MT Chapter 01

LEE KIAN KEONG STPM MATHEMATICS (T) 1: Functions

1 Functions

1. [STPM ]Find the value of x, with 0 < x < 360, which satisfies equation secx+ tanx = 4. Give your answers correct tothe nearest 0.1. [6 marks]

[Answer : 62]

2. [STPM ]Sketch the graph of y = |1 2x|, x R and the graph of y = x, x 0 on the same coordinate system. [3 marks]Solve the inequality |1 2x| > x. [4 marks]

[Answer : {x : 0 x < 14, x > 1}]

3. [STPM ]Function f is defined as

f(x) =

{x(x pi), 0 x < 2pi;pi2 sin(x pi), 2pi x 3pi.

(a) Sketch the graph of f . [4 marks]

(b) Find the range of f . [3 marks]

(c) Determine whether f is a one-to-one function. Give reasons for your answer. [2 marks]

[Answer : (b) {y : pi2 y < 2pi2} ; (c) f is not one-to-one function.]

4. [STPM ]Solve the equation

2 logx 3 log3x =

3

2.

[6 marks]

[Answer : x = 3,1

81]

5. [STPM ]The function f is defined as follows:

f : x 4 + (x 1)2, x R.


(b) State the range of f . [1 marks]

(c) Determine if f1 exist. [2 marks]

[Answer : (b) {y : y 4} ; (c) No]

6. [STPM ]Given that x+ 2 is a factor of f(x) = x3 + (a+ 2b)x2 + (a 3b)x+ 8. Find a in terms of b, and find q(x) so thatf(x) = (x+ 2)q(x) holds for all values of b. [5 marks]

Determine the values of b so that f(x) = 0 has at least two distinct real roots. [6 marks]

Sketch on different diagram, the graph of y = f(x) when b = 65

and b =2

5. [4 marks]

[Answer : a = 7b ; q(x) = x2 (5b+ 2)x+ 4 ; {b : b < 65, b 2

5}]

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7. [STPM ]

Function f is defined by f(x) =1

xwith x R and x 6= 0. Determine the set of values of x so that f(x) > f(x1).

[5 marks]

[Answer : {x : 0 < x < 1}]

8. [STPM ]Given that x3 +mx2 + nx 6 is divisible by x 3 and x+ 2. Find the values of m and n. [5 marks]

[Answer : m = 0, n = 7]

9. [STPM ]Given that f(x) = log2(15 2x x2). Find the range of x so that f(x) is defined. [3 marks]Find the maximum value of 15 2x x2 and hence deduce the maximum value of f(x). [4 marks]

[Answer : {x : 5 < x < 3} ; 16, 4]

10. [STPM ]Express sinx 3 cosx in the form r sin(x), with r > 0 and 0 < < 90, giving the value of correct to thenearest 0.1. Sketch the curve y = sinx 3 cosx for 0 x 360. [8 marks]Find the range of values of x between 0 and 360 which satisfies the inequality sinx 3 cosx 2. [4 marks]Find the largest and the smallest value for

1

sinx 3 cosx+ 5. [3 marks]

[Answer :

10 sin(x 71.6) ; {x : 110.3 < x < 212.9} ; 1510 ,

1

5 +

10]

11. [STPM ]Solve the equation

4x

1x = 3.

[5 marks]

[Answer : x =9

16]

12. [STPM ]Determine the values of k so that the quadratic equation x2 + 2kx + 4k 3 = 0 has two distinct real roots.

[4 marks]

[Answer : {k : k < 1, k > 3}]

13. [STPM ]The function f is defined as follows:

f : x 5x+ 2x 5 , x 6= 5

(a) Find f2 and hence deduce f1. [3 marks]

(b) Find f13(2). [3 marks]

[Answer : (a) f2(x) = x , f1 =5x+ 2

x 5 ; (b) -4]

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14. [STPM ]Show that the roots of x2 + bx+ c = 0, a 6= 0 are given by

x =bb2 4ac

2a.

Deduce that if m+ ni, with m,n R, is a root of this equation, then m ni is another root. [5 marks]

(a) Show that 2+i is a root of f(x) = 0 where f(x) = 2x3 5x2 2x+ 15, and find its other roots. [5 marks](b) Find a polynomial g(x) so that f(x) xg(x) = 15 7x. Express g(x) in the form p(x q)2 + r, with

p, q, r R, find the maximum of 1g(x)

. [5 marks]

[Answer : (a) 2-i , 32

; (b) g(x) = 2

(x 5

4

)2+

15

8,

8

15]

15. [STPM ]The function f is defined by

f(x) =

{2 |x 1|, x < 3,x2 9x+ 18, x 3.


(b) Determine the set of x so that f(x) > 1 x6

. [5 marks]

[Answer : (b) {x : 0 < x < 125, x > 6}]

16. [STPM ]Express 9 sin 6 cos in the form r sin( ), with r > 0 and 0 < < 90. Hence, find the smallest and thelargest value for 9 sin 6 cos 1. [6 marks]

[Answer : 3

13 sin( 33.7) , 3

13 1 , 3

13 1]

17. [STPM ]Given that f(x) = x3 + px2 + 7x+ q where p, q are constants. When x = 1, f (x) = 0. When f(x) is dividedby (x+ 1), the remainder is 16. Find the values of p and q. [4 marks]

(a) Show that f(x) = 0 only has one real roots. Find the set of values of x such that f(x) > 0. [6 marks]

(b) Expressx+ 9

f(x)in partial fraction. [5 marks]

[Answer : p = 5, q = 13 ; (a) {x : x > 1} ; (b) 12(x 1)

x+ 5

2(x2 + 6x+ 13)]

18. [STPM ]

Express1 2x

x2(1 + 2x2)as partial fractions. [5 marks]

[Answer : 2x

+1

x2+

4x 21 + 2x2

]

19. [STPM ]

Express the function f : x |12x 1| + |1

2x + 1|, x R, in the form that does not involve the modulus sign.

Sketch the graph of f and determine its range. [7 marks]

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[Answer : f : x

x, x < 22, 2 x < 2x, x 2

,

R={y : y 2}]

20. [STPM ]Function f is defined by f(x) = x2n (p+ 1)x2 + p, where n and p are positive integers.Show that x 1 is a factor of f(x) for all values of p. [3 marks]When p = 4, x 2 is a factor of f(x). Find the value of n and factorise f(x) completely. [5 marks]With the value of n you have obtained, find the set of values of p such that f(x) + 2x2 2 = 0 has roots whichare distinct and real. [7 marks]

[Answer : n = 2 , (x 2)(x+ 2)(x 1)(x+ 1) ; {p : p > 2, p 6= 3}]

21. [STPM ]Solve the simultaneous equations

log4(xy) =1

2, (log2 x)(log2 y) = 2.

[6 marks]

[Answer : x =1

2, y = 4, x = 4, y =

1

2]

22. [STPM ]The functions f and g are defined by

f : x 2x, x R;g : x cosx | cosx|,pi x pi.

(a) Find the composite function f g and state its domain and range. [4 marks](b) Show, by definition, that f g is an even function. [2 marks](c) Sketch the graph of f g. [2 marks]

[Answer : (a) f g : x 2(cosx | cosx|), D = {x : pi x pi}, R = {y : 4 y 0}]

23. [STPM ]

The function f is defined by f : x 3x+ 1, x R, x 13

.

Find f1 and state its domain and range. [4 marks]

[Answer : f1 : x x2 13

, Df1 = {x : x 0}, Rf1 = {x : x 13}]

24. [STPM ]

Express

59 24

6 as p

2 + q

3 where p and q are integers. [7 marks]

[Answer : 4

2 3

3]

25. [STPM ]Show that polynomial 2x3 9x2 + 3x+ 4 has x 1 as factor. [2 marks]Hence,

(a) find all the real roots of 2x6 9x4 + 3x2 + 4 = 0. [5 marks](b) determine the set of values of x so that 2x3 9x2 + 3x+ 4 < 12 12x. [6 marks]

[Answer : x = 1, x = 1, x = 2, x = 2 ; x < 1]

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26. [STPM ]

Express cosx+

3 sinx in the form r cos(x ), with r > 0 and 0 < < pi2

. [4 marks]

Hence, find the set of values of x with 0 x 2pi, which satisfies the inequality 0 < cosx+

3 sinx < 1.[5 marks]

[Answer : 2 cos(x pi

3

), {x : 2pi

3< x

2 + 1x. [4 marks]

[Answer : {x : x < 2

5}]

30. [STPM ]Express cos + 3 sin in the form r cos( ), where r > 0 and 0 < < 90. [4 marks]

[Answer :

10 cos( 71.6)]

31. [STPM ]Find all values of x, where 0 < x < 360, which satisfy the equation tanx+ 4 cotx = 4 secx. [5 marks]

[Answer : 41.8, 138.2]

32. [STPM ]

Find the solution set of inequality |x 2| < 1x

where x 6= 0. [7 marks]

[Answer : {x : 0 < x < 1 +

2, x 6= 1}]

33. [STPM ]The functions f and g are given by

f(x) =ex exex + ex

and g(x) =2

ex + ex.

(a) State the domains of f and g, [1 marks]

(b) Without using differentiation, find the range of f , [4 marks]

(c) Show that f(x)2 + g(x)2 = 1. Hence, find the range of g. [6 marks]

[Answer : (a) {x : x R}, {x : x R} ; (b) {y : 1 < y < 1} ; (c) {y : 0 < y 1}]

34. [STPM ]

Express2x+ 1

(x2 + 1)(2 x) in the formAx+B

x2 + 1+

C

2 x where A, B and C are constants. [3 marks]

[Answer :x

x2 + 1+

1

2 x ]

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35. [STPM ]Functions f , g and h are defined by

f : x xx+ 1

; g : x x+ 2x

; h : x 3 + 2x.

(a) State the domains of f and g. [2 marks]

(b) Find the composite function g f and state its domain and range. [5 marks](c) State the domain and range of h. [2 marks]

(d) State whether h = g f . Give a reason for your answer. [2 marks]

[Answer : (a) {x : x Rx 6= 1}, {x : x R, x 6= 0} ;(b) g f(x) = 3 + 2

x, D={x : x R, x 6= 0, x 6= 1}, R={y : y R, y 6= 1, y 6= 3} ;

(c) D={x : x R, x 6= 0}, R={y : y R, y 6= 3} ; (d) h 6= g f ]

36. [STPM ]The polynomial p(x) = x4 + ax3 7x2 4ax + b has a factor x + 3 and when divided by x 3, has remainder60. Find the values of a and b and factorise p(x) completely. [9 marks]

Using the substitution y =1

x, solve the equation 12y4 8y3 7y2 + 2y + 1 = 0. [3 marks]

[Answer : a = 2, b = 12, (x+ 3)(x 1)(x+ 2)(x 2) , y = 13, 1,1

2,

1

2]

37. [STPM ]Express 4 sin 3 cos in the form R sin( ), where R > 0 and 0 < < 90. Hence, solve the equation4 sin 3 cos = 3 for 0 < < 360. [6 marks]

[Answer : 5 sin( 36.9), = 73.7, 180.0]

38. [STPM ]Find the domain and the range of the function f defined by

f(x) = sin12(x 1)x+ 1

.

[4 marks]

Sketch the graph of f . [3 marks]

[Answer : D={x : 13 x 3} , R={y : pi

2 y pi

2}]

39. [STPM ]

If loga

( xa2

)= 3 loga 2 loga(x 2a), express x in terms of a. [6 marks]

[Answer : x = 4a]

40. [STPM ]Simplify

(a)(

73)22(

7 +

3), [3 marks]

[Answer : 2

7 3

3]

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41. [STPM ]Find the constants A, B, C and D such that

3x2 + 5x

(1 x2)(1 + x)2 =A

1 x +B

1 + x+

C

(1 + x)2+

D

(1 + x)3.

[8 marks]

[Answer : A = 1, B = 1, C = 1, D = 1]

42. [STPM ]

Using the substitution y = x+1

x, express f(x) = x3 4x 6 4

x+

1

x3as a polynomial in y. [3 marks]

Hence, find all the real roots of the equation f(x) = 0. [10 marks]

[Answer : y3 7y 6 ; x = 1, 3

5

2]

43. [STPM ]Find, in terms of pi, all the values of x between 0 and pi which satisfies the equation

tanx+ cotx = 8 cos 2x.

[4 marks]

[Answer :1

24pi,

5

24pi,

12

24pi,

17

24pi]

44. [STPM ]The function f and g are defined by

f : x 1x, x R \ {0};

g : x 2x 1, x R.Find f g and its domain. [4 marks]

[Answer : f g(x) = 12x 1 , D={x : x R, x 6=

1

2}]

45. [STPM ]

The polynomial p(x) = 2x3 + 4x2 +1

2x k has factor (x+ 1).

(a) Find the value of k. [2 marks]

(b) Factorise p(x) completely. [4 marks]

[Answer : (a) k = 32

; (b)1

2(x+ 1)(2x+ 3)(2x 1)]

46. [STPM ]

Find the solution set of the inequality

4x 1 > 3 3x. [10 marks]

[Answer : {x : 0 < x < 1, 1 < x < 3}]

47. [STPM ]

If t = tan

2, show that sin =

2t

1 + t2and cos =

1 t21 + t2

. [4 marks]

Hence, find the values of between 0 and 360 that satisfy the equation 10 sin 5 cos = 2. [3 marks]

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[Answer : = 36.9, 196.3]

48. [STPM ]

Determine the set of values of x satisfying the inequalityx

x+ 1 1x+ 1

. [4 marks]

[Answer : {x : x < 1, x 1}]

49. [STPM ]

Given that loga(3x 4a) + loga 3x =2

log2 a+ loga(1 2a), where 0 < a 0 and 0 < < 90. Hence, find the maximum andminimum values of the expression

1

5 sin + 12 cos + 15.

[7 marks]

[Answer : 13 sin( + 67.4) ,1

2,

1

28]

57. [STPM ]Solve the equation lnx+ ln(x+ 2) = 1. [4 marks]

[Answer : 1 +1 + e]

58. [STPM ]Find the set of values of x satisfying the inequality 2x 1 |x+ 1|. [6 marks]

[Answer : {x : x 2}]

59. [STPM ]Functions f and g are defined by

f : x x2x 1 for x 6=

1

2;

g : x ax2 + bx+ c, where a, b and c are constants.(a) Find f f , and hence, determine the inverse function of f . [4 marks]

(b) Find the values of a, b and c if g f(x) = 3x2 + 4x 1

(2x 1)2 . [4 marks]

(c) Given that p(x) = x2 2, express h(x) = x2 2

2x2 5 in terms of f and p. [2 marks]

[Answer : (a) f f(x) = x , f1(x) = x2x 1 ; (b) a = 1, b = 0, c = 1 ; (c) h = f p]

60. [STPM ]The polynomial p(x) = ax3 + bx2 4x + 3, where a and b are constants, has a factor (x + 1). When p(x) isdivided by (x 2), it leaves a remainder of 9.(a) Find the values of a and b, and hence, factorise p(x) completely. [6 marks]

(b) Find the set of values of x which satisfiesp(x)

x 3 0. [4 marks]

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(c) By completing square, find the minimum value ofp(x)

x 3 , x 6= 3, and the value of x at which it occurs.[4 marks]

[Answer : (a) a = 2, b = 5, (x 3)(2x 1)(x+ 1) ; (b) {x : x 1, 12 x < 3, x > 3} ; (c) Minimum value=9

8,

x = 14

]

61. [STPM ]

The expression cosx

3 sinx may be written in the form r cos(x + ) for all values of x, where r is positiveand is a acute.

(a) Determine the values of r and . [3 marks]

(b) State the minimum and maximum values of cosx

3 sinx, and determine the corresponding values of xin the interval 0 x 2pi. [3 marks]

(c) Sketch the curve y = cosx

3 sinx for 0 x 2pi. [3 marks]By drawing an appropriate line on the graph, determine the number of roots of the equation

cosx

3 sinx =

(3

4pi

)x

in the interval 0 x 2pi. [3 marks](d) Solve the equation cosx

3 sinx = 1 for 0 x 2pi. [3 marks]

[Answer : (a) r = 2, =pi

3; (b) minimum=-2 when x =

2pi

3, maximum=2 when x =

5pi

3; (c) 3 roots ; (d)

pi

3, pi]

62. [STPM ]Given that 2 x x2 is a factor of p(x) = ax3 x2 + bx 2. Find the values of a and b. Hence, find the set ofvalues of x for which p(x) is negative. [6 marks]

[Answer : a = 2, b = 5 , {x : 2 < x < 12, x > 1}]

63. [STPM ]Functions f and g f are defined by f(x) = ex+2 and (g f)(x) = x, for all x 0.(a) Find the function g, and state its domain. [5 marks]

(b) Determine the value of (f g)(e3). [2 marks]

[Answer : (a) g(x) =

lnx 2 , D={x : x e2} ; (b) e3]

64. [STPM ]

Solve the simultaneous equations log9

(x

y

)=

3

4and (log3 x)(log3 y) = 1. [8 marks]

[Answer : x = 9, y =

3 or x =

3

3, y =

1

9]

65. [STPM ]The function f is defined by

f : x x2 x, for x 12.

(a) Find f1, and state its domain. [4 marks]

(b) Find the coordinates of the point of intersection of graph f and f1. [3 marks]

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(c) Sketch, on the same coordinates axes, the graph of f and f1. [3 marks]

[Answer : (a) f1(x) =1

2+

x+

1

4, D={x : x 1

4} ; (b) (2,2)]

66. [STPM ]Sketch a graph of y = cos 2 in the range of 0 pi. [2 marks]Hence, find the set of values of , where 0 pi, satisfying the inequality 4 sin2 2

3. [5 marks]

[Answer : {x : pi12 x 11pi

12}]

67. [STPM ]The functions f and g are defined by

f : x 7 e2x, x R;g : x 7 (lnx)2, x > 0.

(a) Find f1 and state its domain. [3 marks]

(b) Show that g

(1

2

)= g(2), and state, with a reason, whether g has an inverse. [4 marks]

[Answer : (a) f1 =1

2lnx , D={x : x > 0}]

68. [STPM ]

Express cosx + sinx in the form r cos(x ), where r > 0 and 0 < < 12pi. Hence, find the minimum and

maximum values of cosx+ sinx and the corresponding values of x in the interval 0 x 2pi. [7 marks]

(a) Sketch the graph of y = cosx+ sinx for 0 x 2pi. [3 marks](b) By drawing appropriate lines on your graph, determine the number of roots in the interval 0 x 2pi of

each of the following equations.

i. cosx+ sinx = 12

, [1 marks]

ii. cosx+ sinx = 2, [1 marks]

(c) Find the set of values of x in the interval 0 x 2pi for which | cosx+ sinx| > 1. [3 marks]

[Answer :

2 cos(x pi4

) , minimum value=

2 when x =5pi

4, maximum value=

2 when x =

pi

4; (b) (i) two roots ,

(ii) no roots ; (c) {x : 0 < x < pi2, pi < x 0} ; (c) No]

70. [STPM ]

Sketch the graph of y = sin 2x in the range 0 x pi. Hence, solve the inequality | sin 2x| < 12

, where 0 x pi.[6 marks]

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[Answer : {x : 0 x < pi12,

5pi

12< x 3x+ 1. [3 marks]

[Answer : (a) h = 2, k = 1 ; (b) (x2 1)(2x2 + x+ 2) + (3x+ 1) ;(c) q(x) = 2

(x+

1

4

)2+

15

8(ii) minimum value=

15

8, when x = 1

4; (d) {x : x < 1, x > 1}]

72. [STPM ]

The function f is defined as f(x) =1

2

(ex ex), where x R.

(a) Show that f has an inverse. [3 marks]

(b) Find the inverse function of f , and state its domain. [7 marks]

[Answer : (b) f1(x) = ln(x+x2 + 1), D={x : x R}]

73. [STPM ]Sketch, on the same axes, the graphs of y = |2x+1| and y = 1x2. Hence, solve the inequality |2x+1| 1x2.

[8 marks]

[Answer : {x : x 1

3, x 0}]

74. [STPM ]

Determine the set of values of x satisfying the inequality x+ 4 3x

. [6 marks]

[Answer : {x : x 2

7, 0 < x 2 +

7}]

75. [STPM ]Functions f and g are defined by

f(x) = x2 + 4x+ 2, x R,

g(x) =3

x+ 3, x 6= 3, x R

(a) Sketch the graph of f , and find its range. [4 marks]

(b) Sketch the graph of g, and show that g is a one-to-one function. [3 marks]

(c) Give a reason why g1 exists. Find g1, and state its domain. [4 marks]

(d) Give a reason why g f exists. Find g f , and state its domain. [4 marks]

[Answer : (a) R={y : y 2} ; (c) g1(x) = 3x 3 , D={x : x R, x 6= 0} ;

(d) g f(x) = 2x2 + 4x+ 5

, D={x : x R}]

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1 Functions2 Sequences and Series3 Matrices4 Complex Numbers5 Analytic Geometry6 Vectors7 Limits and Continuity8 Differentiation9 Integration10 Differential Equations11 Maclaurin Series12 Numerical Methods13 Data Description14 Probability15 Probability Distributions16 Sampling and Estimation17 Hypothesis Testing18 Chi-squared Tests

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MT Chapter 01