, 1) (C) (1, ) (D) (0, ) 63. If the roots of the equation x2 –px + q = 0 differ by unity, then (A) p2 = 1 –4q (B) p2 = 1 + 4q (C) q2 = 1 –4p (D) q2 = 1 + 4p 64. If y = 2[x] + 3 = 3[x – 2] + 5, then [x + y] is ([x] denotes the greatest integer function) (A) 10 (B) 15 (C) 12 (D) none of these
65. The set of solutions of 0
x1e2x1x
3x
23
is
(A) (1, ) (B) (– 2, 1) (C) (– , – 2) [1, ) (D) none of these 66. If the roots of x2 + (a – 2)x + a2 = 0 are equal in magnitude but opposite in signs, then
(A) a
3
131,3
131 (B) a
,3
131
(C) a
,3
131 (D) none of these
67. Total number of real roots of sinx = x2 + x + 1 is /are to ; (A) 1 (B) 2 (C) 3 (D) none of these
68. The equation sin2x – 2 sinx + a = 0 will have atleast one real root if, (A) a [– 3, 1] (B) a [– 1, 1] (C) a [0, 1] (D) none of these 69. The number of real solutions of the equation (x –1)2 –4|x –1| + 3 = 0 is (A) 4 (B) 2 (C) 1 (D) 3 70. If the equations ax2 + bx + c = 0 and cx2 + bx + a = 0 a c have negative common root then
the value of a –b + c is (A) 0 (B) 2 (C) 1 (D) none of these
71. The number of integral solutions of 21
1x2x
2 is
(A) 4 (B) 5 (C) 3 (D) none of these 72. If ax2 + bx + 9 = 0 does not have distinct real roots. a, b R, then the greatest value of
b –3a is (A) 3 (B) –3 (C) 6 (D) –6 73. If x2 – 3x + 2 is a factor of x4 – px2 + q = 0 then p, q are (A) 2, 3 (B) 4, 5 (C) 5, 4 (D) 0, 0 74. The inequality |2x – 3| < 1 is valid when x lies in the interval (A) (3, 4) (B) (1, 2) (C) (–1, 2) (D) (–4, 3) 75. If sin and cos are the roots of the equation lx2 + mx + n = 0, then (A) l2 – m2 + 2ln = 0 (B) l2 + m2 + ln = 0 (C) l2 – m2 – ln = 0 (D) l2 + m2 – ln = 0
76. If 1x43x2
2x2x
then values of x are
(A) ( –, –2)
1,
41 (4, ) (B)
41,2 (1, 4)
(C)
1,
21 (D) none of these
77. If roots of the equation 9x2 + 4ax +4 = 0 are imaginary, then (A) a ( –3, 3) (B) a ( –, –3) (3, ) (C) a (2 , 3) (D) none of these 78. If (2 + –2)x2 + ( +2)x < 1 for all x R then belongs to the interval (A) ( –2, 1) (B) (–2, 2/5) (C) (2/5, 1) (D) none of these 79. If , , be the roots of the equation x(1+ x2) + x2 (6 +x) + 2 = 0. Then the value of –1
+ –1 + –1 is (A) –3 (B) 1/2 (C) –1/2 (D) none of these 80. If the roots of 4x2 + 5k = (5k+1)x differ by unity then the negative value of k is
(A) –3 (B) –1/5 (C) –3/5 (D) none of these 81. The solution set of the inequation log1/3(x2 + x+1) + 1 > 0 is (A) ( –, –2) (1, ) (B) [–1, 2] (C) (–2, 1) (D) (–, ) 82. Let and are the roots of equation x2 + x + 1 = 0, the equation whose roots are 19, 17
is (A) x2 – x –1 = 0 (B) x2 –x +1 = 0 (D) x2 + x –1 = 0 (D) x2 + x + 1 = 0 83. If p and q are non–zero constants, the equation x2 + px + q = 0 has roots and , the
equation qx2 + px +1 = 0 has roots (A) and 1/ (B) 1/ and (C) 1/ and 1/ (D) none
84. The solution set of 11x
4x3x2
, x R, is
(A) (3, ) (B) (–1, 1) (3, ) (C) [–1, 1] [3, ) (D) none 85. If the quadratic equation x2 + x + a2 + b2 + c2 – ab – bc – ca = 0 has imaginary roots, then (A) 2 ( - ) +(a - b)2 +(b - c)2 + (c - a)2 > 0 (B) 2 ( - ) +(a - b)2 +(b - c)2 + (c - a)2 < 0 (C) 2 ( - ) +(a - b)2 +(b - c)2 + (c - a)2 = 0 (D) none of these . 87. If x2 +ax +b is an integer for every integer x then (A) ‘a’ is always an integer but ‘b’ need not be an integer (B) ‘b’ is always an integer but ‘a’ need need not be an integer (C) a+b is always an integer (D) a and b are always integers. 88. The value of ‘p’ for which the sum of the square of the roots of
2x2 - 2(p -2)x - p -1= 0 is least, is (A) 1 (B) 11/4 (C) 2 (D) –1 89. If x2 –4x +
21log a = 0 does not have two distinct real roots, then maximum value of a is
(A) 41 (B)
161
(C) 41
(D) none of these
90. The largest negative integer which satisfies 3x2x1x2
> 0 is
(A) –4 (B) –3 (C) –1 (D) –2
91. The number of real solutions of 4x
24x
x 22
is
(A) 0 (B) 1 (C) 2 (D) infinite
92. If the roots of 4x2 + 5k = (5k + 1) x differ by unity then the negative value of k is
(A) –3 (B) 51
(C) 53
(D) none of these
93. If the absolute value of the difference of roots of the equation x2 + px + 1 = 0 exceeds 3 p
then (A) p < -1 or p > 4 (B) p > 4 (C) –1 < p < 4 (D) 0 p < 4 94. If a, b, c, d are positive reals such that a + b + c + d = 2 and m = (a + b) (c + d), then (A) 0 m 1 (B) 1 m 2 (C) 2 m 3 (D) 3 m 4
95. If 1n1n
nn
baba
be the geometric mean between two distinct positive reals a and b, then the
value of n is (A) 0 (B) 1/2 (C) –1/2 (D) 1 96. Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and
the second term is 3/4, then (A) a = 7/4, r = 3/7 (B) a = 2, r = 3/8 (C) a = 3/2, r = 1/2 (D) a = 3, r = 1/4 97. If a + b + c = 0 then ab/cca/bbc/a 222
x.x.x is equal to …….. 98. If ax2 + bx + c = 0 and bx2 + cx + a = 0 have a common root and a 0, then
abccba 333 is equal to ………..
99. If a, b, c are positive real numbers, then the number of real roots of the equation
ax2 + b|x| + c = 0 is …………..
100. The solution set of 11x
4x3x2
, x R, is
(A) (3, ) (B) (–1, 1) (3, ) (C) [–1, 1] [3, ) (D) none of these
LEVEL−II 1. A quadratic equation whose roots are sec2
and cosec2 can be; (A) x2 –2x + 2 = 0 (B) x2 –3x + 3 = 0 (C) x2 –4x + 4 = 0 (D) none of these 2. If x1, x2 are roots of x2 –3x + a = 0, a R and x1 < 1 < x2 then;
(A) a (-, 2) (B)
49,
(C)
49,2 (D) none of these
3. If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to sum of the
squares of the reciprocals then 2
2
abc
acb
is equal to;
(A) 2 (B) –2 (C) 1 (D) –1 4. If discriminant of a quadratic equation ax2 + bx + c = 0 is a perfect square then roots are
always (A) rational (B) integers (C) imaginary (D) none of these 5. The values of ‘a’ for which the quadratic expression x2 ax + 4 is nonnegative for all real
values of x; is given by (A) ( 4, 4) (B) [ 4, 4] (C) ( , 4) (4, ) (D) none of these 6. If a, b, c are odd integers, then roots of the quadratic equation ax2 + bx + c = 0 (A) are always rational (B) cannot be rational (C) are imaginary (D) none of these 7. If x be real, then maximum value of the expression 7 + 10x 5x2 is given by (A) 7 (B) 10 (C) 12 (D) none of these
8. The number of solutions of 2
2xlog1xlog5log 2
is
(A) 2 (B) 3 (C) 1 (D) none of these 9. The roots of the equation (a + c − b)x2 − 2cx + (b + c − a) = 0 are
(A) 1, bca
c2
(B) 1, bcaacb
(C) 1, c2
acb (D) 1,
acbbca
10. If the product of the roots of the equation x2 − 3kx + 2e2logk − 1 = 0 is 7, then the roots are
real for k equal to (A) 2 (B) 4 (C) −2 (D) none of these 11. If sin and cos are the roots of the equation ax2 + bx + c = 0, then
(A) (a − c)2 = b2 − c2 (B) (a − c)2 = b2 + c2 (C) (a + c)2 = b2 − c2 (D) (a + c)2 = b2 + c2 12. If the roots of x2 + ax + b = 0 are non−real, then the value of a2 − 4b − 1 is always (A) negative (B) positive (C) zero (D) nothing can be said 13. If , are the roots of the equation ax2 + bx + c = 0, then the roots of the equation cx2 + bx +
a = 0, are (A) −, − (B) , −
(C) , 1 (D)
1,1
14. If ax2
+ bx + c = 0 is satisfied by every value of x, then (A) b = 0, c = 0 (B) c = 0 (C) b = 0 (D) a = b = c = 0 15. Let S be the set of values of ‘a’ for which 2 lie between the roots of quadratic equation x2 +
(a + 2) x – (a + 3) = 0. Then S is given by (A) (-, -5) (B) (5, ) (C) (-, -5] (D) [5, ) 16. If , , are the roots of the equation, x3 + P0x2 + P1x + P2 = 0, then (1 - 2) (1 - 2)
(1 - 2) is equal to (A) (1 + P1)2 - (P0 + P2)2 (B) (1 + P1)2 + (P0 + P2)2 (C) (1 - P1)2 - (P0 - P2)2 (D) None of these 17. The set of values of ‘a’ for which the inequality x2 – (a + 2)x-(a + 3) < 0 is satisfied for at least
one positive real x is _________. 18. Consider the equation x3 – nx + 1 =0, n N , n 3 . Then (A) Equation has atleast one rational root . (B) Equation has exactly one rational root. (C) Equation has atleast one root belonging to (0, 1). (D) Equation has no rational root. 19. The real values of x which satisfy x2 – 3x + 2 0 and x2 – 3x – 4 0 are given by (A) -1 x 1 (B) 1 x 2 (C) 2 x 4 (D) none of these
20. If x is real, then c3x4xcx2x
2
2
can take all real values if
(A) 0 < c < 2 (B) –1 < c < 1 (C) –1 < c < 1 (D) none of these 21. If and are the roots of the equation x2 + px + q = 0 and 4, 4 are the roots of x2 -
rx + s = 0 then equation x2 – 4qx + 2q2 –r = 0 has always (A) two real roots (B) two positive roots (C) roots of positive sign (D) two negative roots 22. If one root of equation x2 − 3ax + f(a) = 0, is double of the other then f(x) = (A) 2x (B) x2 (C) 2x2 (D) x 23. If ax2 + bx + c = 0; a, b, c R; a 0, has no real roots then (a + b + c)c is (A) < 0 (B) = 1
(C) = 0 (D) > 0 24. f(x) = ax3 + bx2 + cx + d has only one real root at x = − 2. If a + b + c + d > 0, then the value
of 8a + 4b + 2c + d is (A) = 0 (B) > 0 (C) < 0 (D) can’t determine 25. The equation (x − 3)9 + (x − 32)9 + … +(x − 39)9 = 0 has (A) all the roots real (B) one real and rest imaginary roots (C) at least one real root (D) none of these 26. If both roots of the equation x2 − 2ax + a2 − 1 = 0 lies between − 3 and 4, then [a] is, where
[.] denotes greatest integer function. (A) 0, 1, 2 (B) −1, 0, 1, 2 (C) 0, 1, 2, 3 (D) −3, −2, −1, 0
27. The expression
1
acacbxax
abcbaxcx
cabacxbx
represents
(A) quadratic expression (B) quadratic equation (C) identity (D) none of these 28. The number of solutions of the equation |x2 − x− 6| = x + 2, x R is (A) 2 (B) 3 (C) 4 (D) none of these
29. Solutions of 2
x1x
= 4 +
x1x
23 are
(A) −1, −2 (B) 1, 2
(C) 1, 21
(D) none of these
30. If f(x) is a quadratic expression such that f(x) > 0 x R and if g(x) = f(x) + f(x) + f(x) then
g(x) is (A) negative (B) positive (C) zero (D) none of these
31. If x is real, then the expression 7x2x71x34x
2
2
can have no value between
(A) 3 and 7 (B) 4 and 8 (C) 5 and 9 (D) 6 and 10 32. The set of values of m for which both roots of the equation x2 − (m + 1)x + m + 4 = 0 are real
and negative consists of all m such that (A) − 3 < m −1 (B) − 4 < m − 3 (C) −3 m 5 (D) − 3 m or m 5 33. Give that ax2 + bx + c = 0 has no real solution and a + b + c < 0 then (A) c = 0 (B) c > 0 (C) c < 0 (D) none of these 34. The equation x2 + ax + b = 0 and x2 + bx + a = 0 will have a common root. The common root
is (A) −2 (B) 1 (C) 2 (D) none of these
35. If b > a, then the equation (x – a) (x – b) = 1 has (A) both roots in [a, b] (B) both roots in (– , a) (C) both roots in (b, ) (D) one root in (– , a) and other in (b, ) 36. If and ( < ), be the roots of x2 + bx + c = 0, (where c < 0 < b), then (A) 0 < < (B) < 0 < < || (C) < < 0 (D) < 0 < || < 37. If p and q be roots of x2 – 2x + A = 0 and r, s be the roots of x2 – 18x + B = 0, if p < q < r < s
are in A.P. Then (A) A = – 3, B = 77 (B) A = 77, B = – 3 (C) A = 3, B = – 77 (D) none of these 38. The set of values of ‘a’ for which all the solutions of the equation (log1/2x)2 + 4a log1/2x + 1 = 0
are positive and distinct (A) (– 1, 0) (B) R (C) (– , – 1/2)(1/2, ) (D) none of these 39. The set of positive integral values of ‘a’ for which at least one of the roots of the equation
x2 + (a + 10)x + 10a – 33 = 0 is a positive integer, is (A) {2} (B) N (C) {1, 3} (D) none of these 40. Sum of the real roots of the equation x2 + |x| - 6 = 0 (A) 1 (B) 0 (C) -1 (D) none of these
41. Find the interval in x for which 3x1x
8x4xe 22x
0 = ………….
42. If the expression
x11mx
x1 is non-negative x R then minimum value of m must be
(A) - 21 (B) 2 (C)
41 (D)
21
43. If , be the roots of 4x2 – 16x + = 0 , R, such that 1 < < 2 and 2 < < 3 then
number of integral solutions of is (A) 5 (B) 6 (C) 2 (D) 3 44. If a is an integer and the equation (x – a) (x – 10) + 1 = 0 has integral roots then the value
of a are (A) 10, 8 (B) 12, 10 (C) 12 , 8 (D) none of these 45. The quadratic equations x2 + ax + 12 = 0, x2 + bx + 15 = 0 and x2 + (a + b)x + 36 = 0 have a
positive common root (), given by (A) = 4 (B) = 5 (C) = 10 (D) = 3
46. The greatest value of 24
4x 4x 9 is
(A) 49
(B) 4
(C) 94
(D) 12
47. The set of values of a for which 1 lies between the roots of x2 – ax – a + 3 = 0 is (A) (–, –6) (B) (–, +6) (C) (–, –6) (2, ) (D) (2, ) 48. Maximum value of 5 + 4x – x2, is (A) 5 (B) 6 (C) 9 (D) 1 49. The equation (ax2 + bx + c)(ax2 – dx – c) = 0, x 0, has (A) four real roots (B) at least two real roots (C) at most two real roots (D) no real roots 50. If the equation x2 + 5bx + 8c = 0, does not have two distinct real roots, then minimum value
of 5b + 8c is (A) 1 (B) 2 (C) –2 (D) –1 51. If a + b + c = 0, then one root of the equation ax2 + bx + c = 0 is (a 0) (A) –1 (B) 2 (C) 1 (D) 3 52. If the bigger root of x2 +2ax – 6 + 5a = 0 is negative then exhaustive set of values of a is; (A) a(6/5 , 2] [3, ) (B) a(6/5 , 3] (C) [2, ) (D) none of these 53. If f (x) = ax2 + bx + 8 does not have distinct real roots, then the least value of 4a – b is (A) –4 (B) –8 (C) –6 (D) –2 54. If the roots of the equation x2 –2ax + a2 + a –3 = 0 are less than 3, then (A) a < 2 (B) 2 a 3 (C) 3 < a 4 (D) a > 4 55. If roots of the equation x2 –(a + 3)x + 3a –1 = 0 are integral, then the value of a is (A) 3 (B) 2 (C) 1 (D) –2 56. If ax2 + bx + c = 0 has non real roots and c R+, then (A) a – 2b + 4c < 0 (B) a – 2b + 4c > 0 (C) a – 2b + 4c = 0 (D) none of these 57. If x3 + ax + b = 0, (a, b R) has a repeated non– zero root, then (A) ‘a’ has to be necessarily a positive real number.
(B) ‘a’ has to be necessarily a negative real number. (C) ‘a’ can be any real number. (D) None of these
58. If x2 − 3ax + 2 < 0 x [1, 3] then exhaustive set of values of ‘a’ is (A) a (1, ) (B) a (1, 11/9) (C) a (11/9, ) (D) none of these
59. If 3x3
+ x2 –3x + c = 0 is of the form (x –)2 (x –) then c =
(A) –5/3 (B) 9 (C) –9 (D) 0
60. If a, a1, a2, ….., an R then
n
1i
2iax is the least if x is equal to
(A) a1 + a2 + …..+ an (B) 2(a1 + a2 + …..+ an) (C) n(a1 + a2 + …..+ an) (D) none of these 61. The number of real roots of the equation (x –1)2 + (x –2)2 + (x –3)2 = 0 is (A) 3 (B) 2 (C) 1 (D) 0 62. If p and q are the roots of the equation x2 + px +q = 0 then (A) p =1 (B) p =1 or 0 (C) p = –2 (D) p = –2 or 0 63. The roots and of the quadratic equation ax2 +bx +c = 0 are real and of opposite sign.
Then the roots of the equation (x - )2 + (x - )2 = 0 are (A) positive (B) negative (C) real and of opposite sign (D) imaginary
64. If the inequality 52x2x4x3mx
2
2
is satisfied for all x R , then
(A) 1 < m < 5 (B) -1 < m < 5 (C) 1< m < 6 (D) m < 2471 .
65. Given real numbers a, b, c and a 0. If is a root of a2x2 + bx + c = 0, is a root of
a2x2 - bx – c = 0, and 0<<, then the equation a2x2 + 2bx + 2c = 0 has a root that always satisfies
)D(, )C(2
)B(,2
)A(
66. The equation ax2 + bx + a = 0, x3 − 2x2 + 2x − 1 = 0 have two roots in common. Then a + b
must be equal to (A) 1 (B) −1 (C) 0 (D) none of these 67. If a, b, c are in G.P. then the equation ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have a
common root if d e f, ,a b c
are in
(A) A.P. (B) G.P. (C) H.P. (D) none of these 68. If c > 0 and 4a + c < 2b, then ax2 –bx + c = 0 has a root in the interval (A) (0, 2) (B) (2, 4) (C) (0, 1) (D) (-2, 0) 69. The number of real solutions of the equations ex = x is (A) 0 (B) 1 (C) 2 (D) infinite
70. The number of real solutions of the equation 2xx
2x
226123 is
(A) 1 (B) 2 (C) 4 (D) infinite 71. The number of real solutions of the equations eI xI = I x I is (A) 0 (B) 1 (C) 2 (D) 4 72. The number of numbers between n and n2 which are divisible by n is (n I) (A) n (B) n –1 (C) n –2 (D) none of these 73. If the ratio of the roots of the equation x2 + px + q = 0 be equal to the ratio of the roots
of x2 + lx + m = 0, then (A) p2 m = q2 l (B) pm2 = q2 l (C) p2 l = q2 m (D) p2 m = l2 q 74. The number of solutions of the equation 5x + 5 –x = log1025, x R is …….. 75. If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 has (A) at least one root in (0, 1) (B) one root in (2, 3) and other in (-2, -1) (C) imaginary root (D) none of these
LEVEL −III 1. If the roots of x2 − bx + c = 0 are the two consecutive integers, then b2 − 4c is (A) 0 (B) 1 (C) 2 (D) none of these 2. If a 2 + b2 +c2 + d2 = 1, then the maximum value of ab + bc + cd +da is
(A) zero (B) One (C) Two (D) None of these
3. The number of real solutions of the equation cos5 x+sin3x=1 in the interval [0,2] is (A) 2 (B) 1 (C) 3 (D) Infinite
4. Let f(x) = ax3 + bx2 + x +d has local extrema at x = and such that . < 0, f(), f() > 0;
Then the equation f(x) = 0
(A) has 3 distinct real roots (B) has only one real root, which is positive if a f() < 0 (C) has only one real root, which is negative if a f() > 0 (D) has 3 equal real roots
5. If sin, sin and cos are in GP, then roots of x2 + 2xcot + 1 = 0 are always
(A) equal (B) real (C) imaginary (D) greater than 1
6. Let a, b,c, R such that 2a + 3b + 6c = 0. Then the quadratic equation ax2 + bx + c = 0 has (A) at least one root in (0,1) (B) at least one root in ( -1, 0) (C) both roots in (1,2) (D) imaginary roots
7. If ax2 + bx + 1=0 does not have 2 distinct real roots then least value of 2a– b is
____________
8. If x is real, then least value of expression 1x2x5x6x
2
2
is ;
(A) –1 (B) –1/2 (C) –1/3 (D) none of these 9. If a, b, c are real and a + b + c = 0, then quadratic equation 4ax2 + 3bx +2c = 0 has; (A) two real roots (B) two imaginary roots (C) one real root only (D) none of these
10. If x is real, then expression cx
bxax
will assume all real values provided
(A) a> b> c (B) a< b < c (C) a > c > b (D) b > a > c 11. If x2 + 2bx + c = 0 and x2 + 2ax − c = 0 are two quadratic equation then (A) at least one has real roots (B) both have real roots (C) both have imaginary roots (D) at least one has imaginary root. 12. If the roots of ax2 + bx + c = 0, lies between 1 and 2. Then 9a2 + 6ab + 4ac is (A) < 0 (B) = 0 (C) > 0 (D) can’t say
13. For the equation 3x2 + px + 3 = 0, p > 0 if one of the root is square of the other, then p is equal to
(A) 31 (B) 1
(C) 3 (D) 32
14. If the equation ax2 – bx + 5 = 0 doesn’t have two distinct real roots then the minimum value
of a + b is (A) – 5 (B) 5 (C) 0 (D) none of these 15. If a > 1, roots of the equation (1 – a)x2 + 3ax – 1 = 0, are (A) one positive (B) both negative (C) both positive (D) both complex roots 16. If f(x) = ax2 + bx + c , g(x) = – ax2 + bx + c where ac 0 then f(x). g(x) = 0 has (A) at least three real roots (B) no real roots (B) at least two real roots (D) exactly two real roots 17. The number of real solutions of the equation 3x + x2 = 5 is (A) 1 (B) 2 (C) 3 (D) 0
18. The number of real solutions of the equation 2x
xx3109
is
(A) 0 (B) 1 (C) 2 (D) none of these
19. The equation 11x68x1x43x has (A) no solution (B) only one solution (C) only two solutions (D) more than two solutions 20. Let a > 0, b > 0, c > 0. Then both the roots of the equation ax2+bx+c=0 (A) are real and negative (B) have negative real parts (C) are rational numbers (D) none of these 21. x4 - 4x - 1 = 0 has (A) exactly one positive real root (B) exactly one negative real root (C) exactly two real roots (D) All the above. 22. Let a, b, c be non-zero real numbers, such that
2
0
281
0
28 dxcbxaxxcos1dxcbxaxxcos1 . Then the quadratic equation
ax2 +bx+c =0 has (A) no root in (0, 2) (B) at least one root in (1, 2) (C) two roots in (0, 2) (D) two imaginary roots. 23. If the two roots of the equation ( -1) ( x2 + x + 1)2 – ( + 1) (x4 + x2 +1) = 0 are real and
distinct, then lies in the interval < −2, > 2.
ANSWERS
LEVEL −I 1. B 2. A 3. D 4. C 5. B 6. C 7. C 8. B 9. A 10. A 11. A 12. B 13. D 14. D 15. C 16. A 17. B 18. D 19. A 20. A 21. C 22. A 23. D 24. A 25. C 26. B 27. A 28. D 29. A 30. B 31. B 32. A 33. C 34. C 35. B 36. B 37. B 38. B 39. C 40. D 41. C 42. B 43. A 44. A 45. B 46. C 47. A 48. C 49. C 50. B 51. B 52. C 53. D 54. B 55. 0 56. B 57. B 58. A 59. A 60. C 61. A 62. B 63. B 64. B 65. D 66. D 67. D 68. A 69. A 70. A 71. C 72. A 73. C 74. B 75. A 76. A 77. A 78. B 79. C 80. B 81. C 82. D 83. C 84. B 85. A 87. D 88. B 89. B 90. D 91. A 92. B 93. B 94. A 95. B 96. D 97. 1 98. 3 99. 0 100. B LEVEL −II 1. C 2. A 3. A 4. D 5. B 6. B 7. C 8. D 9. B 10. A 11. B 12. A 13. D 14. D 15. A 16. A 17. (−2, ) 18. A 19. A, C 20. D 21. A 22. C 23. D 24. B 25. B 26. B 27. C 28. B 29. B 30. B 31. C 32. B 33. C 34. B 35. D 36. D 37. A 38. C 39. A 40. B 41. (−3, −2](−1, 2] 42. C 43. D 44. C 45. D 46. D 47. D 48. C 49. B 50. D 51. C 52. A 53. D 54. A 55. A 55. B 57. B 58. C 59. C 60. D 61. D 62. B 63. C 64. D 65. D 66. C 67. A 68. A 69. A 70. A 71. A 72. C 73. D 74. 0 75. C LEVEL −III 1. B 2. B 3. C 4. B, C 5. B 6. A 7. −1/2 8. C
9. A 10. C 11. A 12. A 13. C 14. A 15. C 16. C 17. A 18. A 19. D 20. B 21. D 22. B 23. (−, −2)(2, )
1
ST
LEVEL-I
1. If the bisector of angle A of ABC makes an angle with BC, then sin is equal to
(A) cos
2
CB (B) sin
2
CB
(C) sin
2AB (D)sin
2AC
2. If the radius of the circumcircle of an isosceles triangle ABC is equal to AB = AC then the angle A is (A) /6 (B) /3 (C) /2 (D) 2/3
3. In a triangle ABC, if ,cos2coscos2cab
bca
cC
bB
aA
then the value of the
angle A is (A) 300 (B)450 (C)600 (D) 900
4. If A = 450, B =750 then a + c 2 is equal to (A) 2b (B) 3b (C) 2 b (D) b 5. The sides of a triangle inscribed in a given circle subtend angle , and at the
centre. The minimum value of the arithmetic mean of cos( + /2), cos( + /2) and cos( +/2) is equal to
(A) 0 (B) 1/ 2 (C) –1 (D) - 3 /2 6. A regular polygon of nine sides, each of length 2, is inscribed in a circle. The
radius of the circle is
(A)sec9
(B)sin9
(C) cosec9
(D) tan9
7. In an acute angled triangle ABC, the least value of secA + secB + secC is (A) 6 (B)3 (B) 9 (D) 4 8. A circle is inscribed in an equilateral triangle of side a. The area of any square
inscribed in the circle is (A) a2/4 (B) a2/6
2
(C) a2/9 (D) 2a2/3
9. If 3 sin2A + 2sin2B =1 and 3 sin2A – 2 sin2B = 0, where A and B are acute angles, then A + 2B is equal to
(A) /3 (B) /4 (C) /2 (D) none of these. 10. If in a ABC, cos(A - C)cosB + cos2B = 0, then a2, b2, c2 are in (A) A.P. (B) G.P. (C) H. P. (D) none of these 11. If tan(A+B), tanB, tan(B+C) are in A.P., then tanA, cotB, tanC are in (A) A.P. (B) G.P. (C) H.P. (D) none of these 12. If twice the square of the diameter of a circle is equal to the sum of the squares
of the sides of the inscribed triangle ABC, then sin2A + sin2B + sin2C is equal to
(A) 2 (B) 3 (C) 4 (D) 1 13. Consider a triangle ABC, with given A and side ‘a’. If bc = x2, then such a
triangle would exist if, ( x is a given positive real number) .
(A) a < x sin2A (B) a >2x sin
2A
(C) a < 2 x sin2A (D) None of these .
14. If in ABC a, b, c are in geometric progression then, (A) cot2A, cot2B, cot2C are in G.P.
(B) cosec2A, cosec2B, cosec2C are in A.P. (C) cosec2A, cosec2B, cosec2C are in G.P. (D) none of these.
15. If in a ABC, 8R2 = a2 + b2 + c2, then the triangle is (A) Equilateral (B) Right angled (C) Isosceles (D) None of these
16. In a triangle ABC, angle B is greater than angle A, B –A < 32 . If the values of A
and B satisfy the equation 3sinx – 4sin3x - k = 0 (0 < k < 1), then angle C is equal to
(A) 3 (B)
6
(C) 32 (D) None of these
3
17. If in a triangle ABC, b + c = 4a. Then 2Ccot
2Bcot is equal to
(A) 35 (B)
53
(C) 85 (D) None of these
18. The ex-radai of a triangle r1, r2, r3 are in Harmonic progression, then the sides a,
b, c are in (A) A.P (B) G.P (C) H.P (D) none of these 19. In a ABC A = 300, B = 600, then a : b : c is (A) 1 : 2 : 3 (B) 1 : 3 : 2 (C) 1 : 2 : 3 (D) 1 : 2 : 3 20. In a ABC, the value of a (cos B + cos C) + b (cos A + cos C) + c (cos A + cos B)
is (A) a + b (B) a + b + c (C) b + c (D) b + c –a 21. In a triangle a = 13, b = 14, c = 15, r = (A) 4 (B) 8 (C) 2 (D) 6
22. In a triangle ABC, If b + c = 3a, then the value of cot 2B cot
2C is
(A) 1 (B) 2 (C) 3 (D) 3
23. In a triangle ABC, then 2ac sin 21 (A –B + C) is
(A) a2 + b2 –c2 (B) c2+ a2 –b2
(D) b2 –c2 –a2 (D) c2 –a2 –b2 24. The angle A of the triangle ABC, in which (a + b + c) (b + c –a) = 3bc is (A) 300 (B) 450 (C) 600 (D) 1200
25. In a triangle ABC, Let C = 2 , if r is the inradius and R is the circumradius of the
triangle, then 2 (r + R) is equal to (A) a + b (B) b + c (C) c + a (D) a + b + c
26. In a triangle ABC, 2Atan.
bcbc
is equal to
4
(A)
B2Atan (B)
B2Acot
(C)
2BAtan (D) none of these
27. In a ABC, a = 2b and |A –B| = 3 , the measure of angle C
…………………………………….. 28. In a ABC, the sides a, b and c are such that they are the roots of x3 –11x2 + 38x
–40 = 0 then the value of c
Ccosb
Bcosa
Acos =
……………………………………… 29. If AD, BE and CF are the medians of a ABC, then (AD2 + BE2 + CF2) : (BC2 +
CA2 + AB2) = ……………………………………………….. 30. sin A, sin B, sin C are in A.P for the ABC then (A) altitudes are in A.P (B) sides are in A.P (C) altitudes are in H.P (D) medians are in A.P 31. In a triangle ABC, tan C< 0, then (A) tan A . tan B < 1 (B) tan A . tan B > 1 (C) tan A + tan B + tan C < 0 (D) tan A + tan B + tan C > 0
32. If in a triangle ABC, b + c = 4a. Then 2Ccot
2Bcot is equal to
(A) 35 (B)
53 (C)
85 (D) None of
these
33. If in a triangle ABC, cosA = CsinBsin
AsinBsinCsin
CsinBsin 2
, then the triangle is
(A) right angled (B) isosceles (C) scalene (D) None of these
34. In a triangle, the lengths of the two larger sides are 10 and 9 respectively. If the
angles are in A.P., then the length of third side can be (A) 5 – 6 (B) 3 (C) 5 (D) 3 3 35. In a ABC, maximum value of c cos (A - ) + a cos(C + ), equals (A) a (B) b (C) c (D) 22 ca
5
36. In a triangle ABC, a2 ( cos2B - cos2C) + b2 ( cos2C – cos2A) + c2 ( cos2A- cos2B) equals
(A) 0 (B) 1 (C) -1 (D) none of these 37. In a ABC, the angles A and B are two values of satisfying 3 sin+ cos =
, || < 2. Then C equals (A) 60 (B) 90 (C) 120 (D) none of these 38. If the ex-radii of a triangle ABC are in H.P., then the sides a, b, c are in (A) A.P. (B) G.P. (C) H.P. (D) None of these
6
LEVEL-II
1. The expression 224))()()((
cbcbabacacbcba
is equal to
(A) cos2A (B) sin2A (C) cosA cosB cosC (D) None of these 2. The perimeter of a triangle ABC is 6 times the arithmetic mean of the sines of its
angles. If the side a is 1, then the angle A is (A) /6 (B) /3 (C) /2 (D) 3. If a2, b2,c2 are in A.P , then cotA, cotB, cotC are in (A) A.P (B) G.P (C) H.P (D) None of these 4. The area of the circle and the regular polygon of n sides and of equal perimeter
are in the ratio of (A) tan(/n) : /n (B) cos (/n) : /n (C) sin(/n) : /n (D) cot(/n) : /n 5. In a triangle ABC, (a+b+c)(b+c-a) = bc if (A) < 0 (B) > 0 (C) 0 < < 4 (D) > 4 6. In a triangle ABC, AD is the altitude from A. Given b > c, C =230 and AD
= 22 cbabc
then B is equal to
(A) 230 (B) 1130 (C) 670 (D) 900 7. In any triangle ABC, a3cos(B-C) + b3 cos(C-A) + c3cos(A-B) is equal to (A) 6abc (B) 9abc (C) 3abc (D) None 8. In a triangle ABC, cba is (A) always positive (B) always negative (C) positive only when c is smallest (D) none of these . 9. In a triangle with sides a,b, and c, a semicircle touching the sides AC and CB is
inscribed whose diameter lies on AB. Then , the radius of the semicircle is (A) a/ 2 (B) / s
(C) ba
2 (D) bas
abc2
cos2A cos
2B cos
2C
10. A triangle is inscribed in a circle. The vertices of the triangle divide the circle
in to three arcs of length 3, 4 and 5 units, then area of the triangle is equal to,
7
(A) 2
3139 (B)
21339
(C) 22
3139 (D)
221339
11. If a sinx + bcos(C + x) + bcos (C –x) = , then the minimum value of |cosC| is
(A) 2
22
ba (B) 2
22
b4a
(C) 2
22
b16a (D) none of these
12. In a ABC, the point D divides BC in the ratio 1:2 . Also AD is perpendicular to
AB. Then the value of the expression tanB(1+2tanA tanC) – 2tanC is (A) 0 (B) 1 (C) –1 (D) none of these 13. If in ABC , secA , secB, secC are in Harmonic progression, then (A) a, b, c, are in harmonic progression.
(B) cot2A , cot
2B , cot
2C are in harmonic progression
(C) r1, r2, r3 are in arithmetic progression
(D) cot2A , cot
2B , cot
2C are in arithmetic progression .
14. In a triangle ABC a = 7, b = 8 and c= 9. Then the length of median from B to AC
is given by (A) 9 (B) 8 (C) 7 (D) 6 15. If sinA and sinB of a triangle ABC satisfy c2x2 – c(a+b)x + ab = 0, then the
triangle is (A) Equilateral (B) Isosceles (C) Right angled (D) Acute angled 16. The number of triangles that can be made with the given data: b = 2cm, c = 6 cm
and B = 30°, is
(A) 1 (B) 2 (C) zero (D) None of these 17. In ABC, if AB = c , AC = b, BC = a and A : B : C = 1 : 2 : 5, then (A) b2 = a(c + a) (B) b2 = a( c – a) (C) b2 = a( a – c) (D) None of these.
18. In ABC, if 18
cb14
ba12
ac
, then
8
(A) r711r1 (B) r2 = 11r
(C) r112r3 (D) None of these
19. If a cos A = b cos B, the triangle is (A) equilateral (B) right angled (C) isosceles (D) right angled or isosceles 20. The sides of a triangle are a, b and 22 baba , then the greatest angle is
(A) 3 (B)
2
(C) 32 (D) none of these
21. Two sides of a are given by the roots of the equation x2 –2 3 x + 2 = 0. The
angle between the sides is 3 . The perimeter of the triangle is
22. In a triangle ABC, R = circumradius and r = inradius. The value of
cbaCcoscBcosbAcosa
is equal to
(A) rR (B)
r2R
(C) Rr (D)
Rr2
23. In a triangle ABC, 2 cos acca
ca2
CA22
, then
(A) B = 3 (B) B = C
(D) A, B, C are in A.P (D) B + C = A 24. The distance of the circumcentre of the acute angled ABC from the sides BC,
CA and AB are in the ratio (A) a sin A : b sin B : c sin C (B) cos A : cos B : cos C (C) a cot A : b cot B : c cot C (D) none of these 25. If twice the square of the diameter of a circle is equal to the sum of the squares
of the sides of the inscribed triangle ABC, then sin2A + sin2B + sin2C is equal to
(A) 2 (B) 3 (C) 4 (D) 1
26. In ABC, if 18
cb14
ba12
ac
, then
9
(A) r711r1 (B) r2 = 11r (C) r
112r3 (D) None of
these
27. In a triangle ABC, 2 sinA cosC = 1 and 21
CtanAtan then triangle is
(A) right angled at A (B) right angled at B (C) right angled at C (D) none of these
28. In a triangle ABC,
2133221
Rsrrrrrr
is equal to
(A) 4 (B) 4 abc (C) 2abc4
(D)
29. In a ABC, 2CcoscBcosbAcosa
is equal to
(A) abc
8 (B) R2
(C) abc8 3 (D) None of
these 30. If p1, p2 and p3 are respectively the lengths of perpendiculars from the vertices of
a triangle ABC to the opposite sides, then the value of p1p2p3 is
(A) 2
222
R8cba (B) 3
222
R8cba (C) 4
222
R8cba (D) 2
222
R4cba
31. If in a triangle cos2A + cos2B – cos2C = 1, then the triangle is (A) Right angled at A (B) Right angled at B (C) Right angled at C (D) not a right triangle
32. If in a triangle ABC, 0CosC
CosACosBSinC
SinASinB
then the triangle is
(A) right angled (B) equilateral (C) isosceles (D) None of these
33. If sin and - cos are the roots of the equation ax2 – bx – c = 0, where a, b, c are
the sides of a triangle ABC then
(A) cosB = 1 - a2
c (B) cosB = 1- a2
b (C) cosB = 1 +a2
c (D) cosB = 1
+ a2
b
34. In a right angled triangle ABC, with right angle at B, 23
22
21
2 r1
r1
r1
r1
=
(A) 2
2R8
(B) 2
2R2
(C)
2R4 (D) None of
these
10
35. If in a triangle ABC, C = 1350, then value of tan A + tan B + tan A tan B equals (A) 0 (B) 1 (C) –1 (D) none of these 36. Suppose the angles of a triangle ABC are in A.P. and sides b and c satisfy b : c =
2:3 then the angle A equals (A) 450 (B) 600 (C) 750 (D) 900 37. If a2, b2, c2 are the roots of the equation x3 –Px2 + Qx – R = 0 where a, b, c be
the sides of a triangle ABC then the value of c
Ccosb
Bcosa
Acos equals
(A) RP (B)
R2P
(C) R4
P (D) none of these
38. In a triangle ABC, ACsinbac
CBsinacb 2222
equals
(A) R (B) R21
(B) 2R (D) none of these
11
ANSWERS
LEVEL −I 1. A 2. D 3. D 4. A 5. D 6. C 7. A 8. B
9. C 10. A 11. C 12. A 13. D 14. C 15. B 16. C 17. A 18. A 19. B 20. B 21. A 22. B 23. B 24. C
25. 26. D 27. 28. 916
29. 30. B 31. C 32. A 33. A 34. A 35. B 36. A 37. C 38. A LEVEL −II 1. B 2. A 3. A 4. A 5. C 6. B 7. C 8. A
9. C 10. A 11. B 12. A 13. B,C 14. C 15. C 16. C 17. A 18. A 19. D 20. C 21. 6 1 2 22. C 23. 24. C
25. 26. 27. 28. A 29. 30. 31. 32. 33. C 34. A 35. B 36. C 37. B 38. D
SL
LEVEL-I 1. If a2 + b2 – c2 + 2ab = 0, then family of straight lines ax + by + c = 0 is concurrent at the
points. (A) (–1, 1) (B) (1, –1) (C) (1, –2) (D) (–1, –1), (1, 1) 2. The pair of straight lines perpendicular to the pair of lines ax2 + 2hxy + by2 = 0 has the
equation. (A) ax2 – 2hxy + by2 = 0 (B) ay2 + 2hxy + bx2n = 0 (C) bx2 + 2hxy + ay2 (D) bx2 – 2hxy + ay2 = 0 3. If x1, x2, x3 as well as y1, y2, y3 are in G.P with same common ratio ( 1) then the points (x1,
y1), (x2, y2) and (x3, y3). (A) lie on a straight line (B) lie on an ellipse (C) lie on a circle (D) are the vertices of a triangle 4. If a, c, b are in A.P the family of line ax + by + c = 0 passes through the point.
(A)
21,
21 (B) (1, – 2) (C) (1, 2) (D)
21,
21
5. The image of the point (3, –8) in the line x+ y = 0 is (A) (–8, 3) (B) (–3, 8) (C) (8, – 3) (D) (3, 8) 6. The nearest point on the line 2x + 3y = 5 from the origin is.
(A) (3, –1/3) (B)
1315,
1310 (C) (0, 5/3) (D) (1, 1)
7. A straight line through A(2, 1) is such that its intercept between the axis is bisected at A. its
equation is. (A) 2x + y – 4= 0 (B) x + 2y – 4 = 0 (C) x + 2y – 4 = 0 (D) x + 2y – 2 = 0 8. The incentre of the triangle with vertices (1, 3 ), (0, 0) and (2, 0) is.
(A)
23,1 (B)
31,
32 (C)
23,
32 (D)
31,1
9. It is desired to construct a right angled triangle ABC (C = /2) in xy plane so that it’s sides
are parallel to coordinates axis and the medians through A and B lie on the lines y = 3x+1 and y = mx +2 respectively. The values of ‘m’ for which such a triangle is possible is /are ,
(A) 12 (B) 3/4 (C) 4/3 (D) 1/12 10. The equation of the line bisecting the obtuse angle between y –x =2 and 3 y +x =5 is
(A) 2
5xy32
2xy
(B) 2
5xy32
2xy
(C) 2
5xy32
2xy
(D) none of these
11. If the intercept made on the line y = mx by the lines x = 2 and x =5 is less then 5, then the
range of m is
(A) (−4/3 ,4/3) (B) ( , −4/3) (4/3 , ) (C) [−4/3, 4/3) (D) none of these.
12. The equations of three sides of a triangle are x = 5, y – 2 = 0 and x + y = 9. The coordinates of the circumcentre of the triangle are
(A) ( 6, 3) (B) (6, -3)
(C) ( -6, 3) (D) none of these.
13. The equation of a straight line passing through the point (–2, 3) and making intercepts of
equal length on the axes is (A) 2x + y + 1 = 0 (B) x –y = 5 (C) x –y + 5 = 0 (D) none of these 14. If the intercept made on the line y = mx by the lines y = 2 and y = 6 is less than 5 then the
range of values of m is
(A)
,34
34, (B)
34,
34
(C)
43,
43 (D) none of these
15. If a, c, b are in G.P then the line ax + by + c= 0 (A) has a fixed direction (B) always passes through a fixed point (C) forms a triangle with axes whose area is constant (D) none of these 16. If a ray travelling along the line x = 1 gets reflected from the line x + y = 1, then the equation
the line along which the reflected ray travels is (A) y = 0 (B) x –y = 1 (C) x = 0 (D) none of these 17. The equations of the lines representing the sides of a triangle are 3x – 4y =0, x+y =0
and 2x –3y =7. The line 3x +2y =0 always passes through the (A) incentre (B) centroid (C) circumcentre (D) orthocentre 18. If the lines x = a + m, y = -2 and y = mx are concurrent, the least value of |a| is (A) 0 (B) 2 (C) 22 (D) None of these 19. Equation of a line passing through the intersection of the lines 2x +y =3 and x + y = 1 and
perpendicular to the line y = 2x +k is (A) x - 2y =0 (B) x+ 2y =0 (C) y – x =0 (D) y +x = 0 20. If the sum of the reciprocals of the intercepts made by a line on the coordinate axes is 1/5,
then the line always passes through (A) ( 5, -5) (B) ( -5, 5) (C) (-5, -5) (D) ( 5, 5) 21. If 4a2 + 9b2 – c2 + 12ab = 0, a, b, c R+, then the family of straight lines ax + by + c = 0 is
concurrent at (A) (2, 3) (B) (– 2, – 3) (C) 2, – 3) (D) (– 3, 2)
22. Point P (2, 4) is translated parallel to the line y – x – 1 = 0, through a distance 3 2 so that its ordinate is decreased and it reaches at Q. If R is the mirror image of Q in the line y – x – 1 = 0, its coordinate are
(A) (– 1, 1) (B) (0, 0) (C) (6, 6) (D) none of these 23. If the line y = 3 x cuts the curve x3 + y2 + 3x2 + 9 = 0 at the points A, B, C, then OA.OB.OC
( O being origin) equals (A) 36 (B) 72 (C) 108 (D) none of these 24. Let O be the origin, and let A(1, 0), B(0, 1) be two points. If P(x, y) is a point such that xy > 0
and x + y < 1, then (A) P lies either inside the OAB or in the third quadrant
(B) P cannot be inside the OAB (C) P lies inside the OAB (D) none of these 25. Let ABC be a triangle with equation of sides AB, BC, CA respectively x – 2 =0,
y – 5 = 0 and 5x + 2y – 10 = 0, then the orthocentre of triangle lies on the line (A) x – y = 0 (B) 3x + y =1 (C) 4x + y = 13 (D) x – 2y =1 26. The foot of the perpendicular on the line 3x + y = drawn from the origin is C if the line cuts
the x–axis and y–axis at A and B respectively then BC : CA is (A) 1: 3 (B) 3 : 1 (C) 1: 9 (D) 9 : 1 27. A straight line is drawn through the centre of the circle x2 + y2 – 2ax = 0, parallel to the
straight line x + 2y = 0 and intersecting the circle at A and B. Then the area of AOB is
(A) 5
a2
(B) 5
a3
(C) 3
a2
(D) 3
a3
28. In what ratio does the point (3, –2) divide the line segment joining the points (1, 4) and
(–3, 16)? (A) 1 : 3 (externally) (B) 3 : 1 (externally) (C) 1 : 3 (internally) (D) 3 : 1 (internally) 29. For what value of x will the points (x, –1), (2, 1) and (4, 5) lie on a line? (A) 1 (B) 0 (C) 2 (D) none of these 30. The angle between straight lines x2 –y2 –2y –1 = 0 is (A) 900 (B) 600 (C) 750 (D) 360 31. The distance between the lines 4x + 3y = 11 and 8x + 6y = 15 is (A) 7/2 (B) 7/10 (C) 4 (D) none of these 32. Find the length of the perpendicular from origin to the straight line 3x –y + 2 = 0 (A) 2 (B) –2/ 10 (C) 2/ 10 (D) none of these 33. If the sum of the slopes of the lines given by 4x2 + 2kxy –7y2 = 0 is equal to the product of
the slopes then k = (A) –4 (B) 4 (C) –2 (D) 2 34. Find the value of k, so that the equation –2x2 + xy + y2 –5x + y + k = 0 may represent a pair
of straight lines
(A) –2 (B) 2 (C) 0 (D) none of these 35. The image of the point (1, 3) in the line x + y –6 = 0 is (A) (3, 5) (B) (5, 3) (C) (1, –3) (D) (–1, 3) 36. The lines joining the origin to the points of intersection of 2x2 + 3xy –4x + 1 = 0 and
3x + y = 1 given by (A) x2–y2 –5xy = 0 (B) x2 –y2 +5xy = 0 (C) x2 + y2 –5xy = 0 (D) x2 + y2 + 5xy = 0 37. The distance between the lines 3x + 4y =9 and 6x + 8y +15 =0 (A) 3 /10 (B) 33 /10 (C) 33 /5 (D) None of these 38. The equations of the three sides of a triangle are x =2, y +1=0 and x +2y =4. The
coordinates of the circumcentre of the triangle are (A) (4, 0) (B) (2, – 1) (C) (0, 4) (D) None of these 39. If the lines y – x =5, 3x +4y =1 and y =mx +3 are concurrent then the value of m is (A) 19/5 (B) 1 (C) 5/19 (D) None of these 40. A line passing through the origin and making an angle /4 with the line y – 3x =5 has the
equation (A) x + 2y =0 (B) 2x =y (C) x =2y (D) y – 2x =0 41. The points (– 1, 1) and ( 1, – 1) are symmetrical about the line (A) y +x =0 (B) y =x (C) x +y =1 (D) None of these 42. The member of the family of lines ( p +q)x + (2p +q)y = p + 2q, where p 0, q 0, pass
through the point (A) (3, – 1) (B) – 3 ,1) (C) (1, 1) (D) None of these 43. The equation of straight line which passes through the point (1, 2) and makes an angle
cos– 1
31 with the x– axis is
(A) 22 x + y – 2 12 = 0 (B) 2x + 2 y – 2 = 0
(C) x + 2 2 y – 2 2 2 1 = 0 (D) none of these
44. The quation of the line joining the points (– 1, 3) and (4, – 2) is (A) x + y – 1 =0 (B) x + y +1 =0 (C) x + y +2 =0 (D) x + y – 2 =0 45. The equation of the line through (3, 4) and parallel to the line y =3x +5 is (A) 3x – y – 5 =0 (B) 3x + y – 5 = 0 (C) 3x + y + 5 = 0 (D) 3x – y + 5 = 0 46. Locus of the point of intersection of lines x cos+ y sin = a and x sin – y cos =a ( R ) is (A) x2 + y2 =a2 (B) x2 + y2 = 2a2 (C) x2 + y2 + 2x + 2y = a2 (D) none of these
47. The quadratic equation whose roots are the x and y intercepts of the line passing through
(1, 1) and making a triangle of area A with axes is (A) x2 + Ax + 2A = 0 (B) x2 – 2Ax +2A = 0 (C) x2 – Ax + 2A = 0 (D) None of these
48. The area of the quadrilateral formed by y = 1 – x, y = 2 – x and the coordinate axes is (A) 1 (B) 2 (C) 3/2 (D) None of these 49. The incentre of the triangle formed by the lines y = |x| and y = 1 is (A) (0, 2 - 2 ) (B) (2 - 2 , 0) (C) (2 + 2 , 0) (D) (0, 2 + 2 ) 50. If one vertex of an equilateral triangle is at (1, 2) and the base is x + y + 2 = 0, then the
length of each side is
(A)23
(B) 32
(C) 32 (D)
23
51. Points on the line x + y = 4 that lie at a unit distance from the line 4x+ 3y10=0 are (A) (3, 1) and (7, 11) (B) (3, 7) and (2, 2) (C) (3, 7) and (7, 11) (D) none of these 52. The locus of the mid-point of the portion intercepted between the axes by the line
x cos + y sin = p, where p is a constant is
(A) x2 + y2 = 4p2 (B) 222 p4
y1
x1
(C) 222
p4yx (D) 222 p
2y1
x1
53. The straight lines of the family x(a+b) + y (a-b) = 2a (a and b being parameters) are (A) not concurrent (B) Concurrent at (1, -1) (C) Concurrent at (1, 1) (D) None of these 54. If the sum of the distances of a point from two perpendicular lines in a plane is 1, then
its locus is (A) square (B) a circle (C) straight line (D) two intersecting lines
55. If the line y = mx meets the lines x + 2y – 1= 0 and 2x – y + 3 = 0 at the same point, then m is equal to
(A) 1 (B) -1 (C) 2 (D) -2
56. The area inclosed by 3|x| + 4|y| 12 is (A) 6 squar units (B) 12 sq. units (C) 24 square units (D) 36 square units 57. If a, b, c are in A.P. then line 2ax + 3by + 3c = 0 always passes through fixed point (A) (2, –2) (B) (3/2, 2) (C) (3/2, –2) (D) none of these
58. Equation (3a – 2b)x2 + (c –2a)y2 + 2hxy = 0 represents pair of straight lines which are perpendicular to each other then (a – b) is equal to
(A) b + c (B) b – c (C) c – b (D) 2c 59. ax + by + c = 0 represents a line parallel to x–axis if (A) a = 0, b = 0 (B) a = 0, b 0 (C) a 0, b = 0 (D) c = 0 60. If the angle between the two straight lines represented by 2x2+ 5xy+ 3y2+7y+4= 0 is tan–1m
then m equals to (A) 1/5 (B) 1 (C) 7/5 (D) 7 61. The diagonals of a parallelogram PQRS are along the straight lines ax + 2by = 50 and
4bx – 2ay =100. Then PQRS must be a (A) rhombus (B) rectangle (C) square (D) none of these 62. The area enclosed by |x| + |y| = 1 is (A) 1 (B) 2 (C) 3 (D) 4 63. If the line 6x –y + 2 + k(2x + 3y + 13) = 0 is parallel to x-axis, then the value of k is
(A) 31
(B) 31 (C) –3 (D) 3
64. The straight line passing through the point of intersection of the straight lines x –3y + 1 = 0
and 2x + 5y –9 = 0 and having infinite slope has the equation (A) x = 2 (B) 3x + y –1 = 0 (C) y = 1 (D) none of these 65. The equations of the lines through (1, 1) and making angle 45° with the line x + y = 0 are
given by (A) x2 xy + x y = 0 (B) xy y2 + x y = 0 (C) xy + x + y = 0 (D) xy + x + y + 1 = 0 66. If a line is perpendicular to the line 5x y = 0 and forms a triangle with coordinate axes of
area 5 sq. units, then its equation is (A) x + 5y 5 2 = 0 (B) x 5y 5 2 = 0 (C) 5x + y 5 2 = 0 (D) 5x y 5 2 = 0 67. The co-ordinates of foot of the perpendicular form the point (2, 4) on the line x + y = 1 are
(A)
23,
21
(B)
23,
21
(C)
21,
34
(D)
21,
43
68. The distance of the line 2x –3y = 4 from the point (1, 1) in the direction of the line x + y = 1 is
……………………………………………. 69. If the point (2, a) lies between the lines x + y = 1 and 2(x + y) = 5, then a lies between
……………………….. and ………………………………….. 70. If mn = 1, then the lines mx + y = 1 and y – nx = 2 will be ………………………………………
71. If the point (2a –3, a2 –1) is on the same side of the line x + y –4 = 0 as that of the origin, then the set of values of a is ……………………………
72. The set of lines ax + by + c = 0 where 3a + 2b + 4c = 0 is concurrent at the point
……………………………………………….. 73. If the image of the point (–2, 1) by a line mirror be (2, –1) then the equation of the line mirror
is ……………………………………..………… 74. If the point (–2, 0), (–1, 3/1 ) and (cos, sin) are collinear then the cumber of values of
[0, 2]. (A) 0 (B) 1 (C) 2 (D) infinite 75. If ‘a’ and ‘b’ are real numbers between 0 and 1 such that the points (a, 1), (1, b) and (0, 0)
from an equilateral triangle then the values of ‘a’ and ‘b’ respectively (A) 2 − 3 , 2 − 3 (B) −2 + 3 , −2 + 3 (C) 2 3 , 2 3 (D) none of these
76. If f(x) =
0x,c
0x,x
bx1logax1log
is continuous at x = 0, then the line ax + by + c = 0 passes through the point (A) (1, −1) (B) (−1, 1) (C) (1, 1) (D) (0, 0)
LEVEL-II
1. The centroid (1, 2), circumcentre (–2, 1) then co– ordinate of orthocentre is. (A) (4, 7) (B) (–4, 7) (C) (7, 4) (D) (5/2, 5/2) 2. It the co– ordinates of vertices of a triangle are (0, 5), (1, 4) and (2, 5) then the co– ordinate
of circumcentre will be.
(A) (1, 5) (B)
29,
23 (C) (1, 4) (D) none of these
3. The equation of the image of pair of rays y = |x| by the line x = 1 is (A) |y| = x + 2 (B) |y| + 2 = x (C) y = |x – 2| (D) none of these 4. If the line segment on lx + my = n2 intercepted by the curve y2 = ax subtends a right angle at
the origin, then (A) a, n, l are in G.P (B) l, m, n are in G.P (C) l, m, n2 are in G.P (D) l, n2, m are in G.P 5. If the line y = 3 x cuts the curve x4 + ax2y + bxy + cx + dy + 6 = 0 at A, B, C and D, then
OA.OB.OC.OD ( where O is the origin) is (A) a – 2b +c (B) 2c2d (C) 96 (D) 6 6. A ray of light travelling along the line x + y = 1 is inclined on the x-axis and after
refraction it enters the other side of the x-axis by turning 15 away from the x-axis. The equation of the line along which the refraction ray travels is
(A) 3 y - x +1 = 0 (B) 3 y + x +1 = 0 (C) 3 y + x -1 = 0 (D) none of these . 7. The coordinates of the point(s) on the line x + y = 5, which is/are equidistant from the
lines |x| = |y|, is/are
(A) (5, 0) (B) (1, 4)
(C) (-5, 0) (D) (0, -5)
8. If the point (a, a) falls between the lines |x + y| = 2, then (A) |a| = 2 (B) |a| =1 (C) |a| < 1 (D) |a| < 1/2 9. A line has intercepts a and b on the coordinate axes. If keeping the origin fixed, the
coordinate axes are rotated through 90, the same line has intercepts p and q, then (A) p =a, q = b (B) p = b, q = a (C) p = -b, q = -a (D) p = b, q = -a 10. Two sides of a rhombus OABC ( lying entirely in first quadrant or fourth quadrant ) of area
equal to 2 sq. units, are y = 3x , y = 3 x . Then possible coordinates of B is / are (‘O’
being the origin) (A) 31,31 (B) 31,31
(C) 13,13 (D) none of these
13. Equation of the bisector of angle B of the triangle ABC is y = x. If A is (2, 6) and B is (1, 1); equation of side BC is
(A) 2x + y – 3 = 0 (B) x – 5y + 4 = 0 (C) x – 6y + 5 = 0 (D) none of these 14. Vertex opposite to the side x + y – 2 = 0 of the equilateral triangle, with centroid at the origin;
is (A) (– 1, 1) (B) (2, 2) (C) (– 2, – 2) (D) none of these
15. A =
0,tt1 2 and B =
t2,tt1 2 are two variable points where t is a parameter,
the locus of the middle point of AB is (A) a straight line (B) a pair of straight line (C) circle (D) none of these 16. The ends of a diagonal of a square are (2 ,– 3) and (– 1 ,1). Another vertex of the square
can be (A) (– 3/2, – 5/2) (B) (– 5/2, 3/2) (C) (1/2 , 5/2) (D) None of these 17. If the equations of the three sides of a triangle are 2x + 3y =1, 3x–2y +6 = 0 and x + y =1,
then the orthocentre of the triangle lies on the line (A) 13x +13 y = 1 (B) 169x +26 y = -178 (C) 169x + y = 0 (D) none of these. 18. The orthocentre of the triangle formed by the lines 2x2 + 3xy – 2y2 – 9x + 7y – 5 = 0
4x + 5y – 3 = 0 lies at (A) ( 3/5 , 11/5) (B) (6/5, 11/5) (B) (5/6, 11/5) (D) None of these
19. The number of lines that can be drawn from the point (2, 3), so that its distance from (-1,
6) is equal to 6, is
(A) 1 (B) 2 (C) 0 (D) infinite
20. If OAB is an equilateral triangle (O is the origin and A is a point on the x-axis), then centroid
of the triangle will be (A) always rational (B) rational if B is rational (C) rational if A is rational (D) never rational (a point P(x, y) is said to be rational if both x and y are rational) 21. Equation of a straight line passing through the point (4, 5) and equally inclined to the lines
3x = 4y + 7 and 5y = 12x + 6 is (A) 9x –7y = 1 (B) 9x + 7y = 71 (C) 7x – 9y = 73 (D) 7x – 9y + 17 = 0 22. Two vertices of a triangle are (5, -1) and (-2, 3). If the orthocentre of the triangle is the origin,
then the third vertex is (A) (-4, 7) (B) (-4, -7) (C) (4, -7) (D) (4, 7) 23. Drawn from the origin are two mutually perpendicular lines forming an isosceles triangle with
the straight line 2x + y = a. Then the area of this triangle is ………………………….
24. Two particles start form the same point (2, –1), one moving 2 units along the line x + y = 1 and the other 5 units along the line x –2y = 4. If the particles move towards increasing y, then their new positions are ……………….., ……………………………
25. The points (, ), (, ), (, ) and (, ) where , , , are different real numbers, are (A) collinear (B) vertices of square (C) vertices of rhombus (D) concyclic 26. A ray travelling along the line 3x – 4y = 5 after being reflected from a line l travels along the
line 5x + 12y = 13. Then the equation of line l is (A) x + 8y = 0 (B) x = 8y + 3 (C) 32x + 4y = 65 (D) 32x – 4y + 65 = 0 27. A light ray emerging from the point source placed at P(2, 3) is reflected at a point ‘Q’ on the
y−axis and then passes through the point R(5, 10). Co−ordinates of ‘Q’ is (A) (0, 3) (B) (0, 2) (C) (0, 5) (D) none of these 28. Equation ax2 + 2hxy + by2 = 0 represents a pair of lines combined equation of lines that can
be obtained by reflecting these lines about the x−axis is (A) ax2 − 2hxy + by2 (B) bx2 − 2hxy + ay2 = 0 (C) bx2 + 2hxy + ay2 (D) none of these 29. Let A(x1, y1), B(x2, y2) and C(x3, y3) be three points such that abscissae and ordinates form 2
different A.P.’s . Then these points (A) form an equilateral triangle (B) are collinear (C) are concyclic (D) none of these 30. a, b, c are in A.P. and ax + by + c = 0 represents the family of line. Equation of line of this
family passing through P(, ); where = values of ‘x’ where 1x1x
2
2
has the least value and
= dxxx1
1
; is
(A) 3x + y − 1 = 0 (B) x + y + 1 = 0 (C) 3x − 2y − 7 = 0 (D) none of these 31. The co-ordinates of the vertices of rectangle ABCD; where A(0, 0), B(4, 0), C(4, 2), D(0, 2)
undergoes following ‘3’ successive tranformations a. (x, y) (y, x) b. (x, y) (x + 3y, y)
c. (x, y)
2
yx,2
yx
Then the final figure formed will be (A). a square (B) a rhombus (C) a rectangle (D) a parallelogram
LEVEL-III 1. If the straight lines ax + by + p = 0 and x cos + y sin = p are inclined at an angle /4 and
concurrent with the straight line x sin - y cos = 0, then the value of a2 +b2 is (A) 0 (B) 1 (C) 2 (D) none of these . 2. If one vertex of an equilateral triangle of side 2 is the origin and another vertex lies on
the line x = y3 , then the third vertex can be (A) (0, 2) (B) 3, 1
(C) (–2, –2) (D) 1,3 3. The locus of a point which divides a line segment AB = 4cm in 1 : 2, where A lies on the line
y = x and B lies on the y = 2x is (A) 234x2 + 153y2- 378xy – 32 = 0 (B) 234x2 + 153y2- 378xy + 32 = 0 (C) 234x2 + 153y2 + 378xy + 32 = 0 (D) None of these 4. All points lying inside the triangle formed by the points (1, 3), (5, 0) and (–1, 2) satisfy (A) 3x + 2y 0 (B) 2x + y –13 0 (C) 2x – 3y –12 0 (D) –2x + y 0 5. A family of lines is given by (1 + 2)x + (1 – )y + = 0, being the parameter. The line
belonging to this family at the maximum distance from the point (1, 4) is (A) 4x – y + 1 (B) 12x + 33y = 7 (C) 13x + 12y + 9 = 0 (D) none of these 6. If A (0, 1) and B(2, 0) be two points and ‘P’ be a point on the line 4x + 3y + 9 = 0. Co-
ordinates of the point ‘P’ such that |PA − PB| is minimum is
(A)
5
14,203 (B)
5
14,203
(C)
5
12,203 (D) 24 17,
5 5
7. Consider the points A (0, 1) and B (2, 0). ‘P’ be a point on the line 4 x + 3 y + 9 = 0 Coordinates of
the point ‘P’ such that PA PB is maximum, is
(A) 12 17,5 5
(B) 24 17,5 5
(C) 24 17,5 5
(D) 12 17,5 5
8. A straight line passing through P (3, 1) meet the coordinate axes at ‘A’ and ‘B’. It is given that
distance of this straight line from the origin ‘O’ is maximum. Area of OAB is equal to
(A) 503
sq. units (B) 100
3 sq. units
(C) 253
sq. units (D) 1 sq. units
9. Consider the points A (0, 1) and B (2, 0) P be a point on the line y = x. Coordinates of the point ‘P’
such that PA+ PB is minimum, is (A) (2/3, 2/3) (B) (3/2, 3/2) (C) (1, ½) (D) (2, 2)
10. Consider the points A (3, 4) and B (4, 13). If ‘P’ be a point on the line y = x such that PA + PB is
minimum, then ‘P’ is
(A) 31 31,7 7
(B) 31 31,7 7
(C) 13 13,7 7
(D) 23 23,7 7
11. Equation ax2 + 2bxy + by2 = 0 represents a pair of lines. Combined equation of lines that can be
obtained by reflecting these lines about the x axis is (A) b x2 2 b x y + a y2 = 0 (B) a x2 + 2 b x y + b y2 = 0 (C) b x2 + 2 b x y + a y2 = 0 (D) a x2 2 b x y + b y2 = 0 12. If the point P (a, a2 ) lies completely inside the triangle formed by the lines x = 0, y = 0 and x + y = 2,
then exhaustive range of ‘a’ is (A) a (0, 2) (B) a (0, 1) (C) a (1, 2 ) (D) a ( 2 , 1) 13. Equation of the straight line belonging to the family of lines (x + y) + (2x y + 1) = 0 , that is farthest
from (1, 3) is (A) 13 y 6 x = 7 (B) 13 y + 6 x = 0 (C) 15 y + 6 x = 7 (D) 15 y 6 x = 7 14. If a < b < c < d and ‘k’ is the number of real roots of the equation (x a) (x c) + 2 (x b) (x d) = 0,
then equation of the line parallel to yaxis and cutting an intercept ‘k’ on xaxis is, (A) x = 0 (B) x = 1 (C) x = 2 (D) None of these 15. If a, b, c are in A. P., then the straight lines a x + 2 y + 1 = 0, b x + 3 y + 1 = 0 and c x + 4 y + 1 = 0 (A are concurrent (B) form a triangle (C) are parallel (D) Can’t say 16. If a, b, c are in A. P. then the image of the point of intersection of the family of lines ax + b y + c = 0 in
the line y = 0 lies on the line (A) x + 2 y 5 = 0 (B) 2 x = y = 0 (C) 3 x + 4 y + 5 = 0 (D) 3 x + 4 y 11 = 0
17. If f (x) = log 1 ax log 1 bx
x
, x 0 and is continuous at x = 0,
= c , x = 0 then the line a x + b y + c = 0 passes through the point (A) (1, 1) (B) (1, 1) (C) (1, 1) (D) (1,1)
18. If m = 200 200
i 3 i 32 2
, then equation of the image of the line having slope ‘m’ and passing
through (0, 0) in the xaxis is (A) x y = 0 (B) x + y = 0 (C) 2 x 3 y = 0 (D) 2 x + 3 y = 0 19. If 3 a + 4 b + 2 c = 0, then the point of concurrent of the family of lines a x + b y + c = 0 and (1, 2) are (A) on the same sides of the line 4 x y + 1 = 0 (B) on the opposite side of the line 4 x y + 1 = 0 (C) are at equal distances from the origin. (D) None of these
20. If a, b, c are three consecutive integers, then the family of lines a x + b y + c = 0 are concurrent at the point,
(A) (1, 2) (B) (2, 1) (C) (1, 2) (D) None of these
ANSWERS LEVEL −I 1. D 2. D 3. A 4. D 5. C 6. B 7. C 8. D 9. B 10. A 11. A 12. A 13. C 14. A 15. C 16. A 17. D 18. C 19. B 20. D 21. B 22. B 23. B 24. A 25. C 26. D 27. A 28. A 29. A 30. A 31. B 32. C 33. C 34. D 35. A 36. A 37. B 38. A 39. C 40. C 41. B 42. A 43. A 44. D 45. A 46. B 47. B 48. C 49. A 50. B 51. A 52. B 53. C 54. A 55. B 56. C 57. C 58. B 59. B 60. A 61. A 62. B 63. C 64. A 65. D 66. A 67. B 68. 2 69. -1, 1/2 70. 1 71. a (-4, 2)
72. 3 1,2 2
73. y = 2x 74. B 75. A
76. C LEVEL −II 1. C 2. A 3. C 4. A 5. C 6. D 7. A 8. C 9. D 10. A, B 13. B 14. C 15. D 16. A 17. B 18. A 19. C 20. D 21. A 22. B
23. 2a
5 24. 2 2, 2 1 and 4 22 , 1
5 5
25. D 26. 27. C 28. A 29. B 30. A 31. D 32. D LEVEL −III 1. C 2. A 3. A 4. C 5. B 6. D 7. d 8. A 9. A 10. B 11. D 12. B 13. D 14. C 15. a 16. A 17. D 18. b 19. A 20. C
TRIGONOMETRIC EQUATION
LEVEL-I 1. If sin(cos) = cos( sin), then sin2 may take value (A) 3/4 (B) –3/4 (C) 1/4 (D) None of these 2. General solution to the equation tan2 + cos2 - 1 = 0 will be given by
(A) = n (B) = 2n +/4 (C) = n +/4 (D) = 2n -/4
3. If sin = p then the equation whose solution is tan 2 is
(A) px2 + 2xp – 1 = 0 (B) px2 + 2x – p =0 (B) x2 + 2x – p =0 (D) None of these
4. If tan(cot x) = cot(tan x) then sin 2x is equal to
(A) 1n22 (B) 1n2
4
(C) 1nn2 (D) 1nn
4
*5. If sin-1x + tan-1x =2
, then 2x2 + 1 =
(A) 5 (B) 2
15
(C) 2 (D) none of these 6. Solution set of the equation sin2x + cos23x = 1 is given by
(A)
In,4
n (B)
In,2
n
(C) In,n (D) none of these *7. The difference between the roots in the first quadrant (0 x /2) of the equation
4 cosx (2 – 3 sin2x) + (cos2x + 1) = 0 is (A) /6 (B) /4 (C) /3 (D) /2
8. The value of
451tan2tan 1 is equal to
(A) 177
(B) - 177
(C) -177
(D) none of these
*9. The set of values of a for which x2 – ax – sin-1(sin3) > 0 for all x R is
(A) R (B)
2,
2
(C) (D) none of these 10. If sin-1 (sinx) = - x, then x belongs to (A) (-, ) (B) [ 0, ]
(C)
23,
2 (D) [, 2]
*11. If cos-1 x+ cos-1 y+ cos-1z = 3 then x2+ y2 + z2 – xy – yz – zx equals to (A) 0 (B) 1 (C) 2 (D) 3
12. The number of real solutions of the equation tan-1 1xx + sin-1 1xx2 = 2 is
(A) zero (B) one (C) two (D) infinite
*13. If x 1 , then 2 tan-1x+ sin-12x1
x2
is equal to
(A) 4 tan-1x (B) (C) 0 (D) None of these 14. If sinx + siny + sinz = 3, x, y, z [ 0, 2], then (A) x2 + y2 + z2 – xy – yz – zx = 0 (B) x3 + y3 + z3 = 3/3 (C) x3 + y3 + z3 = 0 (D) x + y + z = 0 15. If cos1 + 2 cos2 + 3 cos3 = 6 then tan1+ tan2+ tan3 equals to (A) 1/2 (B) 6 (C) 0 (D) 3 *16. The equation esinx + e-sinx = 2sinx will have (A) no solution (B) one solution (C) two solution (D) none of these 17. If 1+ tan = 2 then cos - sin equals to ( (2n+1)/2) (A) 2 sin (B) 2 sin (C) 2 cos (D) 2 cos 18. Value of cos(2 cos-1(4/5)) equals to (A) 6/25 (B) 7/25 (C) 4/25 (D) 8/25 19. If 4 cos-1x + sin-1x = then x equals to (A) 1/2 (B) 1/ 2 (C) 1 (D) 3 /2
*20. Number of solution to the equation sin-1x – cos-1x = cos-1
23 is
(A) one (B) two (C) four (D) none of these.
*21. The solutions of the equation (cos2x – 4sinx + 6) (1-sinx) = cos2x are (A) 2n, n I (B) (4n – 1) /2, n I
(C) (4n + 1) ,2 n I (D) None of these
22. If 2xtan2 = secx – cosx, then
(A) x = 2n, n I (B) x = (2n + 1), n I
(C) x = (2n + 1)2 , n I (D) None of these
23. The inequality log2x < sin–1 (sin5) holds if (A) x (0, 25-2) (B) x (25-2 , ) (C) x (22 -5 , ) (D) None of these 24. The value(s) of y for which the equation 4 sinx+3cosx= y2– 6y +14 has a real solution, is
(are) (A) 3 (B) 5 (C) –3 (D) None of these. 25. The most general values of x for which sinx + cosx = min.{ y2 – 10 y +26, y2 –6y +12},
y R, are given by
(A) [2n, (2n+1) ] (B) [ n + (-1)n
44
, ( n+1) ]
(C) [2n, ( 4n+1)2 ] (D) None of these
26. If cos-1x + cos-1y + cos-1z =3 , then x3 +y3 +z3 is equal to (A) –3 (B) 3 (C) 0 (D) None of these *27. The solution(s) x, of the equation 3 cosx – sinx = (cos10y+ sec10y), is (are)
(A) /6 (B) - /6 (C) -/3 (D) /3
28. If cotx coty = k and x + y =/3, then tanx, tany satisfy the equation (A) kt2 – 3 (k –1)t +1= 0 (B) kt2 + 3 ( k –1)t +1 = 0 (C) kt2 – 3 ( k +1)t +1= 0 (D) kt2 + 3 ( k +1)t +1 = 0
29. If 4z3sin3y2tan1xcos
1xcos 22
2
, then
(A) x may be a multiple of (B) x can not be an even multiple of (C) z can be a multiple of (D) y can be a multiple of /2 .
30. tan + tan2 + tan tan2 = 1. Then is equal to (A) /12 (B) 5/12 (C) -3/12 (D) -7/12 31. If –1 < x < 0 then tan-1x equals
(A)
21 x1cos (B)
2
1
x1
xsin
(C)
xx1cot
21 (D) cosec-1 x
32. The set of all x in ( -, ) satisfying |4sinx-1| < 5 is given by
(A)
103,
10 (B)
103,
10 (C)
103,
10 (D) none of these.
33. The number of roots of the equation x+ 2tanx = /2 in the interval [0, 2] is (A) 1 (B) 2 (C) 3 (D) infinite 34. The general solution of the equation sinx + cosx = 1, for n = 0, 1, 2,...... is
(A) x = 2n (B) x = 2n + 21
(C) x = n + (-1)n 4 4 (D) none of these
35. The solution set of (2cosx - 1) (3 + 2cosx) = 0 in the interval 0 x 2 is
these. of none D2/3cos,3
5,3
C
35,
3 B
3 A
1
36. The number of solutions of the equation tanx + secx = 2cosx lying in the interval [0, 2]
is (A) 0 (B) 1 (C) 2 (D) 3 37. The general solution of the equation tan2 + 23 tan = 1 is given by
12n D
121+6n= C
21+n= B
2= A
38. The general solution of sinx - 3sin2x + sin3x = cosx - 3cos2x + cos3x is
2/3cos+2n D82
n1- C
8+
2n B
8+n A
1-n
*39. The value of tan[cos-1 4/5 + tan-1 2/3] or tan [sin-1(3/5) + cot-1 3/2] is (A) 6/17 (B) 7/16 (C) 17/6 (D) none of these.
40. The principal value of sin-1(sin32 ) is
(A) -2/3 (B) 2/3 (C) 4/3 (D) None of these *41. If 1 + |sinx| + sin2x + |sin3x| + .... = 4 + 2 3 , then
(A) x = 6 (B)
3
(C) x = 32 (D) x =
65
42. The number of ordered pair (x, y), where x and y satisfy x + y = 2/3 and cosx + cosy =
3/2 is (A)0 (B)1 (C)2 (D) infinity
43. The number of solutions of cos2 + sin + 1 = 0, is ( [0, 2]) (A) 0 (B)1 (C) 2 (D) infinity 44. If sin-1x > cos-1x, then
(A) x
21,1 (B) x
21,0
(C) x
1,21 (D) x
0,
21
45. The set of all values of x in the interval [0, ] for which 2sin2x - 3sinx + 1 0 contains (A)[0, /6] (B)[0, /3] (C)[2/3, ] (D) [0, /6] {/2} [5/6, ]
*46. If the expression 2/xsini21
xtani2/xcos2/xsin
is real If x belong to the set
(A) {n : nI} (B) {2n : nI} (C) {n+/4 : nI} (D) {2n+/4 : nI} 47. sinx, sin2x, sin3x are in A.P. if (for n I)
(A) x = 2
n (B) x = n
(C) x = 2n (D) x = 3
n
*48. sinx cosx cos 2x = k has a solution, if k belong to the interval (A)[0, 1] (B)[-1,0] (C) [-/2, /2] (D) [-1/4, ¼]
LEVEL-II 1. The values of x in [0, 2] which satisfy the equation 21+ |sinx| + | sin2x| + |sin3x| + .. . = 2 are
(A) 0 (B) (C) 2 (D) 3/2
2. The values of in the interval (-/2, /2) satisfying the equation 2sec3 = tan4 + 2
tan2 is (A) /4 (B) -/4 (C) (D) none of these
*3. equals1cos11sintan 1
(A) 0 (B) 2
1
(C) 12
(D) 42
1
*4. The value of x that satisfies the equation tan2x = tan-1(tan3) is (A) /3 (B) - /3 (C) 3tan 1 (D) none of these
5. If sin-1x + sin-1y = 32
, cos-1x – cos-1y = 3
, then the number of ordered pairs (x, y) is
(A) 0 (B) 1 (C) 2 (D) none of these 6. The number of real solutions of cos-1x + cos-1 2x = - is
(A) 0 (B) 1 (C) 2 (D) infinitely many
7. sinx + cos x = y2 – y +a has no value of x for any y if ‘a’ belongs to
(A) ( 0, 3 ) (B) (- 3 , 0 ) (C) ( - , - 3 ) (D) ( 3 , )
8. The values of k, for which the system of equations cosx cos2y = (k2 – 4)2 +1 and
sinx sin2y = k +2 holds, is (are) given by (A) k = 2 (B) k = - 2 (C) k = 2 (D) none of these
9. The value of tan[sin-1(cos(sin-1x))] tan[ cos-1 (sin(cos-1x) )], (x ( 0, 1)) is equal to (A) 0 (B) 1 (C) –1 (D) none of these.
10. The value of tan-1
A2tan
21 + tan-1(cotA) + tan-1(cot3A), for 0 < A < /4, is
(A) tan-1 2 (B) tan-1(cotA) (C) 4 tan-1(1) (D) 2 tan-1(2) *11. The value of a for which the equation 4cosec2( (a + x)) + a2 – 4a = 0 has a real solution, is (A) a = 1 (B) a = 2 (C) a = 10 (D) None of these
*12.
bacos
21
4cos 1 +
bacos
21
4cos 1 is equal to
(A) b
ba (B) ba
b
(C) b
ba (D) None of these
13. If 2 sin-1x = cos-1(1 – 2x2), then (A) -1 x 1 (B) -1 x 0 (C) x = 1/2 (D) 0 x 1
14. If Asin1 =sin2A –cos
2A , then
42A could lie in quadrant
(A) first (B) second (C) third (D) fourth
15. General solution to the equation tan2 + cos2 -1 =0 will be given by (A) = n (B) = 2n +/4 (C) = n +/4 (D) = 2n -/4
16. If sinx + cosx = y1y , x [ 0, ], then
(A) x =/4 (B) y =0 (C) y = 1 (D) x= 3/4. 17. The minimum value of 2sinx + 2cosx is
(A) 1 (B) 2 – 21 (C) 2-1/ 2 (D) 2
112
*18. The number of solutions of the equation
tan-1
1x21
+ tan-1
1x41
= tan-12x
2 is
(A) 1 (B)2 (C)3 (D) 4
*19. The value of tan-1
xyayxa
1
1 +tan-1
21
12
aa1aa
+ tan-1
32
23
aa1aa
+ ... +tan-1
1nn
1nn
aa1aa
+ tan-1
na1 is
(A) 0 (B) 1
(C) tan-1
yx (D) tan-1
xy
*20. If sinx + cosx = 1 + sinx cosx, then
(A) sin
4x =
21 (B) sin
21
4x
(C) cos2
14
x
(D) cos
21
4x
21. If tan-1x + cot-1x + si n-1x x (0, 1] then (A) =0 , = /2 (B) = 0, = (C) = /2, = (D) = /2, =
LEVEL-III 1. If all the solutions ‘x’ of acosx + a–cosx = 6 (a > 1) are real, then set of values of a is (A) [3+2 2 , ) (B) (6, 12) (C) (1, 3 + 2 2 ) (D) none of these.
2. The value of
2sec412cos
432sincotsin 1111 is
(A) 0 (B) /4 (C) /6 (D) /2 3. The number of integral values of p for which the equation cos (psinx) = sin(p cosx) has a
solution in [0, 2] is (A) 1 (B) 2 (C) 3 (D) none of these
4. If (tan-1x)2 + (cot-1x)2 = 8
5 2, then x equals to
(A) -1 (B) 1 (C) 0 (D) none of these *5. The number of points inside or on the circle x2 + y2 = 4 satisfying
tan4x + cot4x + 1 = 3sin2y is (A) one (B) two (C) four (D) infinite
*6. If cos 03
xcos6
xsin
, then x is
(A) n + /4, n I. (B) n - /2, n I. (C) n - /4, n I. (D) none of these 7. Indicate the relation which is true (A) tan | tan-1 x | = | x | (B) cot | cot-1 x | = x (C) tan-1 | tan x | = | x | (D) sin|sin-1x |=|x| 8. The values of x between 0 and 2 which satisfy the equation sinx xcos8 2 =1 are in
A.P. with common difference (A) /4 (B) /8 (C) 3/8 (C) 5/8
9. In a triangle ABC, the angle B is greater than angle A. If the values of angles A and B
satisfy the equation 3sinx - 4sin3x - k = 0, 0 < k < 1, then the value of C is
6
5 D32 C
2 )B(
3 A
10. If A = 2 tan-1(2 2 -1) and B = 3 sin-1(1/3) + sin-1 (3/5), then (A) A = B (B) A < B
(C) A > B (D) none of these
*11. The equation (cosp - 1)x2 + (cosp)x + sinp = 0, where x is a variable, has real roots. Then the interval of p may be
(A) (0, 2) (B) (-, 0)
(C)
2
,2
(D) (0, )
ANSWERS LEVEL −I 1. A 2. A,C 3. D 4. B 5. A 6. A 7. A 8. C 9. C 10. C 11. A 12. C 13. B 14. A 15. C 16. A 17. 18. B 19. D 20. A 21. C 22. A 23. A 24. A 25. D 26. A 27. B 28. A 29. A,D 30. A, B, C, D 31. B 32. A 33. C 34. C 35. B 36. C 37. C 38. 39. C 40. D 41. B, C 42. A 43. B 44. C 45. D 46. B,C 47. A, C 48. D LEVEL −II 1. A, B, C 2. A, B 3. D 4. D 5. B 6. A 7. D 8. B 9. B 10. C 11. C 12. C 13. D 14. A,B 15. 16. A, C 17. D 18. B 19. C 20. A, D 21. LEVEL −III 1. A 2. A 3. D 4. A 5. C 6. 7. A, B, D 8. A 9. C 10. C 11. D
TRI
LEVEL-I 1. If sin + cosec = 2, then the value of sinn + cosecn, n 2, n N equals (A) 2 (B) 2n (C) 1 (D) none of these
2. The maximum value of 1 + sin
4
+ 2 cos
4
, R, equals
(A) 3 (B) 5 (C) 4 (D) none of these 3. The least value of cos2 – 6 sin cos + 3 sin2 + 2 is (A) 4 + 10 (B) 4 – 10 (C) 0 (D) none of these
4. If 0 < < 4 , cos( + ) =
53 and cos( – ) =
54 , then sin2 is equals
(A) 1 (B) 0 (C) 2 (D) none of these
5. The numerical value of sin18 . sin
185 . sin
187 is equal to
(A) 1 (B) 81
(C) 41 (D) none of these
6. If tan. tan
3
. tan
3
= -1, ( 0 < < /2), then value of 3 sin - 4 cos3 =
(A) 1 (B) -1 (C) 1/ 2 (D) -1/ 2 7. If in a ABC, sin2 A + sin2 B + sin2 C = 2, then the triangle is
(A) isosceles triangle (B) right angled triangle (C) acute angle triangle (D) obtuse angled triangle
8. Minimum value of the expression 2 sin x + 4 cos x + 3 5 is
(A) 5 5 (B) 2 5 + 3 (C) 2 5 -3 (D) none of these 9. The maximum value of 4 sin2 x + 3 cos2 x + sin x/2 + cos x/2 is (A) 4 + 2 (B) 3 + 2 (C) 9 (D) 4
10. If tan = 21 , tan =
31 , then + = ____________________________
(A) 0 (B) /2
(C) /4 (D)
11. The value of tan 150 = ________________________________________
12. If 2 sin . sec 3 = tan 3 -tan , then 2[sin . sec 3 + sin 3 . sec 32 + …..+sin 3n –1 . sec 3n] = ______________________________
13. If tan = ab , then a cos 2 + b sin 2 =________________________
14. Maximum value of 2 cos + 3 sin + 4 is _____________________
15. If sec -tan = 5, then sec = ___________________________
16. If < 2 < 2
3 , then 4cos222 equals to
(A) –2 cos (B) -2 sin (C) 2 cos (D) 2 sin 17. If tan = n for some non-square natural number n then sec 2 is (A) a rational number (B) an irrational number (C) a positive number (D) none of these. 18. If and are two distinct roots of the equation a tan x + b sec x = c, then tan ( + ) is equal
to
(A) 22
22
caca
(B) 22
22
caca
(C) 22 ca
ac2
(D) 22 caac2
19. If sin = 3 sin ( + 2) then value of tan( + ) + 2 tan is (A) 3 (B) 2 (C) 1 (D) 0 20 In a ABC, if cotA cotB cotC > 0, then the is (A) acute angled (B) right angled (C) obtuse angled (D) does not exist
21 If sinx = cos2x, then cos2x (1 + cos2x) equals to (A) 0 (B) 1 (C) 2 (D) none of these
22 The value of sin 150 = ______________________________________________ 23 Maximum value of 2 cos + 3 sin + 5 = _______________________________ 24 If sin sin -cos cos = 1, then tan + tan = _________________________
25 If tan = yx , then x cos 2 + y sin 2 = __________________________________
26 The value of cos 100 –sin 100 is (A) positive (B) negative (C) 0 (D) 1
27 00
00
66tan69tan166tan69tan
=
(A) 1 (B) –1 (C) 0 (D) none of these 28. The value of sin 120 sin 280 sin 540 = _______________________________ 29. If sin sin -cos cos + 1 = 0, then 1 + cot tan = ____________________________
30. The equation sin2 = xy2
yx 22 is possible if
(A) x = y (B) x = -y (C) 2x = y (D) none of these 31. 3 sin x + cos x is maximum when x is (A) 300 (B) 450 (C) 600 (D) 900 32. The minimum value of 3tan2 +12 cot2 is (A) 6 (B) 15 (C) 24 (D) none of these .
33. If
sin3sinthen,4
tan3tan equals
(A) 3/5 (B) 4/5 (C) 3/4 (D) none of these.
34. For any real , the maximum value of cos2( cos) + sin2(sin)
(A) is 1 (B) is 1 + sin21 (C) is 1+ cos21 (D) does not exist
35. If cosecA + cotA = 11/2, then tanA is equal to (A) 111/44 (B) 44/117 (C) 44/125 (D) 117/125 36. If in ABC, A = sin–1(x), B = sin–1(y) and C = sin–1(z), then 222222 y1x1zz1x1yz1y1x is equal to (A) xyz (B) x+y+z
(C) z1
y1
x1
(D) None of these
37. If Tn = sinn + cosn, then mT
TT
6
46
holds for values of m satisfying
(A)
31,1m (B)
31,0m
(C) 0,1m (D) None of these 38. If 4 sinA + secA =0 then tanA equals to (A) 4 2 (B) – 2 3 (C) 2 4 3 (D) 4 2 3
39. Value of the expression 2sinx – cos2x is always (A) greater than or equal to -3/2 (B) less than or equal to 3/2 (C) greater than or equal to -1/2 (D) none of these
40. If cos25 + sin 25 = k , then cos20 is equal to
(A) 2
k (B) – 2
k
(C) 2
k (D) None of these
41. If 2n = /2 , then tan tan2 tan3 tan(2n – 1) is equal to (A) 1 (B) – 1 (C) 0 (D) None of these
42. If –2 < cos + sec 2, then cosn + secn is equal to (n N) (A) 2 (B) 2n (C) 0 (D) None of these. 43. If tan = n tan , then maximum value of tan2 ( - ) is equal to
(A) n4
)1n( 2 (B) n4
)1n( 2
(C) n2
)1n( (D) n2
)1n(
44. If cos - sin = 2 sin ; Then the value of cos + sin is equal to; (A) 2 cos (B) - 2 cos (C) - 2 cos (D) none of these
45. If sec2 = 2yx
xy4
, then x and y ;
(A) are always equal (B) can be any real number (C) can assume finite number (D) none of these. 46. If cosx +secx = -2, then for a positive integer n, cosnx + secnx is
(A) always 2 (B) always –2 (C) 2, if n is odd (D) 2, if n is even
47. If | sinx + cosx| = | sinx| + | cosx| , then x belongs to the quadrant
(A) I or III (B) II or IV (C) I or II (D) III or IV
48. sinx + cos x = y2 – y +a has no value of x for any y if ‘a’ belongs to
(A) ( 0, 3 ) (B) (- 3 , 0 ) (C) ( - , - 3 ) (D) ( 3 , )
49. If tanA + cotA = 4, then tan4A + cot4A is equal to (A) 110 (B) 191 (C) 80 (D) 194
50. cos2
12
+ cos2
4
+ cos2
125
is
(A) 33
2
(B) 32
(C) 2
33 (D)
23
51. If A lies in the second quadrant and 3 tanA + 4 = 0 then the value of 2 cotA – 5 cosA + sinA
is equal to
(A) 1037
(B) 1023
(C) - 1053
(D) none of these
52. The minimum value of sec2 + cos2 is (A) 1 (B) 0 (C) 2 (D) none of these
53. If sin = p then the equation whose solution is tan 2
is
(A) px2 + 2xp – 1 = 0 (B) px2 + 2x – p =0 (B) x2 + 2x – p =0 (D) None of these
LEVEL-II
1. sin2 = xy4yx 2 , where x, y R, gives real if and only if
(A) x + y = 0 (B) x = y (C) |x| = |y| 0 (D) none of these
2. Let a = cosA + cosB – cos(A + B) and b = 4 sin2A
. sin2B
. cos2
BA . Then a – b is equal to
(A) 1 (B) 0 (C) – 1 (D) none of these 3. If 3 sin + 4 cos = 5, then 4 sin – 3cos is equal to (A) 0 (B) 5 (C) 1 (D) none of these 4. If in ABC C = 900, then the maximum value of sin A sin B is (A) 1/2 (B) 1 (C) 2 (D) 3/4
5. If lies in fourth quadrant, then
22cos42sincos4 224 is equal to
(A) 1 (B) 2 (C) –2 (D) 0 6. If ( + + + ) = then cos cos -sin sin = (A) 4 (B) 2 (C) 0 (D) none of these
7. If x + y = 2 then minimum value of sec x + sec y is, x, y
2
,0
(A) 2 cos (B) cos 2 (C) 2 sec (D) none of these
8. 0
00
50tan420tan70tan =
(A) 1 (B) 1/2 (C) –1 (D) –1/2 9. In a triangle ABC maximum value of sin A + sin B + sin C is
(A) 233 (B)
232 (C) 33 (D)
23
10. If 1 + sin + sin 2 + sin3 + …..to = 4 + 2 3 , 0 < < , /2 then
(A) = 6 (B) =
3 (C) =
6 or
3 (D) =
3 or
32
11. The value of tan 10 tan 20 tan 30 ……..tan 890_______________________________
12. Value of 94sin
93sin
92sin
9sin
is ________________________ __
13. If sinx + sin2 x = 1, then cos12 x + 3 cos10 x + 3 cos8 x + cos6 x –1 =
____________________________
14. If sin ( + ) = 1, sin ( -) = 21 where ,
2
,0 , then tan ( + 2) tan (2 + )
is______________ 15. If in a ABC, sin2 A + sin2 B + sin2 C = 2, then the triangle is
(A) isosceles triangle (B) right angled triangle (C) acute angle triangle (D) obtuse angled triangle
16. If cot + tan = x and sec -cos = y then
(A) sin cos = 1/x (B) sin tan = y (C) 1xyyx3/223/22 (D) 1xyyx
3/223/22 17. The minimum value of cos(cosx) is (A) 0 (B) –cos1 (C) cos1 (D) –1 18. If sin, sin and cos are in G.P, then roots of the equation
x2 + 2x cot + 1 = 0 are always. (A) equal (B) real (C) imaginary (D) greater than 1
19. If A + B = 450, then (1 + tan A) (1 + tan B) = _____________________________ 20. If sin , cos , tan are in G.P, then cot6 -cot2 is (A) 1 (B) –1 (C) 0 (D) 2 21. If sin x + sin2x = 1, then cos8 x + 2 cos6 x + cos4 x is
(A) 0 (B) –1 (C) 2 (D) 1
22. If tan =
sinsinsin2 then cot , cot , cot are in
(A) AP (B) GP (C) HP (D) none of these 23. If tan2 = 2 tan2 + 1 , then cos 2 + sin2 = (A) 1 (B) 2 (C) 0 (D) –1 24. The value of expression 3 cosec 200 –sec 200 is equal to (A) 2 (B) 4
(C) 0
0
40sin20sin2 (D) 0
0
40sin20sin4
25. If sinx + cosx + tanx + cot x + secx + cosecx = 7 and sin2x =a – b 7 , then
ordered pair ( a, b) can be, (A) ( 6, 2) (B) (8, 3) (C) (22, 8) (D) (11, 4) 26. If tanx – tan2x = 1, then the value of tan4x – 2tan3x – tan2x + 2tanx + 1 is (A) 1 (B) 2 (C) 3 (D) 4 27. The minimum value of the expression xcosxsin 66
33 is (A) 2.31/8 (B) 2.37/8 (C) 3.21/8 (D) None of these 28. If sin + sin = 3 (cos - cos), then sin3 + sin3 is equal to (A) 3 (B) 0 (C) 1 (D) None of these 29. The minimum value of (asec - btan)2 , |a| < |b|, is (A) 0 (B) a2 + b2 (C) ab (D) (1/2) (a2 + b2) 30. If tanx + tan2x + tan3x =1 then the value of 2 cos6x –2 cos4x + cos2x equals to (A) 1/2 (B) 2 (C) 1 (D) none of these 31. If a 16 sinx cosx + 12 cos2x – 6 b for all x R then (A) a = -5, b = 5 (B) a = -4, b = 4 (C) a = -10, b =10 (D) none of these
32. If ksin2x + k1 cosec2x = 2, x (0, /2), then cos2x +5 sinx cosx + 6 sin2x is equal to
(A) 2
2
k6k5k (B) 2
2
k6k5k
(C) 6 (D) none of these
33. Value of sin4
8 + sin4
83 + sin4
85 + sin4
87 is equal to;
(A) 3/2 (B) 2/3 (C) 2/3 (D) 3/2
34. The minimum value of 6xcos32xsin2
1
is equal to
(A) - 101
(B) - 31
(C) 101
(D) 61
LEVEL-III 1. In a ABC, cos2A + 4cos(B + C) sinB sinC is equal to (A) 2 cos2A + cos2B (B) cos2B - 2 sin2A (C) cos2B + 2 cosA (D) none of these
2. The value of cot2 36 cot2 72 is (A) 1/2 (B) 1/3 (C) 1/4 (D) 1/5
3. If x = , satisfy both the equations a cos2x +b cosx +1 = 0 and a sin2x + psinx+1=0,
then (A) 2a( a+2) = b2 – p2 (B) 2a(a - 2)= b2+p2 (C) 2a( a+2) = b2 +p2 (D) None of these
4. The number of points inside or on the circle x2 + y2 = 4 satisfying
tan4x + cot4x + 1 = 3sin2y is (A) one (B) two (C) four (D) infinite 5. The number of ordered 4-tuple (x, y, z, w) (x, y, z, w [0, 10]) which satisfied the inequality,
wcoszsinycosxsin 22225432 120 is
(A) 0 (B) 144 (C) 81 (D) infinite 6. If all the solutions ‘x’ of acosx + a–cosx = 6 (a > 1) are real, then set of values of a is (A) [3+2 2 , ) (B) (6, 12) (C) (1, 3 + 2 2 ) (D) none of these. 7. A quadrilateral ABCD is circumscribed about a circle, then
(A) AB.sin2A . sin
2B = CD sin
2C sin
2D (B) AB. sin
2C sin
2A . = CD sin
2B sin
2D
(C) AB.sin2A
. sin2D
= CD sin2B
sin2C
(C) None of these .
ANSWERS LEVEL −I 1. A 2. C 3. B 4. A 5. B 6. C 7. 8. D 9. A 10. C 11. 2 3 12. tan3n - tan
13. A 14. 13 4 15. 135
16. A
17. A 18. D 19. D 20. A 21. D 22. 3 1 23. 13 5 24. 0
25. 2 2
2 2
x 3y x
x y
26. A
27. B 28. 0sin18
4
29. 0 30. A 31. C 32. D 33. D 34. B 35. B 36. A 37. C 38. B 39. A 40. A 41. A 42. A 43. A 44. A 45. A 46. D 47. A 48. D 49. D 50. D 51. B 52. C 53. D LEVEL −II 1. C 2. A 3. B 4. A 5. B 6. C 7. C 8. B
9. A 10. D 11. 1 12. 316
13. 0 14. 1 15. B 16. C 17. C 18. B 19. 2 20. A 21. D 22. A 23. C 24. B 25. C 26. D 27. A 28. B 29. A 30. A 31. C 32. D 33. A 34. C LEVEL −III 1. D 2. D 3. C 4. C 5. B 6. A 7. A
VECTOR LEVEL−I
1.
OBandOA are two vectors such that |OBOA|
= |OB2OA|
. Then (A) BOA = 90 (B) BOA > 90 (C) BOA < 90 (D) 60 BOA 90 2. If candb
are two non-collinear vectors such that 4cb.a
and
cysinb6x2xcba 2 , then the point ( x, y) lies on
(A) x =1 (B) y =1 (C) y = (D) x + y = 0 3. The scalar cbacb.a
equals
(A) 0 (B) 2 c b a
(C) c b a
(D) None of these 4. If c,b,a are three unit vectors, such that cba is also a unit vector, and 1, 2, 3 are
angle between the vectors, a,candc,b;b,a respectively then cos1 + cos2 + cos3 equals
(A) 3 (B) -3 (C) 1 (D) -1
5. If angle between 3
isbanda , then angle between b3anda2
is
(A) /3 (B) -/3 (C) 2/3 (D) -2/3 6. The vectors km3jmi2 and kjm2im1 include an acute angle for (A) all real m (B) m < –2 or m > –1/2 (C) m = –1/2 (D) m [–2, –1/2] 7. a 3, b 4, c 5
such that each is perpendicular to sum of the other two, then
a b c
=
(A) 5 2 (B) 2
5 (C) 10 2 (D) 5 3
8. If x
and y
are two vectors and is the angle between them, then 1 x y2
is equal to
(A) 0 (B) 2 (C)
2sin (D)
2cos
9. If ˆ ˆ ˆ ˆ ˆ ˆu i a i j ( a j ) k ( a k )
, then
(A) u is unit vector (B) u = a + i + j + k (C) u = 2a (D) none of these 10. Let banda be two unit vectors such that ba is also a unit vector. Then the angle
between banda is (A) 30 (B) 60 (C) 90 (D) 120
11. If kjia
, k4j3i4b
and kjic
are linearly dependent vectors and c
=
3 . (A) =1, = -1 (B) = 1, 1 (C) = -1, 1 (D) = 1, = 1 12. Let k2ji2a
and jib
. If c
is a vector such that ca
= c
, 22ac
and the
angle between ba
and c
is 30, then cba
=
(A) 32 (B)
23
(C) 2 (D) 3 13. Let kia , kx1jixb and kyx1jxiyb . Then cba depends on (A) only x (B) only y (C) NEITHER x NOR y (D) both x and y 14. If |a||ba| , then ba2.b equals (A) 0 (B) 1 (C) 2a.b (D) none of these 15. If |a| = 3, |b| = 5, |c| = 7 and cba = 0, then angle between a and b is
(A) 4 (B)
3
(C) 2 (D) none of these
16. Given that angle between the vectors kj3ia and kji2b is acute, whereas
the vector b makes with the co-ordinate axes on obtuse angle then belongs to (A) (-, 0) (B) (0, ) (C) R (D) none of these 17. If candb,a
are unit coplanar vectors then the scalar triple product
ac2,cb2,ba2
= (A) 0 (B) 1 (C) 3 (D) 3 18. If baba
, then the angle between a
and b
is
(A) acute (B) obtuse (C) /2 (D) none of these
19. If the lines
|c|
c
|b|
bxr and bcyb2r intersect at a point with position vector
|c|
c
|b|
bz , then
(A) z is the AM between |b| and |c| (B) z is the GM between |b| & |c|
(C) z is the HM between |b| and |c| (D) z = |b| + |c|
20. Let ABCDEF be a regular hexagon and AB a b c
, , BC CD then AE
is (A)
a b c (B)
a b (C)
b c (D) c a 21. The number of unit vectors perpendicular to vectors
a 11 0, , and b = 0,1,1 is (A) One (B) Two (C) Three (D) Infinite 22. If p and d are two unit vectors and is the angle between them, then
(A) 12 2
2 sinp d
(B) p d = sin
(C) 12
12
cosp d (D) 2cos1ˆˆ21 2
dp
23. The value of k for which the points A(1, 0, 3) , B(-1, 3,4) ,C(1, 2, 1) and D(k, 2, 5) are coplanar is (A) 1 (2)2 (C) 0 (D) -1
24. If
a a a
b b b
c c c
2 3
2 3
2 3
1
1
1
0
and the vectors A = (1, a, a2), B = (1, b, b2), C = (1,c,c2) are
non - coplanar, then the value of abc will be (A) –1 (B) 1 (C) 0 (D) None of these 25. Let a, b, c be distinct non-negative numbers. If the vectors kbjcic ,k i ,kc ja ia lie in
a plane, then c is (A) the arithmetic mean of a and b (B) the geometric mean of a and b (C) the harmonic mean of a and b (D) equal to zero ` 26. The unit vector perpendicular to the plane determined by P(1, -1, 2), Q(2, 0, -1), R(0, 2, 1) is
(A) 6
2 kji (B)
62kji
(C) 6
2 kji (D) None of these
27. If
CBA ,, are non-coplanar vectors then
BAC
CAB
BAC
CBA
.
.
.
. is equal to
(A) 3 (B) 0 (B) 1 (D) None of there
28. If the vector ,ˆˆˆ kjia kjbi ˆˆˆ and kcji ˆˆˆ (a b c1) are coplanar, then the value of
cba
1
11
11
1 is equal to
(A) 1 (B) 0 (C) 2 (D) None of these 29. If cba ,, are vectors such that ba
. =0 and cba . Then
(A) 222 cba (B) 222 cba
(C) 222cab
(D) None of these
30. The points with position vector 60i + 3j, 40i – 8j and ai –52j are collinear if (A) a = -40 (B) a = 40 (C) a = 20 (D) none of these . 31. Let banda be two unit vectors such that ba is also a unit vector. Then the angle
between banda is (A) 30 (B) 60 (C) 90 (D) 120 32. If vectors ax k5j3i and x kax2j2i make an acute angle with each other, for all x
R, then a belongs to the interval
(A)
0,
41 (B) ( 0, 1) (C)
256,0 (D)
0,
253
33. A vector of unit magnitude that is equally inclined to the vectors ji , kj and ki is;
(A) kji31
(B) kji31
(C) kji31
(D) none of these
34. Let a, b, c be three distinct positive real numbers. If r,q,p lie in plane, where
kbjaiap , kiq and kbjcicr then b is (A) A.M of a, c (B) the G.M of a, c (C) the H.M of a, c (D) equal to c 85. The scalar CBACB.A is equal to ______________________ 36. If c,b,a are unit coplanar vectors, then the scalar triple product ac2,cb2,ba2 is
equal to _____________________ 37. The area of a parallelogram whose diagonals represent the vectors k2ji3 and
k4j3i is
(A) 10 3 (B) 5 3 (C) 8 (D) 4
38. The value of accbba is equal to
(A) 2 cba (B) 3 cba
(C) cba (D) 0
LEVEL−II
1. If a
is any vector in the plane of unit vectors candb , with cb = 0, then the
magnitude of the vector cba
is (A) |a|
(B) 2
(C) 0 (D) none of these . 2. If banda
are two unit vectors and is the angle between them, then the unit vector
along the angular bisector of banda
will be given by
(A)
2cos2
ba
(B)
2cos2
ba
(C)
2sin2
ba
(D) none of these.
3. If a is a unit vector and projection of x along a is 2 units and xbxa , then x is given by
(A) baba21
(B) baba221
(C) baa (D) none of these. 4. If 0c9b5a4 , then )ba( [ )cb( )ac( ]is equal to (A) A vector perpendicular to plane of candb,a (B) A scalar quantity (C) 0
(D) None of these
5. The shortest distance of the point (3, 2, 1) from the plane, which passes through a(1, 1, 1)
and which is perpendicular to vector k3i2 , is
(A) 34
(B) 2 (C) 3 (D) 131
6. Let kji2a
, kj2ib
and a unit vector c
be coplanar. If c
is perpendicular to a
then c
=
(A) kj21
(B) kji31
(C) j2ˆi51
(D) kji21
7. Let a
and b
be the two non–collinear unit vector. If bbaau
and bav
, then v
is
(A) u
(B) auu
(C) bauu
(D) none of these
8. If b,a and c are unit vectors, then 222
accbba does NOT exceed
(A) 4 (B) 9 (C) 8 (D) 6 9. If kji2awhere,3r.aandatbra and kj2ib then r equals
(A) j52i
67
(B) j31i
67
(C) k31j
32i
67
(D) none of these
10. If accbba
= 0 and at least one of the numbers , and is non-zero,
then the vectors candb,a
are (A) perpendicular (B) parallel (C) co-planar (D) none of these 11. The vectors a
and b
are non-zero and non-collinear. The value of x for which vector
c
= (x –2)a
+ b
and d
= (2x +1)a
– b
are collinear. (A) 1 (B) 1/2 (C) 1/3 (D) 2 12 cba
, acb , then
(A) a = 1, cb (B) c = 1, a = 1
(C) b
= 2, ab 2 (D) b
= 1, ab
13. If a
, b
, c
are three non - coplanar vectors and p, q, r
are vectors defined by the
relations b cpa b c
, c aqa b c
, a bra b c
then the value of expression
(a + b).p + (b + c).q + (c + a).r
is equal to (A) 0 (B) 1 (C) 2 (D) 3 14. The value of 2 2 2ˆ ˆ ˆ|a i | + |a j| + |a k|
is
(A) a2 (B) 2a2 (C) 3a2 (D) None of these 15. If kj2b,jia and babr,abar , then a unit vector in the direction of r is;
(A) kj3i111
(B) kj3i111
(C) kji31
(D) none of these
16. kak.ajaj.aiai.a is equal to;
(A) 3 a (B) r (C) 2 r (D) none of these
17. If the vertices of a tetrahedron have the position vectors kiandkj2,ji,0 then the
volume of the tetrahedron is (A) 1/6 (B) 1 (C) 2 (D) none of these 18. A = (1, -1, 1), C = (-1, -1, 0) are given vectors; then the vector B which satisfies CBA
and 1B.A is ___________________________________
19. If c,b,a are given non-coplanar unit vectors such that 2
cb)cb(a , then the angle
between a and c is ________________________________ 20. Vertices of a triangle are (1, 2, 4) (3, 1, -2) and (4, 3, 1) then its area is_______________ 21. A unit vector coplanar with k2ji and kj2i and perpendicular to kji is
_______________________
LEVEL−III 1. If c,b,a are coplanar vectors and a is not parallel to b then bbacaababc is
equal to (A) cbaba (B) cbaba (C) cbaba (D) none of these 2. The projection of kji on the line whose equation is r = (3 + ) i + (2 -1) j + 3 k ,
being the scalar parameter is;
(A) 141 (B) 6
(C) 146 (D) none of these
3. If q,p are two non-collinear and non-zero vectors such that (b –c) qp +(c –a)p + (a –b)q= 0
where a, b, c are the lengths of the sides of a triangle, then the triangle is (A) right angled (B) obtuse angled (C) equilateral (D) isosceles L−I 1. B 2. A 3. A 4. D 5. C 6. B 7. A 8. 9. C 10. D 11. B 12. B 13. C 14. A 15. B 16. A 17. A 18. A 19. C 20. C 21. B 22. C 23. D 24. A 25. B 26. C 27. B 28. A 29. A 30. A 31. D 32. C 33. C 34. C 35. O 36. O 37. B 38. A L−II 1. A 2. B 3. B 4. C 5. A 6. A 7. A 8. B 9. D 10. C 11. C 12. D 13. D 14. B 15. A 16. D 17. A 18. K 19. / 3 20. 5 5 / 2
21. − J K2
ON J K2
L−III 1. 2. C 3. C
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