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01.01. FUNCTIONS FUNCTIONS

Some questions (Assertion–Reason type) are given below. Each question contains Statement – 1 (Assertion) and Statement – 2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE is correct. So select the correct choice :Choices are :

(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1.

(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for Statement – 1.

(C) Statement – 1 is True, Statement – 2 is False.(D) Statement – 1 is False, Statement – 2 is True.

1. Let f(x) = cos3px + sin . Statement – 1 : f(x) is not a periodic function.Statement – 2 : L.C.M. of rational and irrational does not exist

2. Statement – 1: If f(x) = ax + b and the equation f(x) = f –1(x) is satisfied by every real value of x, then aÎR and b = –1. Statement – 2: If f(x) = ax + b and the equation f(x) = f –1(x) is satisfied by every real value of x, then a = –1 and bÎR.

3. Statements-1: If f(x) = x and F(x) = , then F(x) = f(x) always

Statements-2: At x = 0, F(x) is not defined.

4. Statement–1 : If f(x) = x ¹ 0, 1, then the graph of the function y = f (f(f(x)), x > 1 is a

straight line Statement–2 : f(f(x)))) = x

5. Let f(1 + x) = f(1 – x) and f(4 + x) = f(4– x) Statement–1 : f(x) is periodic with period 6 Statement–2 : 6 is not necessarily fundamental period of f(x)

6. Statement–1 : Period of the function f(x) = does not exist Statement–2 : LCM of rational and irrational does not exist

7. Statement–1 : Domain of f(x) = is (–¥, 0)

Statement–2 : | x | – x > 0 for x Î R–

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8. Statement–1 : Range of f(x) = is [0, 2] Statement–2 : f(x) is increasing for 0 £ x £ 2 and decreasing for – 2 £ x £ 0.

9. Let a, b Î R, a ¹ b and let f(x) = .

Statement–1 : f is a one–one function. Statement–2 : Range of f is R – {1}

10. Statement–1 : sin x + cos (px) is a non–periodic function. Statement–2 : Least common multiple of the periods of sin x and cos (px) is an irrational number.

11. Statement–1: The graph of f(x) is symmetrical about the line x = 1, then, f(1 + x) = f(1 – x). Statement–2 : even functions are symmetric about the y-axis.

12. Statement–1 : Period of f(x) = sin is 2(n)!

Statement–2 : period of |cos x| + |sin x| + 3 is p.

13. Statement–1 : Number of solutions of tan(|tan–1x|) = cos|x| equals 2 Statement–2 : ?

14. Statement–1 : Graph of an even function is symmetrical about y–axis Statement–2 : If f(x) = cosx has x (+)ve solution then total number of solution of the above equation is 2n. (when f(x) is continuous even function).

15. If f is a polynomial function satisfying 2 + f(x).f(y) = f(x) + f(y) + f(xy) x, yÎRStatement-1: f(2) = 5 which implies f(5) = 26 Statement-2: If f(x) is a polynomial of degree 'n' satisfying f(x) + f(1/x) = f(x). f(1/x), then f(x) = 1 xn + 1

16. Statement-1: The range of the function sin-1 + cos-1x + tan-1x is [p/4, 3p/4] Statement-2: sin-1x, cos-1x are defined for |x| £ 1 and tan-1x is defined for all 'x'.

17. A function f(x) is defined as f(x) =

Statement-1 : f(x) is discontinuous at xll xÎR Statement-2 : In the neighbourhood of any rational number there are irrational numbers and in the vincity of any irrational number there are rational numbers.

18. Let f(x) = sin Statement-1 : f(x) is a periodic function Statement-2: LCM of two irrational numbers of two similar kind exists.

19. Statements-1: The domain of the function f(x) = cos-1x + tan-1x + sin-1x is [-1, 1]Statements-2: sin-1x, cos-1x are defined for |x| £ 1and tan-1x is defined for all x.

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20. Statement-1 : The period of f(x) = = sin2x cos [2x] – cos2x sin [2x] is 1/2 Statements-2: The period of x – [x] is 1, where [] denotes greatest integer function.

21. Statements-1: If the function f : R R be such that f(x) = x – [x], where [] denotes the greatest integer less than or equal to x, then f-1(x) is equals to [x] + x Statements-2: Function ‘f ’ is invertible iff is one-one and onto.

22. Statements-1 : Period of f(x) = sin 4p {x} + tan p [x] were, [] & {} denote we G.I.F. & fractional part respectively is 1. Statements-2: A function f(x) is said to be periodic if there exist a positive number T independent of x such that f(T + x) = f(x). The smallest such positive value of T is called the period or fundamental period.

23. Statements-1: f(x) = is one-one function

Statements-2: is monotonically decreasing function and every decreasing function is

one-one.

24. Statements-1: f(x) = sin2x (|sinx| - |cosx|) is periodic with fundamental period p/2 Statements-2: When two or more than two functions are given in subtraction or multiplication form we take the L.C.M. of fundamental periods of all the functions to find the period.

25. Statements-1: ex = lnx has one solution. Statements-2: If f(x) = x f(x) = f1(x) have a solution on y = x.

26. Statements-1: F(x) = x + sinx. G(x) = -x H(x) = F(X) + G(x), is a periodic function. Statements-2: If F(x) is a non-periodic function & g(x) is a non-periodic function then h(x) = f(x) g(x) will be a periodic function.

27. Statements-1: is an odd function.

Statements-2: If y = f(x) is an odd function and x = 0 lies in the domain of f(x) then f(0) = 0

28. Statements-1: is one to one and non-monotonic function.

Statements-2: Every one to one function is monotonic.

29. Statement–1 : Let f : [1, 2] [5, 6] [1, 2] [5, 6] defined as then

the equation f(x) = f1(x) has two solutions. Statements-2: f(x) = f1(x) has solutions only on y = x line.

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30. Statements-1: The function (ps qr ¹ 0) cannot attain the value p/r.

Statements-2: The domain of the function g(y) = is all real except a/c.

31. Statements-1: The period of f(x) = sin [2] xcos [2x] – cos2x sin [2x] is 1/2 Statements-2: The period of x – [x] is 1.

32. Statements-1: If f is even function, g is odd function then (g ¹ 0) is an odd function.

Statements-2: If f(–x) = –f(x) for every x of its domain, then f(x) is called an odd function and if f(–x) = f(x) for every x of its domain, then f(x) is called an even function.

33. Statements-1: f : A B and g : B C are two function then (gof)–1 = f–1 og–1. Statements-2: f : A B and g : B C are bijections then f–1 & g–1 are also bijections.

34. Statements-1: The domain of the function is (4n + 1) , n Î N.

Statements-2: Expression under even root should be 0

35. Statements-1: The function f : R R given a > 0, a ¹ 1 is invertible.Statements-2: f is many one into.

36. Statements-1: (x) = sin (cos x) is a one-one function.

Statements-2:

37. Statements-1: For the equation kx2 + (2 k)x + 1 = 0 k Î R {0} exactly one root lie in (0, 1).

Statements-2: If f(k1) f(k2) < 0 (f(x) is a polynomial) then exactly one root of f(x) = 0 lie in (k1, k2).

38. Statements-1: Domain of

Statements-2: when x > 0 and when x < 0.

39. Statements-1: Range of f(x) = |x|(|x| + 2) + 3 is [3, ¥)Statements-2: If a function f(x) is defined x Î R and for x 0 if a £ f(x) £ b and f(x) is even function than range of f(x) f(x) is [a, b].

40. Statements-1: Period of {x} = 1.Statements-2: Period of [x] = 1

41. Statements-1: Domain of f = . If f(x) =

Statements-2: [x] £ x xÎ R

42. Statements-1: The domain of the function sin–1x + cos–1x + tan–1x is [–1, 1]Statements-2: sin–1x, cos–1x are defined for |x| £ 1 and tan–1x is defined for all ‘x’

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02.02. LIMITS, CONTINUITY & DIFFERENTIABILITYLIMITS, CONTINUITY & DIFFERENTIABILITY

43. Statements-1: The set of all points where the function f(x) = is differentiable

is (–¥, ¥).

Statements-2: Lf(0) = 1, Rf(0) = 0 and f(x) = , which exists x ¹ 0.

44. Statements-1: f(x) = then f(x) is differentiable at x = 1

Statements-2: A function y = f(x) is said to have a derivative if

45. Consider the function f(x) = (|x| – |x – 1|)2 Statement – 1: f(x) is continuous everywhere but not differentiable at x = 0 and 1. Statement – 2: f (0–) = 0, f (0+) = –4, f (1–) = 4, f (1+) = 0.

46. Statement – 1: does not exist

Statement – 2: L.H.L. = 1 and R.H.L. = –1

47. Statement–1 : cos–1 (cos2x) does not exist Statement–2 : cosec–1x is well defined for |x| 1.

48. Let f : [0, 2] [0, 2] be a continuous function Statement–1 : f(x) = x for at least one 0 £ x £ 2 Statement–2 : f(x) = – x for at least one 0 £x £ 2

49. Let h(x) = f(x) + g(x) and f(a), g(a) are finite and definite Statement–1 : h(x) is continuous at x = 9 and hence h(x) = x2 + 1 cosx| is continuous at x = 0 Statement–2 : h(x) is differentiable at x = a and hence h(x) = x2 + |cosx| is differentiable at x = 0

50. Statement–1 : f(x) = e|x| is non differentiable at x = 0. Statement–2 : Left hand derivative of f(x) is – 1 and right hand derivative of f(x) is 1.

51. Statement–2 : , where [x] = G.I.F

Statement–2 : as x 0, cos x lies between 0 and 1.

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52. Statement–1 : does not exist.

Statement–2 : sec–1 t is defined for those t, whose modulus value is more than or equal to 1.

53. Suppose [] and {} denotes the greatest integer function and fractional part function respectively. Let f(x) = {x} + . Statement–1 : f is not differentiable at integrable points. Statement–2 : f is not continuous at integral points.

54. Statement–1 : .

Statement–2 : .

55. Statement–1 : The number of points of discontinuity of f(x) is all 0. Where f(x) = .

Statement–2 : The function h(x) = max {- x, 1, x2} b x Î R, is not differnetiable at two values of x.

56. Statement–1 : If p, q, r all are positive , then is es/q

Statement–2 : = e.

57. Statement–1 : For f(x) = ||x2| – 4|x||, the number of points of non differentiability is 3. Statement–2 : A continuous function is always differentiable

58. Statement–1 : If f(x) = x (1 – logx) then for 0 < a < c < b (a – b logc = b (1 – log b) – a (1 – loga) Statement–2 : If f(x) is diff. (a , b) and cont. in [a, b] then for at least one a < c < b f(c) =

59. Statement – 1 : Let {x} denotes the fractional part of x. Then

Statement – 2 :

60. Statement – 1 : = 1 - cost

Statement – 2 : sinx is continuous in any closed interval [0, t]

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61. Statement – 1 : where [] G.I.F.

Statement – 2 : = 1

62. Statement – 1 : The function f(x) = is continuous at a point x = a ¹4.

Statement – 2 : For x = a, f(x) has a definite value and as x a, f(x) has a limit which is also equal to its definite value of x = a ¹ 4.

63. Statements-1: x sin

Statements-2: y sin

64. Statements-1: f(x) = (sinx)2n , then the set of points of discontinuities of f is {(2n + 1) p/2, nÎI} Statements-2: Since -1 < sinx < 1, as n ¥, (sinx)2n 0, sinx = 1 (1)2n 1, n ¥.

65. Statements-1: f(x) = (cosx)2n, then f is continuous everywhere in (-¥, ¥) Statements-2: f(x) = cosx is continuous everywhere i.e., in (-¥, ¥)

66. Statements-1: For the graph of the function y = f(x) the valid statement is

f(x) is differentiable at x = 0 Statements-2: Lf (c) = R f (c), we say that f (c) exists and Lf (c) = Rf (c) = f (c).

67. Statements-1:

Statements-2: f(g(x)) = f(L) where . Also function ‘f’ must be continuous at L.

68. Statements-1: f(x) = max (1, x2, x3) is differentiable xÎR except x = -1, 1 Statements-2: Every continuous function is differentiable

69. Statements-1:

Statements-2: Since sinx has a range of [-1, 1] xÎR

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70. Statements-1: f(x) = , is a continuous function at x = 0

Statements-2: If left hand limit = right hand limit & both the limits exists finitely then function can be made continuous.

71. Statements-1: f(x) = x|x| is differentiable at every point in its domain. Statements-2: If f(x) is as a derivative at every point & g(x) has a derivative at every point in their domains, then h(x) = f(x).g(x) is differentiable at every point in its domain.

72. Statements-1: x = cosx for some x Î (0, p/2) Statements-2: If f(x) is a continuous in an interval I and f(a) & f(b) are two values at a & b & c is any value in between f(a) & f(b), then there is some x in (a, b) where f(x) = c.

73. Statements-1: f : R R and f(x) = ex ex the range of f(x) is R Statements-2: If f(x) is a continuous function in [a, b] then f(x) will take all values in between f(a) and f(b).

74. Statements-1: If a < b < c < d then (x a) (x c) (x b) (x d) = 0 will have real for all Î R.

Statements-2: If f(x) is a function f(x1) f(x2) < 0 then f(x) = 0, for at least one x Î (x1, x2).

75. Statements-1:

Statements-2: then for every positive number G arbitrarily assign (however

large) there exist a > 0 such that for all x Î (a , a) (a, a + ) f(x) a > 0.

76. Statements-1: The maximum and the minimum values of the function

exists.

Statements-2: If domain of a continuous function is in closed interval then its range is also in a closed interval.

77. Statements-1: For any function y = f(x)

Statements-2: If f(x) is a continuous function at x = a then

78. Statements-1:

Statements-2: If y = f(x) is continuous in (a, b) then

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79. Statements-1: If f is finitely derivable at c, then f is continuous at c. Statements-2: If at x = c both LHD and RHD exist finitely but LHD ¹ RHD then f(x) is continuous at x = c.

80. Statements-1: If f(x + y) = f(x) + f(y), then f is either differentiable everywhere or not differentiable everywhereStatements-2: Any function is either differentiable everywhere or not differentiable everywhere.

81. Statements-1: The function f(x) = |x3| is differentiable at x = 0 Statements-2: At x = 0 f(x) = 0

82. Statements-1: : When |x| < 1

Statements-2: For –1 < x < 1, as n ¥,x2n 0.

83. Statements-1: : f(x) = is discontinuous for integral values of x. where [.] denotes

greatest integer function. Statements-2: For integral values of x, f(x) is undefined.

84. Statements-1: : f(x) = xn sin is differentiable for all real values of x (n 2)

Statements-2: for n 2 right hand derivative = Left hand derivative (for all real values of x).

85. Statements-1: The function is discontinuous at x = 0.

Statements-2: f(0) = 0.

86. Statements-1: The function f(x) defined by is differentiable at

x = 2.Statements-2: L.H.D. at x = 2 = R.H.D. at x = 2

87. Statements-1: [.] denotes greatest integer function.

Statements-2: [.] denotes greatest integer function.

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88. Statements-1:

Statements-2: f (1) = 2 f (1+) = 3, f (1) = 5 f (2+) = 6.

89. Statements-1: does not exist

Statements-2: Right hand limit as x 0 does not exist

90. Statements-1:

Statements-2: since (1 + x)1/x = e

91. Statements-1: sinx = 0 has atleast one roots between (– p/2, p/2) Statements-2: Since sinx is continuous in [-p/2, p/2] and sin (-p/2) = -1, sin (p/2 = 1 i.e. sinx has opposite sign is at x = -p/2, x = p/2, by intermediate theorem

92. Statements-1: Let f(x) = = 0, x = 0 then f(x) has a jump discontinuity at

x = 0. Statements-2: Since f(x) = 1

and f(x) = 1

93. Statements-1: The set of all points where the function

f(x) = is differentiable (-¥, ¥) – {0}

Statements-2: Lf(0) = 1, Rf(0) = 0 is

f(x) = . which exists x¹0

94. Statements-1: f(x) = , where [] denotes greatest integer function, then f(x) is

differentiable at x = 1

Statements-2: L f (1)

=

f(1) does not exist.

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03.03. APPLICATION OF DERIVATIVESAPPLICATION OF DERIVATIVES95. Statements-1: For the circle (x – 1)2 + (y – 1)2 = 1, the tangent at the point (1, 0) is the x-axis.

Statements-2: the derivative of a single valued function y = f(x) at x = a is the slope of the tangent drawn to the curve at x = a.

96. Statements-1: Both sin x, and cos x are decreasing functions in [ Good ]

Statements-2: If a differentiable function decreases is an interval (a, b) then its derivative also decreases in (a, b).

97. Statements-1: [ Good ]

Statements-2: The function has a local maximum at x = e

98. Statements-1: Conditions of LMVT fail in f(x) = |x – 1| (x – 1)Statements-2: |x – 1| is not differentiable at x = 1

99. Let f(x) =

Statement–1 : Minimum value of f(x) occurs at x =

Statement–2 : Minimum of f(x) = ax2 + bx + c (a > 0) occurs at x = –b/2a.

100. Statement–1 : ab > ba, for 2.91 < a < b

Statement–2 : f(x) = is a decreasing function for x > e.

101. Statement–1 : Total number of critical points of f(x) = max. {1/2, sinx, cox} – p £ x £ p are 5 Statement–2 : Total number of critical points of f(x) = max {1/2, x, cosx} – p £ x £ p are 2

102. Let f(x) = 5p2 + 4(x – 1) – x2, xÎR and p is a real constant Statement–1 : If the maximum values of f(x) is 20, then p = –2. Statement–2 : If the maximum value of f(x) is 20, then p = 2.

103. Let f(x) = sin–1 x + cos–1 x + tan–1x and x Î [– 1, 1]

Statements-1: Range of f(x) is .

Statements-2: f(x) is an increasing function.

104. Let f(x) = x3 Statements-1: x = 0, in the point of inflexion for f(x) Statements-2: f (x) < 0 for x < 0 and f (x) > 0 for x > 0.

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105. Suppose f(x) = +

Statements-1: f is an increasing function. Statements-2: derivative of f(x) with respect to x is always greater than zero.

106. Let 0 < x £ and f(x) =

Statements-1: The minimum value of f is , attained at x = .

Statements-2: 0 < sin x < x, x Î .

107. Statements-1: The equation x2 = x sin x + cos x has only one solution. Statements-2: The derivative of the function x2 – x sin x – cos x is x(2- cos x).

108. Statement–1 : Angle of intersects in between y = x2 and 6y = 7 – x3 at (1, 1) is p/4 Statement–2 : Angle of intersection between any two curve is angle between the tangents at the point of intersection.

109. Statement – 1 : The curve y = x1/3 has a point of inflection at x = 0 Statement – 2 : A point where y fails to exist can be a point of inflection

110. Let f(x) and g(x) are two positive and increasing functionStatement – 1 : If (f(x)) g(x) is decreasing then f(x) < 1 Statement – 2 : If f(x) is decreasing then f(x) < 0 and increasing then f(x) > 0 for all x.

111. Statement – 1 : If f(0) = 0, f(x) = ln (x + ), then f(x) is positive for all xÎR0 Statements-2: f(x) is increasing for x > 0 and decreasing for x < 0.

112. Statements-1: The two curves y2 = 4x and x2 + y2 – 6x + 1 = 0 at the point (1, 2) intersect orthogonally. Statements-2: Two curves y = f(x) & y=g(x) intersect orthogonally at (x1 y1) if (f (x1).g((x1)) = 1.

113. Statements-1: If 27a + 9b + 3c + d = 0, then the equation 4ax3 + 3bx2 + 2cx + d = 0 has atleast one real root lying between (0, 3) Statements-2: If f(x) is continuous in [a, b], derivable in (a, b), then at least one point cÎ (a, b) such that f(c) = 0.

114. Statements-1: f(x) = {x} has local minima at x = 1. Statements-2: x = a will be local minima for y = f(x) provided also

f(x) > f(a).

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115. Statements-1: f(x) =

= . Mean value theorem is applicable in the interval [0, 1].

Statements-2: For application of mean value theorem, f(x) must be continuous in [0, 1] and differentiable in (0, 1).

116. Statements-1: For some 0 < x1 < x2 < p/2, tan-1x2 – tan-1x1 < x2 – x1 Statements-2: If f(x) >f(x1) x2 > x1 function is always increasing

117. Statements-1: The graph of a continuous function y = f(x) has a cusp at point x = c if f (x) has same sign on both sides of c. Statements-2: The concavity at any point x = c depends upon f (x). If f (x) < 0 or f (x) > 0 the function is either concave up or concave down.

118. Statements-1: If f be a function defined for all x such that |f(x) f(y)| < (x y)2 then f is constantStatements-2: If a(x) < b(x) < (x) for all x and

119. Statements-1: f : R R be a function such that f(x) = x3 + x2 + 3x + sinx. Then f is one-one. Statements-2: f(x) is neither increasing nor decreasing.

120. Statements-1: If a & b are any two roots of equation ex cosx = 1, then the equation ex sinx – 1 = 0 has at least one root in (a, b)

Statements-2: f is continuous in [a, b]. f is derivable in (a, b). f(a) = f(b) then these exists x Î (ab)such that f(x) = 0

121. Statements-1: The minimum value of the expression x2 + 2bx + c is c – b2. Statements-2: The first order derivative of the expression at x = –b is zero and second derivative is always positive.

122. Statements-1: Let (x) = sin (cosx) in then (x) is decreasing in

Statements-2: (x) £ 0 xÎ

123. Statements-1: The function f(x) = x4 8x3 + 22x2 24x + 21 is decreasing for every x Î (2, 3) (¥, 1)Statements-2: f (x) > 0 for the given values of x.

124. Statements-1: For the function f(x) = xx, x = 1/e is a point of local minimum.Statements-2: f (x) changes its sign from ve to positive in neighbourhood of x = 1/e.

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125. Statements-1: Consider the function f(x) = (x3 – 6x2 + 12x – 8) ex is neither maximum nor minimum let x = 2 Statements-2: f(x) = 0, f(x) = 0, f(x) ¹ 0 at x = 2

126. Statements-1: Consider the function f(x)

Statements-2: f(x) > 0, f(x) > 0 where x1 < x2

127. Consider the following function with regard to the functionf(x) = (x3 – 6x2 + 12x – 8) ex

Statement-1: f(x) is neither maximum nor minimum at x = 2Statement-2: f (x) = 0, at x = 2.

128. Statements-1: Equation f(x) = x3 + 9x2 + 2ax + a2 + a + 1 = 0 has at least one real negative root.Statements-2: Every equation of odd degree has at least one real root whose sign is opposite to that of its constant term.

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04. INDEFINITE & DEFINITE INGEGRATION04. INDEFINITE & DEFINITE INGEGRATION

129. Let F(x) be an indefinite integral of cos2x. Statement-1: The function F(x) satisfies F(x + p) = F(x) real x Statement-2: cos2(x + p) = cos2x.

130. Statement-1: ò|x| dx can not be found while can be found.

Statement-2: |x| is not differentiable at x = 0.

131. Statement-1: dx = tan–1 (x2) + C

Statement-2: = tan–1x + C

132. Statement-1: If y is a function of x such that y(x – y)2 = x then

Statement-2: = log (x – 3y) + c

133. Statement–1 : f(x) = logsecx –

Statement–2 : f(x) is periodic

134. Statement–1 :

Statement–2 :

135. Statement–1 : ; where [x] = G.I.F.

Statement–2 : [tan–1 x] = 0 for 0 < x < tan 1 and [tan–1 x] = 1 for tan 1 £ x < 10.

136. Statement–1 :

Statement–2 :

.

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137. Statement–1 :

Statement–2 : .

138. Statement–1 :

Statement–2 : .

139. Statement–1 : If f(x) satisfies the conditions of Rolle's theorem in [a, b], then

Statement–2 : If f(x) satisfies the conditions of Rolle's theorem in [a, b], then

140. Statement–1 : , where [] denotes G.I.F. equals 8p.

Statement–2 : If f(x) = |sinx| + |cosx|, then 1 £ f(x) £ .

141. Let f(x) be a continuous function such that nÎI

Statement–1 :

Statement–2 :

142. Let In = dx, n Î N

Statement–I : I1. I2, I3 . . . is an increasing sequence. Statement–II : x is an increasing function.

143. Let f be a periodic function of period 2. Let g(x) = dt and h(x) = g(x + 2) – g(x).

Statement–1 : h is a periodic function. Statement–2 : g(x + 2) – g(x) = g(2).

144. Statement–1 :

Statement–2 : .

Bansal Classes []

16

Bansal Classes] []

145. Statement–1 : If I1 = and then I1 = I2.

Statement–2 :

146. Statement–1 : 8 < .

Statement–2 : If m is the smallest and M is the greatest vlaue of a function f(x) in an interval

(a, b), then the vlaue of the integral is such that for a < b, we have M(b – a) £

.

147. Statement–1 : (asinbx – bcosbx)+c

Then A is

Statement–2 : = ex tanx + c

148. Statement–1 : is equal to

Statement–2 : is 2/11 ln |x + | + c

149. Statement–1 : is p/12

Statement–2 : dx

150. Statement–1 : If f satisfies f(x + y) = f(x) + f(y) x , y ÎR then = 0

Statement–2 : If f is an odd function then = 0

151. Statement–1 : If f(x) is an odd function of x then is an even function of (n)

Statement–2 : If graph of y = f(x) is symmetric about y–axis then f(x) is always an even function.

Bansal Classes []

17

Bansal Classes] []

152. Statement–1 : Area bounded by y = {x}, {x} is fractional part of x = 0, x = 2 and x–axis is 1. Statement–2 : Area bounded by y = |sinx|, x = 0, x = 2p is 2 sq. unit.

153. Statement-1:

Statement-2: , symbols have their usual meaning.

154. Statement-1: If In = òtann x dx, then 5 (I4 + I6) = tan5x .

Statement-2: If In = ò tan4x dx, then - In-2 = In, nÎN.

155. Statement-1: If a > 0 and b2 – 4ac < 0, then the value of the integral will be of

the type tan-1 , where A, B, C, are constants. Statement-2: If a > 0, b2 – 4ac < 0 then ax2 + bx + c can be written as sum of two squares.

156. Statements-1:

Statements-2: = ex f(x) + c

157. Statements-1:

= log |tan-1 (x + 2/x)| + c

Statements-2:

158. Statements-1:

Statements-2: òex (f(x) + f(x)) dx = ex f(x) + c.

159. Statements-1:

Statements-2: For integration by parts we have to follow ILATE rule.

160. Statements-1: A function F(x) is an antiderivative of a function f(x) if F (x) = f(x) Statements-2: The functions x2 + 1, x2 p, x2 + are all antiderivatives of the function 2x.

Bansal Classes []

18

Bansal Classes] []

161. Statements-1: dx = – , a < b

Statements-2: If f(x) is a function continuous every where in the interval (a, b) except x = c

then

162. Statements-1:

Statements-2: m and M be the least and the maximum value of a continuous function

y = f(x) in [a, b] then

163. Statements-1:

Statements-2: if f(x) £ g(x) £ h(x) in (a, b) then

164. Statements-1:

Statements-2: For any functions f(x) and g(x), integrable on the interval (a,b), then

165. Statements-1:

Statements-2: If F(x) is antiderivative of a continuous function (a, b) then

166. Statements-1: can be integrated by substitution it sinx = t.

Statements-2: All integrands are integrated by the method of substitution only.

167. Statement-1 : =

Statement-2 : òex (f(x) + f (x)dx = ex f(x) + c

168. Statements-1:

Statements-2: equals .

Bansal Classes []

19

Bansal Classes] []

169. Statements-1:

Statements-2:

170. Statements-1:

Statements-2:

171. Statements-1:

Statements-2:

172. Statements-1: The value of can not exceed

Statements-2: If m £ f(x) £ M x Î [a, b] then

173. Statements-1:

Statements-2: Area bounded by y = 3x and y = x2 is sq. units

174. Statements-1: dx = log|10 x + x10| + c

Statements-2:

175. Statements-1: = tan (xex) + c

Statements-2:

176. Statement-1 : f(x) = then f(x) = -

Statements-2: f(x) = , then f(x) + (ln x)2

Bansal Classes []

20

Bansal Classes] []

177. Statement-1 : .

Statements-2: Since is an odd function. So, that .

178. Statements-1 : = (2n + 1) – COSt (0 £ t £ p)

Statements-2:

and if f(a + x) = f(x)

179. Statements-1: The value of the integral belongs to [0, 1]

Statements-2: If m & M are the lower bound and the upper bounds of f(x) over [a, b] and f is

integrable, then m (b a) £ £ M(b – a).

180. Statements-1: = cot1, where [] denotes greatest integer function.

Statements-2: is defined only if f(x) is continuous in (a, b) [] function is

discontinuous at all integers

181. Statements-1: = 0

Statements-2: if f(x) is an odd function.

182. Statements-1: All continuous functions are integrable Statements-2: If a function y = f(x) is continuous on an interval [a,b] then its definite integral over [a, b] exists.

183. Statements-1: If f(x) is continuous on [a, b], a ¹ b and if , then f(x) = 0 at least

once in [a, b] Statements-2: If f is continuous on [a, b], then at some point c in [a, b]

f(c) =

Bansal Classes []

21

Bansal Classes] []

184. Statements-1:

Statements-2: where CÎ (A, B)

185. Statements-1:

Statements-2: If f is an odd function

186. Statement-1 If then

Statement-2 : = kn ekx and

187. Statement-1 :

Statements-2:

188. Statements-1:

Statements-2: where a < c < b.

189. Statements-1:

Statements-2:

190. Statements-1:

Statements-2:

191. Statements-1:

Statements-2:

Bansal Classes []

22

Bansal Classes] []

05. 05. STRAIGHT LINES STRAIGHT LINES

192. Let the equation of the line ax + by + c = 0Statement-1: a, b, c are in A.P.which force ax + by + c = 0 to pass through a fixed point (1, -2) Statement-2: Any family of lines always pass through a fixed point

193. Statement-1: The area of the triangle formed by the points A(1000, 1002), B(1001, 1004) C(1002, 1003) is same as the area formed by A (0, 0), B (1, 2), C (2, 1)Statement-2: The area of the triangle is constant with respect to translation of coordinate axes.

194. Statement-1: The lines (a + b)x + (a – 2b) y = a are concurrent at the point .

Statement-2: : The lines x + y – 1 = 0 and x – 2y = 0 intersect at the point .

195. Statement-1: Each point on the line y – x + 12 = 0 is equidistant from the lines 4y + 3x – 12 = 0, 3y + 4x – 24 = 0.

Statement-2: : The locus of a point which is equidistant from two given lines is the angular bisector of the two lines.

196. Statement-1: If A(2a, 4a) and B(2a, 6a) are two vertices of a equilateral triangle ABC and the vertex C is given by . Statement-2: : An equilateral triangle all the coordinates of three vertices can be rational

197. Statement-1: If the Point (2a – 5, a2) is on the same side of the line x + y – 3 = 0 as that of the origin, then the set of values of aÎ (2, 4) Statement-2: : The points (x1, y1) and (x2 , y2) lies on the same or opposite side of the line ax+by+c=0, as ax1 + by1 + c and ax2 + by2 + c have the same or opposite signs.

198. Statement-1: If a, b, c are in A.P. then every line of the form of ax + by + c = 0 where a, b, c are arbitrary constants pass through the point (1,-2) Statement-2: : Every line of the form of ax + by + c = 0 where a, b, c are arbitrary constants pass through a fixed point if their exist a linear relation between a, b & c.

199. Statement-1: If the vertices of a triangle are having rational co-ordinate then its centroid, circumcenter & orthocenter are rational Statement-2: : In any triangle, orthocenter, centroid and circum center are collinear and centroid divides the line joining orthocenter and circumcenter in the ratio 2 : 1.

200. Statement-1: If line y = , makes an angle with positive direction of x-axis, then

tan = -1/3, cos =

Statement-2: : The parametric equation of line passing through (x1, y1) is given by

where r is parameter & Î [0, p)

Bansal Classes []

23

Bansal Classes] []

201. Statement-1: In ABC, A(1, 2) is vertex & line x – y – 5 = 0 is equation of bisector of ABC, then (7 , – 4) is a point lying on base BC. Statement-2: : Bisector between two lines is locus of points equi-distant from both the lines.

202. Statement-1: Area of the triangle formed by 4x + y + 1 = 0 with the co-ordinate axes is

sq. units.

Statement-2: : Area of the triangle made by the line ax + by + c = 0 with the co-ordinate axes

is .

203. Statement-1: If (a1x + b1y + c1) + (a2x + b2y + c2) + (a3x + b3y + c3) = 0 then lines a1x + b1y +c1= 0, a2x + b2y + c2 = 0 and a3x + b3y + c3 = 0 cannot be parallel Statement-2: : If sum of three straight lines equations is identically zero then they are either concurrent or parallel.

204. Statement-1: The three non-parallel lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0, a3x + b3y + c3 =

0 are concurrent if = 0

Statement-2: : The area of the triangle formed by three concurrent lines must be zero.

205. Statement-1: The point (a, a2) lies inside the formed by the lines 2x + 3y 1 = 0,

x + 2y 3 = 0, and 5x 6y 1 = 0 for every aÎ

Statement-2: : Two points (x1, y1) and (x2, y2) lie on the same side of straight line ax + by + c = 0 if ax1 + by1 + c & ax2 + by2 + c are of opposite sign.

206. Statement-1: The equation of the straight line which passes through the point (2, 3) and the point of the intersection of the lines x + y + 4 = 0 and 3x y 8 = 0 is 2x y 7 = 0Statement-2: : Product of slopes of two perpendicular straight lines is 1.

207. Statement-1: The incentre of a triangle formed by the lines

a. ; is (0, 0).

Statement-2: : The point (0, 0) is equidistant from the lines

and

208. Statement-1: The combined equation of lines L1 & L2 is 2x2 + 6xy + y2 = 0 and that of L3 & L4

is 4x2 + 18xy + y2 = 0. If the angle between L1 & L4 is a then angle between L2 & L3 is also a. Statement-2: : If the pair of lines L1L2 = 0 & L3L4 = 0 are equally inclined lines then angle between L1 & L2 = angle between L2 and L3.

Bansal Classes []

24

Bansal Classes] []

06.06. AREA UNDER THE CURVES AREA UNDER THE CURVES

209. Let |A1| be the area bounded between the curves y = |x| and y = 1 – |x| ; |A2| be the area bounded between the curves y = –|x| and y = |x| – 1. Statement-1: |A1| = |A2| Statement-2: Area of two similar parallelograms are equal.

210. Statement-1: Area bounded between the curves y = |x – 3p| and y = cos–1 (cosx) is p2/2 Statement-2: |x – 3p| = 3p – x for 5p/2 £ x £ 3p cos–1 (cosx) = x – 2p, 2p £ x £3p

211. Statement-1: Area of the ellipse in the first quadrant is equal to p

Statement-2: Area of the ellipse is pab.

212. Statement-1: Area enclosed by the curve | x | + | y | = 2 is 8 units Statement-2: represents an square of side length unit.

213. Statement-1: The area bounded by y = x(x – 1)2, the y–axis and the line y = 2 is

(x (x – 2)2 – 2) dx is equal to .

Statement-2: The curve y = x(x – 1)2 is intersected by y = 2 at x = 2 only and for 0 < x < 2, the curve y = x(x – 1)2 lies below the line y = 2.

214. Let f be a non–zero odd function and a > 0.

Statement-1: . Because

Statement-2: Area bounded by y = f(x), x = a, x = – a and x–axis is zero.

215. Statement-1: The area of the curve y = sin2 x from 0 to p will be more than that of the curve y = sin x from 0 to p. Statement-2: x2 > x if x > 1.

216. Statement-1: The area bounded by the curves y =x2 – 3 and y = kx + 2 is least if k = 0. Statement-2: The area bounded by the curves y = x2 – 3 and y = kx + 2 is .

217. Statement-1: The area of the ellipse 2x2 + 3y2 = 6 will be more than the area of the circle x2 + y2 – 2x + 4y + 4 = 0. Statement-2: The length of the semi-major axis of ellipse 2x2 + 3y2 = 6 is more than the radius of the circle x2 + y2 – 2x + 4y + 4 = 0.

Bansal Classes []

25

Bansal Classes] []

218. Statement-1: Area included between the parabolas y = x2/4a and the curve

y = is sq. units.

Statement-2: Both the curves are symmetrical about y-axis and required area is

219. Statement-1: The area of the region bounded by y2 = 4x , y = 2x is 1/3 sq. units.

Statement-2: The area of the region bounded by y2 = 4ax, y = mx is sq. units.

220. Statement-1: Area under the curve y = sinx, above ‘x’ axis between two ordinates x = 0 & x = 2p is 4 units.

Statement-2:

221. Statement-1: Area under the curve y = [|sinx| + |cosx|], where [] denotes the greatest integer function. above ‘x’ axis and between the ordinates = 0 & x = p is p units. Statement-2: f(x) = |sinx| + |cosx| is periodic with fundamental period p/2.

222. Statement-1: Area between y = 2 – x2 & y = x is equal to

Statement-2: When a region is determined by curves that intersect, the intersection points give the units of integration.

223. Statement-1: Area of the region bounded by the lines 2y = -x + 8, x-axis and the lines x = 3 and x = 5 is 4 sq. units. Statement-2: Area of the region bounded by the lines x = a, x = b, x-axis and the curve y =

f(x) is .

224. Statement-1: The area of the region included between the parabola and the line

3x 2y + 12 = 0 is 27 sq. units.

Statement-2: The area bounded by the curve y = f(x) the x-axis and x = a, x = b is

where f is a continuous function defined on [a, b].

225. Statement-1: The area of the region sq. units.

Statement-2: The area bounded by the curves y = f(x), x-axis ordinates x = a, x = b is

Bansal Classes []

26

Bansal Classes] []

226. Statement-1: Area bounded by y2 = 4x and its latus rectum = 8/3 Statement-2: Area of the region bounded by y2 = 4ax and it is latus rectum 8a2/3

07. 07. DIFFERENTIAL EQUATION DIFFERENTIAL EQUATION

227. Statement-1: The order of the differential equation whose general solution is y = c1cos2x + cos2sin2x + c3cos2x + c4e2x + c5 is 3 Statement-2: Total number of arbitrary parameters in the given general solution in the statement (1) is 6.

228. Statement-1: Degree of differential equation of parabolas having their axis along x–axis and vertex at (2, 0) is 2. Statement-2: Degree of differential equation of parabola having their axis along x–axis and vertex at (1, 0) is 1.

229. Statement–1 : Solution of the differential equation is xy = .

Statement–2 : Solution of the differential equation is

where P and Q are function of x alone.

230. Let the general solution of a differential equation be y = aebx + c . Statement–1 : Order of the differential equation is 3. Statement–2 : Order of the differential equation is equal to the number of actual constant of the solution

231. Let F be the family of ellipses on the Cartesian plane, whose directrices are x = 2. Statement–1 : The order of the differential equation of the family F is 2. Statement–2 : F is a two parameter family.

232. Consider the differential equation (x2 + 1). .

Statement–1 : For any member of this family y ¥ as x ¥. Statement–2 : Any solution of this differential equation is a polynomial of odd degree with positive coefficient of maximum power.

233. Statement–1 : The solution of the differential equation x is

y = xecx.

Statement–2 : A solution of the differential equation is y = 2.

234. Statement-1: Order of the differential equation of family of parabola whose axis is perpendicular to y–axis and ratus rectum is fix is 2. Statement-2: Order of first equation is same as actual no. of abitrary constant present in diff. equation.

Bansal Classes []

27

Bansal Classes] []

235. Statement-1: Solution of y dy = x – x as is family of rectangular hyperbola

Statement-2: Solution of y is family of parabola

236. Statement-1: Solution of differential equation dy (x2y – 1) + dx (y2x – 1) = 0 is

Statement-2: Order of differential equation of family of circle touching the coordinate axis is 1.

237. Statement-1: Integrating factor of is ex

Statement-2: Integrating factor of is

238. Statement-1: The differential equation of all circles in a plane must be of order 3. Statement-2: There is only one circle passing through three non-collinear points.

239. Statement-1: The degree of the differential equation + is 3.

Statement-2: The degree of the highest order derivative occuring in the D.E. when the D.E. has been expressed as a polynomial of derivatives.

240. Statement-1: Solution of is - tan (x2 + y2) = c

Statement-2: Since the given differential equation is homogenous can be solved by putting y = vx

241. Statement-1: The order of the differential equation formed by the family of curve y = c1ex + (c2 + c3) is ‘1’. Here c1, c2, c3, c4 are arbitrary constant. Statement-2: The order of the differential equation formed by any family of curve is equal to the number of arbitrary constants present in it.

242. Statement-1: The degree of differential equation is not defined.

Statement-2: The degree of differential equation is the power of highest order derivative when differential equation has been expressed as polynomial of derivatives.

243. Statement-1: The order of differential equation of family of circles passing then origin is 2. Statement-2: The order of differential equation of a family of curve is the number of independent parameters present in the equation of family of curves

Bansal Classes []

28

Bansal Classes] []

244. Statement-1: Integrating factor of is x3

Statement-2: Integrating factor of is eòpdx

245. Statement-1: The differentiable equation y3dy + (x + y2) dx = 0 becomes homogeneous if we put y2 = t. Statement-2: All differential equation of first order and first degree becomes homogeneous if we put y = tx.

246. Statement-1: The general solution of + P(x) y = Q(x) is

Statement-2: Integrating factor of + P(x) y = Q(x) is

247. Statement-1: The general solution of is yex = ex + c

Statement-2: The number of arbitrary constants in the general solution of the differential equation is equal to the order of differential equation.

248. Statement-1: Degree of the differential equation is 2.

Statement-2: In the given equation the power of highest order derivative when expressed as a polynomials in derivatives is 2.

249. Statement-1: The differential equation of the family of curves represented by y = A.ex is given

by .

Statement-2: is valid for every member of the given family.

250. Statement-1: The differential equation can be solved by putting y = vx

Statement-2: Since the given differentiable equation is homogenous

251. Statement-1: A differential equation can be solved by finding. If =

= then solution y.x = òx3dx + c Statement-2: Since the given differential equation in of the form dy/dx + py = wherep, are function of x

252. Statement-1: The differential equation of all circles in a plane must be of order 3.Statement-2: There is only on circle passing through three non collinear points.

Bansal Classes []

29

Bansal Classes] []

08. 08. CIRCLES CIRCLES253. Tangents are drawn from the origin to the circle x2 + y2 - 2hx - 2hy + h2 = 0 (h 0)

Statement 1: Angle between the tangents is p/2Statement 2: The given circle is touching the co-ordinate axes.

254. Consider two circles x2 + y2 – 4x – 6y – 8 = 0 and x2 + y2 – 2x – 3 = 0 Statement 1: Both circles intersect each other at two distinct points Statement 2: Sum of radii of two circles in greater than distance between the centres of two circles

255. C1 is a circle of radius 2 touching x–axis and y–axis. C2 is another circle of radius greater than 2 and touching the axes as well as the circle c1. Statement–1 : Radius of circle c2 = Statement–2 : Centres of both circles always lie on the line y = x.

256. From the point P( ), tangents PA and PB are drawn to the circle x2 + y2 = 4. Statement–1 : Area of the quadrilateral OAPB (obeying origin) is 4. Statement–2 : Tangents PA and PB are perpendicular to each other and therefore quadrilateral OAPB is a square.

257. Statement–1 : Tangents drawn from ends points of the chord x + ay – 6 = 0 of the parabola y2 = 24x meet on the line x + 6 = 0 Statement–2 : Pair of tangents drawn at the end points of the parabola meets on the directrix of the parabola

258. Statement–1 : Number of focal chords of length 6 units that can be drawn on the parabola y2 – 2y – 8x + 17 = 0 is zero Statement–2 : Lotus rectum is the shortest focal chord of the parabola

259. Statement–1 : Centre of the circle having x + y = 3 and x – y = 1 as its normal is (1, 2). Statement–2 : Normals to the circle always passes through its centre.

260. Statement–1 : The number of common tangents to the circle x2 + y2 = 4 and x2 + y2 – 6x –8y – 24 = 0, is one Statement–2 : If C1C2 = , then number of common tangents is three. Where C1C2 = Distance between the centres at both the circle and r1, r2 are the radius of the circle respectively

261. Statement–1 : The circle having equation x2 + y2 –2x + 6y + 5 = 0 intersects both the coordinate axes. Statement–2 : The lengths of x and y intercepts made by the circle having equation x2 + y2 + 2gx + 2fy + c = 0 are 2 and respectively.

Bansal Classes []

30

Bansal Classes] []

262. Statement–1 : The number of circles that pass through the points (1, – 7) and (– 5, 1) and of radius 4, is two. Statement–2 : The centre of any circle that pass through the points A and B lies on the perpendicular bisector of AB.

263. The line OP and OQ are the tangents from (0, 0) to the circle x2 + y2 + 2gx + 2fy + c = 0. Statement–1 : Equation of PQ is fx + gy + c = 0. Statement–2 : Equation of circle OPQ is x2 + y2 + gx + fy = 0.

264. Statement–1 : 2 + y2 + 2xy + x + y = 0 represent circle passing through origin.Statement–2 : Locus of point of intersection of perpendicular tangent is a circle

265. Statement–1 : Equation of circle touching x–axis at (1, 0) and passing through (1, 2) is x2 + y2

– 2x – 2y + 1 = 0 Statement–2 : If circle touches both the axis then its center lies on x2 – y2 = 0

266. Statement-1: Let C be any circle with centre (0, ) has at the most two rational points on it Statement-2: A straight line cuts a circle at atmost two points

267. Tangents are drawn from each point on the line 2x + y = 4 to the circle x2 + y2 = 1 Statement-1: The chords of contact passes through a fixed point Statement-2: Family of lines (a1x + b1y + c1) + k (a2x + b2y + c2) = 0 always pass through a fixed point.

268. Statement-1: The common tangents of the circles x2 + y2 + 2x = 0 and x2 + y2 - 6 = 0 form an equilateral triangle Statement-2: The given circles touch each other externally.

269. Statement-1: The circle described on the segment joining the points (-2, -1), (0, -3) as diameter cuts the circle x2 + y2 + 5x + y + 4 = 0 orthogonally Statement-2: Two circles x2 + y2 + 2g1x + 2f1y + c1 = 0 x2 + y2 + 2g2x + 2f2y + c2 = 0 orthogonally if 2g1g2 + 2f1f2 = c1+ c2

270. Statement-1 : The equation of chord of the circle x2 + y2 – 6x + 10y – 9 = 0, which is bisected at (-2, 4) must be x + y – 2 = 0. Statement-2 : In notations, the equation of the chord of the circle S = 0 bisected at (x1, y1) must be T = S1.

271. Statement-1 : If two circles x2 + y2 + 2gx + 2fy = 0 and x2 + y2 + 2gx + 2fy = 0 touch each other, then fg = fg Statement-2 : Two circles touch other, if line joining their centres is perpendicular to all possible common tangents.

272. Statement-1 : Number of circles passing through (1, 2), (4, 7) and (3, 0) is one. Statement-2 : One and only circle can be made to pass through three non-collinear points.

273. Statement-1 : The chord of contact of tangent from three points A, B, C to the circle x2 + y2 = a2 are concurrent, then A, B, C will be collinear. Statement-2 : A, B, C always lies on the normal to the circle x2 + y2 = a2

Bansal Classes []

31

Bansal Classes] []

274. Statement-1 : Circles x2 + y2 = 144 and x2 + y2 – 6x – 8y = 0 do not have any common tangent. Statement-2 : If one circle lies completely inside the other circle then both have no common tangent.

275. Statement-1 : The equation x2 + y2 – 2x – 2ay – 8 = 0 represents for different values of ‘a’ a system of circles passing through two fixed points lying on the x-axis. Statement-2 : S = 0 is a circle & L = 0 is a straight line, then S + L = 0 represents the family of circles passing through the points of intersection of circle and straight line. (where is arbitrary parameter).

276. Statement-1 : Lengths of tangent drawn from any point on the line x + 2y – 1 = 0 to the circles x2 + y2 – 16 = 0 & x2 + y2 – 4x – 8y – 12 = 0 are equal Statement-2 : Director circle is locus of point of intersection of perpendicular tangents.

277. Statement-1 : One & only one circle can be drawn through three given points Statement-2 : Every triangle has a circumcircle.

278. Statement-1 : The circles x2 + y2 + 2px + r = 0, x2 + y2 + 2qy + r = 0 touch if

Statement-2 : Two circles with centre C1, C2 and radii r1, r2 touch each other if r1 r2 = c1c2

279. Statement-1 : The equation of chord of the circle x2 + y2 – 6x + 10y – 9 = 0 which is bisected at (-2, 4) must be x + y – 2 = 0 Statement-2 : In notations the equation of the chord of the circle s = 0 bisected at (x1, y1) must be T = S1.

280. Statement-1 : The equation x2 + y2 – 4x + 8y – 5 = 0 represent a circle. Statement-2 : The general equation of degree two ax2 + 2hxy + by2 – 2gx + 2fy + c = 0 represents a circle, if a = b & h = 0. circle will be real if g2 + f2 – c 0.

281. Statement-1 : The least and greatest distances of the point P(10, 7) from the circle x2 + y2 4x 2y 20 = 0 are 5 and 15 units respectively.Statement-2 : A point (x1, y1) lies outside a circle s = x2 + y2 + 2gx + 2fy + c = 0 if s1 > 0 where s1 = x1

2 + y12 + 2gx1 + 2fy1 + c.

282. Statement-1 : The point (a, a) lies inside the circle x2 + y2 4x + 2y 8 = 0 when ever a Î (1, 4)Statement-2 : Point (x1, y1) lies inside the circle x2 + y2 + 2gx + 2fy + c = 0, if

.283. Statement-1 : If n 3 then the value of n for which n circles have equal number of radical

axes as well as radical centre is 5.Statement-2 : If no two of n circles are concentric and no three of the centres are collinear then number of possible radical centre = nC3.

284. Statement-1 : Two circles x2 + y2 + 2ax + c = 0 and x2 + y2 + 2by + c = 0 touches if

Statement-2 : Two circles centres c1, c2 and radii r1, r2 touches each other if r1 ± r2 = c1c2.

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32

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285. Statement-1 : Number of point a Î I, lying inside the region bounded by the circles x2 + y2 2x 3 = 0 and x2 + y2 2x 15 = 0 is 1.Statement-2 : Sum of squares of the lengths of chords intercepted by the lines x + y = n, n ÎN on the circle x2 + y2 = 4 is 18.

09. 09. PARABOLA PARABOLA

286. Statement-1 : Slope of tangents drawn from (4, 10) to parabola y2 = 9x are .

Statement-2 : Every parabola is symmetric about its directrix.

287. Statement-1 : Though (, + 1) there can’t be more than one normal to the parabola y2 = 4x, if < 2. Statement-2 : The point (, + 1) lies outside the parabola for all ¹ 1.

288. Statement-1 : If x + y = k is a normal to the parabola y2 = 12x, then k is 9. Statement-2 : Equation of normal to the parabola y2 = 4ax is y – mx + 2am + am3 = 0

289. Statement-1 : If b, k are the segments of a focal chord of the parabola y2 = 4ax, then k is equal to ab/b-a. Statement-2 : Latus rectum of the parabola y2 = 4ax is H.M. between the segments of any focal chord of the parabola

290. Statement-1 : Two parabolas y2 = 4ax and x2 = 4ay have common tangent x + y + a = 0 Statement-2 : x + y + a = 0 is common tangent to the parabolas y2 = 4ax and x2 = 4ay and point of contacts lie on their respective end points of latus rectum.

291. Statement-1 : In parabola y2 = 4ax, the circle drawn taking focal radii as diameter touches y-axis. Statement-2 : The portion of the tangent intercepted between point of contact and directix subtends 90° angle at focus.

292. Statement-1 : The joining points (8, -8) & (1/2, 2), which are lying on parabola y2 = 4ax, pass through focus of parabola. Statement-2 : Tangents drawn at (8, -8) & (1/2, -2) on the parabola y2 = 4ax are perpendicular.

293. Statement-1 : There are no common tangents between circle x2 + y2 – 4x + 3 = 0 and parabola y2 = 2x. Statement-2 : Equation of tangents to the parabola x2 = 4ay is x = my + a/m where m denotes slope of tangent.

294. Statement-1 : Three distinct normals of the parabola y2 = 12x can pass through a point (h ,0) where h > 6.Statement-2 : If h > 2a then three distinct nroamls can pass through the point (h, 0) to the parabola y2 = 4ax.

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33

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295. Statement-1 : The normals at the point (4, 4) and of the parabola y2 = 4x are

perpendicular.Statement-2 : The tangents to the parabola at the and of a focal chord are perpendicular.

296. Statement-1 : Through (, + 1) there cannot be more than one-normal to the parabola y2 = 4x if < 2. Statement-2 : The point (, + 1) lines out side the parabola for all ¹ 1.

297. Statement-1 : Slope of tangents drawn from (4, 10) to parabola y2 = 9x are 1/4, 9/4 Statement-2 : Every parabola is symmetric about its axis.

298. Statement-1 : If a parabola is defined by an equation of the form y = ax2 + bx + c where a, b, c ÎR and a > 0, then the parabola must possess a minimum. Statement-2 : A function defined by an equation of the form y = ax2 + bx + c where a, b, cÎR and a ¹ 0, may not have an extremum.

299. Statement-1 : The point (sin a, cos a) does not lie outside the parabola 2y2 + x 2 = 0 when

Statement-2 : The point (x1, y1) lies outside the parabola y2 = 4ax if y12 4ax1 > 0.

300. Statement-1 : The line y = x + 2a touches the parabola y2 = 4a(x + a).Statement-2 : The line y = mx + c touches y2 = 4a(x + a) if c = am + a/m.

301. Statement-1 : If PQ is a focal chord of the parabola y2 = 32x then minimum length of PQ = 32. Statement-2 : Latus rectum of a parabola is the shortest focal chord.

302. Statement-1 : Through (, + 1), there can’t be more than one normal to the parabola y2 = 4x if < 2.

Statement–2 : The point (, + 1) lies outside the parabola for all Î R ~ {1}.

303. Statement–1 : Perpendicular tangents to parabola y2 = 8x meets on x + 2 = 0 Statement–2 : Perpendicular tangents of parabola meets on tangent at the vertex.

304. Let y2 = 4ax and x2 = 4ay be two parabolas Statement-1: The equation of the common tangent to the parabolas is x + y + a = 0 Statement-2: Both the parabolas are reflected to each other about the line y = x.

305. Let y2 = 4a (x + a) and y2 = 4b (x + b) are two parabolas Statement-1 : Tangents are drawn from the locus of the point are mutually perpendicular Statement-2: The locus of the point from which mutually perpendicular tangents can be drawn to the given comb is x + y + b = 0

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34

Bansal Classes] []

10. 10. ELLIPSE ELLIPSE

306. Tangents are drawn from the point (-3, 4) to the curve 9x2 + 16y2 = 144. STATEMENT -1: The tangents are mutually perpendicular. STATEMENT-2: The locus of the points from which mutually perpendicular tangents can be drawn to the given curve is x2 + y2 = 25.

307. Statement–1 : Circle x2 + y2 = 9, and the circle (x – ( + y ( = 0 touches each other internally. Statement–2 : Circle described on the focal distance as diameter of the ellipse 4x2 + 9y2 = 36 touch the auxiliary circle x2 + y2 = 9 internally

308. Statement–1 : If the tangents from the point (, 3) to the ellipse are at right

angles then is equal to 2. Statement–2 : The locus of the point of the intersection of two perpendicular tangents to the

ellipse = 1, is x2 + y2 = a2 + b2.

309. Statement–1 : x – y – 5 = 0 is the equation of the tangent to the ellipse 9x2 + 16y2 = 144.

Statement–2 : The equation of the tangent to the ellipse is of the form y = mx

.

310. Statement–1 : At the most four normals can be drawn from a given point to a given ellipse.

Statement–2 : The standard equation of an ellipse does not change on changing x

by – x and y by – y.

311. Statement–1 : The focal distance of the point on the ellipse 25x2 + 16y2 = 1600 will be 7 and 13.

Statement–2 : The radius of the circle passing through the foci of the ellipse and

having its centre at (0, 3) is 5.

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35

Bansal Classes] []

312. Statement-1 : The least value of the length of the tangents to intercepted between

the coordinate axes is a + b.

Statement-2 : If x1 and x2 be any two positive numbers then

313. Statement-1 : In an ellipse the sum of the distances between foci is always less than the sum of focal distances of any point on it. Statement-2 : The eccentricity of any ellipse is less than 1.

314. Statement-1 : Any chord of the conic x2 + y2 + xy = 1, through (0, 0) is bisected at (0, 0)Statement-2 : The centre of a conic is a point through which every chord is bisected.

315. Statement-1 : A tangent of the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P & Q. The angle between the tangents at P and Q of the ellipse x2 + 2y2 = 6 is p/2 Statement-2 : If the two tangents from to the ellipse x2/a2 + y2/b2 = 1 are at right angle, then locus of P is the circle x2 + y2 = a2 + b2.

316. Statement-1 : The equation of the tangents drawn at the ends of the major axis of the ellipse 9x2 + 5y2 – 30y = 0 is y = 0, y = 7. Statement-1 : The equation of the tangent drawn at the ends of major axis of the ellipse x2/a2 + y2/b2 = 1 always parallel to y-axis

317. Statement-1 : Tangents drawn from the point (3, 4) on to the ellipse will be

mutually perpendicular Statement-2 : The points (3, 4) lies on the circle x2 + y2 = 25 which is director circle to the

ellipse .

318. Statement-1 : For ellipse , the product of the perpendicular drawn from focii on

any tangent is 3.

Statement-2 : For ellipse , the foot of the perpendiculars drawn from foci on any

tangent lies on the circle x2 + y2 = 5 which is auxiliary circle of the ellipse.

319. Statement-1 : If line x + y = 3 is a tangent to an ellipse with foci (4, 3) & (6, y) at the point (1, 2), then y = 17. Statement-2 : Tangent and normal to the ellipse at any point bisects the angle subtended by foci at that point.

320. Statement-1 : Tangents are drawn to the ellipse at the points, where it is

intersected by the line 2x + 3y = 1. Point of intersection of these tangents is (8, 6).

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36

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Statement-2 : Equation of chord of contact to the ellipse from an external point is

given by

321. Statement-1 : In an ellipse the sum of the distances between foci is always less than the sum of focal distances of any point on it. Statement-2 : The eccentricity of any ellipse is less than 1.

322. Statement-1 : The equation x2 + 2y2 + xy + 2x + 3y + 1 = 0 can never represent a hyperbola Statement-2 : The general equation of second degree represent a hyperbola it h2 > ab.

323. Statement-1 : The equation of the director circle to the ellipse 4x2 + 9x2 = 36 is x2 + y2 = 13. Statement-2 : The locus of the point of intersection of perpendicular tangents to an ellipse is called the director circle.

324. Statement-1 : The equation of tangent to the ellipse 4x2 + 9y2 = 36 at the point (3, 2) is

.

Statement-2 : Tangent at (x1, y1) to the ellipse is

325. Statement-1 : The maximum area of PS1 S2 where S1, S2 are foci of the ellipse

and P is any variable point on it, is abe, where e is eccentricity of the ellipse.Statement-2 : The coordinates of pare (a sec , b tan ).

326. Statement-1 : In an ellipse the sum of the distances between foci is always less than the sum of focal distance of any point on it. Statement-2 : The eccentricity of ellipse is less than 1.

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37

Bansal Classes] []

11. 11. HYPERBOLA HYPERBOLA

327. Let Y = xÎ [3, ¥) and Y1 = be xÎ (-¥, -3] two curves.

Statement 1: The number of tangents that can be drawn from to the curve

Y1 = is zero

Statement 2: The point lies on the curve Y = .

328. Statement–1 : If (3, 4) is a point of a hyperbola having focus (3, 0) and (, 0) and length of the transverse axis being 1 unit then can take the value 0 or 3. Statement–2 : , where S and S are the two focus 2a = length of the transverse axis and P be any point on the hyperbola.

329. Statement–1 : The eccentricity of the hyperbola 9x2 – 16y2 – 72x + 96y – 144 = 0 is .

Statement–2 : The eccentricity of the hyperbola is equal to .

330. Let a, b, a Î R – {0}, where a, b are constants and a is a parameter.

Statement–1 : All the members of the family of hyperbolas have the same

pair of asymptotes. Statement–2 : Change in a, does not change the slopes of the asymptotes of a member of the

family .

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38

Bansal Classes] []

331. Statement–1 : The slope of the common tangent between the hyperbola and

may be 1 or – 1.

Statement–2 : The locus of the point of inteeersection of lines and is

a hyperbola (where m is variable and ab ¹ 0).

332. Statement–1 : The equation x2 + 2y2 + xy + 2x + 3y + 1 = 0 can never represent a hyperbola. Statement–2 : The general equation of second degree represents a hyperbola if h2 > ab.

333. Statement–1 If a point (x1, y1) lies in the region II of shown in the figure, then

Statement–2 If (P(x1, y1) lies outside the a hyperbola , then

334. Statement–1 Equation of tangents to the hyperbola 2x2 3y2 = 6 which is parallel to the line y = 3x + 4 is y = 3x 5 and y = 3x + 5. Statement–2 y = mx + c is a tangent to x2/a2 y2/b2 = 1 if c2 = a2m2 + b2.

335. Statement–1 : There can be infinite points from where we can draw two mutually

perpendicular tangents on to the hyperbola

Statement–2 : The director circle in case of hyperbola will not exist because a2 <

b2 and director circle is x2 + y2 = a2 – b2.

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39

Bansal Classes] []

336. Statement–1 : The average point of all the four intersection points of the rectangular hyperbola xy = 1 and circle x2 + y2 = 4 is origin (0, 0). Statement–2 : If a rectangular hyperbola and a circle intersect at four points, the average point of all the points of intersection is the mid point of line-joining the two centres.

337. Statement–1 : No tangent can be drawn to the hyperbola which have slopes

greater than

Statement–2 : Line y = mx + c is a tangent to hyperbola . If c2 = a2m2 – b2

338. Statement–1 : Eccentricity of hyperbola xy – 3x – 3y = 0 is 4/3 Statement–2 : Rectangular hyperbola has perpendicular asymptotes and eccentricity =

339. Statement–1 : The equation x2 + 2y2 + xy + 2x + 3y + 1 = 0 can never represent a hyperbola Statement–2 : The general equation of second degree represent a hyperbola it h2 > ab.

340. Statement–1 : The combined equation of both the axes of the hyperbola xy = c2 is x2 – y2 = 0. Statement–2 : Combined equation of axes of hyperbola is the combined equation of angle bisectors of the asymptotes of the hyperbola.

341. Statement–1 : The point (7, 3) lies inside the hyperbola 9x2 4y2 = 36 where as the point (2, 7) lies outside this.

Statement–2 : The point (x1, y1) lies outside, on or inside the hyperbola according

as < or = or > 0

342. Statement–1 : The equation of the chord of contact of tangents drawn from the point (2, 1) to the hyperbola 16x2 9y2 = 144 is 32x + 9y = 144.

Statement–2 : Pair of tangents drawn from (x1, y1) to is SS1 = T2

343. Statement–1 : If PQ and RS are two perpendicular chords of xy = xe, and C be the centre of hyperbola xy = c2. Then product of slopes of CP, CQ, CR and CS is equal to 1.Statement–2 : Equation of largest circle with centre (1, 0) and lying inside the ellipse x2 + 4y2

16 is 3x2 + 3y2 6x 8 = 0.

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40

Bansal Classes] []

12. 12. COMPLEX NUMBERS COMPLEX NUMBERS

344. Let z = ei = cos + isinStatement 1: Value of eiA .eiB . eiC = –1 if A + B + C = p. Statement 2: arg(z) = and |z| = 1.

345 Let a1, a2, .... , an ÎR+

Statement–1 : Minimum value of

Statement–2 : For positive real numbers, A.M G.M.

346. Let log then A.P., where a, b, c are in G.P. If a, b, c represents the

sides of a triangle. Then Statement–1 : Triangle represented by the sides a, b, c will be an isosceles triangle Statement–2 : b + c < a

347. Let Z1, Z2 be two complex numbers represented by points on the curves |z| = and |z – 3 – 3i| = . Then Statement–1 : min |z1–z2| = 0 and max |z1 – z2| = 6 Statement–2 : Two curves |z| = and |z – 3 –3i| = 2 touch each other externally

348. Statement–1 : If |z – i| £ 2 and z0 = 5 + 3i, then the maximum value of |iz + z0| is 7Statement–2 : For the complex numbers z1 and z2 |z1 + z2| £ |z1| + |z2|

349. Let z1 and z2 be complex number such that

Statement–1 : arg

Statement–2 : z1, z2 and origin are collinear and z1, z2 are on the same side of origin.

350. Let fourth roots of unity be z1, z2, z3 and z4 respectively

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41

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Statement–1 :Statement–2 : z1 + z2 + z3 + z4 = 0.

351. Let z1, z2, . . . , zn be the roots of zn = 1, n Î N. Statement–1 : z1. z2 . . . zn = (– 1)n Statement–2 : Product of the roots of the equation anxn + an – 1xn – 1 + an – 2

xn – 2 + . . . + a1x + a0

= 0, an ¹ 0, is (– 1)n. .

352. Let z1, z2, z3 and z4 be the complex numbers satisfying z1 – z2 = z4 – z3. Statement–1 : z1, z2, z3, z4 are the vertices of a parallelogram

Statement–2 : .

353. Statement–1 : The minimum value of | is 0. Statement–2 : For any two complex number z1 and z2, .

354. Statement–1 : Let z1 and z2 are two complex numbers such that then the

orthocenter of AOB is . (where O is the origin)

Statement–2 : In case of right angled triangle, orthocenter is that point at which triangle is right angled.

355. Statement–1 : If w is complex cube root of unity then (x – y) (xw – y) (xw2 – y) is equal to x3 + y2 Statement–2 : If w is complex cube root of unity then 1 + w + w2 = 0 and w3 = 1

356. Statement-1 : If |z| £ 4, then greatest value of |z + 3 – 4i| is 9. Statement-2 : Z1, Z2 ÎC, |Z1 + Z2| £ |Z1| + |Z2|

357. Statement-1: The slope of line (2 – 3i) z + (2 + 3i) 1 = 0 is

Statement-2:: The slope of line bÎR & a be any non-zero complex.

Constant is

358. Statement-1: The value of is i

Statement-2: The roots of the equation zn = 1 are called the nth roots of unity where

z =

where k = 0, 1, 2, ... (n 1)

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42

Bansal Classes] []

359. Statement-1: |z1 – a| < a, |z2 – b| < b |z3 – c| < c, where a, b, c are +ve real nos, then |z1 + z2 + z3| is greater than 2|a + b + c| Statement-2: |z1 z2| £ |z1| + |z2|

360. Statement-1: (cos2 + isin2)p = 1 Statement-2: (cos +isin)n = cosn + isin n it is not true when n is irrational number.

361. Statement-1 : If a1, a2, a3 …. a 8 be the 8th root of unity, then a116 + a2

16 + a316 + … + a8

16 = 8 Statement-2 : In case of sum of pth power of nth roots of unity sum = 0 if p ¹ kn where p, k, n are integers sum = n if p = kn.

362. Statement-1: Locus of z, satisfying the equation |z – 1| + |z – 8| = 16 is an ellipse of eccentricity 7/16 Statement-2:: Sum of focal distances of any point is constant for an ellipse

363. Statement-1: arg = arg z2 – arg z1 & arg zn = n(argz)

Statement-2: If |z| = 1, then arg (z2 + ) = arg z.

364. Statement-1: If |z z + i| £ 2 then Statement-2: If |z 2 + i| £ 2 the z lies inside or on the circle having centre (2, 1) & radius 2.

365. Statement-1: The area of the triangle on argand plane formed by the complex numbers z, iz

and z + iz is |z|2

Statement-2: The angle between the two complex numbers z and iz is .

366. Statement-1: If (z1, z2 ¹ 0), then locus of z is circle.

Statement-2 : As, represents a circle if, {0, 1}

367. Statement-1: If z1 and z2 are two complex numbers such that |z1| = |z2| + |z1 – z2|, then Im

.

Statement-2: arg (z) = 0 z is purely real.

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43

Bansal Classes] []

368. Statement-1: If a = cos , p = a + a2 + a4, q = a3 + a5 + a6, then the

equation whose roots are p and q is x2 + x + 2 = 0 Statement-2: If a is a root of z7 = 1, then 1 + a + a2 + …. + a6 = 0.

369. Statement-1: If |z| < then |z2 + 2z cosa| is less than one. Statement-2: |z1 + z2| < |z1| + |z2| . Also |cosa| £ 1.

370. Statement-1: The number of complex number satisfying the equation |z|2 + P|z| + q = 0 (p, q, Î R) is atmost 2. Statement-2 : A quadratic equation in which all the co-efficients are non-zero real can have exactly two roots.

371. Statement-1: If is a complex number, then the maximum value of |b| is

.

Statement-2 :: On the locus the farthest distance from origin is .

372. Statement-1: The locus of z moving in the Argand plane such that arg is a

circle. Statement-2: This is represent a circle, whose centre is origin and radius is 2.

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44

Bansal Classes] []

13. 13. BINOMIAL THEOREM BINOMIAL THEOREM

373. Statement-1: The binomial theorem provides an expansion for the expression (a + b)n. where a, b, n Î R.Statement-2: All coefficients in a binomial expansion may be obtained by Pascal’s triangle.

374. Statement-1: If n is an odd prime then integral part of is divisible by 20 n.Statement-2: If n is prime then nC1, nC2, nC3, ….. nCn – 1 must be divisible by n.

375. Statement–1 : 260 when divided by 7 leaves the reminder 1. Statement–2 : (1 + x)n = 1 + n1x, where n, n1 Î N.

376. Statement–1 :Statement–2 : and nCr = nCn – r

377. Let n be a positive integers and k be a whole number, k ≤ 2n. Statement–1 : The maximum value of 2nCk is 2nCn.

Statement–2 : , for k = 0, 1, 2, . . . , n – 1.

378. Let n be a positive integer. Statement–1 : is divisible by 64.

Statement–2 : and in the binomial expansion of (1+8)n+1, sum of first two terms is 8n + 9 and after that each term is a multiple of 82.

379. Statement–1 : If n is an odd prime, then integral part of is divisible by 20n.

Statement–2 : If n is prime, then nc1, nc2, nc3 . . . ncn – 1 must be divisible by n.

380. Statement–1 : The coefficient of x203 in the expression (x – 1)(x2 – 2) (x2 – 3) . . . (x20 – 20) must be 13. Statement–2 : The coefficient of x8 in the expression (2 + x)2 (3 + x)3 (4 + x)4 is equal to 30.

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45

Bansal Classes] []

381. Statement–1 : C02 + C1

2 + C22 + C3

2 + ... + Cn2 =

Statement–2 : nC0 – nC1 + nC2 .... + (–1)n nCn = 0

382. Statement–1 : Some of coefficient (x – 2y + 4z)n is 3n Statement–2 : Some of coefficient of (c0x0 + c1x1 + c2x2 + ..... + cnxn)n is 2n

383. Statement-1: The greatest coefficient in the expansion of (a1 + a2 + a3 + a4)17 is

Statement-2: The number of distinct terms in (1 + x + x2 + x3 + x4 + x5)100 is 501.

384. Statement-1: The co-efficient of x5 in the expansion of (1 + x2)5 (1 + x)4 is 120 Statement-2: The sum of the coefficients in the expansion of (1 + 2x – 3y + 5z)3 is 125.

385. Statement-1: The number of distinct terms in (1 + x + x2 + x3 + x4)1000 is 4001 Statement-2: The number of distinct terms in the expansion (a1 + a2 + ... + am)n is n+m-1Cm-1

386. Statement-1: In the expansion of (1 + x)30, greatest binomial coefficient is 30C15 Statement-2: In the expansion of (1 + x)30, the binomial coefficients of equidistant terms from end & beginning are equal.

387. Statement-1: Integral part of is even where nÎI.

Statement-2: Integral part of any integral power of the expression of the form of p + is even.

388. Statement-1 : = 21C4

Statement-2: 1 + x + x2 + x3 + ... + xn-1 = = sum of n terms of GP.

389. Statement-1: Last two digits of the number (13)41 are 31. Statement-2: When a number in divided by 1000, the remainder gives the last three digits.

390. Statement-1: nC0 + nC1 + nC2 + ….. + nCn = 2n where n Î N.Statement-2: The all possible selections of n distinct objects are 2n.

391. Statement-1 : The integral part of is odd, where n Î N.Statement-2 : (x + a)n (x a)n = 2[nC0xn + nC2xx 2 a2 + nC4 + xn 4 a4 + …..]

392. Statement-1: If n is even than 2nC1 + 2nC3 + 2nC5 + ... + 2nCn-1 = 22n-1 Statement-2: 2nC1 + 2nC3 + 2nC5 + ... + 2nC2n-1 = 22n-1

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46

Bansal Classes] []

393. Statement-1 : Any positive integral power of can be expressed as for some natural number N > 1. Statement-2 : Any positive integral power of can be expressed as A + B where A and B are integers.

394. Statement-1 : The term independent of x in the expansion of .

Statement-2: The Coefficient of xb in the expansion of (1 + x)n is nCb.

395. Statement-1: The coefficient of x8 in the expansion of (1 + 3x + 3x2 + x3)17 is 51C2. Statement-2 : Coefficient of xr in the expansion of (1 + x)n is nCr.

396. Statement-1: If (1 + x)n = c0+c1x + c2x2 + … + cnxn then c0 2.c1 + 3.c2 ….. + (1)n (n + 1)cn = 0

Statement-2: Coefficients of equidistant terms in the expansion of (x + a)n where n Î N are equal.

397. Statement-1:

Statement-2: If 22003 is divided by 15 then remainder is 8.

398. Statement-1: The co-efficient of (1 + x2)5 (1 + x)4 is 120. Statement-2: The integral part of is odd.

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47

Bansal Classes] []

14. PERMUTATION & COMBINATION14. PERMUTATION & COMBINATION

399. Statement-1: 51 × 52 × 53 × 54 × 55 × 56 × 57 × 58 is divisible by 40320 Statement-2: The product of r consecutive natural numbers is always divisible by r!

400. Statement-1: Domain is {d1, d2, d3, d4}, range is {r1, r2, r3}. Number of into functions which can be made is 45. Statement-2: Numbers of into function = number of all functions – number of onto functions. = 34 – 3(4C2 . 2C1) = 81 – 36 = 45 of d1, d2, d3, d4 any two correspond to r1, remaining two to r2, r3 one with each 4C2 × 2C1 = 12, total = 12 × 3 = 36 = number of onto functions.

401. Statement-1: The smallest number which has 24 divisors is 420. Statement-2: 24 = 3 × 2 × 2 = (2 +1) (1 + 1) (1 + 1) (1 + 1), therefore, prime factors of the number are 2, 2, 3, 5, 7 & their product is 420.

402. Consider the word 'SMALL' Statement–1 : Total number of 3 letter words from the letters of the given word is 13. Statement–2 : Number of words having all the letters distinct = 4 and number of words having two are alike and third different = 9

403. Statement–1 : Number of non integral solution of the equation x1 + x2 + x3 = 10 is equal to 34. Statement–2 : Number of non integral solution of the equation x1 + x2 + x3+ . . . xn = r is equal to n + r – 1Cr

404. Statement–1 : 10Cr = 10C4 r = 4 or 6 Statement–2 : nCr = nCn – r

405. Statement–1 : The number of ways of arranging n boys and n girls in a circle such that no two boys are consecutive, is .

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Statement–2 : The number of ways of arranging n distinct objects in a circle is

406. Statement–1 : The number of ways of selecting 5 students from 12 students (of which six are boys and six are girls), such that in the selection there are at least three girls is 6C3 ´ 9C2. Statement–2 : If a work has two independent parts, of which first part can be done in m way and for each choice of first part, the second part can be done in n ways, then the work can be completed in m ´ n ways.

407. Statement–1 : The number of ways of writing 1400 as a product of two positive integers is 12. Statement–2 : 1400 is divisible by exactly three prime numbers.

408. Statement–1 : The number of selections of four letters taken from the word ‘PARALLEL’ must be 15. Statement–2 : Coefficient of x4 in the expansion of (1 – x)-3 is 15.

409. Statement–1 : Total number of permutation of n things of which p are alike of one kind, q are

alike of 2nd kind, r are alike of 3rd kind and rest are all difference is .

Statement–2 : Total number of selection from n identical object is n.

410. Statement–1 : A polygon has 44 diagonals and number of sides are 11. Statement–2 : From n distinct object r object can be selected in nCr ways.

411. Let y = x + 3, y = 2x +3, y = 3x + 2 and y + x = 3 are four straight lines Statement-1 : The number of triangles formed is 4C3 Statement-2 : Number of distinct point of intersection between various lines will determine the number of possible triangle.

412. Statement-1 : The total number of positive integral solutions (zero included) of x + y + z + w = 20 without restriction is 23C20 Statement-2 : Number of ways of distributing n identical items among m persons when each person gets zero or more items = m + n -1Cn

413. Statement-1 : The total ways of selection of 5 objects out of n(n 5) identical objects is one.Statement-2: If objects are identical then total ways of selection of any number of objects from given objects is one.

414. Statement-1: The total number of different 3-digits number of type N = abc, where a < b < c is 84. Statement-2: O cannot appear at any position, so total numbers are 9C3.

415. Statement-1: The number of positive integral solutions of the equation x1x2x3x4x5 = 1050 is 1875. Statement-2: The total number of divisor of 1050 is 25.

416. Statement-1:

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Statement-2 : nCr + nCr-1 = n+1Cr

417. Statement-1 : is a natural number for all nÎN

Statement-2 : The number of ways of distributing mn things in m groups each containing n

things is .

418. Statement-1: The number of divisors of 10, 800 is 60. Statement-2: The number of odd divisors of 10, 800 is 12.

419. Statement-1: Number of onto functions from A B where A contains n elements 2B contains m elements (where n m) = mn – mC1 (m – 1)n + mC2 (m – 2)n + ... Statement-2: Number of ways of putting 5 identical balls in 3 different boxes when empty boxes are not allowed are 6.

420. Statement-1 : 4 persons can be seated in a row containing 12 chairs, such that no two of them are consecutive in 9C4 × 4! ways Statement-2: Number of non-negative integral solutions of equation x1 + x2 + ... + xr = n is = n+r-1Cr-1.

421. Statement-1: The number of selections of four letters taken from the word PARALLEL must be 22. Statement-2: Coefficient of x4 in the expansion of (1 – x)3 is 10.

422. Statement-1: Number of permutations of n dissimilar things taken ‘n’ at a time is nPn. Statement-2: n(A) = n(B) = n then the total number of functions from A to B are n!

423. Statement-1: Number of permutations of n dissimilar things taken n at a time in nPn . Statement-2: n(A) = n(B) = n then the total number of functions from A to B are n!

424. Statement-1: nCr = nCp r = p or r + p = n Statement-2: nCr = nCn–r

425. Statement-1: The total number of words with letters of the word civilization (all taken at a time) is 19958393.Statement-2: The number of permutations of n distinct objects (r taken at a time) is npr+1.

426. Statement-1: The number of ways in which 81 different beads can be arranged to form a

necklace is

Statement-2: Number of circular arrangements of n different objects is (n 1)!.

427. Statement-1: There are 9n, n digit numbers in which no two consecutive digits are same. Statement-2: The n digits number in which no two consecutive digits are equal cannot contain zero.

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50

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428. Statement-1: is divisible by 6.

Statement-2: : Product of three consecutive integer is divisible by 6.

15. 15. PROBABILITY PROBABILITY

429. P(E) = or [ Good ]

Statement-1: Always the probability of an event is a rational number and less than or equal to one Statement-2: The equation P(E) = |sin| is always consistent.

430. Let A and B be two event such that P(AB) 3/4 and 1/8 £ P(AÇB) £ 3/8 Statement–1 : P(A) + P(B) 7/8 Statement–2 : P(A) + P(B) £ 11/8

431. Statement–1 : The probability of drawing either a ace or a king from a pack of card in a

single draw is .

Statement–2 : For two events E1 and E2 which are not mutually exclusive, probability is given by P .

432. Let A and B be two independent events.

Statement–1 : If P(A) = 0.3 and P then P(B) is .

Statement–2 : P where E is any event.

433. Let A and B be two independent events of a random experiment.

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51

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Statement–1 : P(A Ç B) = P(A). P(B) Statement–2 : Probability of occurrence of A is independent of occurrence or non–occurrence of B.

434. A fair die is rolled once.

Statement–1 : The probability of getting a composite number is

Statement–2 : There are three possibilities for the obtained number (i) the number is a prime number (ii) the number is a composite number (iii) the number is 1, and hence probability of

getting a prime number =

435. Let A and B are two events such that P(A) = and P(B) = , then

Statement–1 : £ P .

Statement–2 : .

436. Statement–1 : Three of the six vertices of a regular hexagon are chosen at random. The

probability that the triangle with three vertices equilateral equals to .

Statement–2 : A die is rolled three times. The probability of getting a large number than the

previous number is .

437. Statement-1: A coin is tossed thrice. The probability that exactly two heads appear, is 3/8 Statement-2: Probability of success r times out of total n trials = P(r) = nCr = nCr pr qn-r where p be the probability of success and q be the probability of failure.

438. Statement-1 : For any two events A and B in a sample space P(A/B) , P(B) ¹ 0

is always true Statement-2 : For any two events A and B 0 < P(A B) £ 1.

439. Statement-1: The letters of the English alphabet are written in random order. The probability

that letters x & y are adjacent is .

Statement-2: The probability that four lands deals at random from 94 ordinary deck of 52 cends will contain from an ordinary deck of 52 cends will contain from each suit is 1/4.

440. Statement-1: The probability of being at least one white ball selected from two balls drawn simultaneously from the bag containing 7 black & 4 white balls is 34/35. Statement-2: Sample space = 11C2 = 55, Number of fav. Cases = 4C1 × 7C1 + 4C2 × 7C0

441. Statement-1: If A, B, C be three mutually independent events then A and BC are also independent events. Statement-2: Two events A and B are independent if and only if P(AÇB) = P(A) P(B).

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442. Statement-1: If A and B be two events such that P(A) = 0.3 and P(AB) = 0.8 also A and B are independent events P(B) is 0.5.Statement-2: IF A & B are two independent events then P(AÇB) = P(A).P(B).

443. Statement-1: The probability of occurence of a doublet when two identical dies are thrown is 2/7. Statement-2: When two identical dies are thrown then the total number of cases are 21 in place of 36 (when two distinct dies are thrown) because the cases like (5, 6). (6, 5) are considered to be same.

444. Statement-1: A = {2, 4, 6} , B = {1, 2, 3, } where A & B are the number occuring on a dice, then P(A) + P(B) = 1 Statement-2: If A1, A2, A3 ... An are all mutually exclusive events, then P(A1) + P(A2) + ... + P(An) =1.

445. Statement-1: If P(A/B) P(A) then P(B/A) P(B)

Statement-2:: P(A/B) =

446. Statement-1: Balls are drawn one by one without replacement from a bag containing a white and b black balls, then probability that white balls will be first to exhaust is a/a+b. Statement-2: Balls are drawn one by one without replacement from a bag containing a white and b black balls then probability that third drawn ball is white is a/a+b.

447. Statement-1: Out of 5 tickets consecutively numbers, three are drawn at random, the chance that the numbers on them are in A.P. is 2/15. Statement-2: Out of (2n + 1) tickets consectively numbered, three are drawn at random, the

chance that the numbers on them are in A.P. is .

448. Statement-1: If the odds against an event is 2/3 then the probability of occurring of an event is 3/5. Statement-2: For two events A and B P(A Ç B) = –1 P (A B)

449. Statement-1: A, B, C are events such that P(A) = 0.3, P(B) = 0.4 P(C) = 0.8, P(AÇB) = 0.08, P(AÇC) = 0.28, P(AÇBÇC) = 0.09 then P(BÇC) Î (0.23, 0.48). Statement-2: 0.75 £ P(A B C) £ 1.

450. Statement-1: If P(A) = 0.25, P(B) = 0.50 and P(AÇB) = 0.14 then the probability that neither A nor B occurs is 0.39. Statement-2:

451. Statement-1: For events A and B of sample space if .

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Statement-2:

16. VECTORS & 3 – DIMENSIONS16. VECTORS & 3 – DIMENSIONS452. Let be three non-coplanar vectors then ( = 0

Statement 1: can be expressed as linear combination of and . Statement 2: Given non-coplanar vectors one vector can be expressed as a linear combination of other two.

453. A vector has components p and 1 with respect to a rectangular cartesian system. If the axes are rotated through an angle a about the origin in the anticlockwise sense. Statement–1 : If the vector has component p + 2 and 1 with respect to the new system then p = –1 Statement–2 : Magnitude of vector original and new system remains same

454. Let | and angle between and is p/6 Statement–1 : Statement–2 :

455. Statement–1 :

Statement–2 : If , , are linearly dependent vectors then they are coplanar.

456. Statement–1 : If then is parallel to .

Statement–2 : If = then = 0.

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457. Let be a non-zero vector satisfying for given non–zero vectors and .Statement–1 : and are coplanar vectors. Statement–2 : is perpendicular to the vectors and .

458. Let and be two non–collinear vectors. Statement–1 : vector ´ is a vector in the plane of and , perpendicular to .

Statement–2 : = , for any vector .

459. Statement–1 : If three points P, Q, R have position vectors , , respectively and , then the points P, Q, R must be collinear.

Statement–2 : If for three points A, B, C; , then the points A, B, C must be collinear.

460. Statement–1 : Let and be two non collinear unit vectors. If and

then

Statement–2 : The vector is makes an angle of with the vector

.

461. Statement-1: If are unit vectors inclined at an angle a and is a unit vector bisecting

the angle between them, then

Statement-2: If ABC is an isosceles triangle with AB = AC = 1, then vector representing

bisector of angle A is given by =

462. Statement-1: The direction ratios of line joining origin and point (x, y, z) must be x, y, z. Statement-2: If P is a point (x, y, z) in space and OP = r, then direction cosines of OP are

.

463. Statement-1: If the vectors , and are coplanar, then ||2 is equal to 16. Statement-2: The vectors and are coplanar iff ) = 0

464. Statement-1: A line L is perpendicular to the plane 3x – 4y + 5z = 10

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Statement-2: Direction co-sines of L be < >

465. Statement-1 : The points with position vectors , are collinear. Statement-2: The position vectors 2 are linearly dependent vectors.

466. Statement-1: If are three unit vectors such that then the angle between

is p/2

Statement-2: If then = 0.

467. Statement-1: If cosa, cosb, cos are the direction cosine of any line segment, cos2a + cos2b + cos2 = 1. Statement-2: If cosa, cosb, cos are the direction cosine of line segment, cos2a + cos2b + cos2 = 1.

468. Statement-1: The direction cosines of one of the angular bisector of two intersecting lines having direction cosines as l1 , m1, n1, & l2, m2, n2 is proportional to l1 + l2, m1+ m2, n1 + n2. Statement-2: The angle between the two intersecting lines having direction cosines as l1, m1, n1

& l2, m2, n2 is given by cos = l1 l2 + m1m2 + n1n2.

469. Statement-1: If Statement-2: = 0 either or = 0 or

470. Statement-1: Statement-2: sin , when is angle, when your fingers curls from A to B

471. Statement-1 : A vector r the plane of (1, -1, 0), (2, 1, -1) & (-1, 1, 2) is Statement-2 : always gives a vector perpendicular to plane of

472. Statement-1 : Angle between planes & . (acute angle) is given by cos = Statement-2 : Angle between the planes in same as acute angle formed by their normals.

473. Statement-1: In ABC, = 0 Statement-2 : If then

474. Statement-1: and are parallel vectors it p = 9/2 and q = 2.

Statement-2 : If and are parallel

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475. Statement-1: The direction ratios of line joining origin and point (x, y, z) must be x, y, z Statement-2: If P is a point (x,y, z) in space and OP = r then directions cosines of OP are

476. Statement-1: The shortest distance between the skew lines and is

Statement-2: Two lines are skew lines if three axist no plane passing through them.

477. Statement-1: , are parallel vectors of p = 3/2 and q = 4.

Statement-2: and are parallel if .

478. Statement-1: If and are coplanar then . Statement-2: A set of vectors is said to be linearly independent if every relation of the form l1 + l2 + … + ln = 0 implies that l1 = l2 = …. = ln = 0 (scalars).

479. Statement-1: The shortest distance between the skew lines is

Statement-2: Two lines are skew lines if there exists no plane passing through them.

480. Statement-1: The curve which is tangent to a sphere at a given point is the equation of a plane. Statement-2: Infinite number of lines touch the sphere at a given point.

481. Statement-1: In ABC Statement-2: If then ( law of addition).

482. Statement-1: and are parallel vectors if

Statement-2: If and are parallel then .

483. Statement-1: If are coplanar then

Statement-2: A set of vectors is said to be linearly independent if every relation of the form l1 = l2 = ….. = ln = 0.

484. Statement-1: The shortest distance between the skew lines and is

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Statement-2: Two lines are skew lines if there exists no plane passing through them.

485. Statement-1 : The value of expression Statement-2 :

486. Statement-1: A relation between the vectors is

Statement-2 :

3-Dimension

487. The equation of two straight line are and

Statement–1 : The given lines are coplanarStatement–2 : The equation 2x1 – y1 = 1, x1 + 3y1 = 4, 3x1 + 2y1 = 5 are consistent.

488. Statement–1 : The distance between the planes 4x – 5y + 3z = 5 and 4x – 5y + 3z + 2 = 0 is

.

Statement–2 : The distance between ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is

.

489. Given the line and the plane p : x - 2y - z = 0

Statement-1: L lies in p Statement-2: L is parallel to p

490. The image of the point (1, b, 3) in the

Statement-1: Line will be (1, 0, 7)

Statement-2: Length of the perpendicular from the point A( ) on the line is given

by d =

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58

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17. MATRICES & DETERMINANTS17. MATRICES & DETERMINANTS

491. Statement–1 : = = 0

Statement–2 : = 4

492. Let f(x) =

Statement–1 : f(100) + f(99) + f(98) + ... + f(1) =

Statement–2 : f(x) = 0

493. Let A =

Statement–1 : Inverse of A exists for all ÎR Statement–2 : Inverse of A exists if ÎR – {8}

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494. Let A =

Statement–1 : A–1 = adj (A) Statement–2 : |A| = 1

495. Statement–1 : If A = then A–1 exist if ¹ 8.

Statement–2 : A–1 exists if | A | = 0.

496. Let there be a system of equations 6x + 5y + z = 0 3x – y + 4z = 0 x + 2y – 3z = 0 Statement–1 : System of equations has infinite number of nontrivial solution for ¹ – 5.

Statement–2 : It will have non trivial solution is .

497. Let a, b, be the roots of the equation x3 + ax + b = 0; a, b Î R.

Statement–1 :

Statement–2 : Any cubic equation over reals has at least one real root.

498. Let A be a square matrix of order 3 satisfying AA = I. Statement–1 : AA = I Statement–2 : (AB) = B A

499. Statement–1 : The determinant of a matrix is zero.

Statement–2 : The determinant of a skew symmetric matrix of odd order is zero.

500. Statement–1 : If Ar = , where r is a natural number, then

Statement–2 : If A is a matrix of order 3 and |A| = 2, then .

501. Statement–1 : If matrix A = then A3 – 3A2 – I = 0

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Statement–2 : If B is a symmetric matrix then B–1 will also be symmetric.

502 Statement–1 : Adjoint of a diagonal matrix is diagonal matrix Statement–2 : If |A| = 0 then (adj A) A = A(adjA) = 0

503. Statement-1: The inverse of the matrix does not exist.

Statement-2: The matrix is singular.

504. Statement-1: If A = , then A-1 =

Statement-2 : The inverse of a diagonal matrix is a diagonal matrix.

505. Statement-1: The inverse of the matrix A = does not exist.

Statement-2: The determinant of a skew-symmetric matrix is zero.

506. Consider the following matrix A =

Statement-1: A is involutory matrix Statement-2: A2 = I (identity matrix)

507. Consider the following system of equation ax + y + z = 0, x + by + z = 0, x + y + cz = 0 Statement-1: Above system of equation will have infinitely many solution if

Statement-2: Above system of equation will have infinitely many solution if D=

508. Statement-1: If A is a skew symmetric of order 3 then its determinant should be zero Statement-2: If A is square matrix than detA = detA = det (-A).

509. Statement-1: If A and B are two matrices such that AB = B, BA = A then A2 + B2 = A + B Statement-2: A and B are idempotent matrices

Bansal Classes []

61

Bansal Classes] []

510. Statement-1: The possible dimensions of a matrix containing 32 elements is 6. Statement-2: The number of ways of expressing 32 as a product of two positive integers is 6.

511. Statement-1: The determinants of a matrix is zero.

Statement-2: The determinant of a skew symmetric matrix of odd order is zero.

512. Statement-1: Every square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew symmetric matrix.Statement-2: The elements on the main diagonal of a skew symmetric matrix are all different.

513. Statement-1:

Statement-2: A.M. G.M.

514. Statement-1: The value of = is 59

Statement-2: The sum of products of elements of a row (column) is zero.

515. Statement-1: The system of linear equations x + y + z = 6, x + 2y – 3z = 14 and 2x + 5y - z = 9( ÎR) half unique solution. If ¹ 8. Statement-2: A homogenous system is always is consistent for homogenous system, x = y = z = 0 is a always a solution where determinant ¹ 0 i.e., ¹0.

516. Statement-1: If w is a cube root of unity and A = , then A100 is equal to A

Statement-2: If A, and B are idempotent matrices, then AB is idempotent if A and B commute

517. Statement-1: If A = [aij] is a scalar matrix then trace of A is

Statement-2: If the value of x = y; y = 1

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62

Bansal Classes] []

18. 18. QUADRATIC EQUATIONS QUADRATIC EQUATIONS

518. Statement-1: If x ÎR, 2x2 + 3x + 5 is positive.Statement-2: If < 0, ax2 + bx + c, ‘a’ have same sign x ÎR.

519. Statement-1: If is a root of x2 – 2x – 1 = 0, then will be the other root.Statement-2: Irrational roots of a quadratic equation with rational coefficients always occur in conjugate pair.

520. Statement-1: The roots of the equation 2x2 + 3i x + 2 = 0 are always conjugate pair. Statement-2: Imaginary roots of a quadratic equation with real coefficients always occur in conjugate pair.

521. Consider the equation (a2 – 3a + 2) x2 + (a2 – 5a + 6)x + a2 – 1 = 0 Statement – 1: If a = 1, then above equation is true for all real x. Statement – 2: If a = 1, then above equation will have two real and distinct roots.

522. Consider the equation (a + 2)x2 + (a – 3) x = 2a – 1 Statement–1 : Roots of above equation are rational if 'a' is rational and not equal to –2. Statement–2 : Roots of above equation are rational for all rational values of 'a'.

523. Let f(x) = x2 = –x2 + (a + 1) x + 5 Statement–1 : f(x) is positive for same a < x < b and for all aÎR Statement–2 : f(x) is always positive for all xÎR and for same real 'a'.

524. Consider f(x) = (x2 + x + 1) a2 – (x2 + 2) a –3 (2x2 + 3x + 1) = 0 Statement–1 : Number of values of 'a' for which f(x) = 0 will be an identity in x is 1.

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63

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Statement–2 : a = 3 the only value for which f(x) = 0 will represent an identity.

525. Let a, b, c be real such that ax2 + bx + c = 0 and x2 + x + 1= 0 have a common rootStatement–1 : a = b = c Statement–2 : Two quadratic equations with real coefficients can not have only one imaginary root common.

526. Statement–1 : The number of values of a for which (a2 – 3a + 2) x2 + (a2 – 5a + b) x + a2 – 4 = 0 is an identity in x is 1. Statement–2 : If ax2 + bx + c = 0 is an identity in x then a = b = c = 0.

527. Let a Î (– ¥, 0). Statement–1 : ax2 – x + 4 < 0 for all x Î R Statement–2 : If roots of ax2 + bx + c = 0, b, c Î R are imaginary then signs ofax2 + bx + c and a are same for all x Î R.

528. Let a, b, c Î R, a ≠ 0. Statement–1 : Difference of the roots of the equation ax2 + bx + c = 0

= Difference of the roots of the equation – ax2 + bx – c = 0 Statement–2 : The two quadratic equations over reals have the same difference of roots if product of the coefficient of the two equations are the same.

529. Statement–1 : If the roots of x5 – 40x4 + Px3 + Qx2 + Rx + S = 0 are in G.P. and sum of their reciprocal is 10, then Statement–2 : x1. x2. x3.x4.x5 = S, where x1, x2, x3, x4, x5 are the roots of given equation.

530. Statement–1 : If 0 < a < , then the equation (x – sin a) (x – cos a) – 2 = 0 has both roots in

(sin a, cos a)Statement–2 : If f(a) and f(b) possess opposite signs then there exist at least one solution of the equation f(x) = 0 in open interval (a, b).

531. Statement–1 : If a 1/2 then a < 1 < p where a , b are roots of equation –x2 + ax + a = 0 Statement–2 : Roots of quadratic equation are rational if discriminant is perfect square.

532. Statement-1 : The number of real roots of |x|2 + |x| + 2 = 0 is zero. Statement-2 : xÎR, |x| 0.

533. Statement-1: If all real values of x obtained from the equation 4x – (a – 3) 2x + (a – 4) = 0 are non-positive, then aÎ (4, 5] Statement-2: If ax2 + bx + c is non-positive for all real values of x, then b2 – 4ac must be –ve or zero and ‘a’ must be –ve.

534. Statement-1: If a , b , c , d Î R such that a < b < c < d, then the equation (x – a) (x – c) + 2(x – b) (x – d) = 0 are real and distinct. Statement-2: If f(x) = 0 is a polynomial equation and a, b are two real numbers such that f(a) f(b) < 0 has at least one real root.

535. Statement-1: f(x) = xÎR

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64

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Statement-2: ax2 + bx + c > 0 xÎR if a > 0 and b2 – 4ac < 0.

536. Statement-1: If a + b + c = 0 then ax2 + bx + c = 0 must have ‘1’ as a root of the equation Statement-2: If a + b + c = 0 then ax2 + bx + c = 0 has roots of opposite sign.

537. Statement-1: ax2 + bx + C = 0 is a quadratic equation with real coefficients, if 2 + is one root then other root can be any other real number. Statement-2: If is a real root of a quadratic equation, then P - is other root only when the coefficients of equation are rational

538. Statement-1: If px2 + qx + r = 0 is a quadratic equation (p, q, rÎR) such that its roots are a, b& p + q + r < 0, p – q + r < 0 & r > 0, then 3[a] + 3[b] = 3, where [] denotes G.I.F. Statement-2: If for any two real numbers a & b, function f(x) is such that f(a).f(b) < 0 f(x) has at least one real root lying between (a, b)

539. Statement-1: If is a root of a quadratic equation then another root of this equation must be Statement-2: If ax2 + bx + c = 0, a, b, c Î Q, having irrational roots then they are in conjugate pairs.

540. Statement-1: If roots of the quadratic equation ax2 + bx + c = 0 are distinct natural number then both roots of the equation cx2 + bx + a = 0 cannot be natural numbers.

Statement-2: If a, b be the roots of ax2 + bx + c = 0 then are the roots of cx2 +

bx + a = 0.

541. Statement-1: The (x – p) (x – r) + (x – q) (x – s) = 0 where p < q < r < s has non real roots if > 0. Statement-2: The equation (p, q, r ÎR) bx2 + qx + r = 0 has non-real roots if q2 – 4pr < 0.

542. Statement-1: One is always one root of the equation (l – m)x2 + (m – n) x + (n – l ) = 0, where l, m, nÎR. Statement-2: If a + b + c = 0 in the equation ax2 + bx + c = 0, then 1 is the one root.

543. Statement-1: If (a2 – 4) x2 + (a2 – 3a + 2) x + (a2 – 7a + 0) = 0 is an identity, then the value of a is 2. Statement-2: If a = b = 0 then ax2 + bx + c = 0 is an identity.

544. Statement-1: x2 + 2x + 3 > 0 x Î RStatement-2: ax2 + bx + c > 0 x Î R if b2 4ac < 0 and a > 0.

545. Statement-1: Maximum value of

Statement-2: Minimum value of ax2 + bx + c (a > 0) occurs at .

Bansal Classes []

65

Bansal Classes] []

546. Statement-1: If quadratic equation ax2+ bx 2 = 0 have non-real roots then a < 0 Statement-2: For the quadratic expression f(x) = ax2 + bx + c if b2 4ac < 0 then f(x) = 0 have non real roots.

547. Statement-1: Roots of equation x5 40x4 + Px3 + Qx2 + Rx + S = 0 are in G.P. and sum of their reciprocal is equal to 10 then |s| = 32. Statement-2: If x1, x2, x, x4 are roots of equation

ax4 + bx3 + cx2 + dx + e = 0 (a ¹ 0)x1 + x2 + x3 + x4 = b/a

548. Statement-1: The real values of a form which the quadratic equation 2x2 – (a3 + 8a – 1) + a2 – 4a = 0. Possesses roots of opposite signs are given by 0 < a < 4. Statement-2: Disc 0 and product of root is < 2

19. 19. SEQUENCE & SERIES SEQUENCE & SERIES

549. Statement–1 : In the expression (x + 1) (x + 2) . . . (x + 50), coefficient of x49 is equal to 1275.

Statement–2 : .

550. Let a, b, c, d are four positive number

Statement–1 :

Statement–2 : .

551. Let a, b, c and d be distinct positive real numbers in H.P. Statement–1 : a + d > b + c

Statement–2 :

552. Let a, r Î R – {0, 1, – 1} and n be an even number. Statement–1 : a. ar. ar2 . . . arn – 1 = (a2 rn – 1)n/2. Statement–2 : Product of kth term from the beginning and from the end in a G.P. is independent of k.

553. Statement–1 : Let p, q, r Î R+ and 27pqr (p + q + r)3 and 3p + 4q + 5r = 12, then p3 + q4 + r5 is equal to 4.

Bansal Classes []

66

Bansal Classes] []

Statement–2 : If A, G, and H are A.M., G.M., and H.M. of positive numbers a1, a2, a3, . . . , an

then H £ G £ A.

554. Statement–1 : The sum of series n.n + (n – 1) (n + 1) + (n – 2) (n + 2) + . . . 1. (2n – 1) is

.

Statement–2 : The sum of any series Sn can be given as, Sn = , where Tr is the general

ten of the series.

555. Statement–1 : P is a point (a, b, c). Let A, B, C be images of P in yz, zx and xy plane

respectively, then equation of plane must be .

Statement–2 : The direction ratio of the line joining origin and point (x, y, z) must be x, y, z.

556. Statement–1 : If A, B, C, D be the vertices of a rectangle in order. The position vector of A, B, C, D be a, b, c, d respectively, then . Statement–2 : In a triangle ABC, let O, G and H be the circumcentre, centroid and orthocentre of the triangle ABC, then OA + OB + OC = OH.

557. Statement-1: 1 + 3 7 + 13 + .... upto n terms =

Statement-2: is HM of a & b if n = -

558. Statement-1: 1111 .... 1 (up to 91 terms) is a prime number

Statement-2: If are in A.P., then are also in A.P.

559. Statement-1: For a infinite G.P. whose first term is ‘a’ and common ratio is r, then where |r| 1Statement-2: A, G, H are arithmetic mean, Geometric mean and harmonic mean of two positive real numbers a & b. Then A, G, H are in G.P.

560. Statement-1: 11 11 …… 1 (up to 91 terms) is a prime number.

Statement-2: If

Are in A.P., then are also in A.P.

561. Statement-1: The sum of all the products of the first n positive integers taken two at a time is

(n – 1) (n + 1) n(3n + 2)

Bansal Classes []

67

Bansal Classes] []

Statement-2: = (a1 + a2 + ... + an)2 – (a12 + a2

2 + an2)

562. Statement-1: Let the positive numbers a, b, c, d, e be in AP, then abcd, abce, abde, acde, bcde are in HP Statement-2: If each term of an A.P. is divided by the same number k, the resulting sequence is also

563. Statement-1: If a, b, c are in G.P., are in H.P.

Statement-2: When we take logarithm of the terms in G.P., they occur in A.P.

564. Statement-1: If 3p + 4q + 5r = 12 then p3q4r5 1 here p, q, r ÎR+ Statement-2: If the quantities are positive then weighted arithmetic mean is greater than or equal to geometric mean.

565. Statement-1: S = 1/4 – 1/2 + 1 – 2 + 22 .... =

Statement-2: Sum of n terms of a G.P. with first term as ‘a’ and common ratio as r in given by

, |r| > 1.

566. Statement-1: -4 + 2 – 1 + 1/2 – 1/4 + ... ¥ is a geometric sequence. Statement-2: Terms of a sequence are positive numebrs.

567. Statement-1: The sum of the infinite A.P. 1 + 2 + 22 + 23 + ….. + ….. is given by

Statement-2: The sum of an infinite G.P. is given by where |r| < 1 a is first term and r is

common ratio.

568. Statement-1: If a1, a2, a3, ….. an are positive real numbers whose product is a fixed number C, then the minimum value of a1 + a2 + ….. + an 1 + 2an is n(2c)1/n.

Statement-2: If a1, a2, a3, ….. an Î R+. then

569. Statement-1: If a(b – c) x2 + b (c – a) x + c(a – b) = 0 has equal roots, then a, b, c are in H.P. Statement-2: Sum of the roots and product of the root are equal

570. Statement-1: for every n > 0

Statement-2: Every sequence whose nth term contains n! in the denominator converges to zero.

571. Statement-1: Sum of an infinite geometric series with common ratio more than one is not possible to find out. Statement-2: The geometric series (Infinite) with common ratio more than one becomes diverging and sum is not fixed.

Bansal Classes []

68

Bansal Classes] []

572. Statement-1: If arithmetic mean of two numbers is 5/2, Geometric mean of the numbers is 2 then harmonic mean will be 8/5. Statement-2: for a group of numbers (GM)2 = (AM) × (HM).

573. Statement-1: If a, b, c, d be four distinct positive quantities in H.P. then a + d > b + c, ad > bc.Statement-2: A.M. > G.M. > H.M.

574. Statement-1: The sum of n arithmetic means between two given numbers is n times the single arithmetic mean between them.Statement-2: nth term of the A.P. with first term a and common difference d is a + (n + 1)d.

575. Statement-1: If a + b + c = 3 a > 0, b > 0, c > 0, then greatest value of a2b3c4 = 31024 77.

Statement-2: If ai > 0 i = 1, 2, 3, ….. n, then

20. TRI , TE & ITF20. TRI , TE & ITF

576. Statement-1: The value of

Statement-2: If x > 0, y > 0, xy > 1, then tan–1x + tan–1y = .

577. Statement-1: is the principal value of

Statement-2: cos–1(cos x) = x if xÎ[0, p]

578. Statement-1: The value of cot–1(–1) is

Statement-2:

579. Statement-1: If then the principal value of sin–1x is

Statement-2: sin–1(sin x) = x xÎR.

580. Statement-1: If A, B, C are the angles of a triangle such that angle A is obtuse then tan B tan C > 1.

Statement-2: In any triangle .

Bansal Classes []

69

Bansal Classes] []

581. Let f() = sin.sin (p/3 + ) . sin (p/3 – ) Statement-1: f() £ 1/4 Statement-2: f() = 1/4 sin2

582. Statement–1 : Number of ordered pairs (, x) satisfying 2sin = ex + e–x, Î[0, 3p] is 2. Statement–2 : Number of values of x for which sin2x + cos4x = 2 is zero.

583. Statement–1 : The number of values of xÎ [0, 4p] satisfying | cosx – sinx| 2 is 2. Statement–2 : |cos (x + p/6)| = 1 number of solutions of | cosx – sinx| 2 is 4

584. Statement–1 : Number of solutions of sin–1 (sinx) = 2p – x; xÎ[3p/2, 5p/2] is 1 Statement–2 : sin–1 (sinx) = x, xÎ [–p/2, p/2]

585. Statement–1 : Number of ordered pairs (x, y) satisfying sin–1x = p – sin–1y and cos–1x + cos–1y = 0 simultaneously is 1 Statement–2 : Ordered pairs (x, y) satisfying sin–1x = p – sin–1y and cos–1x + cos–1y = 0 will lie on x2 + y2 = 2.

586. Statement–1 : The equation k cos x – 3 sin x = k + 1 is solvable only if k belongs to the interval

Statement–2 : .

587. Statement–1 : The equation 2 sec2x – 3 sec x + 1 = 0 has no solution in the interval (0, 2p) Statement–2 : sec x £ – 1 as sec x 1.

588. Statement–1 : The number of solution of the equation is only one. Statement–2 : The number of point of intersection of the two curves y = |sin x| and y = |x| is three.

589. Statement–1 : The equation sin x = 1 has infinite number of solution. Statement–2 : The domain of f(x) = sin x is (– ¥, ¥).

590. Statement–1 : There is no solution of the equation .

Statement–2 : 0 £ and tan2 x + cot2x 2.

591. Statement–1 : The equation sin2x + cos2y = 2 sec2 z is only solvable when sin x = 1, cos y = 1 an sec z = 1 where x, y , z Î R. Statement–2 : Maximum value of sin x and cos y is 1 and minimum value of sec z is 1.

592. Statement–1 : If cot–1x < n, nÎ R then x < cot (n) Statement–2 : cot–1 (x) is an decreasing function.

593. Statement–1 : If sin = a for exactly one value of Î , then a can take infinite value

in the interval [– 1, 1]. Statement–2 : – 1 £ sin £ 1.

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70

Bansal Classes] []

594. Statement–1 : tan 5° is an irrational number. Statement–2 : tan 15° is an irrational number.

595. Let be an acute angle Statement–1 : sin6 + cos6 ≤ 1. Statement–2 : sin + cos ≤ 1

596. Statement–1 : sin is a root of 8x3 – 6x + 1 = 0.

Statement–2 : For any Î R, sin 3 = 3 sin – 4 sin3 .

597. Let f be any one of the six trigonometric functions. Let A, B Î R satisfying f(2A) = f(2B). Statement–1 : A = np + B, for some n Î I. Statement–2 : 2p is one of the period of f.

598. Let x Î [-1, 1]

Statement–1 : 2 sin-1 x = sin-1 .

Statement–2 : - 1 £ 2x .

599. Let f(x) = cos–1 x Statement–1 : f is a decreasing function. Statement–2 : f(– x) = p – f(x).

600. Statement–1 : The total number of 2 real roots of the equation x2 tan x = 1 lies between the interval (0, 2p). Statement–2 : The total number of solution of equation in [0, 2p] is 3.

601. Statement–1 : The number of real solutions of equation sin ex cos ex = 2x – 2 + 2- x – 2 is 0.

Statement–2 : The number of solutions of the eqution 1 + sin x sin2 n [- p, p] is 0.

602. Statement–1 : Equation has 3 real roots.

Statement–2 : the number of real solution of ; x Î is 2.

603. Statement–1 : If

tan-1 = tan-1 , then = .

Statement–2 : The sum of series cos-1 2 + cot-1 8 + cot-1 18 + . . . is .

604. Statement-1: If tan + sec = , 0 < < p, then = p/6 Statement-2: General solution of cos = cosa is = a, if 0 < a < p/2

605. Statement-1: If x < 0, tan-1x + tan-1 = p/2

Statement-2: tan-1x + cos-1x = p/2,xÎR

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71

Bansal Classes] []

606. Statement-1: sin-1 (sin10) = 10 Statement-2: For principal value sin-1 (sinx) = x

607. Statement-1: cos

Statement-2: cos cos2 cos23 .... cos2n-1 = - if = n 2.

TRI

608. Statement-1: sin3 < sin1 < sin2 is true Statement-2: sinx is positive in first and second quadrants.

609. Statement-1: The equation 2sin2x – (P + 3) sinx + (2P – 2) = 0 possesses a real solution if PÎ[-1, 3] Statement-2 : -1 £ sinx £ 1

610. Statement-1: The maximum value of 3sin + 4cos is 5 here ÎR.

Statement-2:: - £ asin + bcos £

611. Statement-1: If A + B + C = p, cosA + cosB + cosC £ 3/2

Statement-2:: If A + B + C = p, sin

612. Statement-1: The maximum & minimum values of the function f(x) =

does not exists. Statement-2: The given function is an unbounded function.

613. Statement-1: If x < 0 tan-1x +tan-1 = p/2

Statement-2: tan-1x + cot-1x = p/2 xÎR.

614. Statement-1: In any triangle square of the length of the bisector AD is bc

Statement-2: In any triangle length of bisector AD =

615. Statement-1: If in a triangle ABC, C = 2acosB, then the triangle is isosceles. Statement-2: Triangle ABC, the two sides are equal i.e. a = b.

616. Statement-1: If the radius of the circumcircle of an isosceles triangle pqR is equal to pq = PR then the angle p = 2p/3. Statement-2: OPQ and oPR will be equilateral i.e., OPq = 60°, OPR = 60°

Bansal Classes []

72

Bansal Classes] []

617. Statement-1: The minimum value of the expression sina + sinb + sin is negative, where a, b, are real numbers such that a + b + = p.

Statement-2: If a, b, are angle of a triangle then sina + sinb + sin = 4cos .

618. Statement-1: If in a triangle sin2A + sin2B + sin2C = 2 then one of the angles must be 90°. Statement-2: In any triangle sin2A + sin2B + sin2C = 2 + 2cosA cosB cosC.

619. Statement-1: If in a ABC a 2c and b 3c then cosB must tend to –1.

Statement-1: In a ABC cosB = .

620. Statement-1: cos(45 A) cos(45 B) sin(45 A) × sin (45 B) = sin(A + B).Statement-2: cos(90 ) = sin .

621. Statement-1: The maximum and minimum values of 7cos + 24sin are 25 and 25 respectively.Statement-2: for all .

622. Statement-1: If

Statement-2:

TE

623. Statement-1: The numbers sin 18° and –sin54° are roots of same quadratic equation with integer coefficients. Statement-2: If x = 18°, then 5x = 90°, if y = -54°, then 5y = -270°.

Inverse Trigonometry

624. Statement-1: The number of solution of the equation cos( is one. Statement-2: cosx = cosa x = 2np a nÎI

625. Statement-1: If sin2x + cos2y + sin2z = 3 where -p/2 £ x £ p/2, 0 £ y £ p, - p/2 £ z £ p/2 then

x = p/2, y = 0, z = p/2. Statement-2: |sinx| £ 1, |cosy| £ 1, |sinz| £ 1 so sin2x + cos2y + sin2z £ 3 where x, y, z ÎR.

626. Statement-1: The number of values of ‘x’ satisfying is zero.

Statement-2: tanx is periodic which period p whereas tan2x is periodic with period p/2.

Inverse Trigonometric Function

Bansal Classes []

73

Bansal Classes] []

627. Statement-1: The range of sin-1x + cos-1x + tan1x is

Statement-2: sin-1x + cos-1x = p/2 for every xÎR.

628. Statement-1: sin-1 (sin10) = 10 Statement-2: sin-1 (sinx) = x for - p/2 £ x £ p/2

629. Statement-1: If sin–1x + sin–1y = , the value of cos–1x + cos–1y is p/3.

Statement-2: sin–1x + cos–1x = p/2 xÎ [–1, 1].

630. Statement-1: 7p/6 is the principal value of cos–1

Statement-2: cos–1 (cosx) = x, if xÎ [0, p].

631. Statement-1: has no solution.

Statement-2: a cos + b sin = c has solution if

632. Statement-1: The equation sin4x + cos4x + sin2x + a = 0 is valid if

Statement-2: If discriminant of a quadratic equation is ve. Then its roots are real.

633. Statement-1: In a ABC cosAcosB + sinAsinBsinC = 1 then ABC must be isosceles as well as right angled triangle.

Statement-2: In a ABC if tanA tanB = k. then k must satisfy k2 6k + 1 0

634. Statement-1: If r1, r2, r3 in a ABC are in H.P. then sides a, b, c are in A.P.

Statement-2:: .

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74

Bansal Classes] []

21. PROPERTIES OF SOLUTION OF TRIANGLES21. PROPERTIES OF SOLUTION OF TRIANGLES

635. In a ABC, a = 6, b = 3 and cos(A – B) = 4/5. Statement–1 : C = p/2

Statement–2 : sinA =

636. The angles of a right angled triangle ABC are in A.P.

Statement–1 : r/R =

Statement–2 :

637. Statement – 1 : If tan-12, tan-13 are two angles of a triangle, then the third angle is p/4

Statement – 2 : tan-1x + tan-1y = p + tan-1 , if x > 0, > 0, xy > 1

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75