New STPM Mathematics (T) Chapter Past Year Question
March 6, 2017 | Author: Lee Kian Keong | Category: N/A
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NEW STPM Mathematics (T) Topical Past Year Questions Compiled by: KK LEE (Lee Kian Keong) August 8, 2016
Abstract
This documents contains all the questions from STPM Past Year Papers and i sorted all the questions according to the chapters (New STPM syllabus). Download the file from scribd is not allowed. If you willing to download the file, please contact me by facebook me or download directly from my website http://kkleemaths.com. I used more than 5 years to make this file. Please appreciate my hard work.
Contents
1 Functions
2 Sequences and Series 3 Matrices
4 Complex Numbers 5 Analytic Geometry 6 Vectors
7 Limits and Continuity 8 Differentiation 9 Integration
10 Differential Equations 11 Maclaurin Series
12 Numerical Methods 13 Data Description 14 Probability
15 Probability Distributions 16 Sampling and Estimation 17 Hypothesis Testing 18 Chi-squared Tests
2 15 24 33 36 38 44 48 64 79 89 96 99
120
131 166 175 178
1
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1: Functions
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Functions
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1
STPM MATHEMATICS (T)
1. [STPM ] Find the value of x, with 0◦ < x < 360◦ , which satisfies equation sec x + tan x = 4. Give your answers correct to the 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. Solve the inequality |1 − 2x| >
√
x.
3. [STPM ] Function f is defined as
[3 marks] [4 marks]
1 4
[Answer : {x : 0 ≤ x < , x > 1}]
( x(x − π), 0 ≤ x < 2π; f (x) = 2 π sin(x − π), 2π ≤ x ≤ 3π.
(a) Sketch the graph of f . (b) Find the range of f .
[4 marks] [3 marks]
(c) Determine whether f is a one-to-one function. Give reasons for your answer.
[Answer : (b) {y : π 2 ≤ y < 2π 2 } ; (c) f is not one-to-one function.]
4. [STPM ] Solve the equation
2 logx 3 − log3
5. [STPM ] The function f is defined as follows:
√
3 x= . 2
[6 marks]
[Answer : x = 3,
f : x → 4 + (x − 1)2 , x ∈ R.
(a) Sketch the graph of f . (b) State the range of f . (c) Determine if f
[2 marks]
−1
exist.
1 ] 81
[2 marks] [1 marks] [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 that f (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 2 Sketch on different diagram, the graph of y = f (x) when b = − and b = . 5 5 2
[6 marks]
[4 marks]
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STPM MATHEMATICS (T)
1: Functions 6 5
2 5
Function f is defined by f (x) = f (x) > f (x − 1).
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7. [STPM ]
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[Answer : a = −7b ; q(x) = x2 − (5b + 2)x + 4 ; {b : b < − , b ≥ }]
1 with x ∈ R and x 6= 0. Determine the set of values of x so that x [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. 2
[3 marks]
Find the maximum value of 15 − 2x − x and hence deduce the maximum value of f (x).
[4 marks]
[Answer : {x : −5 < x < 3} ; 16, 4]
10. [STPM ] Express sin x − 3 cos x in the form r sin(x − α), with r > 0 and 0◦ < α < 90◦ , giving the value of α correct to the nearest 0.1◦ . Sketch the curve y = sin x − 3 cos x for 0◦ ≤ x ≤ 360◦ . [8 marks] Find the range of values of x between 0◦ and 360◦ which satisfies the inequality sin x − 3 cos x ≥ 2. Find the largest and the smallest value for [Answer :
11. [STPM ] Solve the equation
√
1 . sin x − 3 cos x + 5
[4 marks] [3 marks]
10 sin(x − 71.6◦ ) ; {x : 110.3◦ < x < 212.9◦ } ;
s
4x √ = 3. 1− x
5−
1 √
10
,
5+
1 √
10
]
[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]
13. [STPM ] The function f is defined as follows:
f :x→
5x + 2 , x 6= 5 x−5
(a) Find f 2 and hence deduce f −1 .
[Answer : {k : k < 1, k > 3}]
[3 marks]
3
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STPM MATHEMATICS (T)
1: Functions
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[3 marks]
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(b) Find f 13 (2). [Answer : (a) f 2 (x) = x , f −1 (x) =
14. [STPM ] Show that the roots of x2 + bx + c = 0, a 6= 0 are given by √ −b ± b2 − 4ac x= . 2a
5x + 2 ; (b) −4] x−5
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, 1 . [5 marks] with p, q, r ∈ R, find the maximum of g(x) 3 2
[Answer : (a) 2-i , − ; (b) g(x) = 2 x −
15. [STPM ] The function f is defined by
( 2 − |x − 1|, f (x) = x2 − 9x + 18,
(a) Sketch the graph of f .
x < 3, x ≥ 3.
5 4
2
+
15 8 , ] 8 15
[5 marks]
x (b) Determine the set of x so that f (x) > 1 − . 6
[5 marks]
[Answer : (b) {x : 0 < x <
12 , x > 6}] 5
16. [STPM ] Express 9 sin θ − 6 cos θ in the form r sin(θ − α), with r > 0 and 0◦ < α < 90◦ . Hence, find the smallest and the largest value for 9 sin θ − 6 cos θ − 1. [6 marks] √
√
√
[Answer : 3 13 sin(θ − 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 0 (x) = 0. When f (x) is divided by (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. x+9 (b) Express in partial fraction. f (x)
[Answer : p = 5, q = −13 ; (a) {x : x > 1} ; (b)
4
[6 marks]
[5 marks]
x+5 1 − ] 2(x − 1) 2(x2 + 6x + 13)
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STPM MATHEMATICS (T)
1: Functions
19. [STPM ]
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1 − 2x as partial fractions. x2 (1 + 2x2 )
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Express
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18. [STPM ] [5 marks]
2 x
[Answer : − +
1 4x − 2 + ] 2 x 1 + 2x2
1 1 Express the function f : x → | x − 1| + | x + 1|, x ∈ R, in the form that does not involve the modulus 2 2 sign. Sketch the graph of f and determine its range. [7 marks] −x, [Answer : f : x → 2, x,
x < −2 −2 ≤ x < 2 , 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]
2
With the value of n you have obtained, find the set of values of p such that f (x) + 2x − 2 = 0 has roots which are 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
1 log4 (xy) = , (log2 x)(log2 y) = −2. 2
22. [STPM ] The functions f and g are defined by
[6 marks]
f : x → 2x, x ∈ R;
g : x → cos x − | cos x|, −π ≤ x ≤ π.
(a) Find the composite function f ◦ g and state its domain and range. (b) Show, by definition, that f ◦ g is an even function. (c) Sketch the graph of f ◦ g.
1 2
[4 marks]
[2 marks] [2 marks]
[Answer : (a) f ◦ g : x → 2(cos x − | cos x|), D = {x : −π ≤ x ≤ π}, R = {y : −4 ≤ y ≤ 0}]
23. [STPM ]
√
1 3x + 1, x ∈ R, x ≥ − . 3 and state its domain and range.
The function f is defined by f : x → Find f −1
1 2
[Answer : x = , y = 4, x = 4, y = ]
5
[4 marks]
STPM MATHEMATICS (T)
x2 − 1 1 , Df −1 = {x : x ≥ 0}, Rf −1 = {x : x ≥ − }] 3 3
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[Answer : f −1 : x →
1: Functions
24. [STPM ]q √ √ √ Express 59 − 24 6 as p 2 + q 3 where p and q are integers.
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[7 marks]
√
√
[Answer : 4 2 − 3 3]
25. [STPM ] Show that polynomial 2x3 − 9x2 + 3x + 4 has x − 1 as factor. Hence,
[2 marks]
(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]
26. [STPM ] √ π Express cos x + 3 sin x in the form r cos(x − α), with r > 0 and 0 < α < . [4 marks] 2 √ Hence, find the set of values of x with 0 ≤ x ≤ 2π, which satisfies the inequality 0 < cos x+ 3 sin x < 1. [5 marks]
[Answer : 2 cos x −
5π 11π π 2π 2 + . x
[4 marks]
[4 marks]
√
[Answer : {x : x < 2 − 5}]
30. [STPM ] Express cos θ + 3 sin θ in the form r cos(θ − α), where r > 0 and 0◦ < α < 90◦ .
[4 marks]
[Answer :
√
10 cos(θ − 71.6◦ )]
31. [STPM ] Find all values of x, where 0◦ < x < 360◦ , which satisfy the equation tan x + 4 cot x = 4 sec x. [5 marks]
6
[Answer : 41.8◦ , 138.2◦ ]
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STPM MATHEMATICS (T)
1: Functions
33. [STPM ] The functions f and g are given by f (x) =
1 where x 6= 0. x
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Find the solution set of inequality |x − 2| <
ex − e−x ex + e−x
(a) State the domains of f and g,
[7 marks]
√
[Answer : {x : 0 < x < 1 + 2, x 6= 1}]
and g(x) =
ex
2 . + e−x
[1 marks]
(b) Without using differentiation, find the range of f , 2
2
[4 marks]
(c) Show that f (x) + g(x) = 1. Hence, find the range of g.
(x2
[6 marks]
[Answer : (a) {x : x ∈ R}, {x : x ∈ R} ; (b) {y : −1 < y < 1} ; (c) {y : 0 < y ≤ 1}]
34. [STPM ] Express
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32. [STPM ]
Ax + B C 2x + 1 in the form 2 + where A, B and C are constants. + 1)(2 − x) x +1 2−x
35. [STPM ] Functions f , g and h are defined by f :x→
x ; x+1
(a) State the domains of f and g.
g:x→
x+2 ; x
[Answer :
h:x→3+
(b) Find the composite function g ◦ f and state its domain and range. (c) State the domain and range of h.
(d) State whether h = g ◦ f . Give a reason for your answer.
2 . x
[3 marks]
1 x + ] x2 + 1 2 − x
[2 marks]
[5 marks]
[2 marks]
[2 marks]
[Answer : (a) {x : x ∈ R, x 6= −1}, {x : x ∈ R, x 6= 0} ; 2 (b) g ◦ f (x) = 3 + , D={x : x ∈ R, x 6= 0, x 6= −1}, R={y : y ∈ R, y 6= 1, y 6= 3} ; x (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 remainder 60. Find the values of a and b and factorise p(x) completely. [9 marks] 1 Using the substitution y = , solve the equation 12y 4 − 8y 3 − 7y 2 + 2y + 1 = 0. [3 marks] x 1 3
1 1 2 2
[Answer : a = 2, b = 12, (x + 3)(x − 1)(x + 2)(x − 2) , y = − , 1, − , ]
37. [STPM ] Express 4 sin θ − 3 cos θ in the form R sin(θ − α), where R > 0 and 0◦ < α < 90◦ . Hence, solve the equation 4 sin θ − 3 cos θ = 3 for 0◦ < α < 360◦ . [6 marks] 7
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STPM MATHEMATICS (T)
1: Functions
38. [STPM ] Find the domain and the range of the function f defined by
Sketch the graph of f .
f (x) = sin−1
2(x − 1) . x+1
[4 marks] [3 marks]
[Answer : D={x :
π π 1 ≤ x ≤ 3} , R={y : − ≤ y ≤ }] 3 2 2
39. [STPM] x If loga 2 = 3 loga 2 − loga (x − 2a), express x in terms of a. a
40. [STPM ] Simplify √ √ ( 7 − 3)2 √ , (a) √ 2( 7 + 3)
[6 marks]
[Answer : x = 4a]
[3 marks]
√
√
[Answer : 2 7 − 3 3]
41. [STPM ] Find the constants A, B, C and D such that
42. [STPM ]
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[Answer : 5 sin(θ − 36.9◦ ), θ = 73.7◦ , 180.0◦ ]
A B C D 3x2 + 5x = + + + . 2 2 2 (1 − x )(1 + x) 1 − x 1 + x (1 + x) (1 + x)3
[8 marks]
[Answer : A = 1, B = 1, C = −1, D = −1]
1 4 1 Using the substitution y = x + , express f (x) = x3 − 4x − 6 − + 3 as a polynomial in y. [3 marks] x x x Hence, find all the real roots of the equation f (x) = 0. [10 marks] √ 3± 5 ] [Answer : y − 7y − 6 ; x = −1, 2 3
43. [STPM ] Find, in terms of π, all the values of x between 0 and π which satisfies the equation tan x + cot x = 8 cos 2x.
[4 marks]
[Answer :
8
1 5 12 17 π, π, π, π] 24 24 24 24
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44. [STPM ] The function f and g are defined by
Find f ◦ g and its domain.
45. [STPM ]
f :x→
1: Functions
1 , x ∈ R \ {0}; x
g : x → 2x − 1, x ∈ R.
[4 marks]
[Answer : f ◦ g(x) =
1 1 , D={x : x ∈ R, x 6= }] 2x − 1 2
1 The polynomial p(x) = 2x3 + 4x2 + x − k has factor (x + 1). 2 (a) Find the value of k. (b) Factorise p(x) completely.
46. [STPM ]
[2 marks] [4 marks]
3 2
1 2
[Answer : (a) k = − ; (b) (x + 1)(2x + 3)(2x − 1)]
4 > 3 − 3. Find the solution set of the inequality x − 1 x
47. [STPM ]
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STPM MATHEMATICS (T)
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[10 marks]
[Answer : {x : 0 < x < 1, 1 < x < 3}]
θ 2t 1 − t2 If t = tan , show that sin θ = and cos θ = . [4 marks] 2 1 + t2 1 + t2 Hence, find the values of θ between 0◦ and 360◦ that satisfy the equation 10 sin θ − 5 cos θ = 2.[3 marks]
48. [STPM ]
Determine the set of values of x satisfying the inequality
49. [STPM ]
Given that loga (3x − 4a) + loga 3x =
x 1 ≥ . x+1 x+1
[Answer : θ = 36.9◦ , 196.3◦ ]
[4 marks]
[Answer : {x : x < −1, x ≥ 1}]
2 1 + loga (1 − 2a), where 0 < a < , find x. log2 a 2
50. [STPM ] Find the values of x if y = |3 − x| and 4y − (x2 − 9) = −24.
[7 marks]
[Answer :
2 ] 3
[9 marks]
[Answer : x = 7, x = −9]
51. [STPM ] The polynomial p(x) = 6x4 − ax3 − bx2 + 28x + 12, where a and b are real constants, has factors (x + 2) and (x − 2). 9
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STPM MATHEMATICS (T)
1: Functions
(a) Find the values of a and b, and hence, factorise p(x) completely. 3
[7 marks]
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(b) Give that p(x) = (2x − 3)[q(x) − 41 + 3x ], find q(x), and determine its range when x ∈ [−2, 10]. [8 marks]
[Answer : (a) a = 7, b = 27 , (x + 2)(x − 2)(2x − 3)(3x + 1) ; (b) q(x) = (x − 6)2 + 1 , R=[1,65]]
52. [STPM ] Find the values of x, where 0 ≤ x ≤ π, which satisfy the equation sin3 x sec x = 2 tan x.
[Answer : x = 0, π]
53. [STPM ] Solve the following simultaneous equations:
log3 (xy) = 5
54. [STPM ] The graph of a function f is as follows:
(a) State the domain and range of f .
and
log9
[4 marks]
x2 y
= 2.
[4 marks]
[Answer : x = 27, y = 9]
[2 marks]
(b) State whether f is a one-to-one function or not. Give a reason for your answer.
[2 marks]
[Answer : (a) D={x : −3 ≤ x < −1 − 1 < x ≤ 2} , R={y : −1 < y < 2} ; (b) f is not one to one function.]
55. [STPM ] The polynomial p(x) = 2x4 − 7x3 + 5x2 + ax + b, where a and b are real constants, is divisible by 2x2 + x − 1. (a) Find a and b.
(b) For these values of a and b, determine the set of values of x such that p(x) ≤ 0.
[4 marks] [4 marks]
1 2
[Answer : (a) a = 9, b = −5 ; (b) {x : −1 ≤ x ≤ }] 10
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STPM MATHEMATICS (T)
1: Functions
1 . 5 sin θ + 12 cos θ + 15
57. [STPM ] Solve the equation ln x + ln(x + 2) = 1.
[7 marks]
[Answer : 13 sin(θ + 67.4◦ ) ,
f :x→
√
[Answer : −1 + 1 + e]
[6 marks]
[Answer : {x : x ≤ 2}]
1 x for x 6= ; 2x − 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 . −3x2 + 4x − 1 (b) Find the values of a, b and c if g ◦ f (x) = . (2x − 1)2 x2 − 2 in terms of f and p. (c) Given that p(x) = x2 − 2, express h(x) = 2 2x − 5 [Answer : (a) f ◦ f (x) = x , f −1 (x) =
1 1 , ] 2 28
[4 marks]
58. [STPM ] Find the set of values of x satisfying the inequality 2x − 1 ≤ |x + 1|.
59. [STPM ] Functions f and g are defined by
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56. [STPM ] Express 5 sin θ + 12 cos θ in the form r sin(θ + α), where r > 0 and 0◦ < α < 90◦ . Hence, find the maximum and minimum values of the expression
[4 marks]
[4 marks]
[2 marks]
x ; (b) a = 1, b = 0, c = −1 ; (c) h = f ◦ p] 2x − 1
60. [STPM ] The polynomial p(x) = ax3 + bx2 − 4x + 3, where a and b are constants, has a factor (x + 1). When p(x) is divided 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] p(x) (b) Find the set of values of x which satisfies ≥ 0. [4 marks] x−3 p(x) (c) By completing square, find the minimum value of , x 6= 3, and the value of x at which it x−3 occurs. [4 marks] [Answer : (a) a = 2, b = −5, (x − 3)(2x − 1)(x + 1) ; (b) {x : x ≤ −1,
1 ≤ x < 3, x > 3} ; (c) Minimum 2 9 1 value=− , x = − ] 8 4
61. [STPM ] √ The expression cos x − 3 sin x may be written in the form r cos(x + α) for all values of x, where r is positive and α is a acute. 11
STPM MATHEMATICS (T)
(a) Determine the values of r and α.
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(b) State the minimum and maximum values of cos x − values of x in the interval 0 ≤ x ≤ 2π. √ (c) Sketch the curve y = cos x − 3 sin x for 0 ≤ x ≤ 2π.
√
1: Functions [3 marks]
3 sin x, and determine the corresponding
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[3 marks] [3 marks]
By drawing an appropriate line on the graph, determine the number of roots of the equation √ 3 x cos x − 3 sin x = 4π in the interval 0 ≤ x ≤ 2π. (d) Solve the equation cos x − [Answer : (a) r = 2, α =
√
[3 marks]
3 sin x = −1 for 0 ≤ x ≤ 2π.
[3 marks]
π 2π 5π π ; (b) minimum=-2 when x = , maximum=2 when x = ; (c) 3 roots ; (d) , π] 3 3 3 3
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 of values of x for which p(x) is negative. [6 marks] 1 2
[Answer : a = −2, b = 5 , {x : −2 < x < , 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. 3
(b) Determine the value of (f ◦ g)(e ).
64. [STPM ]
[5 marks]
[2 marks]
[Answer : (a) g(x) =
√
ln x − 2 , D={x : x ≥ e2 } ; (b) e3 ]
3 x Solve the simultaneous equations log9 = and (log3 x)(log3 y) = 1. y 4
65. [STPM ] The function f is defined by
(a) Find f −1 , and state its domain.
[8 marks]
[Answer : x = 9, y =
1 f : x → x2 − x, for x ≥ . 2
[Answer : (a) f
−1
66. [STPM ] Sketch a graph of y = cos 2θ in the range of 0 ≤ θ ≤ π.
√
3 or x =
−1
1 (x) = + 2
−1
.
.
[3 marks]
[3 marks]
r x+
1 1 , D={x : x ≥ − } ; (b) (2,2)] 4 4
Hence, find the set of values of θ, where 0 ≤ θ ≤ π, satisfying the inequality 4 sin2 θ ≥ 2 − 12
1 3 ,y = ] 3 9
[4 marks]
(b) Find the coordinates of the point of intersection of graph f and f (c) Sketch, on the same coordinates axes, the graph of f and f
√
√
[2 marks]
3.[5 marks]
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STPM MATHEMATICS (T)
1: Functions
f : x 7→ e2x , x ∈ R;
g : x 7→ (ln x)2 , x > 0.
π 11π ≤x≤ }] 12 12
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67. [STPM ] The functions f and g are defined by
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[Answer : {x :
(a) Find f −1 and state its domain. 1 = g(2), and state, with a reason, whether g has an inverse. (b) Show that g 2 [Answer : (a) f −1 (x) =
68. [STPM ]
[3 marks]
[4 marks]
1 ln x , D={x : x > 0}] 2
1 Express cos x + sin x in the form r cos(x − α), where r > 0 and 0 < α < π. Hence, find the minimum 2 and maximum values of cos x + sin x and the corresponding values of x in the interval 0 ≤ x ≤ 2π. [7 marks]
(a) Sketch the graph of y = cos x + sin x for 0 ≤ x ≤ 2π.
[3 marks]
(b) By drawing appropriate lines on your graph, determine the number of roots in the interval 0 ≤ x ≤ 2π of each of the following equations. 1 [1 marks] i. cos x + sin x = − , 2 ii. cos x + sin x = 2, [1 marks] (c) Find the set of values of x in the interval 0 ≤ x ≤ 2π for which | cos x + sin x| > 1. [Answer :
√
2 cos(x −
√ √ π 5π π ) , minimum value=− 2 when x = , maximum value= 2 when x = ; (b) (i) two 4 4 4 π 3π roots , (ii) no roots ; (c) {x : 0 < x < , π < x < }] 2 2
69. [STPM ] The function f is defined by f (x) = ln(1 − 2x), x < 0. (a) Find f −1 , and state its domain.
[3 marks]
(b) Sketch, on the same axes, the graphs of f and f −1 .
(c) Determine whether there is any value of x for which f (x) = f
70. [STPM ]
[3 marks]
[4 marks]
−1
(x). 1 2
[3 marks]
[Answer : (a) f −1 (x) = (1 − ex ) ; D={x : x > 0} ; (c) No]
1 Sketch the graph of y = sin 2x in the range 0 ≤ x ≤ π. Hence, solve the inequality | sin 2x| < , where 2 0 ≤ x ≤ π. [6 marks] [Answer : {x : 0 ≤ x <
13
π 5π 7π 11π , 3x + 1.
[1 marks]
[2 marks]
[3 marks]
[Answer : (a) h = 2, k = 1 ; (b) (x2 − 1)(2x2 + x + 2) + (3x + 1) ;
(c) q(x) = 2 x +
1 4
2
72. [STPM ]
The function f is defined as f (x) = (a) Show that f has an inverse.
+
15 1 15 (ii) minimum value= , when x = − ; (d) {x : x < −1, x > 1}] 8 8 4
1 x e − e−x , where x ∈ R. 2
[3 marks]
(b) Find the inverse function of f , and state its domain.
[7 marks]
[Answer : (b) f −1 (x) = ln(x +
p
x2 + 1), D={x : x ∈ R}]
73. [STPM ] Sketch, on the same axes, the graphs of y = |2x + 1| and y = 1 − x2 . Hence, solve the inequality |2x + 1| ≥ 1 − x2 . [8 marks]
74. [STPM ]
Determine the set of values of x satisfying the inequality x + 4 ≤
75. [STPM ] Functions f and g are defined by
√
[Answer : {x : x ≤ 1 − 3, x ≥ 0}]
3 . x
[6 marks]
√
√
[Answer : {x : x ≤ −2 − 7, 0 < x ≤ −2 + 7}]
f (x) = x2 + 4x + 2, x ∈ R, g(x) =
3 , x 6= −3, x ∈ R x+3
(a) Sketch the graph of f , and find its range.
(b) Sketch the graph of g, and show that g is a one-to-one function. (c) Give a reason why g −1 exists. Find g −1 , and state its domain. (d) Give a reason why g ◦ f exists. Find g ◦ f , and state its domain.
14
[4 marks]
[3 marks]
[4 marks] [4 marks]
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STPM MATHEMATICS (T)
1: Functions
3 − 3 , D={x : x ∈ R, x 6= 0} ; x 2 (d) g ◦ f (x) = 2 , D={x : x ∈ R}] x + 4x + 5
76. [STPM ] Find the value of x such that (3 − log3 x) log3x 3 = 1.
77. [STPM ] π Express 12 cos θ − 5 sin θ in the form r cos(θ + α), where r > 0 and 0 < α < . 2 Hence,
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[Answer : (a) R={y : y ≥ −2} ; (c) g −1 (x) =
[4 marks]
[Answer : x = 3]
[4 marks]
(a) state the minimum and maximum values of 12 cos θ − 5 sin θ for real values of θ,
[1 marks]
(b) solve the equation 12 cos θ − 5 sin θ = 0, 0 ≤ θ ≤ 2π,
[3 marks]
(c) sketch the graph of y = 12 cos θ − 5 sin θ for 0 ≤ θ ≤ 2π and determine the range of values of θ in this interval satisfying the inequality −5 ≤ 12 cos θ − 5 sin θ ≤ 0. [7 marks] [Answer : 13 cos(θ + 0.395) ; (a) 13, −13 ; (b) 1.176, 4.317 ; (c) {θ : 1.176 ≤ θ ≤
π 5π , ≤ θ ≤ 4.318}] 2 4
78. [STPM ] The polynomial p(x) = ax4 + x3 + bx2 − 10x − 4, where a and b are constants, has a factor (2x + 1). When p(x) is divided by (x − 1), the remainder is −15. (a) Determine the values of a and b. (b) Factorise p(x) completely.
(c) Find the set of values of x which satisfies the inequality p(x) < 0
[4 marks] [3 marks]
[4 marks]
1 2
[Answer : (a) a = 2 b = −4 ; (b) (2x + 1)(x − 2)(x2 + 2x + 2) ; (c) {x : − < x < 2}]
79. [STPM ] Solve the equation cos x − 2 sin x = 2 for 0◦ ≤ x ≤ 360◦ .
[7 marks]
[Answer : 270◦ , 323.1◦ ]
80. [STPM ] √ Functions f and g are defined by f (x) = ln(x − 1), where x > 1 and g(x) = x − 2, where x ≥ 2. (a) Sketch, on separate diagrams, the graphs of f and g. (b)
−1
i. Explain why f exists. ii. Hence, determine f −1 and state its domain.
(c) Find the composite function f ◦ g and state its domain and range. p (d) Express ln(x − 1) − 2 as a composition of functions which involves f and g.
[3 marks]
[6 marks]
[4 marks] [2 marks]
[Answer : (b)(ii) f −1 (x) = ex + 1 , Domain={x : x ∈ R} ;
(c) f ◦ g(x) = ln(
p
(x − 2) − 1) , Domain={x : x > 3} , Range={y : y ∈ R} ; (d) g ◦ f (x)]
15
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81. [STPM ] The function f is defined by
1: Functions
f (x) = 2x2 + ax + 7, x ∈ R.
31 Without using differentiation, find the values of a if the range of f is ,∞ . 8
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STPM MATHEMATICS (T)
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[5 marks]
[Answer : a = ±5]
82. [STPM ] The polynomial p(x) = x4 + mx3 + nx2 + 2x + 2, where m and n are real constants, has a quadratic factor x2 − 1. (a) Find the values of m and n.
[4 marks]
p(x) ≥ −3 for all x. [4 marks] x2 − 1 (c) By using long division, obtain the remainder when p(x) is divided by x2 − x − 6. Hence, deduce the remainder when p(x) is divided by (x + 2). [5 marks]
(b) Find the other quadratic factor of p(x) and hence, show that
(d) Find g(x) in terms of p(x) such that (x − 2) is a factor of g(x).
[2 marks]
[Answer : (a) n = −3, m = −2 ; (b) x2 − 2x − 2 ; (c)14 − 2x , 18 ; (d) g(x) = (x − 2)p(x)]
16
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2: Sequences and Series
Evaluate
∞ X 1 . 102r r=1
[2 marks]
¯ as a rational number in its lowest form. Express 0.18
2. [STPM ]
Find the expansion of
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1. [STPM ]
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Sequences and Series
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2
STPM MATHEMATICS (T)
[2 marks]
[Answer :
(1 + x2 )p in ascending powers of x until the term in x3 . (1 − x)q
2 1 , ] 99 11
[5 marks]
r 13 1 to be estimated using the above (a) If p = q = , suggest a suitable value of x that enables 2 10 r 13 correct to four decimal places. [7 marks] expansion. Hence, estimate 10 1 [3 marks] (b) If p = − and q lies in the interval [0,9], and the largest possible coefficient of x3 . 3
[Answer : 1 + qx + p +
q(q + 1) 2
q(q + 1)(q + 2) 1 x2 + pq + x3 + . . . ; (a) x = , 1.1395 ; (b) 162] 6 5
3. [STPM ] √ √ Express ( 2 − 1)5 in the form a 2 + b, where a, b are integers.
[3 marks]
√
[Answer : 29 2 − 41]
4. [STPM ] The sum of the first 2n terms of a series P is 20n − 4n2 . Find in terms of n, the sum of the first n terms of this series. Show that this series is an arithmetic series. [4 marks] Series Q is an arithmetic series such that the sum of its first n even terms is more than the sum of its first n odd terms by 4n. Find the common difference of the series Q. [5 marks] If the first term of series Q is 1, determine the minimum value of n such that the difference between the sum of the first n terms of series P and the sum of the first n terms of series Q is more than 980.
5. [STPM ] √ √ Simplify (1 + 2 3)5 − (1 − 2 3)5 .
6. [STPM ]
[6 marks]
[Answer : Sn = 10n − n2 ; d = 4 ; n = 21]
[4 marks]
√
[Answer : 1076 3]
1 − . . . and S∞ denotes the 3 sum to infinity of this series. Find the smallest n such that |S∞ − Sn | < 0.0001. [7 marks]
(a) Sn denotes the sum of the first n terms of a geometric series 3 − 1 +
17
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STPM MATHEMATICS (T)
2: Sequences and Series
(a) the n-th term of the series is log pq 2n ,
8. [STPM ]
1
[2 marks]
[3 marks]
Expand (1+x) 5 in ascending power of x until the term in x3 . By taking x = 1
3 2
[Answer : (a) 10 ; (b) a = 12 , d = 6 , r = ]
7. [STPM ] Given that the sum of the first n terms of a series is n log pq n+1 . Show that
(b) this is an arithmetic series.
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(b) The first and the second term of an arithmetic series equal respectively the first and the second term of a geometric series. The third term of the geometric series exceeds the third term of the arithmetic series by 3. The arithmetic series has a positive common difference and the sum of its first three terms equals 54. Find the first term of both series. Find also the common difference of the arithmetic series and the common ratio of the geometric series. [8 marks]
1 , find the approximation 40
for 32.8 5 correct to four decimal places. [7 marks] 1 1 + ax and (1 + x) 5 are the same until the term in x2 , find the values of a and b. If the expansion of 1 + bx 203 Hence, show that 32.8 ≈ . [8 marks] 101 1 5
[Answer : 1 + x −
9. [STPM ]
2 2 6 3 3 2 x + x + . . . , 2.0101 ; a = , b = ] 25 125 5 5
√ 1 1 Expand (1 + 8x) 2 in the ascending power of x until the term in x3 . By taking x = , find 3 correct 100 to five decimal places. [4 marks]
10. [STPM ]
[Answer : 1 + 4x − 8x2 + 32x3 + . . . ; 1.73205]
a(1 − rn ) . 1−r Give the condition on r such that lim Sn exists, and express this limit in terms of a and r. [5 marks] Given that Sn = a + ar + ar2 + . . . + arn−1 , with a 6= 0. Show that Sn = n→∞
(a) Determine the smallest integer n such that
(b) Find the sum to infinity
4 1+ + 3
2 n 4 4 + ... + > 21. 3 3
32 (1 − x)2 + 33 (1 − x)3 + . . . + 3r (1 − x)r + . . .
and determine the set of x such that this sum exists. [Answer : |r| < 1,
18
[5 marks]
[5 marks]
a 9(1 − x)2 2 4 ; (a) 7 ; (b) , {x : < x < }] 1−r 3x − 2 3 3
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STPM MATHEMATICS (T)
2: Sequences and Series
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11. [STPM ] The sum and product of three consecutive terms of an arithmetic progression are −3 and 24 respectively. Determine the three possible terms of the arithmetic progression. [5 marks] [Answer : 4,-1,-6 or -6,-1,4]
12. [STPM ] x n Expand 1 − where n is a positive integer in ascending powers of x until the term in x3 . If the n 1 coefficient of x3 is − , find n. [6 marks] 27 1 x n (1 − x) 2 in ascending powers of x until the With this value of n, obtain the expansion of 1 − n term in x3 . [5 marks] √ 1 [4 marks] Hence, by taking x = − , find the approximation of 10 accurate to 3 decimal places. 10 [Answer : 1 − x +
√ n − 1 2 (n − 1)(n − 2) 3 3 17 61 3 x − x + . . . , n = 3 ; 1 − x + x2 − x . . . ; 10 ≈ 3.162] 2 2n 6n 2 24 432
13. [STPM ] Express
1 in partial fractions. Hence show that (4r − 3)(4r + 1) n X r=1
1 1 = (4r − 3)(4r + 1) 4
14. [STPM ]
Given that y = √
1−
1 4n + 1
.
[6 marks]
[Answer :
1 1 − ] 4(4r − 3) 4(4r + 1)
1 1 √ , where x > − , show that, provided x 6= 0, 2 1 + 2x + 1 + x y=
√ 1 √ ( 1 + 2x − 1 + x). x
[3 marks]
Using the second form for y, express y as a series of ascending powers of x as far as the term in x2 . Hence, by putting x =
1 , show that 100 √
[6 marks]
79407 10 √ ≈ . 160000 102 + 101
[3 marks]
[Answer : y =
7 1 3 − x + x2 + . . .] 2 8 16
15. [STPM ] Determine the set of x such that the geometric series 1 + ex + e2x + . . . converges. Find the exact value of x so that the series converges to 2. [6 marks] 19
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STPM MATHEMATICS (T)
2: Sequences and Series
Express
1 as partial fraction. −1
4k 2
Hence, find a simple expression for Sn =
n X k=1
[4 marks]
1 and find lim Sn . n→∞ −1
[4 marks]
4k 2
1 1 1 − ; Sn = [Answer : 2(2k − 1) 2(2k + 1) 2
17. [STPM ] 1 1+x 2 Express as a series of ascending powers of x up to the term in x3 . 1 + 2x √ 1 By taking x = , find 62 correct to four decimal places. 30
18. [STPM ]
1 2
7 8
[Answer : 1 − x + x2 −
r2
1−
1 2n + 1
;
1 ] 2
[6 marks]
[3 marks]
√ 25 3 x + . . . ; 62 = 7.8740] 16
2 in partial fractions. + 2r Using the result obtained, Express ur =
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16. [STPM ]
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[Answer : {x : x < 0} ; x = − ln 2]
[3 marks]
1 1 1 1 (a) show that u2r = − + 2 + + , [2 marks] r r r + 2 (r + 2)2 n ∞ ∞ X X X 1 1 3 1 (b) show that − and determine the values of ur = − ur and ur+1 + r . 2 n+1 n+2 3 r=1
r=1
r=1
[9 marks]
[Answer :
1 3 4 1 − ; (b) , ] r r+2 2 3
19. [STPM ] √ 1 Expand (1 − x) 2 in ascending powers of x up to the term in x3 . Hence, find the value of 7 correct to five decimal places. [5 marks] 1 2
1 8
[Answer : 1 − x − x2 −
20. [STPM ] Prove that the sum of the first n terms of a geometric series a + ar + ar2 + . . . is a(1 − rn ) . 1−r
1 3 √ x ; 7 ≈ 2.64609] 16
[3 marks]
(a) The sum of the first five terms of a geometric series is 33 and the sum of the first ten terms of the geometric series is -1023. Find the common ratio and the first term of the geometric series. 3 − . . . are 2 respectively. Determine the smallest value of n such that |Sn − S∞ | < 0.001. [7 marks]
(b) The sum of the first n terms and the sum to infinity of the geometric series 6 − 3 + Sn and S∞
[5 marks]
20
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STPM MATHEMATICS (T)
2: Sequences and Series
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[Answer : (a) r = −2, a = 3 ; (b) n = 12] 21. [STPM ] For the geometric series 7 + 3.5 + 1.75 + 0.875 + ..., find the smallest value of n for which the different between the sum of the first n terms and the sum to infinity is less than 0.01. [6 marks]
22. [STPM ]
Express f (x) =
x2 − x − 1 in partial fractions. (x + 2)(x − 3)
1 1 up to the term in 3 . x x Determine the set of values of x for which this expansion is valid. Hence, obtain an expansion of f (x) in ascending powers of
[Answer : 1 +
[5 marks]
[6 marks] [2 marks]
1 5 5 1 − , 1 + 2 + 3 + . . . , {x : x < −3, x > 3}] x−3 x+2 x x
23. [STPM ] If x is so small that x2 and higher powers of x may be neglected, show that x 10 ≈ 29 (2 − 7x). (1 − x)6 2 + 2
24. [STPM ]
[Answer : 11]
[4 marks]
10−Tn 5 The nth term of an arithmetic progression is Tn , show that Un = (−2)2( 17 ) is the nth term of a 2 geometric progression. [4 marks] ∞ X 1 Un . [4 marks] If Tn = (17n − 14), evaluate 2
n=1
[Answer : −
10 ] 3
25. [STPM ] Express the infinite recurring decimal 0.72˙ 5˙ (= 0.7252525 . . . ) as a fraction in its lowest terms.[4 marks] [Answer :
359 ] 495
26. [STPM ] At the beginning of this year, Mr. Liu and Miss Dora deposited RM10 000 and RM2000 respectively in a bank. They receive an interest of 4% per annum. Mr Liu does not make any additional deposit nor withdrawal, whereas, Miss Dora continues to deposit RM2000 at the beginning of each of the subsequent years without any withdrawal. (a) Calculate the total savings of Mr. Liu at the end of n-th year. (b) Calculate the total savings of Miss Dora at the end of n-th year.
[3 marks]
[7 marks]
(c) Determine in which year the total savings of Miss Dora exceeds the total savings of Mr. Liu. [5 marks]
[Answer : (a) 10000(1.04)n ; (b) 52000[1.04n − 1]; (c) 6] 21
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STPM MATHEMATICS (T)
2: Sequences and Series
27. [STPM ]
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3 + . . ., obtain the smallest value of n if the difference between the sum 2 45 . [6 marks] of the first n + 4 terms and the sum of first n terms is less than 64 For geometric series 6 + 3 +
[Answer : 5]
28. [STPM ] Determine the set of values of x such that the geometric series e−x + e−2x + e−3x + . . . converges. Find the exact value of x if the sum to infinity of the series is 3.
29. [STPM ]
Given f (x) =
x3 − 3x − 4 , (x − 1)(x2 + 1)
[6 marks]
4 3
[Answer : {x : x > 0} ; x = ln ]
Cx + D B + 2 , [5 marks] x−1 x +1 (b) when x is sufficiently small such that x4 and higher powers can be neglected, show that f (x) ≈ 4 + 7x + 3x2 − x3 . [4 marks] (a) find the constants A, B, C and D such that f (x) = A +
30. [STPM ]
Show that
n X r2 + r − 1 r=1
r2 + r
=
n2 . n+1
[Answer : (a) A = 1, B = −3, C = 4, D = 0]
[4 marks]
31. [STPM ] The sum of the first n terms of a progression 3n2 . Determine the n-th term of the progression, and hence, deduce the type of progression. [4 marks]
32. [STPM ] Express in partial fractions
Show that
and hence, find
[Answer : 6n − 3, Arithmetic Progression]
3 . (3r − 1)(3r + 2)
n X r=1
1 1 3 = − , (3r − 1)(3r + 2) 2 (3n + 2)
∞ X r=1
1 . (3r − 1)(3r + 2)
[4 marks]
[2 marks]
[2 marks]
[Answer : 22
1 1 1 − , ] 3r − 1 3r + 2 6
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STPM MATHEMATICS (T)
2: Sequences and Series
A sequence is defined by ur = e−(r−1) − e−r for all integers r ≥ 1. Find deduce the value of
∞ X
ur .
r=1
34. [STPM ] The sequence u1 , u2 , u3 , . . . is defined by un+1 = 3un , u1 = 2. (a) Write down the first five terms of the sequence. (b) Suggest an explicit formula for ur .
35. [STPM ]
n X r=1
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33. [STPM ] ur in terms of n, and [5 marks]
[Answer : 1 − e−n ; 1]
[2 marks]
[2 marks]
[Answer : 2, 6, 18, 54, 162 , ur = 2(3)r−1 ]
1 + ax , where |b| < 1, in ascending powers of x up to the term in x3 . Determine 1 + bx the set of values of x for which both the expansions are valid. [7 marks] 2
Expand (1 + x) 3 and
If the two expansions are identical up to the term in x2 ,
(a) determine the values of a and b, [3 marks] √ 212 1 3 . [3 marks] (b) use x = to obtain the approximation 81 ≈ 8 49 (c) find, correct to five decimal places, the difference between the terms in x3 for the two expansions 1 with x = . [2 marks] 8 2 3
1 9
[Answer : 1 + x − x2 +
4 3 5 1 x + . . . , 1 + (a − b)x + b(b − a)x2 + b2 (a − b)x3 + . . ., |x| < 1 ; (a) a = , b = ; 81 6 6 (c) 0.00006]
36. [STPM ] A sequence a1 , a2 , a3 , . . . is defined by an = 3n2 . The difference between successive terms of the sequence forms a new sequence b1 , b2 , b3 , . . .. (a) Express bn in terms of n.
[2 marks]
(b) Show that b1 , b2 , b3 , . . . is an arithmetic sequence, and state its first term and common difference. [3 marks]
(c) Find the sum of the first n terms of the sequence b1 , b2 , b3 , . . . in terms of an and bn .
[2 marks]
[Answer : (a) 6n + 3 , (b) 9, 6 ; (c) an + bn − 3]
37. [STPM ] ˙ Write the infinite recurring decimal 0.131˙ 8(= 0.13181818 . . .) as the sum of a constant and a geometric series. Hence, express the recurring decimal as a fraction in its lowest terms. [4 marks] [Answer :
23
29 ] 220
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STPM MATHEMATICS (T)
2: Sequences and Series
1 in partial fractions, and deduce that (r2 − 1) 1 1 1 1 . ≡ − r(r2 − 1) 2 r(r − 1) r(r + 1)
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(a) Express
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38. [STPM ]
[4 marks]
Hence, use the method of differences to find the sum of the first (n − 1) terms, Sn−1 , of the series 1 1 1 1 + + + ... + + ..., 2 2 × 3 3 × 8 4 × 15 r(r − 1)
and deduce Sn .
[6 marks]
1 (b) Explain why the series converges to , and determine the smallest value of n such that 4 1 − Sn < 0.0025. 4
1 1 1 − ; (b) Sn−1 = [Answer : (a) 2(r − 1) 2(r + 1) 2
[5 marks]
1 1 1 1 1 − − , Sn = ; (c) 13] 2 n(n + 1) 2 2 (n + 1)(n + 2)
39. [STPM ] √ √ √ √ 6 6 6 3 + 2) and ( 3 − 2) to evaluate ( 3 + 2) + ( 3 − 2)6 . Hence, Use the binomial expansions of ( √ 6 show that 2701 < ( 3 + 2) < 2702. [7 marks]
40. [STPM ]
[Answer : 2702]
(a) Show that for a fixed number x 6= 1, 3x2 + 3x3 + . . . + 3xn is a geometric series, and find its sum in terms of x and n. [4 marks] (b) The series Tn (x) is given by
Tn (x) = x + 4x2 + 7x3 + . . . + (3n − 2)xn , for x 6= 1.
By considering Tn (x) − xTn (x) and using the result from (a), show that
Hence, find the value of
Tn (x) =
20 X r=1
x + 2x2 − (3n + 1)xn+1 + (3n − 2)xn+2 . (1 − x)2
[5 marks]
2r (3r − 2), and deduce the value of
[Answer : (a) sum=
24
19 X r=0
2r+2 (3r + 1).
[6 marks]
3x2 (xn−1 − 1) ; (b) 115343370 , 230686740] x−1
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STPM MATHEMATICS (T)
2: Sequences and Series
41. [STPM ]
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1 8
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1 − 5x in ascending powers of x are 1 − 3x the same. Determine the range of values of x for which both expansions are valid. [6 marks] 1 1 − 5x [3 marks] Use the result (1 − 8x)4 ≈ to obtain an approximation of (0.84) 4 as a fraction. 1 − 3x Show that the first three terms in the expansions of (1 − 8x)4 and
1 8
1
[Answer : {x : x < − , x > } , 0.84 4 ≈
42. [STPM ] The r-th term, ur , of an infinite series is given by
2r+1 2r−1 1 1 + , ur = 3 3
45 ] 47
A , where A is a constant. [2 marks] 32r+1 (b) Find the sum of the first n terms of the series, and deduce the sum of the infinite series. [6 marks] (a) Express ur in the form
[Answer : (a) ur =
10 32r+1
; (b)
n 1 5 5 ] 1− , 12 9 12
43. [STPM ] A recursive formula for the general term of a sequence is given by ur+1 = ur + 2r + 3, where u0 = 1. (a) Write down the first four terms of the sequence.
[2 marks]
(b) Suggest an explicit formula for the general term and verify your answer.
44. [STPM ]
[Answer : (a) 1, 4, 9, 16 ; (b) ur = (r + 1)2 ]
1 A convergent sequence is defined by ur+1 = 1 + ur and u1 = 1. 3 (a) Write down each of the terms u2 , u3 and u4 in the form r 3 1 formula for ur is given by ur = 1− . 2 3 (b) Determine the limit of ur , as r tends to infinity.
45. [STPM ]
Show that
∞ k k−2 X 1 1 k=1
2
3
r−1 k X 1 k=0
3
, and show that an explicit
is a convergent series. Give a reason for your answer.
Hence, determine the sum of the convergent series.
[3 marks]
[5 marks]
[2 marks]
[Answer : (b)
[3 marks] [2 marks]
[Answer :
25
3 ] 2
9 ] 5
Work Smart, Not Hard
Do not share this past year paper.
Refer the full solution at kkleemaths.com.
26
2: Sequences and Series
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STPM MATHEMATICS (T)
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3: Matrices
2 3 1 (a) Given M = −1 0 4. 1 −1 1 1 2 (b) Given matrices A = 2 3 3 1 neous equation
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1. [STPM ]
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Matrices
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3
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Show that M3 − 3M2 + 8M − 24I3 = 0. Deduce M−1 .
[7 marks]
3 −5 1 7 1, B = 1 7 −5. Find AB, and hence solve the simulta2 7 −5 1 −5x + y + 7z = 8, x + 7y − 5z = −16, 7x − 5y + z = 14.
[Answer : (a) A−1
4 1 5 = 24 1
2. [STPM ]
−4 1 5
[8 marks]
12 18 0 −9 ; (b) AB = 0 18 3 0 0
1 −2 −6 9 , find A2 and A3 . Hence, find A100 . If A = −3 2 2 0 −3 −5 [Answer : A2 = 9 −4
3. [STPM ]
1 1 2 Given P = 1 2 1. 2 1 1
(a) Find R so that R = P2 − 4P − I3 . (b) Show that PR + 4I3 = 0.
4. [STPM ] The matrices A and B are given by
−6 10 −4
−6 1 9 , A3 = −3 −3 2
[4 marks]
−2 2 0
−6 −5 9 , A100 = 9 −3 −4
−6 10 −4
−6 9 ] −3
[3 marks] [2 marks]
1 [Answer : (a) R = 1 −3
5 0 0 −2 0 0 A = 1 8 0 , B = − −5 0 . 1 3 5 −1 −3 −2
(a) Determine whether A and B commute.
0 0 , x = 1, y = −1, z = 2] 18
1 −3 1
−3 1 ] 1
[3 marks]
(b) Show that there exist numbers m and n such that A = mB + nI, where I is the 3 x 3 identity matrix, and find the values of m and n. [6 marks] 27
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3: Matrices
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5. [STPM ]
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[Answer : (a) Commute. (b) m = −1, n = 3] 2b − 1 a2 b2 Determine the values of a, b, c so that the matrix 2a − 1 a bc is a symmetric matrix. b b + c 2c − 1
6. [STPM ]
[5 marks]
[Answer : a = 1, b = 0, c = 0]
−10 4 9 2 3 4 Matrix M and N is given by M = 15 −4 −14 , and N = 4 3 1 . −5 1 6 1 2 4 −1 Find MN and deduce N . [4 marks] Product X, Y , Z are assembled from three components A, B, C according to different proportions. Each product X consists of two components of A, four components of B, and one component of C; each product of Y consists of three components of A, three components of B, and two components of C; each product of Z consists of four components of A, one component of B, and four components of C. A total of 750 components of A, 1000 components of B, and 500 components of C are used. With X, Y , Z representing the number of products of X, Y , and Z assembled, obtain a matrix equation representing the information given. [4 marks] Hence, find the number of products of X, Y , and Z assembled. [4 marks]
7. [STPM ]
[Answer : x=200, y=50, z=50.]
1 2 −3 The matrix A is given by A = 3 1 1 . 0 1 −2
(a) Find the matrix B such that B = A2 − 10I, where I is the 3× identity matrix. (b) Find (A + I)B, and hence find (A + I)21 B. −3 [Answer : (a) 6 3
8. [STPM ]
3 3 Matrix A is given by A = 5 4 1 2 Find the adjoint of A. Hence, find
4 1 . 3 A−1 .
1 −2 −1
5 −3 −10 ; (b) 6 −5 3
10 −1 [Answer : −14 5 6 −3
1 −2 −1
[3 marks] [6 marks]
5 −3 −10 , 6 −5 3
1 −2 −1
5 −10] −5
[6 marks]
5/6 −13 17 ; −7/6 1/2 −3
9. [STPM ] The matrices P and Q, where PQ = QP, are given by 2 −2 0 −1 1 0 0 −1 P = 0 0 2 and Q = 0 a b c 0 −2 2
−1/12 5/12 −1/4
−13/12 17/12 ] −1/4
Determine the values of a, b and c. [5 marks] Find the real numbers m and n for which P = mQ + nI, where I is the 3 × 3 identity matrix.[5 marks] 28
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3: Matrices
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[Answer : a = 0, b = 4, c = −4 ; m = −2, n = 0] 10. [STPM ] A, B, C are square matrices such that BA = B−1 and ABC = (AB)−1 . Show that A−1 = B2 = C.
1 2 0 If B = 0 −1 0 , find C and A. 1 0 1
11. [STPM ] The matrix A is given by
[3 marks] [7 marks]
1 [Answer : A = 0 −2
k 1 5 A = 2 k 8 . 8 −3 2
0 1 −2
0 1 0 , C = 0 1 2
0 1 2
0 0] 1
Determine all values of k for which the equation AX = B, where B is a 3 × 1 matrix, does not have a unique solution. [3 marks] For each of these values of k, find the solution, if any, of the equation 1 AX = −2 . 4
[7 marks]
[Answer : k = 3, k = 5 ; For k = 3, no solution. For k = 5, x =
12. [STPM ]
k 1 3 Determine the values of k such that the determinant of the matrix 2k + 1 −3 2 is 0. [4 marks] 0 k 2
13. [STPM ]
17 12 30 7 − t, y = − − t, z = t] 23 23 23 23
1 4
[Answer : k = − , k = 2]
5 2 3 a 1 −18 −1 12 and PQ = 2I, where I is the 3 × 3 identity matrix, If P = 1 −4 3 , Q = b 3 1 2 −13 −1 c determine the values of a, b and c. Hence find P−1 . [8 marks] Two groups of workers have their drinks at a stall. The first group comprising ten workers have five cups of tea, two cups of coffee and three glasses of fruit juice at a total cost of RM11.80. The second group of six workers have three cups of tea, a cup of coffee and two glasses of fruit juice at a total cost of RM7.10. The cost of a cup of tea and three glasses of fruit juice is the same as the cost of four cups of coffee. If the costs of a cup of tea, a cup of coffee and a glass of fruit juice are RM x, RM y and RM z respectively, obtain a matrix equation to represent the above information. Hence determine the cost of each drink. [6 marks]
11/2 1/2 [Answer : a = 11, b = −7, c = 22 ; −7/2 −1/2 −13/2 −1/2
29
−9 6 ; x=RM 1, y=RM 1.30, z=RM 1.40] 11
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14. [STPM ] Consider the system of equations
x + y + pz = q, 3x − y − 2z = 1, 6x + 2y + z = 4,
for the two cases: p = 2, q = 1 and p = 1, q = 2.
3: Matrices
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STPM MATHEMATICS (T)
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For each case, find the unique solution if it exists or determine the consistency of the system if there is no unique solution. [7 marks] [Answer : For p = 2, q = 1, unique solution, x = 1/2, y = 1/2, z = 0. For p = 1, q = 2, no unique solution. The system is not consistent.]
15. [STPM ] The matrices A and B are given by −35 19 18 −1 2 1 A = −3 1 4 , B = −27 −13 45 . −3 12 5 0 1 2 Find the matrix A2 B and deduce the inverse of A. Hence, solve the system of linear equations
[5 marks]
x − 2y − z = −8, 3x − y − 4z = −15, y + 2z = 4.
121 0 [Answer : 0 121 0 0
16. [STPM ] Consider the system of equations
[5 marks]
0 −2/11 −3/11 7/11 0 , 6/11 −2/11 1/11 ; x = −3, y = 2, z = 1] 121 −3/11 1/11 5/11
λx + y + z = 1, x + λy + z = λ, x + y + λz = λ2 ,
where λ is a constant.
(a) Determine the values of λ for which this system has a unique solution, infinitely many solutions and no solution. [5 marks] (b) Find the unique solution in terms of λ.
[5 marks]
[Answer : (a) For unique solution, λ 6= 1, −2. For infinitely many solution, λ = 1.
17. [STPM ]
1 0 0 Matrix A is given by A = 1 −1 0 . 1 −2 1
30
For no solution, λ = −2. λ+1 λ+1 (λ + 1)2 (b) x = − ,y = − ,z = ] λ+2 λ+2 λ+2
STPM MATHEMATICS (T) matrix, and deduce A−1 . 4 3 2 1 . 0 2
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(a) Show that A2 = I, where I is the 3 × 3 identity 1 (b) Find the matrix B which satisfies BA = 0 −1
3: Matrices
1 [Answer : (a) 1 1
18. [STPM ]
(a) The matrix P, Q and R 1 5 2 −2 P= 1 −3
6x + 10y + 8z = 4500 x − 2y + z = 0 x + 2y + 3z = 1080
32 72 4 , 0 12 0
19. [STPM ] The matrix A is given by
0 72 0
0 1/18 7/72 0 , 1/72 −5/72 72 −1/36 1/72
−13/72 −1/72 ; (b) x = 220, y = 190, z = 160 11/72 ]
1 1 c A = 1 2 3 1 c 1
(a) Find the values of c for which the equation AX = B does not have a unique solution. (b) For each value of c, find the solutions, if any, of the equation 1 AX = −3 . −11
20. [STPM ]
3 1] 2
[5 marks]
and B is a 3 × 1 matrix.
−10 −4 −4
[5 marks]
(b) Using the result in (a), solve the system of linear equations
24 40 [Answer : (a) 4 −8 4 8
8 0 0 −1 0 ; (b) 3 1 −2 1
[4 marks]
are given by 6 −13 −50 −33 4 7 −13 4 , Q = −1 −6 −5 , R = 1 −5 −1 2 7 20 15 −2 1 11
Find matrices PQ and PQR and hence, deduce (PQ)−1 .
[4 marks]
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[3 marks]
[5 marks]
[Answer : (a) c 6= 1, 4 ; (b) For c = 1, no solution. For c = 4, x = 5 − 5t, y = −4 + t, z = t]
1 2 1 3 . Matrix P is given by P = 2 1 2 −1 −1
31
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STPM MATHEMATICS (T)
3: Matrices
(a) Find the determinant and adjoint of P. Hence, find P−1 .
[6 marks]
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(b) A factory assembles three types of toys Q, R and S. The total time taken to assemble one unit of R and one unit of S exceeds the time taken to assemble two units of Q by 8 minutes. One unit of Q, two units of R and one unit of S take 31 minutes to be assembled. The time taken to assemble two units of Q, one unit of R and three units of S is 48 minutes. If x, y and z represent the time, in minutes, taken to assemble each unit of toys Q, R and S respectively, i. write a system of linear equations to represent the above information, ii. using the results in (a), determine the time taken to assemble each type of toy.
2 [Answer : (a) 14, 8 −4
21. [STPM ] A and B are two matrices such that −4 −3 6 A = −2 −2 4 2 2 −3
1 −3 5
(b) Using A
obtained in (a), find B.
−2 6 0 and A2 B = 2 0 4 . 0 4 2
22. [STPM ] Matrix A is given by
−2 [Answer : (a) 2 , 2 0
[6 marks]
[4 marks]
3 0 −1 0 4 , 1 2 2 0
3/2 0 1
1 x 1 A = −1 −1 0 1 0 0
and A2 = A−1 . Determine the value of x.
23. [STPM ] Consider the system of equations
[5 marks]
1/14 5/14 −3/14 −1/14 ; (b)(ii) 5, 8, 10 ] 5/14 −3/14
1/7 5 −1, 4/7 −2/7 −3
(a) Find the determinant and adjoint of A. Hence, determine A−1 . −1
[2 marks]
0 −8 2 ; (b) 9 1 0
27 2 18
0 18] 10
[7 marks]
[Answer : 2]
x + 3y + 2z = −2, 3x + ay + 2z = 2a − 1, 2x + 6y + az = b.
(a) If a 6= 9 and a 6= 4, show that the system has a unique solution. (b) If a = 5 and b = 6, find the unique solution.
[5 marks] [3 marks]
(c) If a = 4, show that the system does not have a solution unless b = −4.
[3 marks]
(d) If a = 9, determine whether there are any values of b for which the system has a solution.[3 marks] [Answer : (b) x = 32
77 55 131 , y = − , z = 10 ; (d) b = − ] 4 4 4
STPM MATHEMATICS (T)
3: Matrices
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24. [STPM ] The matrices P and Q, where PQ = QP, are given by 2 −2 0 −1 1 0 0 −1 P = 0 0 2 and Q = 0 a b c 0 −2 2
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Determine the values of a, b and c. [5 marks] Find the real numbers m and n for which P = mQ + nI, where I is the 3 × 3 identity matrix.[5 marks] [Answer : a = 0, b = 4, c = −4 ; m = −2, n = 0]
25. [STPM ] Using an augmented matrix and elementary row operations, find the solution of the system of equations 3x − 2y − 5z = −5, x + 3y − 2z = −6, 5x − 4y + z = 11.
26. [STPM ] A system of linear equations is given by
x + y + z = k, x − y + z = 0, 4x + 2y + λz = 3.
[9 marks]
[Answer : x = 1, y = −1, z = 2]
where λ and k are real numbers. Show that the augmented matrix for the system may be reduced to k 1 1 1 0 −2 −k . 0 0 0 λ − 4 3 − 3k [5 marks]
Hence, determine the values of λ and k so that the system of linear equations has (a) a unique solution, (b) infinitely many solutions, (c) no solution.
[1 marks]
[1 marks]
[1 marks]
[Answer : (a) λ 6= 4 ; (b) λ = 4, k = 1 ; (c) λ = 4, k 6= 1]
27. [STPM ]
5 4 −2 5 −2. Given that matrix M = 4 −2 −2 2
Show that there exist non-zero constants a and b such that M2 = aM + bI, where I is the 3 × 3 identity matrix. [6 marks] Hence, find the inverse of the matrix M.
[3 marks]
33
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STPM MATHEMATICS (T)
3: Matrices
28. [STPM ] A matrix P is given by
−5 0 2 P = 0 2 −1 . −1 4 −2
By using elementary row operations, find the inverse of P.
29. [STPM ] A system of linear equations is given by
x − 3y = 2, px + qz = −1, py + z = −1.
−0.4 0.6 0.2
0.2 0.2] 0.9
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0.6 [Answer : a = 11, b = −10 ; −0.4 0.2
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[5 marks]
0 2 −1 [Answer : 1/4 3 −5/4] 1/2 5 −5/2
(a) Write the augmented matrix for the system of linear equations, and show that it may be reduced to 1 −3 0 2 0 p 1 −1 . 0 0 3 − q 2p − 2 [5 marks]
(b) Determine the values of p and q such that the system has infinitely many solutions, and find the general solution. [4 marks] [Answer : (b) p = 1 an q = 3 ,x = −1 − 3t, y = −t − 1, z = t]
30. [STPM ] The matrices M and N are given by 1 b ca 1 a bc M = 1 b ca , N = 1 a bc . 3 3c 3ab 1 c ab Show that det M = (a − b)(b − c)(c − a). Deduce det N.
31. [STPM ] The variables x, y and z satisfy the system of linear equations
where k is a real constant.
2x + y + 2z = 1, 4x + 2y + z = k, 8x + 4y + 7z = k 2 ,
(a) Write a matrix equation for the system of linear equations. 34
[4 marks]
[2 marks]
[1 marks]
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3: Matrices
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(b) Reduce the augmented matrix to row-echelon form, and show that the system of linear equations does not have a unique solution. [6 marks] (c) Determine all the values of k for which the system of linear equations has infinitely many solutions, and find the solutions in the case when k is positive. [6 marks] (d) Find the set of values of k for which the system of linear equations is inconsistent. 5 3
[Answer : (c) k = , 2 , x =
1−t 5 , y = t, z = 0 ; (d) {k : k ∈ R, k 6= − , k 6= 2}] 2 3
32. [STPM ] Using Gaussian elimination, solve the system of linear equations x + y − z = 0 2x − y − 2z = 4 . 5x − y + z = 2
33. [STPM ]
[8 marks]
2
(b) Show that P(P − 6P + 11I) = 6I, where I is 3 x 3 identity matrix, and deduce P
34. [STPM ] A system of linear equations is given by
x + y + z = 1, 2x + 3y + 2z = 3, 2x + 3y + mz = m2 − 1.
(a) Use Gaussian elimination to reduce the augmented matrix for the system above. (b) Determine the value of m for which the system of linear equations i. has a unique solutions, ii. has infinitely many solutions, iii. has no solution.
35
4 3
1 3
[Answer : x = , y = − , z = −1]
1 1 2 A matrix P is given by P = 0 2 2. −1 1 3 (a) Find P2 − 6P + 11I.
[2 marks]
[3 marks]
−1
.
[5 marks]
[Answer : ]
[4 marks]
[2 marks] [3 marks]
[3 marks]
[Answer : ]
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4: Complex Numbers
√ 2 + ai √ is a real number and find this real number. Determine the value of a if 1 + 2i
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1. [STPM ]
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Complex Numbers
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4
STPM MATHEMATICS (T)
[4 marks]
[Answer : a = 2,
2. [STPM ] If (x + iy)2 = i, find all the real values of x and y.
√
2]
[6 marks]
1 2
1 2
1 2
1 2
[Answer : x = √ , y = √ ; x = − √ , y = − √ ]
3. [STPM ] √ The complex numbers z1 and z2 satisfy the equation z 2 = 2 − 2 3i. (a) Express z1 and z2 in the form a + bi, where a and b are real numbers. (b) Represent z1 and z2 in an Argand diagram.
(c) For each of z1 and z2 , find the modulus, and the argument in radians. [Answer : (a) z1 =
√
[6 marks] [1 marks] [4 marks]
√ 5π π 3 − i, z2 = − 3 + i ; (c) |z1 | = 2, |z2 | = 2 , arg(z1 )=− , arg(z2 )= ] 6 6
4. [STPM ] √ Let zl = 1, z2 = x + iy, z3 = y + ix, where x, y ∈ R, x > 0 and i = −1. If z1 , z2 , . . ., zn is a geometric progression, (a) find x and y,
(b) express z2 and z3 in the polar form,
(c) find the smallest positive integer n such that z1 + z2 + . . . + zn = 0, (d) find the product z1 z2 z3 . . . zn , for the value of n in (c). √
[Answer : (a) x =
√ 2(1 + 3i) , where i = −1. 2 (1 − 3i)
6. [STPM ]
If z = cos θ + i sin θ, show that
[2 marks]
[5 marks]
[3 marks]
π π 1 π 3 π , y = ; (b) z2 = cos + i sin , z3 = cos + i sin ; (c) 12 ; (d) -1] 2 2 6 6 3 3
5. [STPM ] Simplify (a)
[3 marks]
[3 marks]
[Answer : (a) −
13 9 − i] 25 25
1 1 1 = (1 − i tan θ) and express in a similar form. [4 marks] 1 + z2 2 1 − z2
7. [STPM ] Find the roots of the equation (z − iα)3 = i3 , where α is a real constant. 36
[Answer :
1 (1 + i cot θ)] 2
[3 marks]
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STPM MATHEMATICS (T)
4: Complex Numbers
[2 marks]
3
(b) Solve the equation [z − (1 + i)] = (2i) . 2
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3
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(a) Show that the points representing the roots of the above equation form an equilateral triangle. [5 marks]
(c) If ω is a root of the equation ax + bx + c = 0, where a, b, c ∈ R and a 6= 0, show that its conjugate ω ∗ is also a root of this equation. Hence, obtain a polynomial equation of degree six with three of its roots also the roots of the equation (z − i)3 = i3 . [5 marks] [Answer : (1 + α)i , −
√
√ √ √ 3 3 1 1 + (α − )i , + (α − )i ; (b) 1+3i , 1 − 3 1 + 3 ; (c) x6 + 3x4 − 3x2 + 4 = 0] 2 2 2 2
8. [STPM ]
1 . 1−z
If z is a complex number such that |z| = 1, find the real part of
[6 marks]
[Answer :
1 ] 2
9. [STPM ] The equation z 4 − 2z 3 + kz 2 − 18z + 45 = 0 has an imaginary root. Obtain all the roots of the equation and the value of the real constant k. [8 marks]
10. [STPM ]
[Answer : Roots=1 − 2i, 1 + 2i, 3i, −3i , k = 14]
(a) Find the roots of ω 4 = −16i, and sketch the roots on an Argand diagram.
11. [STPM ]
[Answer : (a) 2 cos
4k − 1 8
π + i sin
4k − 1 8
(a) Find the fifth roots of unity in the form cos θ + i sin θ, where −π < θ ≤ π. [Answer : (a) cos
2kπ 5
+ i sin
[5 marks]
2kπ 5
π , k = 0, 1, 2, 3]
[4 marks]
, k = −2, −1, 0, 1, 2]
12. [STPM ] √ The complex number z is such that z − 2z ∗ = 3 − 3i, where z ∗ denotes the conjugate of z. (a) Express z in the form a + bi, where a and b are real numbers. (b) Find the modulus and argument of z.
[3 marks]
(c) Represent z and its conjugate in an Argand diagram.
13. [STPM ]
Show that
1 + cos 2θ + i sin 2θ = i cot θ. 1 − cos 2θ − i sin 2θ
[3 marks]
[3 marks]
√
π 3
[Answer : (a) z = − 3 − i ; (b) modulus=2, argument=− ]
π Hence, show that the roots of the equation (z + i)5 = (z − i)5 are ± cot and ± cot 5 2 π 2 2π 2 π 2 2π Deduce the values of cot + cot and cot cot . 5 5 5 5 37
[2 marks]
2π .[6 marks] 5 [4 marks]
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4: Complex Numbers
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2i . (1 + 3i)2
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Given that z 2 =
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14. [STPM ]
(a) Find the real and imaginary parts of z 2 . Hence, obtain z1 and z2 which satisfy the above equation. [10 marks]
2
(b) Given that z1 and z3 are roots of 5x + ax + b = 0, where a and b are integers. i. Find the values of a and b. ii. Determine z3 and deduce the relationship between z1 and z3 .
[Answer : (a) real part=
[3 marks]
[3 marks]
4 2 1 2 1 3 , imaginary part=− , z1 = − i, z 2 = − + i 25 25 5 5 5 5 2 z3 = + 5
; (b)(i) a = −4, b = 1 (ii) 1 i. z3 is conjugate of z1 .] 5
15. [STPM ] In an Argand diagram the points R and S represent the complex numbers w = u + iv and z = x + iy z−i respectively which are related by w = . 1 − iz (a) Express u and v in terms of x and y.
[Answer : u =
[3 marks]
x2 + y 2 − 1 2x , v = ] x2 + y 2 + 2y + 1 x2 + y 2 + 2y + 1
16. [STPM ] √ Express the complex number z = 1 − 3i in polar form. 1 1 Hence, find z 5 + 5 and z 5 − 5 . z z
[4 marks]
[4 marks]
h π π i 1025 1023√3 1023 1025√3 + i, + i] [Answer : 2 cos − +i − , 3 3 64 64 64 64
17. [STPM ] √ The complex number z is given by z = 1 + 3i. (a) Find |z| and arg z.
[3 marks]
√ (b) Using de Moivre’s theorem, show that z = 16 − 16 3i. (c) Express
z4
z∗
5
[3 marks]
in the form x + yi, where z ∗ is the conjugate of z and x, y ∈ R.
[Answer : (a) 2,
[3 marks]
√ π ; (c) 4 − 4 3i] 3
18. [STPM ] √ √ √ √ Express the complex number 6−i 2 in polar form. Hence, solve the equation z 3 = 6−i 2.[9 marks]
19. [STPM ]
√
π π 6 6 √ 11π 11π √ 13π 13π √ π π z = 2[cos( ) + i sin( )], 2[cos(− ) + i sin(− )], 2[cos(− ) + i sin(− )]] 18 18 18 18 18 18
The complex numbers z and w are given by z = −1 + i and w = 38
[Answer : 2 2[cos(− ) + i sin(− )] ;
i+z . 1 − iz
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STPM MATHEMATICS (T)
4: Complex Numbers
[3 marks]
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(b) Express w in polar form.
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[3 marks]
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(a) Find w in the form x + yi, where x, y ∈ R. State the real and imaginary parts of w.
(c) Using de Moivre’s theorem, determine the cube roots of w. Give your answer in cartesian form. [5 marks]
√ √ 3 1 3 1 π π [Answer : (a) i ; (b) w = 1(cos + i sin ) ; (c) + i,− + i , -i] 2 2 2 2 2 2
39
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5: Analytic Geometry
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Analytic Geometry
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5
STPM MATHEMATICS (T)
1. [STPM ] Show that the mid-points of the parallel chords of the parabola y 2 = 4ax with gradient 2 lie on a straight line parallel to the x-axis. [7 marks] 2. [STPM ] The sum of the distance of the point P from the point (4,0) and the distance of P from the origin is (x − 2)2 y2 8 units. Show that the locus of P is the ellipse + = 1 and sketch the ellipse. [7 marks] 16 12 3. [STPM ] Find the perpendicular distance from the centre of the circle x2 + y 2 − 8x + 2y + 8 = 0 to the straight line 3x + 4y = 28. Hence, find the shortest distance between the circle and the straight line. [7 marks] [Answer : 4, 1]
4. [STPM ] Show that x2 + y 2 − 2ax − 2by + c = 0 is the equation of the circle with centre (a, b) and radius p a2 + b2 − c. [3 marks] C3
C1
C2
The above figure shows three circles C1 , C2 and C3 touching one another, where their centres lie on a straight line. If C1 and C2 have equations x2 + y 2 − 10x − 4y + 28 = 0 and x2 + y 2 − 16x + 4y + 52 = 0 respectively. Find the equation of C3 . [7 marks] [Answer : 5x2 + 5y 2 − 74x + 12y + 156 = 0]
5. [STPM ] The equation of a hyperbola is 4x2 − 9y 2 − 24x − 18y − 9 = 0. (a) Obtain the standard form for the equation of the hyperbola. (b) Find the vertices and the equations of the asymptotes of the hyperbola. [Answer : (a)
[3 marks]
[6 marks]
(x − 3)2 (y + 1)2 2 − = 1 ; (b) Vertices are (0,-1) and (6,-1). Asymptotes are y = x − 3 and 32 22 3 2 y = − x + 1] 3
6. [STPM ] c Show that the parametric equations x = ct and y = , where c is a constant, define a point on the t rectangular hyperbola xy = c2 . [2 marks] The points P , Q, R and S, with parameters p, q, r and s respectively, lie on the rectangular hyperbola xy = c2 . 40
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STPM MATHEMATICS (T)
5: Analytic Geometry
(a) Show that pqrs = −1 if the chords P Q and RS are perpendicular.
[4 marks]
[Answer : (b) y = −
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(b) Find the equation of the line passing through the points P and Q. Deduce the equation of the tangent to the rectangular hyperbola at the point P . [4 marks] c c 1 2c 1 x + + , y = − 2x + ] pq p q p p
7. [STPM ] The equation of an ellipse is 3x2 + y 2 + 30x + 10y + 79 = 0. (a) Obtain the standard form for the equation of the ellipse.
[3 marks]
(b) Find the coordinates of the centre C, the focus F1 , and the focus F2 of the ellipse.
[4 marks]
(c) Sketch the ellipse, and indicate the points C, F1 and F2 on the ellipse.
[2 marks]
[Answer : (a)
√ √ (x + 5)2 (y + 5)2 + = 1 ; (b) C(−5, −5) , F1 (−5, −5 + 14) , F2 (−5, −5 − 14)] 7 21
8. [STPM ] The parametric equations of a conic are x = a cos θ − 3 and y = b sin θ + 4, where a and b are positive constants and 0 ≤ θ ≤ 2π. (a) Find the standard form of the equation of the conic, and identify the type of conic.
[3 marks]
(b) If a = b = 5, determine and sketch the conic.
[3 marks]
[Answer : (a)
41
(x + 3)2 (y − 4)2 + = 1 ; (b) Circle] 2 a b2
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6: Vectors
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Vectors
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6
STPM MATHEMATICS (T)
1. [STPM ] Find a vector r that is perpendicular to the vectors p = i + j and q = 2j − k. Express vector w = 2j + k in terms of p, q, and r. [8 marks] 2 3
1 3
2 3
[Answer : r = −i + j + 2k ; w = p + q + r]
2. [STPM ] Equation of plane π is given by r = j + 2k + λ(i − 3j − 2k) + µ(2i + j + k). Find the equation of the plane that contains the point (0, 3, −3) and parallel to π. [6 marks] [Answer : x + 5y − 7z = 36]
3. [STPM ] The position vector of points A and B respect to O are a = 2i − 2j − 9k and b = −2i + 4j + 15k respectively. Find the vector equation for the line passes through the midpoint of AB and perpendicular to the plane OAB. [3 marks] [Answer : r = j + 3k + λ(3i − 6j + 2k)]
4. [STPM ] Given that origin, O and position vectors of P , Q, R, and S are 4i + 3j + 4k, 6i + j + 2k, 9j − 6k and −i + j + k respectively. Find the equation of the plane OP Q. [2 marks] Show that the point S lies on the plane OP Q.
Show that the line RS are perpendicular to the plane OP Q. Find the acute angle between the line P R and the plane OP Q.
[4 marks] [4 marks] [5 marks]
[Answer : x + 8y − 7z = 0 ; 60◦ ]
5. [STPM ] Forces (4i+3j) N, (3i+7j) N, and (−5i−6j) N act at a point. Calculate the magnitude of the resultant force and the cosine of the angle between the resultant force and the unit vector i. [5 marks] √ [Answer : 2 5 N,
√
5 ] 5
6. [STPM ] Position vectors of the points P and Q relative to the origin O are 2i and 3i + 4j respectively. Find −−→ −−→ the angle between vector OP and vector OQ [Answer : 53.1◦ ]
7. [STPM ] Given that points O, P , and Q non-colinear, R lies on the line P Q. Position vector of P , Q, and R respect to O are p, q, and r respectively. Show that r = µp + (1 − µ)q, where µ is a real number. [2 marks]
8. [STPM ] Prove that the planes ax + by + cz = d and a0 x + b0 y + c0 z = d0 are parallel if and only if a : b : c = a0 : b0 : c0 . Find the equation of the plane π that is parallel to the plane 3x + 2y − 5z = 2 and contains the point (-1, 1, 3). [5 marks] Find the perpendicular distance between the plane π and the plane 3x + 2y − 5z = 2. 42
[2 marks]
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STPM MATHEMATICS (T)
6: Vectors 9√ 38] 19
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[Answer : 3x + 2y − 5z = −16 ;
9. [STPM ] Let u = cos φ i + sin φ j and v = cos θ i + sin θ j, where i and j are perpendicular unit vectors. Show that 1 1 |u − v| = sin (φ − θ). 2 2 [5 marks]
10. [STPM ] The vector equations of two intersecting lines are given by r = 2i + j + λ(i + j + 2k) and r = 2i + 2j − k + µ(i + 2j + k). (a) Determine the coordinates of the point of intersection of the two lines. (b) Find the acute angle between the two lines.
[3 marks]
[4 marks]
[Answer : (a) (1, 0, −2) ; (b) 33.6◦ ]
11. [STPM ] The line l has equation r = 2i + j + λ(2i + k) and the plane π has equation r = i + 3j − k + µ(2i + k) + v(−i + 4j). (a) The points L and M have coordinates (0, 1, −1) and (1, −5, −2) respectively. Show that L lies on l and M lies on π. [3 marks] Determine the sine of the acute angle between the line LM and the plane π and the shortest distance from L to π. [6 marks]
12. [STPM ]
[Answer :
4 1 If the angle between the vectors a = and b = is 135◦ , find the value of p. 8 p
2 2 √ , ] 3 38 3
[6 marks]
[Answer : −3]
13. [STPM ] The planes π1 and π2 with equations x − y + 2z = 1 and 2x + y − z = 0 respectively intersect in the line l. The point A has coordinates (1,0, 1). (a) Calculate the acute angle between π1 and π2 . [2 marks] 1 2 (b) Explain why the vector −1 × 1 is in the direction of l. Hence, show that the equation 2 −1 of l is 0 −1 r = 1 + t 5 . 1 3 where t is a parameter.
[5 marks]
(c) Find the equation of the plane passing through A and containing l.
[3 marks]
(d) Find the equation of the plane passing through A and perpendicular to l.
[2 marks]
43
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STPM MATHEMATICS (T)
6: Vectors
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[3 marks]
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(e) Determine the distance from A to l.
r
◦
[Answer : (a) 80.4 ; (c) 3x + 3y − 4z + 1 = 0 ; (d) −x + 5y + 3z = 2 ; (e)
34 ] 35
14. [STPM ] The position vectors of the points A, B, C and D, relative to an origin, are i + 3j, −5i − 3j, (x − 3)i − 6j and (x + 3)i respectively. (a) Show that, for any value of x, ABCD is a parallelogram. (b) Determine the value of x for which ABCD is a rectangle.
[3 marks]
[4 marks]
[Answer : (b) x = 1]
15. [STPM ] The diagram below shows non-collinear points O, A and B, with P on the line OA such that OP : P A = 2 : 1 and Q on the line AB such that AQ : QB = 2 : 3. The lines P Q and OB produced meet −→ −−→ at the point R. If OA = a and OB = b, R
Q
B
A
P
O
−−→ 1 2 (a) show that P Q = − a + b, 15 5 (b) find the position vector of R, relative to O, in terms of b.
[5 marks]
[5 marks]
[Answer : (b) 4b]
16. [STPM ] Two straight lines l1 and l2 have equations −2x + 4 = 2y − 4 = z − 4 and 2x = y + 1 = −z + 3 respectively. Determine whether l1 and l2 intersect. [7 marks] [Answer : No intersecton.]
17. [STPM ] The position vectors of the points A, B and C, with respect to the origin O, are a, b and c respectively. The points L, M , P and Q are the midpoints of OA, BC, OB, and AC respectively. 1 1 (a) Show that the position vector of any point on the line LM is a + λ(b + c − a) for some scalar 2 2 λ, and express the position vector of any point on the line P Q in terms of a, b and c. [6 marks] (b) Find the position vector of the point of intersection of the line LM and the line P Q.
44
[4 marks]
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STPM MATHEMATICS (T)
6: Vectors 1 2
1 2
1 4
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[Answer : (a) b + µ(a + c − b) ; (b) (a + b + c)] 18. [STPM ] Find the equation of the plane which is parallel to the plane 3x + 2y − 6z − 24 = 0 and passes through the point (1, 0, 0). Hence, determine the distance between these two planes. [6 marks]
19. [STPM ]
[Answer : 3x + 2y − 6z − 3 = 0 , 3]
0 1 The points A and B lie on the line r = 3 + λ −1, and the distance of each point is three units 6 −4 from the origin O. (a) Determine the coordinates of A and B. (b) Find the area of the triangle OAB.
20. [STPM ]
[6 marks]
[3 marks]
[Answer : (a) A(1, 2, 2), B(2, 1, −2) ; (b) 4.5]
(a) Find the equation of line l1 , passing through points A and B, where the position vectors of points A and B are a and b respectively. [1 marks] (b) R is a point on the line l1 in (a). If point C has position vector c, −→ i. find CR in terms of vectors a, b and c. −→ −−→ ii. prove that CR × AB = a × b + b × c + c × a,
21. [STPM ]
22. [STPM ]
The line l has the equation
[3 marks]
[Answer : (a) r = a + λ(b − a) ; (b)(i) (1 − λ)a + λb − c]
Find the coordinates of the point P on the line Q(9, 4, −3).
[1 marks]
y−1 z−3 x = = which is closest to the point 5 1 −2 [6 marks]
[Answer : P (10, 3, −1)]
x+7 y−4 z−5 = = and the plane π has the equation 4x − 2y − 5z = 8. 1 −3 2
(a) Determine whether the line l is parallel to the plane π.
[5 marks]
(b) Find the equation of the plane that is perpendicular to the plane π and contains the points Q(−2, 0, 3) and R(2, 1, 7). [6 marks] [Answer : (a) parallel. (b) x + 12y − 4z = −14]
23. [STPM ] The position vectors of three non-collinear points B, C and D relative to the origin O are b, c and d respectively. If b · (c − d) = 0 and c · (d − b) = 0, show that BC is perpendicular to OD. [4 marks] 45
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STPM MATHEMATICS (T)
6: Vectors
24. [STPM ]
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(a) Determine the values of p and q.
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1 3 5 1 0 The line r = 2 + λ p is perpendicular to the plane r = 0 + s −1 + t 1, where p and 3 1 1 q 2 q are constants. [5 marks]
(b) Using the values of p and q in (a), find the position vector of the point of intersection of the line and the plane. [5 marks]
25. [STPM ]
[Answer : (a) p = −2, q = −5 ; (b) 4i + 4k]
Find the equation of the plane which contains the straight line x − 3 = dicular to the plane 3x + 2y − z = 3.
z+1 y−4 = and is perpen3 2 [6 marks]
[Answer : x − y + z = −2]
26. [STPM ] The position vectors a, b and c of three points A, B and C respectively are given by a = i + j + k,
b = i + 2j + 3k, c = i − 3j + 2k.
(a) Find a unit vector parallel to a + b + c.
[3 marks]
(b) Calculate the acute angle between a and a + b + c.
[3 marks]
(c) Find the vector of the form i + λj + µk perpendicular to both a and b.
[2 marks]
(d) Determine the position vector of the point D which is such that ABCD is a parallelogram having BD as a diagonal. [3 marks] (e) Calculate the area of the parallelogram ABCD.
[4 marks]
1 5
[Answer : (a) √ (i + 2k) ; (b) 39.2◦ ; (c) i − 2j + k ; (d) i − 4j ; (e) 9]
27. [STPM ] Show that the point A(2, 0, 0) lies on both planes 2x − y + 4z = 4 and x − 3y − 2z = 2. Hence, find the vector equation of the line of intersection of both planes. [5 marks] [Answer : r = 2i + λ(14i + 8j − 5k)]
28. [STPM ] −→ −−→ A tetrahedron OABC has a base OAB and a vertex C, with OA = 2i + j + k, OB = 4i − j + 3k and −−→ OC = 2i − j − 3k. −−→ −→ −−→ (a) Show that OC is perpendicular to both OA and OB. ◦
[3 marks]
(b) Calculate, to the nearest 0.1 , the angle between the edge AC and base OAB of the tetrahedron. [5 marks]
(c) Calculate the area of the base OAB and the volume of the tetrahedron. 46
[7 marks]
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STPM MATHEMATICS (T)
6: Vectors √
14,
14 ] 3
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[Answer : (b) 56.8◦ ; (c)
29. [STPM ] Three vectors a = pi + qj, b = −5i + j and c = 4i + 7j are such that a and b are perpendicular and the scalar product of a and c is 78. (a) Determine the values of p and q. (b) Find the angle between a and c.
[4 marks]
[3 marks]
[Answer : (a) p = 2, q = 10 ; (b) 18.4◦ ]
30. [STPM ] A parallelepiped for which OABC, DEF G, ABF E and OCGD are rectangles is shown in the diagram below.
−→ −−→ The unit vectors i and j are parallel to OA and OC respectively, and the unit vector k is perpendicular −→ −−→ −−→ to the plane OABC, where O is the origin. The vectors OA, OB and OD are 4i, 4i + 3j and i + 5k respectively. √ 13 35 (a) Show that cos ∠BEG = . [6 marks] 175 (b) Calculate the area of the triangle AEG. [6 marks] (c) Find the equation of the plane AEG.
[Answer : (b)
47
[3 marks]
1√ 634 ; (c) 15x + 20y − 3z = 60] 2
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7: Limits and Continuity
( (x − 1)2 , x ≤ 1, f (x) = a 1− , x > 1. x If f is continuous at x = 1, determine the value of a and sketch the graph of f .
2. [STPM ] The function f is defined by
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1. [STPM ] Function f is defined by
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Limits and Continuity
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7
STPM MATHEMATICS (T)
[5 marks]
[Answer : a = 1]
x x 1
(a) Find lim f (x) and lim f (x). Hence, determine whether f is continuous at x = 1.
[4 marks]
(b) Sketch the graph of f .
[3 marks]
x→1−
x→1+
3. [STPM ] The function f is defined by
x − 1, f (x) = x + 2 ax2 − 1,
[Answer : (a) 1 + e , 1 + e ; not continuous ]
0≤x 0, is p. x x Show that 0.5 < p < 1. 5. [STPM ] The function f is defined by
( x2 − 1, x ≤ 1, f (x) = k(x + 1), x > 1.
(a) If f is continuous, find the value of k.
6. [STPM ] The function f is defined by
(√
f (x) =
x + 1, |x| − 1, 48
−1 ≤ x < 1, otherwise.
[2 marks]
[Answer : (a) k = 0]
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x→1−
√
[Answer : (a) 0 , 0 ,
7. [STPM ]
√
2 , 0 ; (b) continuous at x = −1 , discontinuous at x = 1]
f (x) − f (x + h) . h→0 h
x, find lim
8. [STPM ] The graph of a function f is as follows:
(a) State the domain and range of f .
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x→1+
(b) Determine whether f is continuous at x = −1 and x = 1.
Given x > 0 and f (x) =
7: Limits and Continuity
lim f (x), lim f (x) and lim f (x).
x→−1+
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lim f (x),
x→−1−
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(a) Find
STPM MATHEMATICS (T)
1
[Answer : − √ ] 2 x
[2 marks]
(b) State whether f is a one-to-one function or not. Give a reason for your answer.
[2 marks]
(c) Determine whether f is continuous or not at x = −1. Give a reason for your answer.
[3 marks]
9. [STPM ] The function f is defined by
(a) Find lim f (x). x→−1
[Answer : (a) D={x : −3 ≤ x < −1, −1 < x ≤ 2} , R={y : −1 < y < 2} ; (b) f is not one to one function ; (c) f discontinuous at x = −1.]
(√
f (x) =
x + 1, |x| − 1,
x ≥ −1; otherwise.
(b) Determine whether f is continuous at x = −1.
10. [STPM ] The function f is defined by
4 √ , x < 0, √ 4 − x f (x) = 2, x = 0, x √ , x > 0. 1+x−1
(a) Show that lim f (x) exists.
[3 marks]
[2 marks]
[Answer : (a) 0 ; (b) Yes.]
[5 marks]
x→0
49
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STPM MATHEMATICS (T)
7: Limits and Continuity
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[2 marks]
[XXAnswer : (a) 2 ; (b) f is not continuous at x = 0.]
11. [STPM ] Evaluate
√
2x2 + 25 − 5 , x→0 x2 √ 9x2 + 1 (b) lim . x→∞ 3x − 1 (a) lim
12. [STPM ]
Given that f (x) =
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(b) Determine whether f is continuous at x = 0.
[3 marks] [3 marks]
[Answer : (a)
3x2 + x . Find lim f (x) and lim f (x). x→∞ 3x2 − 8x − 3 x→− 13
13. [STPM ] The function f is defined by
1 ; (b) 1] 5
[4 marks]
[XXAnswer :
x 1 − e , x < 0, f (x) = 1, x = 0, x e − 1, x > 0.
(a) Determine the existence of the limit of f (x) as x approaches 0.
1 , 1] 10
[4 marks]
(b) State, with a reason, whether f is continuous at x = 0. Hence, give the interval(s) on which f is continuous. [3 marks] (c) Sketch the graph of f .
14. [STPM ] Function f is defined by
[3 marks]
[XXAnswer : (a) 0, (b) (−∞, 0) ∪ (0, ∞)]
2 x − 4 , x 6= 2, f (x) = |x − 2| 4, x = 2.
Determine whether f is continuous at x = 2.
15. [STPM ] A continuous function f is defined by
x3 − 1, −1 ≤ x < 2, f (x) = 1 − (x − 3)2 + c, 2 ≤ x ≤ 8, 2
where c is a constant. 50
[5 marks]
[XAnswer : No]
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STPM MATHEMATICS (T)
7: Limits and Continuity
(a) Determine the value of c. (c) Sketch the graph of f .
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(b) Find the values of x such that f (x) = 0.
(d) Find the maximum and minimum values of f .
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[4 marks] [3 marks] [3 marks]
[3 marks]
(e) State whether f is a one-to-one function or not. Give a reason for your answer. [XXAnswer : (a) c =
16. [STPM ]
Evaluate
6(x − 2) , x3 − 8 x−8 , (b) lim √ √ x→8 6− x−2 (a) lim
x→2
17. [STPM ]
[2 marks]
√ 15 15 ; (b) x = 1, 3 + 15 ; (d) max= , min=−5 ; (e) one-to-one] 2 2
( x2 + 2x + 5, The function g is defined by g((x) = 3ex + k
x < 0, x ≥ 0.
[2 marks]
[3 marks]
[Answer : (a)
√ 1 ; (b) − 6] 2
(a) Find lim g(x) and lim g(x), and determine the value of k such that the function g is continuous x→0−
x→0+
at x = 0.
(b) Describe the continuity of the function g for x = 0, x < 0 and x > 0. 18. [STPM ] Function f is defined by
f (x) =
Determine whether lim f (x) exists. x→3
1 , 2x
x ≤ 3,
x−3 , x2 − 9
x > 3.
51
[5 marks]
[3 marks]
[5 marks]
[Answer : Yes]
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8: Differentiation
1. [STPM ] Differentiate with respect to x and simplify your answer as far as possible: x2 − x + 1 , x2 + x − 1 (b) e−2x [2 cos(3x) − 3 sin(3x)]. (a)
[Answer : (a)
2x(x − 2) ; (b) −13e−2x cos(3x)] (x2 + x − 1)2
2. [STPM ] Differentiate with respect to x and simplify your answer as far as possible: cos x + sin x , cos x − sin x (b) xn loge x. (a)
[Answer : (a)
3. [STPM ]
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Differentiation
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8
STPM MATHEMATICS (T)
2 ; (b) xn−1 (1 + n ln x)] 1 − sin 2x
dy where dx i. y = (2x − 1)3 (3x + 2)4 and express your result in the form of its factors, ii. y = e−3x (2 cos 2x + 3 sin 2x). 1 1 (b) If x = t − and y = 2t + , where t is a non-zero parameter, prove that t t (a) Find
Deduce that −1 <
3 dy =2− 2 . dx t +1
dy < 2. dx
4. [STPM ]
[Answer : (a) (i) 42x(2x − 1)2 (3x + 2)3 ; (ii) −13e−2x sin 2x]
ln x dy Find the x-coordinate of the point on the curve y = 2 (x > 0) such that = 0, and determine if x dx it is a maximum or minimum point. Sketch the curve for x > 0. You can assume that y → 0 when x → ∞.
5. [STPM ]
[Answer : x =
√
e]
1 tan3 x with respect to x, and express your answer in terms of tan x. 3 d2 y dy (b) Given y = ae−mx cos px, prove that + 2m + (m2 + p2 )y = 0. 2 dx dx (a) Differentiate x − tan x +
52
STPM MATHEMATICS (T)
8: Differentiation
6. [STPM ]
Given that y =
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1 dy (c) Given y = ln(1 + x) − x + x2 , show that ≥ 0 for all values of x > −1. 2 dx
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[Answer : (a) tan4 x]
d2 y cos x − sin x dy , show that = 0. + 2y cos x + sin x dx2 dx
7. [STPM ] A curve with the equation y = ax4 + bx3 + cx2 + dx + e, where a, b, c, d, e are constants, has the following characteristics: (a) It is symmetrical about the y-axis,
(b) It passes through the point (2, −18) and has gradient zero at this point, (c) y = 0 when x = 1.
Show that b = d = 0 and find the values of a, c and e. Sketch the curve and give the coordinates of its turning points.
8. [STPM ]
(a) Given that y = (x + 2)2 (3x − 1)3 , find (b) If y =
[Answer : a = 2, c = −16, e = 14]
dy as a product of its factors. dx
e−x dy , show that (1 + x2 ) + (1 + x)2 y = 0. 1 + x2 dx
9. [STPM ] The parametric equation of a curve are
[Answer : (a) (x + 2)(15x + 16)(3x − 1)2 ]
x = a cos3 θ, y = a sin3 θ
where a is a positive constant and 0 ≤ θ < 2π.
Find the equation of the tangent at the point with the parameter θ. This tangent meets the axes at L and M . Prove that the length of LM is independent of θ. [Answer : y cos θ + x sin θ = a sin θ cos θ , LM = a]
10. [STPM ] A spherical balloon is being inflated at a constant rate of 500 m3 s−1 . Find the rate of increase in the total surface area of the balloon when its radius is 20 m.
11. [STPM ] Given that y = αe−2x sin(x + β), where α and β are constants, verify d2 y dy +4 + 5y = 0. 2 dx dx 53
[Answer : 50]
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STPM MATHEMATICS (T)
8: Differentiation
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12. [STPM ] (a) For the curve y = sin x cos3 x, where 0 ≤ x ≤ π, find the x and y coordinates of the points where dy = 0. Sketch this curve. dx 2 1 1 dy 2 3 = cot x(cos2 x − 3 sin2 x)2 (b) For the curve y = sin x cos x, where 0 ≤ x ≤ π, show that 2 dx 4 on the condition that x 6= 0. Sketch the curve. [Answer :
√ ! π π 3 3 , , ,0 , 6 16 2
13. [STPM ] A curve is given by its parametric equations
1 x = t2 , y = 1 − , (t > 0). t
√ ! 5π 3 3 ] ,− 6 16
The curve intersects the x-axis at P . Find the equation of the tangent to the curve at P .
14. [STPM ]
The parametric equations of a curve are x = t2 , y = t3 . Express of the tangent to the curve at the point P (p2 , p3 ).
15. [STPM ]
If y = 3x + sin x − 8 sin Deduce that
dy in terms of t. Find the equation dx
[Answer :
3 dy = t ; 2y = 3px − p3 ] dx 2
1 dy 1 x , find and express your answer in terms of cos x . 2 dx 2
dy ≥ 0 for all values of x. dx
16. [STPM ]
[Answer : 2y − x + 1 = 0]
1 2
2
[Answer : 2 cos x − 1
]
x−3 . (x − 2)(x + 1) (b) Find the points where the curve intersects the axes, and find the stationary points on this curve. (a) Find the equation of the asymptotes of the curve y =
(c) Sketch this curve.
(d) Find the values of k such that the equation (x − 3) = k(x − 2)(x + 1) does not have real roots. 1 9
[Answer : (a) x = 2, x = −1, y = 0 ; (b) (3, 0), (0, 1.5), (1, 1), (5, ) ; (d)
1 < k < 1] 9
17. [STPM ] Find the coordinates of the stationary points of the curve y = x − ln(1 + x) and sketch the curve. 54
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STPM MATHEMATICS (T)
8: Differentiation
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18. [STPM ]
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[Answer : (0, 0)]
1 (a) Find the points on the x-axis intersected by the curve y = x3 − x2 − 2x. Find also the maximum 2 and minimum points, as well as any points of inflection on this curve. (b) Sketch this curve. 1 (c) Find the value of k if the equation x3 − x2 − 2x = k has a repeated root, and state this root. 2 √
√
[Answer : (a) (0, 0), (1 + 5, 0), (1 − 5, 0) ;
2 20 2 44 Maximum point (− , ) , Minimum point (2, −4) , Point of inflection ( , − ) ; 3 27 3 27 20 2 (c) k = −4, repeated root=2; k = , repeated root=− ] 27 3
19. [STPM ] If y = etan x , show that
dy d2 y = (1 + tan x)2 . dx2 dx
20. [STPM ] Show that the equation of the normal to the curve y = tan 2x at the point where the x-coordinate is √ π is 3x + 24y = π − 24 3. 3 21. [STPM ] The parametric equations of a curve are x = t2 − 2, y = t3 − 3. Find the equation of the normal to the curve at the point where the parameter t = 2.
22. [STPM ] Differentiate with respect to x (a) (x2 + 2x)ex 1 − x2 (b) √ , 1 + 2x
2 +2x
,
simplifying your answers.
[Answer : x + 3y = 17]
2
[Answer : (a) 2(x + 1)3 ex
+2x
; (b) −
3x2 + 2x + 1 3
(1 + 2x) 2
]
23. [STPM ] Find the equations of the tangent and normal to the curve x2 y + xy 2 = 12 at the point (1, −4).
24. [STPM ] A curve has the equation y 2 = x2 (x + 3).
[Answer : 8x − 7y = 36 ; 7x + 8y + 25 = 0]
Show that the curve is symmetric about the x-axis. Show that for all points on this curve, x ≥ −3.
Find coordinates of the turning points of this curve.
Sketch the curve. Show clearly the shape of the curve near the origin. Find the area bounded by the loop of this curve. 55
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STPM MATHEMATICS (T)
8: Differentiation 24 √ 3] 5
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[Answer : (−2, 2), (−2, −2) ; Area=
25. [STPM ] Two parallel sides of a rectangle respectively lengthen at a rate of 2 cm per second, while the other two parallel sides shorten such that the area of the rectangle is always 50 cm2 . If, at the time t, the length of each lengthening side is x, the length of each shortening side is y, and the perimeter of the rectangle is p, show that dp y . =4 1− dt x Find the rate of change in the perimeter when (a) x=5 cm, (b) y=5 cm.
√ Show that the perimeter of the rectangle is the least when x = y = 5 2 cm.
26. [STPM ] Differentiate with respect to x 2
(a) (2x3 + 1)ex , (b) ln(x2 e−x ),
27. [STPM ]
[Answer : (a) −4 ; (b) 2]
2
[Answer : (a) 2xex (2x3 + 3x + 1) ; (b)
x2 . x2 − 4 Write the equations of the asymptotes of this curve. A curve has the equation y =
2 − 1] x
Find the coordinates of the turning point on this curve, and determine if this is a maximum or minimum point. Determine if there are any points of inflection on this curve. Sketch this curve.
[Answer : x = −2, x = 2, y = 1 ; (0,0) is a local maximum point ; No points of inflexion]
28. [STPM ] If y = ln(sin px + cos px), show that
d2 y + dx2
dy dx
2
29. [STPM ] Differentiate each of the following with respect to x. (a) (x2 + 1)e−x , √ (b) cos2 ( x).
+ p2 = 0.
2 −x
[Answer : (a) −(x − 1) e 56
√ sin 2 x ; (b) − √ ] 2 x
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30. [STPM ] dy x−1 Find if ey = . dx 3−x
Determine the gradient of the curve y = ln
31. [STPM ]
x−1 3−x
8: Differentiation
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STPM MATHEMATICS (T)
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at the point where it intersects the x-axis.
[Answer :
2 ; 2] (x − 1)(3 − x)
1 1 − . 2 x x Find the coordinates of the turning point of the curve, and determine if it is a maximum or minimum point. A curve has the following equation y =
Sketch this curve.
The tangent to the curve at the point A(1, 0) meets the curve once again at the point B. Find the coordinates of B. 1 4
[Answer : (2, − ) is minimum point, B = (−1, 2)]
32. [STPM ] The equation of a curve is x2 y + xy 2 = 2. Find the equations of both tangents to the curve at the point x = 1.
33. [STPM√] If y = sin x, show that
4y 3
[Answer : x + y = 2 ; y = −2]
d2 y + y 4 + 1 = 0. dx2
34. [STPM ] Differentiate each of the following with respect to x. (a) e−x ln x3 , 2x (b) . 1 + x4
35. [STPM ]
[Answer : (a)
2x [(1 + x4 ) ln 2 − 4x3 ] 3e−x ] (1 − x ln x) ; (b) x (1 + x4 )2
√ dy √ The variables x and y are connected by y x − y − x = 1. Find the values of y and when x = 1. dx
36. [STPM ] The function f is defined by
f (x) = cos x +
1 cos 2x, 0 ≤ x ≤ 2π. 2
57
4 3
[Answer : 4 , − ]
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STPM MATHEMATICS (T)
8: Differentiation
(c) Sketch the graph f . (d) State the range of f . [Answer : (a) 0.4π, 1.6π ; (b)
3 0, 2
2 4 1 3 3 3 3 3 , π, − , 2π, , , ; (d) {y : − ≤ y ≤ }] π, − π, − 2 2 3 4 3 4 4 2
37. [STPM ] If y 2 = ln(x2 y) where x, y > 0, (a) show that (b) find
dy 2y = , dx x(2y 2 − 1)
dy when y = 1. dx
38. [STPM ]
If x = sin3 2θ, y = cos3 2θ, find
dy in terms of θ. dx
39. [STPM ] 1 If y = (2ex − 6x + 5) 2 , show that
40. [STPM ]
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(b) Find all the pairs (x, f (x)) when f 0 (x) = 0.
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(a) Find all values of x in the form of kπ, with k correct to one decimal place when f (x) = 0.
y
d2 y + dx2
dy dx
2
= ex .
2 e
[Answer : (b) √ ]
[6 marks]
[Answer : − cot 2θ]
[4 marks]
Figure above shows a composite solid which consists of a cuboid and a semicylindrical top with a common face ABCD.
58
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8: Differentiation
3
1 [9600 − (8 + 5π)x2 ]. 24x
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y=
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The breadth and length of the cuboid is x cm and 2x cm respectively and its height is y cm. Given that the total surface area of this solid is 2400 cm2 . Show that
[3 marks]
If the volume of this solid is V cm , express V in terms of x. Hence, show that V attains its maximum 40 . [9 marks] when x = √ 4+π Find this maximum volume. [3 marks] [Answer : V =
1 64000 [9600x − 2πx3 − 8x3 ] , √ ] 12 3 π+4
41. [STPM ] Find the gradient of the curve 2x2 + y 2 + 2xy = 5 at the point (2, −1).
42. [STPM ] The parametric equations of a curve are
[3 marks]
[Answer : -3]
x = sec t − tan t; y = cosec t − cot t,
dy 1 + sin t 1 =− . with 0 < t < π. Show that 2 dx 1 + cos t
[4 marks]
−1 3 Tangent to the curve at the point A, with t = tan , meets the tangent to the curve at the point 4 4 B, with t = tan−1 , at point N . Find the coordinates of N . [8 marks] 3
43. [STPM ] The equation of a curve is
[Answer : N =
x2 . x2 − 3x + 2 Find the asymptotes and the stationary points of the curve. Sketch the curve.
y=
[8 marks] [4 marks]
2
2
Determine the number of real roots of the equation k(x − 1) (x − 2) = x where k > 0.
The equation of a curve is y =
[3 marks]
[Answer : Asymptotes are x = 1, x = 2, y = 1 ; 1 root]
44. [STPM ] √ √ dy Find in terms of x if x = e t and y = et . dx
45. [STPM ]
7 7 , ] 17 17
e2kx − 1 where k is a positive constant. e2kx + 1 59
[4 marks]
[Answer :
e
(ln x)2 2
x
ln x]
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STPM MATHEMATICS (T)
8: Differentiation
dy > 0 for all values of x. dx dy d2 y d2 y (b) Show that + ky 2 = k. Hence, show that ≤ 0 for x ≥ 0 and ≥ 0 for x ≤ 0. dx dx2 dx2 (c) Sketch the curve.
Given that y =
sin kx , where k is a positive integer, show that 1 + cos kx sin kx
d2 y = k2 y2 . dx2
[3 marks]
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46. [STPM ]
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(a) Show that
[8 marks]
[4 marks]
[6 marks]
47. [STPM ] The graphs of y = x3 + ax2 + bx + c passes through (3, −21) and has stationary points when x = 2 and x = −2. Find the values of a, b and c. [5 marks] Find the coordinates of these stationary points and determine if they are local extremums. Find also the point of inflexion of the curve. [7 marks] dy < 0. [3 marks] Determine the set of x so that dx [Answer : a = 0 , b = −12 , c = −12 ; (2,-28) is local minimum , (-2,4) is local maximum ; point of inflexion is (0,-12) ; {x : −2 < x < 2}]
48. [STPM ] A curve has parametric equations x = e2t − 2t and y = et + t. Find the gradient of the curve at the point with t = ln 2. [5 marks] [Answer :
1 ] 2
49. [STPM ] A curve with equation y = x3 + px2 + qx + r cuts the y-axis at y = −34 and has stationary points at x = 3 and x = 5. Find the values of p, q, and r. [6 marks] Show that the curve cuts the x-axis only at x = 1, and find the gradient of the curve at that point. Sketch the curve.
50. [STPM ] Given a curve with parametric equation
with a > 0 and t ∈ R.
[7 marks] [2 marks]
[Answer : p = −12 , q = 45 , r = −34 ; 24]
x = a(t − 3t3 ), y = 3at2 ,
Determine the values of t when the curve cuts the y-axis and sketch the curve. 2 2 dx dy Show that + = a2 (1 + 9t2 )2 . dt dt
[4 marks] [3 marks]
√
[Answer : t = 0, 60
√ 3 3 ,− ] 3 3
STPM MATHEMATICS (T)
8: Differentiation
52. [STPM ]
Given that y = e−x cos x, find
53. [STPM ] Function f if defined by
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51. [STPM ] Find the equation of the normal to the curve x2 y + xy 2 = 12 at the point (3, 1).
dy d2 y and when x = 0. dx dx2
f (x) =
2x . (x + 1)(x − 2)
Show that f 0 (x) < 0 for all values of x in the domain of f .
[6 marks]
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[Answer : y =
38 15 x− ] 7 7
[4 marks]
[Answer :
d2 y dy = −1, = 0] dx dx2
[5 marks]
Sketch the graph of y = f (x). Determine if f is a one to one function. Give reasons to your answer. [6 marks]
Sketch the graph of y = |f (x)|. Explain how the number of the roots of the equation |f (x)| = k(x − 2) depends on k. [4 marks]
54. [STPM ]
[Answer : f is not one to one function. If k ≥ 0, 1 root. If k < 0, 3 roots.]
dy √ when y = 1. If y = ln xy, find the value of dx
55. [STPM ]
[5 marks]
[Answer :
1 ] e2
2 A curve is defined by the parametric equations x = 1 − 2t, y = −2 + . Find the equation of the t normal to the curve at the point A(3, −4). [7 marks] The normal of the curve at the point A cuts the curve again at point B. Find the coordinates of B.
56. [STPM ] cos x d2 y dy If y = , where x 6= 0, show that x 2 + 2 + xy = 0. x dx dx 57. [STPM ]
[4 marks]
[Answer : x + y + 1 = 0 ; B(−1, 0)]
[4 marks]
1 Find the coordinate of the stationary point on the curve y = x2 + where x > 0; give the x-coordinate x and y-coordinate correct to three decimal places. Determine whether the stationary point is a minimum point or a maximum point. [5 marks] [Answer : (0.794 , 1.890) , minimum]
58. [STPM ] If y = x ln(x + 1), find an approximation for the increase in y when x increases by δx. Hence, estimate the value of ln 2.01 given that ln 2 = 0.6931. 61
[6 marks]
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STPM MATHEMATICS (T)
8: Differentiation
The function f is defined by f (t) =
4ekt − 1 where k is a positive constant, t > 0. 4ekt + 1
(a) Find the value of f (0). (b) Show that f 0 (t) > 0. 2
0
[1 marks] [5 marks]
00
(c) Show that k[1 − f (t) ] = 2f (t) and hence show that f (t) < 0. (d) Find lim f (t). t→∞
(e) Sketch the graph of f .
60. [STPM ] If y =
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59. [STPM ]
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[Answer : 0.698]
[2 marks] [2 marks]
[Answer : (a)
dy x , show that x2 = (1 − x2 )y 2 . 2 1+x dx
61. [STPM ]
Find the coordinates of the stationary points on the curve y = Sketch the curve.
[6 marks]
3 ; (d) 1] 5
[4 marks]
x3 and determine their nature. x2 − 1 [10 marks] [4 marks]
Determine the number of real roots of the equation x3 = k(x2 − 1), where k ∈ R, when k varies.[3 marks] √ √ √ 3 3 3 3 ) is local min. , (− 3, − ) is local max. [Answer : (0, 0) is inflexion point , ( 3, √ √ √2 √ 2 √ 3 3 3 3 3 3 3 3 3 3 1 real root for − ] 2 2 2 2 2 √
62. [STPM ] sin x − cos x d2 y dy If y = , show that = 2y . 2 sin x + cos x dx dx 63. [STPM ] Show that the curve y =
[6 marks]
x is always decreasing. [3 marks] −1 Determine the coordinates of the point of inflexion of the curve, and state the intervals for which the curve is concave upwards. [5 marks] x2
Sketch the curve.
64. [STPM ] A curve is defined by x = cos θ(1 + cos θ) , y = sin θ(1 + cos θ). (a) Show that
dx dθ
2
+
dy dθ
2
= 2(1 + cos θ).
[3 marks]
[Answer : (0, 0) ; (−1, 0) ∪ (1, ∞)]
[4 marks]
62
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STPM MATHEMATICS (T)
8: Differentiation
[Answer : p = 3, q = 5]
66. [STPM ] A curve is defined by the parametric equations x=t−
where t 6= 0.
2 t
and y = 2t +
1 t
dy 5 1 dy =2− 2 , and hence, deduce that − < < 2. dx t +2 2 dx dy 1 (b) Find the coordinates of points when = . dx 3 (a) Show that
67. [STPM ]
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65. [STPM ] The line y + x + 3 = 0 is a tangent to the curve y = px2 + qx, where p 6= 0 at the point x = −1. Find the values of p and q. [6 marks]
[8 marks]
[3 marks]
[Answer : (b) (-1,3) , (1,-3)]
dy 1 1 , =√ Given that u = (ex + e−x ), where x > 0 and y = f (u) is a differentiable function f . If 2 du u2 − 1 dy show that = 1. [5 marks] dx 68. [STPM ] The functions f and g are defined by
f : x → x3 − 3x + 2, x ∈ R. g : x → x − 1, x ∈ R.
(a) Find h(x) = (f ◦ g)(x), and determine the coordinates of the stationary points of h.
[5 marks]
(b) Sketch the graph of y = h(x).
[2 marks]
1 (c) On a separate diagram, sketch the graph of y = . h(x) Hence, determine the set of values of k such that the equation i. one root, ii. two roots, iii. three roots.
[3 marks]
1 = k has h(x)
[1 marks] [1 marks]
[1 marks]
1 4
1 4 1 {k : k > }] 4
[Answer : (a) h(x) = x3 − 3x2 + 4 , (0,4) , (2,0) ; (c) (i) {k : k < 0, 0 < k < } ; (ii) {k : k = } ; (iii)
69. [STPM ] √ Given that y is differentiable and y x = sin x, where x 6= 0. Using implicit differentiation, show that 2 dy 1 2d y 2 x +x + x − y = 0. dx2 dx 4 [6 marks]
63
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70. [STPM ] The function f is defined by
f (x) =
(a) State all asymptotes of f .
8: Differentiation
ln 2x , where x > 0. x2
[2 marks]
(b) Find the stationary point of f , and determine its nature. (c) Obtain the intervals, where i. f is concave upwards, and ii. f is concave downwards.
[6 marks]
Hence, determine the coordinates of the point of inflexion.
(d) Sketch the graph y = f (x).
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[6 marks] [2 marks]
1 1 2 2 is a maximum point. e , [Answer : (a) x = 0, y = 0 ; (b) 2 e (c)(i) (1.15, ∞) ; (ii) (0, 1.15) ; (1.15, 0.630)]
71. [STPM ]
Given that y = (2x)2x , find
dy in terms of x. dx
72. [STPM ] The function f is defined by
f (x) = √
(a) Show that
[4 marks]
[Answer : (2x)2x (2 + 2 ln(2x))]
e−x , where x ∈ R, 1 + x2
f 0 (x) =
−e−x (x2 + x + 1) 3
(1 + x) 2
(b) Show that f is a decreasing function. (c) Sketch the graph of f .
.
[3 marks] [4 marks] [2 marks]
73. [STPM ] A curve is defined by the parametric equations x = ke−t cos t and y = ke−t sin t, where k is a constant.
(a) Show that
dx dt
74. [STPM ]
2
+
dy dt
2
= 2k 2 e−2t .
[4 marks]
Find the equation of the normal to the curve with parametric equations x = 1 − 2t and y = −2 + at the point (3, −4).
2 t
[6 marks]
[Answer : y = −x − 1]
75. [STPM ] A right circular cone of height a + x, where −a ≤ x ≤ a, is inscribed in a sphere of constant radius a, such that the vertex and all points on the circumference of the base lie on the surface of the sphere. 64
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STPM MATHEMATICS (T)
8: Differentiation
(c) Sketch the graph of V against x.
[2 marks]
[3 marks]
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[6 marks]
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1 (a) Show that the volume V of the cone is given by V = π(a − x)(a + x)2 . 3 (b) Determine the value of x for which V is maximum and find the maximum value of V .
1 1 (d) Determine the rate at which V changes when x = a if x is increasing at a rate of a per 2 10 minute. [4 marks] [Answer : (b) x =
1 a 32 3 , πa ; (d) − πa3 ] 3 81 40
76. [STPM ] Find the gradients of the curve y 3 + y = x3 + x2 at the points where the curve meets the coordinate axes. [6 marks] [Answer : 0 , 1]
77. [STPM ] The parametric equations of a curve are x = θ − sin θ and y = 1 − cos θ. Find the equation of the 1 [7 marks] normal to the curve at a point with parameter π. 2
78. [STPM ] A curve is defined implicitly by the equation x2 + xy + y 2 = 3.
π 2
[Answer : y = −x + ]
dy 2x + y + = 0. [3 marks] dx x + 2y (b) Find the gradients of the curve at the points where the curve crosses the x-axis and y-axis.[5 marks] (a) Show that
(c) Show that the coordinates of the stationary points of the curve are (-1,2) and (1, -2).
[5 marks]
(d) Sketch the curve.
[2 marks]
79. [STPM ] A rectangle with a width 2x is inscribed in a circle of constant radius r.
1 2
[Answer : (b) − , -2]
(a) Express the area A of the rectangle in terms of x and r. √ (b) Show that the rectangle is a square of side r 2 when A has a maximum value.
80. [STPM ] The graph of y = 2 cos x + sin 2x for 0 ≤ x ≤ 2π is shown below.
65
[2 marks]
[5 marks]
p
[Answer : (a) A = 4x r2 − x2 ]
8: Differentiation
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STPM MATHEMATICS (T)
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The points A and C are local extremum points. The points B, D, E and F are points of inflexion. (a) Determine the coordinates of i. the points of local extremum. ii. the points of inflexion.
(b) State the intervals where the graph is concave upward. (c) Calculate the area of the region bounded by the curve and the x-axis.
[5 marks] [5 marks]
[1 marks]
[4 marks]
√ ! √ ! π 3 3 5π 3 3 , ,− [Answer : (a) (i) A = ,C= ; 6 2 6 2 π π 3π 3π , 0 , D = (3.39, −1.45) , E = , 0 , F = (6.03, 1.45) ; (b) , 3.39 ∪ , 6.03 ; (c) 4] (ii) B = 2 2 2 2
81. [STPM ]
The graph of y =
3x − 1 is shown below. (x + 1)3
The graph has a local maximum at A and a point of inflexion at B. 66
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STPM MATHEMATICS (T)
8: Differentiation
(a) Write the equations of the asymptotes of the graph.
[1 marks]
84. [STPM ] Differentiate with respect to t 2
(a) (t2 − 1)et −1 , r 1 (b) ln 1 + . t
[1 marks] [1 marks]
equation
[1 marks]
5 27 ; , 3 128 5 (i) {x : x < −1, −1 < x ≤ 1} ; (ii) (−∞, −1) ∪ ,∞ ; 3 1 (c) (i) {k : 0 < k < } ; (ii) {k : −1 < k ≤ 0}] 4
(a) Find the stationary points on the curve, and determine it’s nature.
83. [STPM ] p If y sin−1 2x = 1 − 4x2 , show that
[9 marks]
[2 marks]
[Answer : (a) x = −1 , y = 0 ; (b) A 1,
82. [STPM ] The equation of a curve is y = x3 e3−2x .
(b) Sketch the curve.
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i. has three distinct real roots, ii. has only one positive root.
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(b) Determine the coordinates of the points A and B. Hence, state dy i. the set of values of x when ≥ 0, dx ii. the intervals where the graph is concave upward. 3x − 1 (c) Using the above graph of y = , determine the set of values of k for which the (x + 1)3 3x − 1 − k(x + 1)3 = 0
,B
[7 marks]
dy + 4xy + 2y 2 = 0. dx
85. [STPM ] For the graph of y = 3x4 + 16x3 + 24x2 − 6,
[3 marks]
[Answer : (a) (0,0)=point of inflexion,
(1 − 4x2 )
1 4
3 27 , =local maximum] 2 8
[5 marks]
[3 marks]
[3 marks]
[Answer : (a) 2t3 et
2
−1
; (b) −
1 ] 2(t2 + 1)
(a) determine the intervals on which the graph is concave upward and concave downward, [6 marks] (b) find the points of inflexion,
[3 marks]
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8: Differentiation
2 3
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(c) determine the extremum point and its nature. Hence, sketch the graph of y = 3x4 + 16x3 + 24x2 − 6.
[Answer : (a) (−∞, −2) ∪ − , ∞ , −2, −
86. [STPM ] A continuous function f is defined by
where c is a constant. (a) Determine the value of c.
2 3
; (b)
[3 marks]
2 14 − , 3 27
and (−2, 10) ; (c) (0, −6) minimum]
x3 − 1, −1 ≤ x < 2, f (x) = 1 − (x − 3)2 + c, 2 ≤ x ≤ 8, 2
(b) Find the values of x such that f (x) = 0. (c) Sketch the graph of f .
[3 marks]
(d) Find the maximum and minimum values of f .
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(e) State whether f is a one-to-one function or not. Give a reason for your answer. [Answer : (a) c =
[4 marks] [3 marks] [3 marks]
[3 marks] [2 marks]
√ 15 15 ; (b) x = 1, 3 + 15 ; (d) max= , min=-5] 2 2
87. [STPM ] A water storage tank ABCDEF GH is a part of an inverted right square based pyramid, as shown in the diagram below.
The complete pyramid OABCD has a square base of sides 12 m and height 15 m. The depth of the 1 tank is 9 m. Water is pumped into the tank at the constant rate of m3 min−1 . 3
68
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STPM MATHEMATICS (T)
8: Differentiation
[3 marks]
(c) Calculate the time taken to fill up the tank if initially the tank is empty.
88. [STPM ] The equation of a curve is y = x(x − 2)3 .
[Answer : (b)
(a) Find the set of values of x for which y ≥ 0.
25 ; (c) 33.7] 3888
[9 marks]
[3 marks]
[Answer : (a) {x : x ≤ 0, x ≥ 2} ; (b) Extremum=
69
[3 marks]
[3 marks]
(b) Determine the extremum point and the points of inflexion on the curve. (c) Sketch the curve.
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(a) Show that the volume of water V m3 when the depth of water in the tank is h m is given by 16 V = h(h2 + 18h + 108). [3 marks] 75 (b) Find the rate at which the depth is increasing at the moment when the depth of water is 3 m.
1 27 ,− 2 16
, Inflexion=(1, −1), (2, 0)]
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9: Integration
1
Z
x dx, give your answer correct to two decimal places. 4 − x2 Z 1 x dx (b) Using the substitution t = tan , or otherwise, find 2 3 − 5 cos x √
(a) Evaluate
1 2
1 [Answer : (a) 0.20 ; (b) ln 4
2. [STPM ]
Z
(a) Find
π 4
tan3 θ dθ, giving your answer correct to two significant figures.
0
2
4
Z
√
(b) Using the substitution u = 2t + 1, or otherwise, evaluate
3. [STPM ]
Z
2a
(a) Find
a
x3 dx. x4 + a4
0
4. [STPM ]
(a) Show that
0
4
x−1 dx = ln 2 2x + 3x + 1
1
Z
Z
(sin x + 3 cos x)2 dx. Z 2 (x − 1)(5x + 2) 1 8 (b) Show that dx = ln . 2 + 2) (2x − 1)(x 2 3 1 (a) Find
0
r
70
2 tan x2 − 1 2 tan x2 + 1
+ c]
[Answer : (a)
Z 1
5
10 ] 3
π 1−x dx = − 1. 1+x 2
25 . 27
(b) Using the substitution u2 = 2x − 1, or otherwise, evaluate
5. [STPM ]
t dt. 2t + 1
[Answer : (a) 0.15 ; (b)
(b) Using the substitution x = cos 2θ, or otherwise, prove that
Z
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1. [STPM ]
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Integration
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9
STPM MATHEMATICS (T)
√ x 2x − 1 dx.
1 ln 4
[Answer : (b)
17 ] 2
428 ] 15
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STPM MATHEMATICS (T)
9: Integration
1
x3 dx, leaving your answer in a form involving logarithms. 2 0 x +2 Z 1p 1 1√ 4 − x2 dx = π + 3. (b) Show that 3 2 0 Z
(a) Evaluate
7. [STPM ]
Z
(a) Find
2 + cos x dx. sin2 x
1 [Answer : (a) + ln 2
Z
(b) Using the substitution x = 2 sin θ, or otherwise, evaluate correct to two significant figures.
3 cos 2x + c] 2
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6. [STPM ]
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[Answer : (a) 5x + 2 sin 2x −
0
1
√
2 ] 3
x2 dx, giving your answer 4 − x2
[Answer : (a) −2 cot x − cosec x + C ; (b) 0.118]
8. [STPM ] Sketch the corresponding curves for the following equations in separate diagrams, showing the turning points and any asymptotes parallel to the axes: (a) y = (x − 1)(x − 3) 1 (b) y = (x 6= 1, x 6= 3) (x − 1)(x − 3)
4 intersects the graphs (b) at the points A and B. 3 Calculate the finite area bounded by the line AB and the portion of the graph between A and B, giving your answer correct to three significant figures. The line y = −
[Answer : (a) (2,-1), minimum point ; (b) (2,-1), maximum point , asymptotes are x = 1, x = 3, y = 0 ;
9. [STPM ]
Z
1
Find the exact value of
√ x x + 3 dx.
−2
10. [STPM ]
Z (a) Evaluate
π 4
tan2 x dx.
0
71
Area=0.235]
8 5
[Answer : − ]
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STPM MATHEMATICS (T)
9: Integration
18 − 4x − x2 B C A + + ≡ , show that A = 2 and determine the 2 (4 − 3x)(1 + x) 4 − 3x 1 + x (1 + x)2 values of B and C.Z 1 7 3 18 − 4x − x2 dx = ln 2 + Hence, show that 2 (4 − 3x)(1 + x) 3 2 0 [Answer : (a) 1 −
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(b) Given that
π ; (b) B = 1, C = 3] 4
11. [STPM ] A bowl in the shape of a hemisphere with a radius a and its rim horizontal, is filled with a liquid to a height of h unit. Show, through integration, that the volume of the liquid in the bowl is 1 2 πh (3a − h). 3
12. [STPM ]
Z
(a) Show that
0
π 6
√ 4π − 3 3 sin x cos x dx = . 192 2
2
(b) Using the substitution u = x2 , or otherwise, evaluate logarithmic form.
13. [STPM ]
Z
√ 2 2
Z 0
x dx, leaving your answer in 1 − x4
[Answer : (b)
2x + 1 √ dx. x+1 Z 4 11x2 + 4x + 12 (b) Show that dx = ln 675. 2 0 (2x + 1)(x + 4) (a) Find
4 3
1 ln 3] 4
3
1
[Answer : (a) (x + 1) 2 − 2(x + 1) 2 + c]
14. [STPM ] Using integration, find te area of the finite region bounded by the curve √ √ √ x+ y = a and the coordinate axes, if a is a positive constant.
[Answer :
1 2 a ] 6
15. [STPM ] Find the coordinates of the points P , Q where the curve 2y = x + 3 meets the parabola y 2 = 4x. Find the area bounded by the arc P Q of this parabola and the line P Q.
8 3
[Answer : P (1, 2) , Q(9, 6) ; ] 72
STPM MATHEMATICS (T)
9: Integration
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16. [STPM ] By using the substitution x = sin θ, or otherwise, evaluate 1 2
Z
1 3
x2
√
1 dx 1 − x2
leaving your answer in the forms of surds.
17. [STPM ]
(a) Express
√
√
[Answer : 2 2 − 3]
17 + x in partial fractions. Hence or otherwise, show that (4 − 3x)(1 + 2x) Z
1 2
− 13
17 + x 1 dx = (19 ln 2 + 9 ln 3). (4 − 3x)(1 + 2x) 6 Z
(b) Using the substitution x = sin θ, or otherwise, find
18. [STPM ]
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0
x2 1 dx. 2 (1 − x2 ) 12
3 1 5 + ; (b) [Answer : (a) 4 − 3x 1 + 2x 2
√ ! π 3 − ] 6 4
8 − 4 that intersects the x-axis at the point A. x The tangent to the curve at the point P (1, 4) intersects the x-axis at the point Q. Find the coordinates of Q. The diagram above shows a portion of the curve y =
Find the area of the shaded region bounded by the curve AP and the line segments P Q and QA.
19. [STPM ]
73
[Answer : Q =
3 , 0 ; A=8 ln 2 − 5] 2
9: Integration
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STPM MATHEMATICS (T)
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The diagram shows a hemispherical bowl with radius a and its rim horizontal. Show that when the depth of water in the bowl is y, the volume V of water in the bowl is given by 1 V = πy 2 (3a − y). 3
Initially, the bowl is empty. Water is then poured into the bowl at a constant rate. The time taken to 1 fill the bowl is T . Find, in terms of T , the time taken for y to become a. Find, in terms of T and a, 2 1 the rate at which the water level rises when y = a. 2 [Answer :
8a 5 T ; ] 16 9T
20. [STPM ] Determine the coordinates of the points of intersection between the graphs y 2 = x and y = −x + 2. Find the area bounded by the two graphs. Find also the volume of the solid generated when this area is rotated through 2π about the y-axis.
21. [STPM ]
(a) Using the substitution x = a tan θ, or otherwise, show that √ Z a x3 8−5 2 , 5 dx = 12a 0 (a2 + x2 ) 2
[Answer : (1, 1) , (4, −2) ;
9 72 ; π] 2 5
with a > 0. Z 8 (x − 1)2 (b) Evaluate , dx, giving your answer correct to three significant figures. x2 − 4 3
22. [STPM ]
[Answer : (b) 3.89]
(a) Find the values of the constants A and B such that cos x ≡ A(3 cos x + 4 sin x) + B(−3 sin x + Z π 2 cos x 4 cos x). Hence or otherwise, find the value of dx giving your answer correct 0 3 cos x + 4 sin x to three decimal places. 74
STPM MATHEMATICS (T)
9: Integration Z
3
r
correct to three significant figures.
0
e2
(b)
e
2
5−x dx giving your answer x−1
[Answer : (a) A =
23. [STPM ] Evaluate Z π 6 tan 2θ dθ, (a) Z
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(b) By using the substitution x = 3 − 2 cos 2θ, or otherwise, find
1 dx, x ln x
4 3 ,B= , 0.235; (b) 1.32] 25 25
[Answer : (a)
24. [STPM ] Find the area bounded by the curves y = 1 − x2 and y = x − 1.
25. [STPM ]
1 ln 2 ; (b) ln 2] 2
[Answer :
Z
(a) By using a suitable substitution or otherwise, find the value of answer correct to three significant figures.
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1
2
9 ] 2
x−1 dx, giving your (2x − 1)2
d (b) Show that (tan3 θ) = 3 tan4 θ + 3 sec2 θ − 3. Hence, determine the value of dx
Z
π 4
tan4 θ dθ.
0
[Answer : (a) 0.108 ; (b)
π 2 − ] 4 3
26. [STPM ] √ π Sketch the area R bounded by the y-axis, x-axis, line x = , and the curve y = 1 + sin x. Find the 2 volume of the solid formed when R is rotated through four right angles about the x-axis.
27. [STPM ] Show that
d (tan3 x) = 3 sec4 x − 3 sec2 x. dx Z
Hence, determine the value of 0
π 4
sec4 x dx.
28. [STPM ]
75
[Answer :
π (π + 2)] 2
[Answer :
4 ] 3
9: Integration
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STPM MATHEMATICS (T)
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1 The shaded area shown in the diagram above is bounded by the curve y = e−x + , the curve y = ex −1, 2 and the y-axis. Find the area of the shaded region. [Answer :
1 (3 ln 2 − 1)] 2
29. [STPM ] dy ln x , where x > 0, determine the set of values for x such that > 0 and the set of values for If y = x dx dy 1 x such that < 0. Hence, show that the maximum value of y is . dx e ln x x Sketch the curve y = , where x > 0. Hence sketch the curve y = , x > 0. Show that the area x ln x 1 ln x , the x-axis, and the line x = is equal to the area bounded by the bounded by the curve y = x e ln x curve y = , the x-axis, and the line x = e. x
30. [STPM ]
Z
Find the value of
1
√ x 1 + x dx.
0
31. [STPM ] Express
x2
[Answer : {x : 0 < x < e} ; {x : x > e}]
1 in the form of partial fractions. Hence, show that −1 Z 1 1 x−1 dx = ln + c, 2 x −1 2 x+1
where c is the constant.
Using integration by parts, show that Z Z 1 x 2x2 dx = + dx. x2 − 1 x2 − 1 (x2 − 1)2 Z Deduce the value of 2
4
[Answer :
x2 dx. Give your answer correct to three decimal places. (x2 − 1)2 76
√ 4 (1 + 2)] 15
STPM MATHEMATICS (T)
9: Integration
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[Answer :
1 1 − ; 0.347] 2(x − 1) 2(x + 1)
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32. [STPM ] Calculate the volume of the solid generated when the area bounded by the curves y = x2 and y 2 = 8x is rotated completely about the x-axis. [Answer :
48 π] 5
33. [STPM ] Sketch the curve y = x(x + 1)(2 − x). Find the area bounded by the curve y = x(x + 1)(2 − x) and the x-axis.
34. [STPM ] Show that
Z
x ln x dx = 1
Z
Hence, find the value of
35. [STPM Z ] e
e
1
e
[Answer : 3
1 2 e +1 . 4
x(ln x)2 dx correct to three decimal places.
[Answer : 1.597]
(2x + 1) ln x dx. Give your answer in terms of e.
Find
1
1 ] 12
[Answer :
1 2 (e + 3)] 2
36. [STPM ] Sketch, on the same axes, the graphs of y 2 = x and y = 2 − x. Show the coordinates of the points of intersection between the graphs. Calculate the area bounded by y 2 = x and y = 2 − x.
If V1 is the volume of the solid formed when the area above the x-axis bounded by y 2 = x, y = 2 − x, and the x-axis is rotated completely about the y-axis, and V2 is the volume of the solid formed when the area under the x-axis bounded by y 2 = x, y = 2 − x, and the x-axis is rotated completely about the y-axis, find V1 : V2 .
37. [STPM ]
Z
By using a suitable substitution, find the value of
3
9 2
[Answer : Area= ; V1 : V2 = 4 : 23]
√ x 1 + x dx.
0
[Answer : 7
11 ] 15
38. [STPM ] Find the coordinates of all the points of intersection between the line y = 12 − 4x and the curve y = 12 − x3 . Show that there is a point of inflection on the curve y = 12 − x3 at x = 0. Sketch, in the same diagram, the line y = 12 − 4x and the curve y = 12 − x3 .
The area bounded by the line y = 12 − 4x and the curve y = 12 − x3 is rotated through four right angles about the x-axis. Calculate the volume of the solid of revolution formed. 77
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STPM MATHEMATICS (T)
9: Integration
Find
√
x dx. 2x − 1
40. [STPM ] Find Z x √ (a) dx, 2 4 − x Z 3 (b) x2 e−x dx
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39. [STPM Z ]
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[Answer : (0, 12), (−2, 20), (2, 4) ; Volume=192π]
1
[Answer :
p
(2x − 1) 2 (x + 1) + C] 3
1 3
3
[Answer : (a) − 4 − x2 + C ; (b) − e−x + C]
41. [STPM ] The gradient of the tangent to a curve at the point (x, y) where x > 3, varies inversely with (x − 3), and the curve passes through the points (4, 0) and (6, ln 9). Show that the equation of the curve is y = 2 ln(x − 3). Sketch the curve.
Find the finite area bounded by the curve, the x-axis and the line x = 6.
42. [STPM ] Show that
43. [STPM ] Function f is defined as
(a) Sketch the graph of f . (b) Find the range of f .
Z 2
3
11 x2 dx = + ln 2. 3 (x − 1) 8
( x(x − π), 0 ≤ x < 2π, f (x) = 2 π sin(x − π), 2π ≤ x ≤ 3π.
[Answer : 2(3 ln 3 − 2)]
[6 marks]
[4 marks] [3 marks]
(c) Determine whether f is a one-to-one function. Give reasons for your answer.
[2 marks]
(d) Find the area of the region bounded by graph f and the x-axis.
[6 marks]
44. [STPM ]
If f (x) =
[Answer : (b) R={y : −π 2 ≤ y < 2π 2 } ; (d) π 3 + 2π 2 ]
x2 , find f 0 (x). Hence, evaluate 2x − 5 Z 2 1
x(x − 5) dx. (2x − 5)2
[5 marks]
78
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STPM MATHEMATICS (T)
9: Integration 11 2x(x − 5) ;− ] (2x − 5)2 6
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45. [STPM ] Sketch the curve y = x(x − 3)(x + 2).
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[Answer : f 0 (x) =
[2 marks]
If A1 and A2 respectively denote the area of the regions bounded by the curve and the x-axis above and below the x-axis, find A1 : A2 . [5 marks]
46. [STPM ]
[Answer :
64 ] 189
1 . [2 marks] x Calculate the volume of the solid of revolution when the region bounded by the above graphs is rotated through 360◦ about the y-axis. [5 marks] Sketch the graphs y = 4, y = 8x2 and y =
47. [STPM ] The function f is defined by
( 2 − |x − 1|, f (x) = x2 − 9x + 18,
(a) Sketch the graph of f . Z 6 f (x)dx. (b) Evaluate 0
48. [STPM ] A curve has equation y 2 = x2 (4 − x2 ).
[Answer :
x < 3, x ≥ 3.
1 π] 2
[5 marks]
[5 marks]
[Answer : (b) 8 ]
Show that for any point (x, y) lying on the curve, then −2 ≤ x ≤ 2 and −2 ≤ y ≤ 2.
[3 marks]
Sketch the curve.
[3 marks]
Calculate the area of the region bounded by this curve.
[4 marks]
Calculate the volume of the solid of revolution when the region bounded by this curve and y = x in the first quadrant is rotated through 360◦ about the x-axis. [5 marks]
49. [STPM ]
Z
Find the value of
4
x 1
0
(5x2 + 1) 2
.
79
√ 6 3 32 π] [Answer : Area= ; Volume= 3 5
[6 marks]
[Answer :
8 ] 5
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STPM MATHEMATICS (T)
9: Integration
50. [STPM ]
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4 Sketch on the same coordinate system the curve y = , and y 2 = 4(x − 1). Find the coordinates of x the points of intersection of the two curves. [6 marks] 4 20 Show that the area of the region bounded by y = , y 2 = 4(x − 1) and y = 4 is − 4 ln 2 .[4 marks] x 3 Calculate the volume of the solid of revolution when this region is rotated through 360◦ about the y-axis. [5 marks] [Answer : Point of intersection=(2,2) ; Volume=
51. [STPM ]
Z
Evaluate the definite integral
π 2
x2 sin x dx.
0
52. [STPM ]
ln x Show that the curve y = has a stationary point at x local minimum point or a local maximum point. Sketch the curve.
296 π] 15
[6 marks]
[Answer : π − 2]
1 e, and determine whether this point is a e [6 marks] [3 marks]
1 ln x , the x-axis, and the line x = is Show that the area of the region bounded by the curve y = x e ln x , the x-axis, and the line x = e. [5 marks] equal to the area of the region bounded by the curve y = x 53. [STPM ]
By using suitable substitution, find
Z
3x − 1 √ dx. x+1
[5 marks]
3
1
[Answer : 2(x + 1) 2 − 8(x + 1) 2 + C]
54. [STPM ] Find the point of intersection of the curves y = −x2 + 3x and y = 2x3 − x2 − 5x. Sketch on the same coordinate system these two curves. [5 marks] Calculate the area of the region bounded by the curves y = −x2 + 3x and y = 2x3 − x2 − 5x. [6 marks]
55. [STPM ]
[Answer : Point of intersection=(0,0), (2,2), (-2,-10) ; Area=16 units2 .]
Z
Using the substitution u = 3 + 2 sin θ, evaluate
0
π 6
cos θ dθ. (3 + 2 sin θ)2
[5 marks]
[Answer :
1 ] 24
56. [STPM ] a The curve y = x(b − x), where a 6= 0, has a turning point at point (2, 1). Determine the values of a 2 and b. [4 marks] Calculate the area of the region bounded bt the x-axis and the curve.
[4 marks]
Calculate the volume of the solid formed by revolving the region about the x-axis.
[4 marks]
80
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STPM MATHEMATICS (T)
9: Integration
Z
Show that
e
ln x dx = 1.
1
58. [STPM ]
1 8 32 ,b=4; ; π] 2 3 15
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57. [STPM ]
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[Answer : a =
[4 marks]
1 Sketch on the same coordinates axis y = x and the curve y 2 = x. Find the coordinate of the points 2 of intersection. [5 marks] 1 Find the area of region bounded by the line y = x and the curve y 2 = x. [4 marks] 2 Find the volume of the solid formed when the region is rotated through 2π radians about the y-axis. [4 marks]
[Answer : (0, 0) , (4, 2) ;
59. [STPM ] Sketch, on the same coordinate axes, the curves y = ex and y = 2 + 3e−x . Calculate the area of the region bounded by the y-axis and the curves.
60. [STPM ]
[2 marks]
[6 marks]
[Answer : 2 ln 3]
Ax + B C 2x + 1 in the form 2 + where A, B and C are constants. + 1)(2 − x) x +1 2−x Z 1 2x + 1 dx. Hence, evaluate 2 0 (x + 1)(2 − x) Express
(x2
61. [STPM ] Find Z 2 x +x+2 (a) dx, x2 + 2 Z x (b) dx. ex+1
62. [STPM ]
[Answer : (a) x +
[Answer :
[4 marks]
1 3 x + ; ln 2] +1 2−x 2
[4 marks]
1 1 x ln(x2 + 2) + C ; (b) − x+1 − x+1 + C] 2 e e
dy 3x − 5 = √ , where x > 0. If dx 2 x
the curve passes through the point (1, −4),
(b) sketch the curve,
x2
[3 marks]
[3 marks]
The gradient of the tangent to a curve at any point (x, y) is given by
(a) find the equation of the curve,
4 64 ; π] 3 15
[4 marks]
[2 marks]
(c) calculate the area of the region bounded by the curve and the x-axis.
[5 marks]
3
1
[Answer : (a) y = x 2 − 5x 2 ; (c) 81
20 √ 5] 3
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STPM MATHEMATICS (T)
9: Integration
sec x(sec x + tan x)2 dx = 1 +
0
64. [STPM ]
Z
Show that
2
3
5 (x − 2)2 dx = + 4 ln 2 x 3
√
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π 4
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Z
Show that
2.
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63. [STPM ] [4 marks]
2 . 3
[4 marks]
65. [STPM ] Sketch, on the same coordinate axes, the curves y = 6 − ex and y = 5e−x , and find the coordinates of the points of intersection. [7 marks] Calculate the area of the region bounded by the curves.
[4 marks]
Calculate the volume of the solid formed when the region is rotated through 2π radians about the x-axis. [5 marks]
66. [STPM ]
Z
Using an appropriate substitution, evaluate
1
[Answer : (ln 5, 1) ; 6 ln 5 − 8 ; π(36 ln 5 − 48)]
1
x2 (1 − x) 3 dx.
[7 marks]
0
[Answer :
67. [STPM ] Given a curve y = x2 − 4 and straight line y = x − 2,
(a) sketch, on the same coordinates axes, the curve and the straight line, (b) determine the coordinate of their points of intersection,
[2 marks]
[2 marks]
(c) calculate the area of the region R bounded by the curve and the straight line, ◦
[4 marks]
(d) find the volume of the solid formed when R is rotated through 360 about the x-axis. [Answer : (b) (−1, 3) , (2, 0) ; (c)
68. [STPM ]
0
[5 marks]
9 108 ; (d) π] 2 5
Z
2e
ln x dx.
Given that f (x) = x ln x, where x > 0. Find f (x), and hence, determine the value of
69. [STPM ]
Use the substitution u = ln x, evaluate
70. [STPM ] 2 Differentiate ex with respect to x.
Z
1
e
(x + 1) ln x dx. x2
Hence, determine integers a, b and c for which Z 2 a 2 x3 ex dx = ec . b 1
27 ] 140
e
[6 marks]
[Answer : 2e ln(2)]
[6 marks]
[Answer :
3 2 − ] 2 e
[9 marks]
82
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STPM MATHEMATICS (T)
9: Integration 2
71. [STPM ] Differentiate tan x with respect to x, and hence, show that π 3
Z 0
72. [STPM ]
π x sec2 xdx = √ − ln 2. 3
Using the substitution x = 4 sin2 u, evaluate
1r
Z
0
73. [STPM ] Show that
Z
e
1
Z
Hence, find the value of 1
e
x(ln x)2 dx.
[6 marks]
[Answer : sec2 x]
x dx. 4−x
x ln x dx =
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[Answer : 2xex , a = 3, b = 2, c = 4]
[6 marks]
[Answer :
1 2 e +1 . 4
√ 2 π − 3] 3
[4 marks]
[3 marks]
1 4
1 4
[Answer : − + e2 ]
74. [STPM ] Sketch, on the same axes, the curve y 2 = x and the straight line y = 2 − x, showing the coordinates of the points of intersection. [4 marks] (a) State whether the curve y 2 = x has a turning point. Justify your answer.
[2 marks]
(b) Calculate the area of the region bounded by the curve y 2 = x and the straight line y = 2 − x. [4 marks]
(c) Calculate the volume of the solid formed by revolving the region bounded by the curve y 2 = x and the straight line y = 2 − x completely about the y-axis. [5 marks]
75. [STPM ]
The equations of two curves are given by y = x2 − 1 and y =
[Answer : (b) 4.5 ; (c) 14.4π]
6 . x2
(a) Sketch the two curves on the same coordinate axes.
[3 marks]
(b) Find the coordinates of the points of intersection of the two curves.
[3 marks]
(c) Calculate the volume of the solid formed when the region bounded by the two curves and the line x = 1 is revolved completely about the y-axis. [6 marks] √
√
[Answer : (b) (− 3, 2), ( 3, 2) ; (c) 2π(−1 + 3 ln 3)] 83
STPM MATHEMATICS (T)
9: Integration
77. [STPM ] Show that
Z
e
1
78. [STPM ]
Z
Find the value of
1
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76. [STPM ] Using the substitution u = x − 2, show that Z 4 1 π x−1 dx = ln 2 + . 2 2 8 2 x − 4x + 8
ln x 1 dx = x5 16
[8 marks]
5 1− 4 . e
[6 marks]
(1 + 2x) ln(1 + x)dx. 0
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[5 marks]
1 2
[Answer : 2 ln 2 − ]
79. [STPM ] The curve y = ln(4x) is shown in the diagram below.
The tangent to the curve at the point P passes through the origin O. e (a) Show that the coordinates of the point P is , 1 , and find the equation of the tangent to the 4 curve. [5 marks] (b) Calculate the area of the shaded region bounded by the curve, the tangent and the x-axis.[5 marks] (c) Calculate the volume of the solid formed when the shaded region is revolved completely about the y-axis. [5 marks]
80. [STPM ]
Show that
4 e
1 8
[Answer : (a) y = x ; (b) e −
d (cosec x) = − cosec x cot x. dx
1 1 ; (c) π(e2 − 3)] 4 96
[2 marks]
84
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STPM MATHEMATICS (T)
9: Integration
x2
81. [STPM ] Evaluate the definite integrals Z
ln 2
(a)
0
Z
ex dx, 1 + ex
3
(b)
√
0
x dx. 1+x
82. [STPM ]
Z
1
Show that
0
2 x2 cos−1 x dx = . 9
83. [STPM ]
The equation of a curve is
dx √ . 4 + x2
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Z
x2 (y − 3)2 + = 1. 4 9
(a) Sketch the curve.
(b) Calculate the area of the region bounded by the curve.
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Using the substitution x = 2 tan θ, find
[6 marks]
[Answer : −
1 p 4 + x2 + C] 4x
[3 marks]
[6 marks]
[Answer : (a) ln
8 3 ; (b) ] 2 3
[6 marks]
[2 marks]
[9 marks]
(c) Calculate the volume of the solid formed when the region is revolved completely about the y-axis.
84. [STPM ]
Z
3
ln x dx = 3 ln 3 − 2.
Show that
1
√
Z
Hence, evaluate
0
2
x ln(1 + x2 ) dx.
85
[4 marks]
[Answer : (b) 6π ; (c) 16π]
[5 marks]
[4 marks]
[Answer :
3 ln 3 − 1] 2
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10: Differential Equations
1. [STPM ] Variables t and y, with t ≥ 0 and y > 0, are related by
1 dy = y(2 − y), dt 2
2et . 1 + et Show that y → 2 when t → ∞. Sketch the graph of y versus t. 6 9 Find the difference in the values of t when y changes from to . 5 5 with the condition y = 1 when t = 0. Show that y =
2. [STPM ]
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Differential Equations
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STPM MATHEMATICS (T)
[8 marks] [4 marks]
[3 marks]
[Answer : 1.7917]
dy A curve passes through point (2, 0) such that its gradient at point (x, y) satisfies equation (x2 − dx dy 3) = 4x(3 + y). Show that the equation of the curve is y = 3(x2 − 2)(x2 − 4). [6 marks] dx Sketch the curve. [3 marks] Find the area of the region bounded by the curve and the x-axis.
3. [STPM ]
[6 marks]
[Answer :
(a) Variables t and v, with 0 < t < 2, is related by the differential equation t
dv = v 2 − v, dt
48 √ (6 2 − 4)] 5
with the condition v = 2 when t = 1. Find v in terms of t and sketch the graph of v versus t.
(b) Show that
[8 marks]
dy y √ = (1 + x2 ) − xy, 2 dx 1+x where y is a function of x. Hence, solve the differential equation d (1 + x ) dx 2
3 2
(1 + x2 )
dy − xy = x(1 + x2 ), dx
with the condition y = 1 when x = 0.
4. [STPM ]
(a) Solve the differential equation
(1 − x)(1 + x2 )
[7 marks]
[Answer : (a) v =
dy + (2 − x + x2 )y = (1 − x)2 , x < 1, dx
with the condition y = 3 when x = 0.
2 ; (b) y = 1 + x2 ] 2−t
[8 marks]
86
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5. [STPM ] Solve the differential equation
10: Differential Equations
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STPM MATHEMATICS (T)
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dy = (x + 1)(y + 1), dx with x > 0 and y > −1, and y = 2 when x = 1. Give your answer in the form y = f (x). x
[Answer : y = 3xex−1 − 1]
6. [STPM ] In a biochemical process, enzyme A changes continuously to enzyme B. Throughout the process, the total amount of A and B is constant. At any time, the rate that B is produced is directly proportional to the product of the amount of A and the amount of B at that time. At the beginning of the process, the amount of A and the amount of B are a and b respectively. If x denotes the amount of B that has been produced at time t after the process has begun, form a differential equation relating x and t to describe the process. [2 marks] Show that the solution of the differential equation is x=
where k is a positive constant.
ab(1 − e−(a+b)kt ) , b + ae−(a+b)kt
[8 marks]
Sketch the graph of x against t. [There is a point of inflection on the graph.]
[2 marks]
7. [STPM ] The rate of change of water temperature is described by the differential equation dθ = −k(θ − θs ) dt
where θ is the water temperature at time t, θs is the surrounding temperature, and k is a positive constant. A boiling water at 100◦ C is left to cool in kitchen that has a surrounding temperature of 25◦ C. The 3 water takes 1 hour to decrease to the temperature of 75◦ C. Show that k = ln . [6 marks] 2 ◦ ◦ When the water reaches 50 C, the water is placed in a freezer at −10 C to be frozen to ice. Find the time required, from the moment the water is put in the freezer until it becomes ice at 0◦ C. [6 marks] [Answer : Time = 4 hours 25 minutes]
8. [STPM ] Find the particular solution for the differential equation
x−2 1 dy + y=− 2 . dx x(x − 1) x (x − 1)
that satisfies the boundary condition y =
3 when x = 2. 4
[8 marks]
[Answer : y =
2x − 1 ] x2
9. [STPM ] The rate of increase in the number of a species of fish in a lake is described by the differential equation dP = (a − b)P dt
where P is the number of fish at time t weeks, a is the rate of reproduction, and b is the mortality rate, with a and b as constants. 87
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10: Differential Equations
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(a) Assuming that P = P0 at time t = 0 and a > b, solve the differential equation and sketch its solution curve. [7 marks] (b) At a certain instant, there is an outbreak of an epidemic of a disease. The epidemic results r in no 1 more offspring of the fish being produced and the fish die at a rate directly proportional to . P There are 900 fish before the outbreak of the epidemic and only 400 fish are alive after 6 weeks. Determine the length of time from the outbreak of the epidemic until all the fish of that species die. [9 marks]
10. [STPM ] The variables t and x are connected by
[Answer : (a) P = P0 e(a−b)t ; (b) 18 weeks]
dx = 2t(x − 1), dt
where x 6= 1. Find x in terms of t if x = 2 when t = 1.
[5 marks] 2
[Answer : x = et
−1
+ 1]
11. [STPM ] A canal of width 2a has parallel straight banks and the water flows due north. The points A and B are on opposite banks and B is due east of A, with the point O as the midpoint of AB. The x-axis and y-axis are taken in the east and north directions respectively with O as the origin. The speed of the current in the canal, vc , is given by x2 vc = v0 1 − 2 , a where v0 is the speed of the current in the middle of the canal and x is the distance eastwards from the middle of the canal. A swimmer swims from A towards the east at speed vr relative to the current inthe canal. Taking y to denote the distance northwards travelled by the swimmer, show that x2 dy v0 [3 marks] 1− 2 . = dx vr a If the width of the canal is 12 m, the speed of the current in the middle of the canal is 10 m s−1 and the speed of the swimmer is 2 m s−1 relative to the current in the canal, (a) find the distance of the swimmer from O when he is at the middle of the canal and his distance from B when he reaches the east bank of the canal, [7 marks] (b) sketch the actual path taken by the swimmer.
12. [STPM ] v Using the substitution y = 2 , show that the differential equation x
may be reduced to
2y dy + y2 = − dx x dv v2 = − 2. dx x
[3 marks]
[Answer : (a) 20 , 40]
[3 marks]
Hence, find the general solution of the original differential equation. 88
[4 marks]
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STPM MATHEMATICS (T)
10: Differential Equations 1 ] Ax2 − x
d 2 (ln tan x) = . dx sin 2x Hence, find the solution of the differential equation
Show that
for which y =
(sin 2x)
[2 marks]
dy = 2y(1 − y) dx
1 1 when x = π. Express y explicitly in terms of x in your answer. 3 4
14. [STPM ] Find the general solution of the differential equation x
dy − 3y = x3 . dx
Find the particular solution given that y has a minimum value when x = 1. Sketch the graph of this particular solution.
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13. [STPM ]
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[Answer : y =
[Answer : y =
[8 marks]
tan x ] 2 + tan x
[4 marks] [3 marks] [3 marks]
1 3
[Answer : y = x3 ln x + Cx3 ; y = x3 ln x − x3 ]
15. [STPM ] Find the general solution of the differential equation x
dy = y 2 − y − 2. dx
[6 marks]
[Answer : y =
2 + Ax3 ] 1 − Ax3
16. [STPM ] A particle moves from rest along a horizontal straight line. At time t s, the displacement and velocity of the particle are x m and v ms−1 respectively and its acceleration, in ms−2 , is given by √ dv = sin(πt) − 3 cos(πt). dt
Express v and x in terms of t.
[7 marks]
Find the velocities of the particle when its acceleration is zero for the first and second times. Find also the distance traveled by the particle between the first and second times its acceleration is zero.[7 marks] [Answer : v =
i √ √ i 1h 1 h√ 1 − cos(πt) − 3 sin(πt) , x = 2 3 cos(πt) + πt − sin(πt) + 3 ; π π √ 1 1 4 3 1 2 3 t= ,v=− ;t= ,v= ; + 2 ] 3 π 3 π 3π π
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10: Differential Equations
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17. [STPM ] The variables x and y, where x > 0, satisfy the differential equation x2
dy = y 2 − xy. dx
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Using the substitution y = ux, show that the given differential equation may be reduced to x
du = u2 − 2u. dx
Hence, show that the general solution of the given differential equation may be expressed in the form 2x y= , where A is an arbitrary constant. [10 marks] 1 + Ax2 Find the equation of the solution curve which passes through the point (1,4) and sketch this solution curve. [4 marks] [Answer : y =
18. [STPM ] Show that the substitution u = x2 + y transforms the differential equation
into the differential equation
(1 − x)
dy + 2y + 2x = 0 dx
(1 − x)
du = −2u. dx
4x ] 2 − x2
[3 marks]
19. [STPM ] A 50 litre tank is initially filled with 10 litres of brine solution containing 20 kg of salt. Starting from time t = 0, distilled water is poured into the tank at a constant √ rate of 4 litres per minute. At the same time, the mixture leaves the tank at a constant rate of k litres per minute, where k > 0. The time taken for overflow to occur is 20 minutes. (a) Let Q be the amount of salt in the tank at time t minutes. Show that the rate of change of Q is given by √ Q k dQ √ . =− dt 10 + (4 − k)t Hence, express Q in terms of t.
[7 marks]
(b) Show that k = 4, and calculate the amount of salt in the tank at the instant overflow occurs. [6 marks]
(c) Sketch the graph of Q against t for 0 ≤ t ≤ 20.
[2 marks]
h
√
[Answer : (a) Q = C 10 + (4 − k)t
20. [STPM ] Find the particular solution of the differential equation ex
dy − y 2 (x + 1) = 0 dx
for which y = 1 when x = 0. Hence, express y in terms of x. 90
√
k+k i− 416−k
; (b) 4kg]
[7 marks]
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STPM MATHEMATICS (T)
10: Differential Equations ex ] 2 + x − ex
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[Answer : y =
21. [STPM ] One of the rules at a training camp of 1000 occupants states that camp activities are to be suspended if 10 of the occupants are infected with a virus. A trainee infected with a flu virus enrolls in the camp causing an outbreak of flu. The rate of increase of the number of infected occupants x at t days is given by differential equation dx = kx(1000 − x), dt where k is a constant. Assume that the outbreak of flu begins at the time the infected trainee enrolls and no one leaves the camp during the outbreak, 1000e1000kt , [9 marks] 999 + e1000kt (b) Determine the value of k if it is found that, after one day, there are five infected occupants,[3 marks] (a) Show that x =
(c) Determine the number of days before the camp activities will be suspended.
22. [STPM ]
Using the substitution z =
may be reduced to
[Answer : (b) k = 0.0016134 or
1 , show that the differential equation y dy 2y − = y2 dx x
2z dz + = −1. dx x
1 ln 1000
[4 marks]
999 199
; (c) 3 days]
[2 marks]
Hence, find the particular solution y in terms of x for the differential equation given that y = 3 when x = 1. [6 marks] Sketch the graph y.
23. [STPM ] Find the general solution of the differential equation
2 ln x 1 dy = . x dx cos y
91
[3 marks]
[Answer : y =
3x2 ] 2 − x3
[5 marks]
1 2
[Answer : sin y = x2 ln x − x2 + c]
STPM MATHEMATICS (T)
10: Differential Equations
may be reduced to
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24. [STPM ] Using the substitution y = vx, show that the differential equation xy
dy − x2 − y 2 = 0 dx vx
dv = 1. dx
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[3 marks]
Hence, find the particular solution that satisfies y = 2 and x = 1.
[6 marks]
[Answer : y 2 = 2x2 (ln x + 2)]
25. [STPM ] The variables x and y, where x > 0 and y > 0, satisfy the differential equation y(y + x) dy = . dx x(y − x)
Show that the substitution y = ux transforms the given differential equation into the differential equation 2u du = . dx x(u − 1) [3 marks]
1 Hence, find the solution of the given differential equation for which y = 2 when x = . 2
[6 marks] y
[Answer : y = 4x + x ln xy or xy = e x −4 ]
26. [STPM ] Differentiate ye−x with respect to x. Hence, find the solution of the differential equation
for which y = 1 when x = 0.
dy − y = ex cos x dx
27. [STPM ] The variables x and y, where x > 0, satisfy the differential equation x2
dy = 2xy + y 2 . dx
[6 marks]
[Answer : e−x
dy ; ye−x = sin x + 1] dx
Using the substitution y = ux, show that the given differential equation can be transformed into x
du = u + u2 . dx
[3 marks]
Show that the general solution of the transformed differential equation can be expressed as u = where A is an arbitrary constant.
x , A−x
[7 marks]
Hence, find the particular solution of the given differential equation which satisfies the condition that y = 2 when x = 1. [3 marks] 92
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STPM MATHEMATICS (T)
10: Differential Equations
28. [STPM ] Find the solution of the differential equation
2xy dy − = ex (1 + x2 ) dx 1 + x2
given that y = 3 when x = 0.
29. [STPM ]
[8 marks]
[Answer : y = (ex + 2)(1 + x2 )]
x2 . x−1 Hence, find the particular solution of the differential equation Show that e
R
x−2 dx x(x−1)
=
x−2 1 dy + y=− 2 dx x(x − 1) x (x − 1)
which satisfies the boundary condition y =
3 when x = 2. 4
30. [STPM ] The variables x and y, where x, y > 0, are related by the differential equation
Using the substitution y =
2y dy + y2 = − . dx x
2x2 ] 3 − 2x
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[Answer : y =
[4 marks]
[4 marks]
[Answer : y =
2x − 1 ] x2
u , show that the differential equation may be reduced to x2 u2 du = − 2. dx x
[3 marks]
Solve this differential equation, and hence, find y, in terms of x, with the condition that y = 1 when x = 1. [6 marks]
31. [STPM ] Show that the substitution y = ux transform the differential equation y dy x = y − 2x cot dx x into the differential equation
x
du = −2 cot u. dx
[Answer : y =
1 ] x(2x − 1)
[3 marks]
Hence, find the solution of the given differential equation satisfying the condition y = 0 when x = 1. Give your answer in the form y = f (x). [5 marks] 93
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STPM MATHEMATICS (T)
10: Differential Equations
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[Answer : y = x cos−1 x2 ] 32. [STPM ] The number of rabbits in a farm increases at a rate proportional to the number of rabbits at a certain time. The number of rabbits doubled to 10000 from the beginning of the year 1985 until the beginning of 1990. At that time, an outbreak of a certain disease occurred in the farm which caused the death of rabbits at the rate of 100 rabbits per month. No vaccine was found until the beginning of the year 1992. Find the remaining number of rabbits that survived just before the vaccine was found. [9 marks] [Answer : unable to solve]
33. [STPM ] Using the substitution u = ln y, show that the non-linear differential equation x
dy + (3x + 1)y ln y = ye−2x dx
can be transformed into the linear differential equation x
du + (3x + 1)u = e−2x . dx
[4 marks]
Solve this linear differential equation, and hence, find the solution of the original non-linear differential equation, given that y = 1 when x = 1. [9 marks] Find the limiting value of y as x → ∞.
[2 marks]
[Answer : y = e
34. [STPM ] The variables x and y, where x, y > 0 are related by the differential equation
Show that the substitution u =
and find u2 in terms of x.
xy
dy + y 2 = 3x4 . dx
e−2x x
−e
1−3x x
; 1]
y transforms the above differential equation into x2 du 1 − u2 =3 , x dx u
[9 marks]
Hence, find the particular solution of the original differential equation which satisfies the condition y = 2 when x = 1. [3 marks] [Answer : u2 = 1 −
A 3 ; y 2 = x4 + 2 ] x6 x
35. [STPM ] Using the substitution z = cos y, find the general solution of the differential equation dy 1 1 + cot y = 2 cosec y. dx x x
[7 marks]
94
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STPM MATHEMATICS (T)
10: Differential Equations
Show that the substitution u =
1 transforms the non-linear differential equation y
into the linear differential equation
y dy + = y 2 ln x dx x du u − = − ln x dx x
1 + Cx] 2x
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36. [STPM ]
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[Answer : cos y = −
[4 marks]
1 Solve this linear differential equation, and hence, obtain y in terms of x, given that y = when x = 1. 2
37. [STPM ] Solve the differential equation
38. [STPM ]
[8 marks]
[Answer : y =
2 dy − y = x2 e2x . dx x
2 ] 4x − x(ln x)2
[6 marks]
1 2
[Answer : y = x2 e2x + Cx2 ]
dy Find the solution of the differential equation x − y = 2 which satisfies the condition y = 0 when dx x = 1. [5 marks] 39. [STPM ] The rate of elimination of a certain drug from a bloodstream is k times the mass, x mg. of the drug still present at time t. The half-life of the drug in the bloodstream is 100 minutes. (a) A dose of x0 mg of the drug is injected directly into the bloodstream.
i. Write down a differential equation relating x and t, and solve this differential equation.[4 marks] ii. Determine the value of k. [2 marks]
(b) The drug is intravenously fed into the bloodstream at an infusion rate of r mg per minute such dx that = −kx + r. Assuming that x = 0 when t = 0. dt i. express x in terms of t. [6 marks] ii. estimate the infusion rate that results in a long-term amount of 50 mg of drug in the bloodstream. [3 marks] 40. [STPM ] By making the change of variable z = x − 3y, show that the differential equation x − 3y + 5 dy = dx x − 3y − 1
may be reduced to a differential equation with separable variables.
[3 marks]
Solve this differential equation, and hence, obtain an equation which defines the relation between x and y. [4 marks] 95
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STPM MATHEMATICS (T)
10: Differential Equations
41. [STPM ] Find the general solution of the differential equation 2x
1 dy + y = x 2 e−2x , x > 0. dx
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[Answer : x − y − 3 ln |x − 3y + 5| = A]
[5 marks]
1 Hence,find the particular solution of the differential equation which satisfies the condition y = e−2 2 when x = 1. [3 marks]
96
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11: Maclaurin Series
1. [STPM ] Given that y = ln(1 + cos x), where −π < x < π, show that d2 y + e−y = 0. dx2
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Maclaurin Series
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11
STPM MATHEMATICS (T)
[4 marks]
6
Using Maclaurin theorem, find the expansion of y in ascending powers of x up to the term x .[7 marks] Hence,
Z
(b) taking x =
1
ln(1 + cos x) dx, correct to three decimal places,
(a) estimate the value of 0
π , estimate the value of ln 2 correct to three decimal places. 2 1 4
[Answer : ln 2 − x2 −
2. [STPM ]
1 1 Given ln y = sin−1 x, where − π < sin−1 x < π, show that 2 2 2 dy (a) (1 − x2 ) − y 2 = 0, dx (b) (1 − x2 )
d2 y dy −x − y = 0. 2 dx dx
−1
[2 marks] [2 marks]
1 4 1 6 x − x + . . . ; (a) 0.608 ; (b) 0.691] 96 1440
[2 marks]
[2 marks]
Hence, find the Maclaurin expansion of esin x in ascending powers of x up to the term in x5 . State the range of values of x for the expansion valid. [7 marks] Using a suitable value of x in the expansion, estimate the value of π correct to four significant figures. Find the percentage error of estimation if π = 3.142. [4 marks]
3. [STPM ] Using Maclaurin expansion, find
4. [STPM ]
1 2
1 3
[Answer : 1 + x + x2 + x3 +
5 4 1 5 x + x + . . . , {x : −1 < x < 1} ; 3.130 ; 0.382%] 24 6
1 − cos2 x . x→0 x(1 − e−x ) lim
[4 marks]
[Answer : 1]
(a) Find the expansion for ex cos x and ex sin x in ascending powers of x up to the term x3 .
(b) Using Maclaurin theorem, find the expansion for sec x in ascending powers of x up to the term x4 . [5 marks] Deduce the first three non zero terms in the expansion sec2 x and tan x in ascending powers of x. [4 marks]
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STPM MATHEMATICS (T) 1 3
1 3
1 2
5. [STPM ]
−1
If y = sin
d2 y x, show that =x dx2
dy dx
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[Answer : (a) 1 + x − x3 + . . . , x + x2 + x3 ; (b) 1 + x2 +
3
d3 y and = dx3
dy dx
11: Maclaurin Series 5 4 2 1 2 x ; 1 + x2 + x4 , x + x3 + x5 ] 24 3 3 15
3 + 3x
2
dy dx
5 .
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[4 marks]
Using Maclaurin theorem, express sin−1 x as a series of ascending powers of x up to the term in x5 . State the range of values of x for the expansion valid. [7 marks] Hence,
(a) taking x = 0.5, find the approximation of π correct to two decimal places, x − sin−1 x (b) find lim . x→0 x − sin x
6. [STPM ]
Use the expansion of x7 .
1 6
[Answer : x + x3 +
[2 marks]
[2 marks]
3 5 x , {x : |x| < 1} ; (a) 3.14 ; (b) −1] 40
1 to express tan−1 x as a series of ascending powers of x up to the term in 1 + x2 [4 marks]
Hence find, in terms of π, the sum of the infinite series
1 1 1 − + − ... 2 3×3 5×3 7 × 33
[2 marks]
√ 3 1 3 1 5 1 7 π] [Answer : x − x + x − x ; 1 − 3 5 7 6
7. [STPM ] If y = (sin−1 x)2 , show that, for −1 < x < 1,
(1 − x2 )
Find the Maclaurin series for (sin
−1
2
d2 y dy −x − 2 = 0. 2 dx dx
[3 marks]
6
x) up to the term in x .
[6 marks]
1 3
[Answer : x2 + x4 +
8 6 x ] 45
8. [STPM ] √ Find the Maclaurin expansion of x cos x up to the term in x3 . State the range of values of x for which the expansion is valid. [7 marks]
9. [STPM ]
Using the result that tan
Hence, find lim
x→0
sin x
x
Z x= 0
1 dt, show that 1 + t2
tan−1 x = x −
where |x| < 1.
tan−1 x
−1
x3 x5 x7 + − + ... 3 5 7
.
1 4
[Answer : x − x3 ; {x : |x| < 1}]
[2 marks] [2 marks]
98
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STPM MATHEMATICS (T)
11: Maclaurin Series
10. [STPM ] Given that y = x − cos−1 x, where −1 < x < 1. d2 y (a) Show that =x dx2
dy −1 dx
3 .
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[Answer : 1]
[2 marks]
(b) Find the Maclaurin series for y in ascending powers of x up to the term in x5 . 1 2
[5 marks]
1 6
[Answer : − π + 2x + x3 +
3 5 x ] 40
11. [STPM ] Using Maclaurin’s theorem, obtain the expansion of ln(1 + cos x) up to the term in x4 . [6 marks] √ π sin x π3 Hence, find the expansion of up to the term in x3 , and show that 3 ≈ 2 − − . 1 + cos x 12 5184
If y = sin
d2 y x, show that =x dx2
1 4
[Answer : ln(2) − x2 −
12. [STPM ]
−1
[5 marks]
dy dx
3
d3 y and = dx3
dy dx
3 + 3x
2
dy dx
1 4 1 1 x + . . . ; x + x3 + . . .] 96 2 24
5 .
[5 marks]
Using Maclaurin’s theorem, express sin−1 x as a series of ascending powers of x up to the term in x5 . State the range of values of x for which the expansion is valid. [7 marks]
13. [STPM ] Using the Maclaurin series for ln(1 + x), evaluate
14. [STPM ] Given that y = sin−1 x. (a) Show that
x − ln(1 + x) x→0 x2 lim
(1 − x2 )
1 6
[Answer : x + x3 +
d2 y dy −x = 0. dx2 dx
3 5 x , {x : |x| < 1}] 40
[3 marks]
[Answer :
1 ] 2
[4 marks]
1 3 (b) Using Maclaurin’s theorem, show that the series expansion for sin−1 x is x + x3 + x5 + . . .. 6 40 State the range of values of x for which the expansion is valid. [8 marks] 1 (c) Using the series expansion in (b), where x = , estimate the value of π correct to three decimal 2 places. [3 marks] 99
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STPM MATHEMATICS (T)
11: Maclaurin Series
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[Answer : (b) {x : |x| < 1} ; (c) 3.139] 15. [STPM ] Using the Maclaurin theorem, find the expansion of ln(1 + sin x) up to the terms in x3 . 2x − ln(1 + sin x)2 . x→0 2x2 − x4
Hence, evaluate lim
16. [STPM ] −1 If y = ecos x , show that (1 − x2 )
17. [STPM ] The function g is defined by
for all values of x.
[2 marks]
1 2
−1
x
up to and including the term in x4 . π
√
g(x) = e
sin
! 3 x . 2
(a) Show that g 00 (x) + g 0 (x) + g(x) = 0. Hence, show that the Maclaurin series for g(x) is √ √ √ 3 3 2 3 4 x− x + x − .... 2 4 48
π
1 2
π
1 3
π
5 π 4 e2x ] 24
[5 marks]
[5 marks]
(b) Use the Maclaurin scries obtained in (a) to g(x) i. find the expansion of ascending powers of x up to the term in x3 . 1 + 2x g(x) ii. find the value of lim . x→0 x
18. [STPM ] If y = (cos−1 x)2 , show that
1 2
[8 marks]
[Answer : y = e 2 − e 2 x + e 2 x2 − e 2 x3 +
− 12 x
1 6
[Answer : x − x2 + x3 ; ]
d2 y dy − y = 0, for − 1 < x < 1. −x 2 dx dx
Hence, find the Maclaurin series for ecos
[5 marks]
[3 marks] [2 marks]
√ √ √ 5 3 2 5 3 3 3 3 [Answer : (b)(i) x− x + x + . . . ; (ii) ] 2 4 2 2
2
(1 − x )
dy dx
2
= 4y.
Show that the Maclaurin’s series for (cos−1 x)2 is
π2 π − πx + x2 − x3 + . . . . 4 6
√
[3 marks]
[7 marks]
100
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STPM MATHEMATICS (T)
11: Maclaurin Series
[6 marks]
Hence, find the first three non-zero terms of the series expansion of e
−2x
tan x.
1 3
4 3
[2 marks]
7 3
[Answer : 1 − 2x + 2x2 − x3 + . . . ; tan x = x + x3 + . . . ; x − 2x2 + x3 + . . .]
20. [STPM ] State the Maclaurin series for cos 2x up to four non-zero terms. 2x2
cos 2x + x→0 6x4
Hence, find the value lim
−1
[1 marks]
.
[3 marks]
2 3
[Answer : 1 − 2x2 + x4 −
21. [STPM ] −1 Show that y = esin x − 1 satisfies the differential equation 2
(1 − x )
Deduce that
(1 − x2 )
and
(1 − x2 )
Find the Maclaurin series for e
sin−1
x
−1
esin x − 1 , (a) determine lim x→2 sin x Z (b) approximate the value of
0.1
(esin
0
dy dx
2
= (y + 1)2 .
d2 y dy −x =y+1 2 dx dx
d4 y d3 y d2 y − 5x − 5 = 0. dx4 dx3 dx2
−1
x
− 1)dx, correct to five decimal places.
(1 + x2 )
(1 + x2 )
1 2
dy =y dx
d2 y dy + (2x − 1) = 0. 2 dx dx −1
Hence, find the Maclaurin expansion of etan
x
[3 marks]
[2 marks]
1 3
[Answer : x + x2 + x3 +
22. Given that ln y = tan−1 x, show that
4 6 1 x + ... ; ] 45 9
[7 marks]
− 1 in ascending powers of x up to the term in x4 .
Hence,
and
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19. [STPM ] State the Maclaurin series for e−2x up to the term in x3 . [1 marks] dy If y = tan x, show that = 1 + y 2 . Obtain the Maclaurin series for tan x up to the term in x3 . dx
[3 marks]
5 4 x + . . . ; (a) 1 ; (b) 0.00518] 24
in ascending powers of x up to the term in x3 .
101
STPM MATHEMATICS (T) dy in terms of tan x, and hence show that dx
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23. Given that y = tan2 x, express
d2 y = 2 + 8y + 6y 2 . dx2
11: Maclaurin Series
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By further differentiation, or otherwise, show that, if x is so small that x7 and higher powers of x can be neglected, 2 17 tan2 x = x2 + x4 + x6 . 3 45 √
24. Given that y = e
1+x
, show that
4(1 + x)
d2 y dy +2 = y. 2 dx dx
By further differentiation, or otherwise, show that Maclaurin series of y up to the term in x4 is 1 1 y = e 1 + x + x2 + kx4 , 2 48 where k is a constant to be determined.
1 1 25. If y = (cos x)x , where − π ≤ x ≤ π, prove that 2 2
dy = y(ln cos x − x tan x). dx
Express y as a series of ascending powers of x up to the term in x3 . 1 1 4 , correct to 3 decimal places. Hence, estimate the value of cos 4
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(b) (1 − x2 )
d2 y dy −x − y = 0. 2 dx dx
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1 1 Given ln y = sin−1 x, where − π < sin−1 x < π, show that 2 2 2 dy (a) (1 − x2 ) − y 2 = 0, dx
−1
11: Maclaurin Series
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[2 marks]
[2 marks]
Hence, find the Maclaurin expansion of esin x in ascending powers of x up to the term in x5 . State the range of values of x for the expansion valid. [7 marks] Using a suitable value of x in the expansion, estimate the value of π correct to four significant figures. Find the percentage error of estimation if π = 3.142. [4 marks]
103
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12: Numerical Methods
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Numerical Methods
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12
STPM MATHEMATICS (T)
1. [STPM ] Use the Newton Raphson method with initial estimate, x0 = 0.5, find the root for equation 10x − 2 sin x = 5 correct to three decimal places. [5 marks] [Answer : 0.615]
2. [STPM ] 1 Draw, on the same axes, the graphs of y = e− 2 x and y = 4 − x2 . State the integer which is nearest to the positive root of the equation 1 x2 + e− 2 x = 4. [3 marks]
Find an approximation for this positive root by using the Newton-Raphson method until two successive iterations agree up to two decimal places; give your answer correct to two decimal places. [5 marks]
3. [STPM ]
Sketch, on the same coordinate axes, the graphs y = ex and y = (1 + x)ex − 2 = 0 has a root in the interval [0, 1].
[Answer : 2 ; 1.90]
2 . Show that the equation 1+x [7 marks]
Use the Newton-Raphson method with the initial estimate x0 = 0.5 to estimate the root correct to three decimal places. [6 marks] [Answer : 0.375]
4. [STPM ] Using the sketch graphs of y = x3 and x + y = 1, show that the equation x3 + x − 1 = 0 has only one real root and state the successive integers a and b such that the real root lies in the interval (a, b). Use the Newton-Raphson method to find the real root correct to three decimal places.
5. [STPM ]
[4 marks] [5 marks]
[Answer : 0.683]
1 Find the coordinate of the stationary point on the curve y = x2 + where x > 0; give the x-coordinate x and y-coordinate correct to three decimal places. Determine whether the stationary point is a minimum point or a maximum point. [5 marks] 1 1 The x-coordinate of the point of intersection of the curves y = x2 + and y = 2 , where x > 0, is x x p. Show that 0.5 < p < 1. Using the Newton-Raphson method to determine the value of p correct to three decimal places and, hence, find the point of intersection. [9 marks]
6. [STPM ]
[Answer : (0.794 , 1.890) , minimum ; p = 0.724 , (0.724 , 1.908)]
Z
Using trapezium rule, with five ordinates, evaluate
0
1p
4 − x2 dx.
[4 marks]
[Answer : 1.91] 104
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STPM MATHEMATICS (T)
12: Numerical Methods
7. [STPM ] x + 4ex = 2 has a root in the interval [-1,0].
4 . Show that the equation 2−x
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Sketch, on the same coordinate axes, the graphs y = e−x and y =
[6 marks]
Estimate the root correct to three decimal places by using Newton-Raphson method with initial estimate x0 = −0.4. [5 marks] [Answer : -0.479]
8. [STPM ] Use the trapezium rule with subdivisions at x = 3 and x = 5 to obtain an approximation to Z 7 x3 dx, giving your answer correct to three places of decimals. [4 marks] 4 1 1+x By evaluating the integral exactly, show that the error of the approximation is about 4.1%. [4 marks]
9. [STPM ]
[XAnswer : 1.701]
Two iterations suggested to estimate a root of the equation x3 − 4x2 + 6 = 0 are xn+1 = 4 − 1 1 xn+1 = (x3n + 6) 2 . 2
(a) Show that the equation x3 − 4x2 + 6 = 0 has a root between 3 and 4.
6 and x2n
[3 marks]
(b) Using sketched graphs of y = x and y = f (x) on the same axes, show that, with initial approximation x0 = 3, one of the iterations converges to the root whereas the other does not. [6 marks]
(c) Use the iteration which converges to the root to obtain a sequence of iterations with x0 = 3, ending the process when the difference of two consecutive iterations is less than 0.05. [4 marks] (d) Determine whether the iteration used still converges to the root if the initial approximation is x0 = 4. [2 marks] 10. [STPM ] Show that the equation x3 − 15x2 + 300 = 0 has a root between 5 and 6.
[3 marks]
Given that x0 = 5 as an initial approximation, use the Newton-Raphson method to find the root correct to three decimal places. [5 marks] [XXAnswer : 5.671]
11. [STPM ] By sketching the graphs of y = 3ex and y = 2x − 8 on the same diagram, show that the equation 3ex + 2x − 8 = 0 has only one real root. [3 marks] Use the Newton-Raphson method, with the initial approximation x0 = 1, to find the root correct to three decimal places. [5 marks] [XAnswer : 0.768]
12. [STPM ] Use the trapezium rule with five ordinates to estimate, to three decimal places, the value of Z 1p 2 − x3 dx. 0
[5 marks]
105
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STPM MATHEMATICS (T)
12: Numerical Methods
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[XXAnswer : 1.310] 13. [STPM ] p The curve y = x x2 + 5 and the straight line y = 3x in the first quadrant is shown in the diagram below.
Using the trapezium rule with four intervals, determine the area of the region bounded curve and the straight line. [5 marks]
14. [STPM ]
[XAnswer : 0.683]
1 − 1 and y = −x3 − 2 on the same coordinate axes. Hence, show Sketch the graphs of y = 2−x that the equation x4 − 2x3 + x − 3 = 0 has two real roots. [7 marks]
Using the Newton-Raphson method with the initial approximation x0 = 2, find the positive real root of the equation x4 − 2x3 + x − 3 = 0, correct to four decimal places. [6 marks] State, with a reason, a situation in which the Newton-Raphson method fails.
[2 marks]
[XAnswer : 2.0977]
15. [STPM ] Show that the equation x4 − 2x3 − x + 1 = 0 has at least a real root in the interval [2, 3].
[2 marks]
2x3n − 1 is more likely to give a x3n − 1 convergent sequence of approximation to a root in the interval [2, 3]. Use your choice with x0 = 2.5 to determine the root corrects to three decimal places. [9 marks] Determine which of the iterations xn+1 = x4n − 2x3n + 1 and xn+1 =
[XAnswer : 2.118]
16. [STPM ] Using the Newton-Raphson method with the initial approximation x0 = 4, find the positive real root of the equation ex = 12.5x + 1, correct to three decimal places. [6 marks] [XAnswer : 3.909]
106
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13: Data Description
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Data Description
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13
STPM MATHEMATICS (T)
1. [STPM ] The following is the systolic blood pressure, in mm Hg. of 10 patients in a hospital. 165 135 151 155 158 146 149 124 162 173
If a patient is selected randomly, find the probability his/her systolic blood pressure exceeds one standard deviation above or below the mean. [6 marks] [Answer :
3 ] 10
2. [STPM ] The following table shows the age, in years, of 121 participants in a conference relating to health. Age 18-24 25-31 32-38 39-45 46-52 53-59
Frequency 6 11 19 26 45 14
(a) Plot a histogram and a frequency polygon for the given grouped data. Give comments on the distribution of the age of the participants. [5 marks] (b) Find the estimates for the median and the semi-interquartile range for the participants ages. [8 marks]
(c) State whether the median is more suitable than the mean as the measure of base (central tendency) for the age distribution of the participants. Give reason(s) for your answer. [2 marks] [Answer : (a) negatively skewed ; (b) 45. , 6.8 ; (c) median]
3. [STPM ] The table below indicates the number of cars belonging to 30 houses in a housing area. Number of cars Frequency
0 2
1 15
2 10
3 2
4 1
(a) Find mode, median, and mean. [3 marks] (b) Determine, if the majority of the houses in the housing area have number of cars exceeding the mean. [2 marks] [Answer : (a) 1 , 1 , 1.5 ; (b) No]
4. [STPM ] The table below shows the number of audiences according to age, in years, that watch a horror film for a session at a mini theater. Age, x 15≤ x E(X)).
[Answer : ]
[3 marks]
[6 marks] [3 marks]
(d) Find the probability that, out of three independent observed values of X, exactly two are greater than E(X). [3 marks]
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15: Probability Distributions
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[Answer : ]
153. A crossword puzzle is published in a newspaper every day except on Sundays. A man can solve, on average, 8 of the 10 puzzles. (a) Find the expected value and standard deviation of the number of puzzles solved in a week.
(b) Show that the probability that the man solves at least 5 puzzles in a week is 0.655 (correct to 3 significant figures). (c) Given that he solves the puzzle on Monday, find to 3 significant figures, the probability that he can solve at least 4 puzzles in the remaining days of the week. (d) Find, to 3 significant figures, the probability that in 4 weeks, he solves 4 or less puzzles in only one of the 4 weeks.
178
[Answer : (a) 4.8 , 0.98 ; (c) 0.737 ; (d) 0.388]
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16: Sampling and Estimation
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1. [STPM ]
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Sampling and Estimation
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16
STPM MATHEMATICS (T)
(a) All the Upper Six students in a state sat for a special English test. The mean and standard deviation of the marks obtained by the students are 100 and 25 respectively. i. Find the number of students from a random sample of 200 students who are expected to obtain not less than 80 marks but not more than 120 marks. ii. If the 50 best students from the sample will be awarded prizes, find the minimum marks obtained by a student to be awarded a prize.
(b) The monthly wages of the workers at a factory are distributed normally with mean RM500 and standard deviation RM100. i. Find the probability that the mean monthly wage of a sample of 25 workers selected randomly is RM450. ii. Find the size of the random sample required so that the mean of the sample is within a range of RM10 from the mean of the population with a probability of 0.9. [Answer : (a) (i) 118 ; (ii) 117 ; (b) (i) 0.00621 ; (ii) 271]
2. [STPM ] A supermarket reports that its daily sale is distributed normally with mean RM20 000 and standard deviation RM4000. (a) Estimate the number of days, in a period of 25 days selected at random, when the daily sale of the supermarket is less than RM19 000. (b) Calculate the probability that the mean daily sale of the supermarket in a period of 25 days selected at random, is within a range of RM 1000 from RM20 000. (c) Find the maximum estimation error that occurs with a probability of 0.9, for the daily sale of the supermarket in the period of 25 days selected at random. Explain your answer. [Answer : (a) 8 ; (b) 0.789 ; (c) 1316]
3. [STPM ] A drink manufacturer intends to estimate the percentage of Malaysians who likes the flavour of its new drink. In a market survey, 224 people from 400 people who tasted the drink say that they like the flavour of the drink. Obtain a 95% confidence interval for the percentage of Malaysians whose likes the flavour of the new drink. [5 marks] [Answer : (51.14%, 60.86%)]
4. [STPM ] The mass of a pill made by a factory is distributed normally with mean p and standard deviation 1.5 g. (a) If µ = 7 g, find the probability that the mean mass of a random sample of 9 pills is less than 6 g. [4 marks] (b) If the mean mass of a random sample of 9 pills is 6.8 g, obtain a 95% confidence interval for µ. State, giving reasons, whether the managements claim that µ = 7.5 g is true or false. [6 marks]
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16: Sampling and Estimation
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(c) Determine the smallest sample size needed so that the value of µ obtained is less than 0.5 g at a 95% confidence level. [5 marks] [Answer : (a) 0.0228 ; (b) (5.82,7.78) ; (c) 35]
5. [STPM ] In a certain bank, the service time for each customer is normally distributed with a mean of 2.5 minutes and a standard deviation of 0.9 minutes. Find the probability that the mean service time for a random sample of 49 customers is at least 2.5 minutes but not more than 2.7 minutes. [4 marks] [Answer : 0.4401]
6. [STPM ] A company sells two brands of batteries A and B. For a random sample of 50 brand A batteries, its 50 50 X X lifespan (in months), are summarised by xi = 1600 and x2i = 51641. For a random sample of i=1
i=1
40 brand B batteries, its lifespan (in months), are summarised by
40 X i=1
yi = 1240 and
40 X
yi 2 = 39064.
i=1
(a) Calculate the unbiased estimate of the mean and variance of the lifespan for brand A battery and brand B battery. Explain what is meant by unbiased estimate. [7 marks] [Answer : (a) 32 , 9 ; 31 , 16]
7. [STPM ] An independent random sample is taken from a normally distributed population with mean 80 and variance 25. Determine the smallest sample size so that the sample mean exceeds the population mean by at least 2 with a probability not exceeding 0.01. [5 marks] [Answer : 34]
8. [STPM ] In a survey of 400 supermarkets throughout the country, it is found that 136 of them sell a daily essential product which contains a certain chemical exceeding the government-approved level. (a) Estimate the percentage of supermarkets in the country which sell the product.
[3 marks]
(b) Obtain a 95% confidence interval for the percentage of supermarkets which sell the product. Give an interpretation of the confidence interval you obtain. [6 marks] (c) Determine the smallest sample size of supermarkets which should be surveyed so that there is a probability of 0.95 that the percentage of supermarkets which sell the product can be estimated with an error of less than 2%. [6 marks] [Answer : (a) 34% ; (b) (29.4%, 38.6%) ; (c) 2156]
9. [STPM ] In a sample of 50 moths from the National Park, there are 27 female moths. Obtain a 95% confidence interval for the proportion of female moths in the National Park. [5 marks] [Answer : (0.4019,0.6781)]
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STPM MATHEMATICS (T)
16: Sampling and Estimation
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10. [STPM ] A normal population has a mean of 75 and a variance of 25. Obtain the smallest sample size needed so that the probability that the error of estimation is less than 1 is at least 0.90. [5 marks] [Answer : 68]
11. [STPM ] The height of a certain type of mustard is distributed normally with mean 21.5 cm and variance 90 cm2 . A random sample of size 10 is taken. (a) State the distribution of the sample mean with its mean and variance.
[2 marks]
(b) Find the probability that the sample mean is located between 18 cm and 24 cm.
[3 marks]
[Answer : (b) 0.676]
12. [STPM ] In a survey of 500 motorists on a certain highway, it is found that 120 of them have exceeded the speed limit. (a) Obtain a 95% confidence interval for the proportion of motorists who have exceeded the speed limit on the highway. [5 marks] (b) Determine the smallest sample size which should be surveyed so that the error of estimation is not more than 0.04 at the 90% confidence level. [5 marks] [Answer : (a) (0.203,0.277) ; (b) 309]
13. [STPM ] A survey carried out in an area to estimate the proportion of people who have more than one house. This proportion is estimated using 95% confidence interval. If the estimated proportion is 0.35, determine the smallest sample size required so that estimation error did not exceed 0.03 and deduce the smallest sample size required so that the estimation error did not exceed 0.01. [7 marks] [Answer : 972 , 8740]
14. [STPM ] The mean and standard deviation of the sleeping period of a sample of 100 students chosen at random in a school are 7.15 hours and 1.10 hours respectively. (a) Estimate the mean and standard deviation of the sleeping period of all the students in the school. (b) Estimate the standard error of the mean.
[3 marks]
[1 marks]
[Answer : (a) 7.15 , 1.106 ; (b) 0.1106]
15. [STPM ] A survey carried out by a manufacturer of decorative lamps finds that 136 out of 400 shops sell the decorative lamps at prices less than the recommended prices. (a) Find the 90% symmetric confidence interval for the proportion of shops selling the decorative lamps at prices less than the recommended prices. [5 marks]
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16: Sampling and Estimation
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(b) Determine the smallest sample size required so that the estimated proportion of shops selling the decorative lamps at prices less than the recommended prices is within 2% of the actual proportion at the 90% confidence level. [5 marks] (c) Calculate the probability that more than 60% of a random sample of 500 shops sell the decorative lamps at prices less than the recommended prices. [3 marks] [Answer : (a) (0.301,0.379) ; (b) 1519 ; (c) 0]
16. [STPM ] A random sample X1 , X2 , . . . , Xn is taken from a normal population with mean µ and variance 1. ¯ lies within 0.2 of Determine the smallest sample size which is required so that the probability that X µ is at least 0.90. [5 marks] [Answer : 68]
17. [STPM ] A factory receives its supply of raw materials in packages. The mass of each package is normally distributed with mean 300 kg and standard deviation 5 kg. A random sample of four packages is selected. Find the probability that the mean mass of the sample lies between 292 kg and 296 kg. [4 marks]
[Answer : 0.0541 or 0.0548]
18. [STPM ] A telecommunications company wants to estimate the proportion of customers who require an additional line. A random sample of 500 customers is taken and it is found that 135 customers require an additional line. (a) Obtain the 99% symmetric confidence interval for the proportion of customers who require an additional line. Interpret the confidence interval obtained. [6 marks] (b) If the company wants to estimate the proportion of customers who require an additional line at a different location, determine the smallest sample size required so that the error of estimation does not exceed 0.03 at the 95% confidence level. [5 marks]
19. [STPM ] A normal population has mean µ and variance σ 2 .
(a) Explain briefly what a 95% confidence interval for µ means.
[Answer : (a) (0.219,0.321) ; (b) 842]
[2 marks]
(b) From a random sample, it is found that the 95% confidence interval for µ is (−1.5, 3.8). State whether it is true that the probability that µ lies in the interval is 0.95. Give a reason. [2 marks] (c) A total of 120 random samples of size 50 are taken from the population and for each sample a 95% confidence interval for µ is calculated. Find the number of 95% confidence intervals which are expected to contain µ. [1 marks] [Answer : (c) 114]
20. [STPM ] A marketing research firm believes that 40% of the subscribers of a magazine will participate in a competition held by the magazine. A preliminary survey of 100 subscribers is conducted to find out their participation in the competition. 182
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16: Sampling and Estimation
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(a) Determine the sampling distribution of the proportion of the subscribers who will participate in the competition, stating its mean and variance. [3 marks] (b) Find the probability that at least 30% of the subscribers will participate in the competition. [3 marks]
[Answer : (b) 0.9839]
21. [STPM ] In a study on the petrol consumption of cars, it is found that the mean mileage per litre of petrol for 24 cars of the same engine capacity is 15.2 km with a standard deviation of 4.2 km. Calculate the standard error of the mean mileage and interpret this standard error. [3 marks] [Answer : 0.876]
22. [STPM ] The lifespan of a type of tyre is normally distributed with mean 70000 km and standard deviation 10 000 km. (a) Determine the probability that a randomly chosen tyre has a lifespan of less than 80 000 km. [2 marks]
(b) Find the probability that the mean lifespan of 10 randomly chosen tyres is more than 68 000 km but less than 75 000 km. [4 marks] (c) Determine the minimum number of tyres to be chosen so that the standard error does not exceed 3500 km at the symmetric 99% confidence interval. [4 marks] [Answer : (a) 0.8413 ; (b) 0.6795 ; (c) 55]
23. [STPM ] A survey is to be carried out to estimate the proportion p of households having personal computers. This estimate must be within 0.02 of the population proportion at a confidence level of 95%. (a) If p is estimated to be 0.12, find the smallest sample size required.
[4 marks]
(b) If the value of p is unknown, determine whether a sample size of 2500 is sufficient.
[4 marks]
[Answer : (a) 1015 ; (b) No]
24. [STPM ] A market survey is conducted at a number of shopping complexes. A random sample of 1250 shoppers are asked whether they consume vitamins and 83% of them say “Yes”. (a) Obtain a symmetric 95% confidence interval for the proportion of shoppers who say “Yes” and interpret this confidence interval. [6 marks] (b) Explain why an interval estimate is more informative than a point estimate.
[2 marks]
[Answer : (a) (0.809,0.851)]
25. [STPM ] The lengths of petals taken from a particular species of flowers have mean 80 cm and variance 30 cm2 . Determine the sampling distribution of the sample mean if 100 petals are chosen at random. [3 marks] Hence, find the probability that the sample mean is at least two standard deviations from the mean. [3 marks]
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[Answer : X¯ ∼ (80, 0.3) ; 0.0456] 26. [STPM ] It is found that 5% of doctors in a particular country play golf. Find, to three decimal places, the probability that, in a random sample of 50 doctors, two play golf. [2 marks] Hence, state the sampling distribution of the proportion of the doctors who play golf, and construct a 98% confidence interval for the proportion. [5 marks] [Answer : 0.2611 , pˆ ∼ (0.05, 0.00095 , (-0.022,0.122)]
27. [STPM ] A researcher wishes to estimate the number of vehicles that pass by a location.
(a) According to a previous study, the standard deviation of the number of vehicles passing by the location per day is 245. Calculate the number of days required so that he is 99% confident that the estimate is within 100 vehicles of the true mean. [3 marks] (b) The standard deviation of the number of vehicles is actually 356. Based on the sample size obtained in (a), determine the confidence level for the estimate to be within 100 vehicles of the true mean. [3 marks] [Answer : (a) 40 days ; (b) 92.4%]
28. [STPM ] In a country, 78% of consumers are in favour of government control over prices. A random sample of 400 consumers is selected. (a) Find the mean and standard deviation of the distribution of the sample proportion.
[3 marks]
(b) Find the probability that the sample proportion is at least 5% lower than the population proportion. [4 marks] [Answer : (a) 0.78 , 0.0207 ; (b) 0.0093 or 0.00787 or 0.00790]
29. [STPM ] A machine is regulated to dispense a chocolate drink into cups. From a random sample of 100 cups of the chocolate drink dispensed, it is found that the cocoa content in one cup of the chocolate drink has mean 5 g and standard deviation 0.5 g. The owner of the machine uses the confidence interval (4.900 g, 5.100 g) to estimate the mean cocoa content in one cup of the chocolate drink. (a) Identify the population parameter under study.
(b) Determine the confidence level for the confidence interval used.
[1 marks]
[5 marks]
[Answer : (b) 95.34%]
30. [STPM ] On the average, a button making machine is known to produce 6% defective buttons. A random sample of 100 buttons is inspected and if eight or more buttons are found to be defective, the operation of the machine will be stopped. (a) State the sampling distribution for the sample proportion of defective buttons.
[1 marks]
(b) Find the probability that the operation of the machine will be stopped.
[3 marks]
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[Answer : (a) pˆ ∼ (0.06, 0.000564) ; (b) 0.264 or 0.2638] 31. [STPM ] A random sample of 200 students of a university is selected and it is found that 120 of them stay in university hostels. (a) Estimate the proportion of the students who stay in the hostels and determine the standard error. [3 marks]
(b) Construct a 95% confidence interval for the proportion of the students who stay in hostels, and interpret your answer. [4 marks] (c) What is the effect on the confidence interval if the confidence level is increased from 95% to 99%? [3 marks]
[Answer : (a) 0.60 , 0.0346 ; (b) (0.532,0.668) ; (c) (0.511,0.689), wider]
32. [STPM ] A census conducted in a school shows that the total hours per week pupils spent watching television has a mean of 16.87 hours and a standard deviation of 5 hours. If a random sample of 100 pupils is taken, find 3 hour of the population mean, 4 (b) the probability that the sample mean is more than 17 hours. (a) the probability that the sample mean is within
[4 marks]
[2 marks]
[Answer : (a) 0.866 ; (b) 0.397]
33. [STPM ] According to a report, 80% of the adult population is in favour of banning cigarettes. A proportion of a random sample of 100 adults is found to be in favour of banning cigarettes. (a) State the sampling distribution.
[2 marks]
(b) Find the probability that the sample proportion in favour of banning cigarettes is i. at least 6% lower than the population proportion, ii. within one standard deviation of the population proportion.
[3 marks] [3 marks]
[Answer : (a) pˆ ∼ (0.8, 0.0016) ; (b) (i) 0.0845 ; (ii) 0.6186 if without continuity correction, (i) 0.0668 ; (ii) 0.683]
34. [STPM ] A food company carries out a market survey in a state on its new flavoured yoghurt. Three hundred randomly chosen consumers taste the yoghurt. Their responses are shown in the table below. Response Number of consumers
Like 195
Dislike 70
Neutral 35
(a) Estimate the proportion of consumers in the state who like the yoghurt. Hence, calculate the probability that the proportion of consumers who like the yoghurt is at least 0.70. [5 marks] (b) Construct a 95% confidence interval for the proportion of consumers in the state who like the yoghurt. [4 marks] 185
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[Answer : (a) 0.65 , 0.0397 ; (b) (0.596,0.704)] 35. [STPM ] In a preliminary sample of 40 postgraduate students in a university, 32 students are satisfied with the services at the main library of the university. (a) Determine the smallest sample size needed to estimate the population proportion with an error not exceeding 0.05 at the 90% confidence level. State any assumption made. [5 marks] (b) State the effect on the sample size
i. if the error is larger than 0.05 with the confidence level unchanged. ii. if the confidence level is higher than 90% with the error unchanged.
[1 marks] [1 marks]
[Answer : (a) 174]
36. [STPM ] The age of women in country A suffering from kidney problems is found to be normally distributed with mean 40 years and standard deviation 5 years. (a) Find the probability that 10 randomly selected women who suffer from kidney problems have the mean age less than 42 years. [3 marks] (b) Find the probability that four randomly selected women who suffer from kidney problems have a total age of more than 145 years. [3 marks] (c) The ages of eight randomly selected women from country B who suffer from kidney problems are as follows: 52, 68, 22, 35, 30, 56, 39, 48. Assuming that the ages of the women who suffer from kidney problems are normally distributed, determine the 95% confidence interval for the mean age of the women. Hence, conclude whether the mean age differs from that of country A, and explain your answer. [6 marks] [Answer : (a) 0.8971 ; (b) 0.9332 ; (c) (33.3,54.2)]
37. [STPM ] A random sample of size n is taken to estimate the mean length of a particular aluminum rod produced by a factory. Assuming that the length of the rod is normally distributed with a standard deviation of 2 mm, determine the smallest value of n so that the width of the confidence interval for the mean length of the rod is 1 mm with a confidence level of at least 90%. [5 marks] [Answer : 44]
38. [STPM ] A random variable X is normally distributed with mean 20 and variance 6.25. The mean of a random ¯ sample of size n is X. ¯ (a) State the sampling distribution of X. ¯ < 18) = 0.0057, find the value of n. (b) If P(X
[1 marks] [4 marks]
6.25 [Answer : (a) X¯ ∼ N (20, ) ; (b) 10] n
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39. [STPM ] According to a recent census, children under 18 years of age spend an average of 16.87 hours per week surfing the Internet with a standard deviation of 5 hours per week. Find the probability that in a random sample of 100 children under 18 years of age, the mean time spent surfing the internet per week is (a) between 16.5 and 17.5 hours, inclusive.
[4 marks]
(b) within 0.75 hour of the population mean,
[3 marks]
(c) at least 0.75 hour lower than the population mean.
[3 marks]
[Answer : (a) 0.6666 ; (b) 0.8664 ; (c) 0.0668]
40. [STPM ] The masses of bags of flour produced in a factory have mean 1.004 kg and standard deviation 0.006 kg. (a) Find the probability that a randomly selected bag has a mass of at least 1 kg. State any assumptions made. [4 marks] (b) Find the probability that the mean mass of 50 randomly selected bags is at least 1 kg. [4 marks] [Answer : (a) 0.7477 ; (b) 1]
41. [STPM ] A population distribution has a mean of 205 and variance of 520. If 25 samples, each of size 40, are taken from this population, (a) calculate the probability that the sample mean is less than 200, (b) determine the number of samples with mean less than 200.
[4 marks]
[2 marks]
[Answer : (a) 0.0827 ; (b) 2]
42. [STPM ] The manufacturer of a closed-circuit television (CCTV) claims that the proportion of households installed with CCTV for security- purposes in a city is 0.15. A random sample of 250 households is taken from the city. (a) Assuming that the claim of the manufacturer is true, calculate the probability that the sample proportion is within 0.05 of the population proportion. [5 marks] (b) If 30 households from the random sample install CCTV. construct a 95% confidence interval for the proportion of households with CCTV. [5 marks] (c) If the number of persons, y, per household for the 30 households from the random sample is 30 30 X X summarised by yi = 130 and yi 2 = 967, construct a 95% confidence interval for the i=1
i=1
average size of households with CCTV.
[5 marks]
[Answer : (a) 0.9664 ; (b) (0.0799, 0.160) ; (c) (2.998,5.669)]
43. [STPM ] A random sample of 100 measurements taken from a population gives the following results: X X x = 2980 and (x − x ¯)2 = 3168. 187
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(a) Determine a 95% confidence interval for the population mean.
[7 marks]
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(b) Suggest two ways to reduce the width of the confidence interval that you obtain.
[2 marks]
44. [STPM ] A random of 15 independent measurements, in Pascal second, of the viscosity of a light machine oil is taken. The values obtained are shown below. 25.2 24.3
24.8 25.1
Find the unbiased point estimates for
25.0 25.3
24.0 25.2
24.5 24.5
24.6 24.4
25.0 24.5
25.6
(a) the mean and variance of the population from which the sample is drawn,
(b) the proportion of population having a viscosity of more than 25.0 Pascal seconds.
[6 marks]
[1 marks]
1 3
[Answer : (a) 24.8 , 0.196 ; (b) ]
45. [STPM ] A health survey is made on daily calcium intake and osteoporosis for senior citizens in a particular area. (a) It is known that the daily calcium intake per person has a mean of 1100 mg and a standard deviation of 450 mg. For a random sample of 50 senior citizens, i. determine the distribution of the sample mean, [3 marks] ii. find the probability that the mean daily calcium intake lies between 970 mg and 1230 mg. [4 marks]
State the effect on the probability in (a)(ii) if the sample size is increased and justify your answer. [2 marks]
(b) It is estimated that 64% of the senior citizens have osteoporosis. If a sample of 125 senior citizens are selected at random, i. determine the distribution of the sample proportion, ii. find the probability that at least 70% of them have osteoporosis.
[3 marks] [3 marks]
[Answer : (a)(i) X¯ N (100, 4050) , (ii) 0.9588 ; (b)(i) pˆ N (0.64, 0.0018432) ; (ii) 0.0962]
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Hypothesis Testing
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STPM MATHEMATICS (T)
1. [STPM ] A random sample of 40 batteries produced by a company is found to have a mean lifespan of 17 months and standard deviation of lifespan of 4 months. Determine, at 5% significance level, whether the mean lifespan of the batteries produced by the company is less than 18 months. [6 marks] [Answer : Test statistic=−1.581 > −1.645]
2. [STPM ] A random sample of 80 voters in an area showed that 57 voters supported party Y . Test, at 2% significance level, the claim by party Y that more than 65% of the voters in that area supported them. [7 marks]
[Answer : Test statistic=1.172 < 2.054]
3. [STPM ] A company sells two brands of batteries A and B. For a random sample of 50 brand A batteries, its 50 50 X X x2i = 51641. For a random sample of xi = 1600 and lifespan (in months), are summarised by i=1
i=1
40 brand B batteries, its lifespan (in months), are summarised by
40 X i=1
yi = 1240 and
40 X
yi2 = 39064.
i=1
(a) Calculate the unbiased estimate of the mean and variance of the lifespan for brand A battery and brand B battery. Explain what is meant by unbiased estimate. [7 marks] [Answer : (a) 32 , 9 ; 31 , 16]
4. [STPM ] A box contains a large number of identical beads of various colours. The proportion of white beads is p. A random sample of size 100 is taken to test the null hypothesis H0 : p = 0.5 against the alternative hypothesis H1 : p < 0.5. If the significance level is fixed as 1%, determine the critical region. [5 marks] [Answer : pˆ < 0.3837]
5. [STPM ] Random variable X is normally distributed with mean µ and variance 36. The significance tests performed on the null hypothesis H0 : µ = 70 versus the alternative hypothesis H1 : µ 6= 70 with a probability of type I error equal to 0.01. A random sample of 30 observations of X are taken and ¯ taken as the test statistic. Find the range of the test statistic lies in the critical region. sample mean X [8 marks]
[Answer : x¯ < 67.18, x¯ > 72.82]
6. [STPM ] The mean and standard deviation of the yield of a type of rice in Malaysia are 960 kg per hectare and 192 kg per hectare respectively. From a random sample of 30 farmers in Kedah who plant this rice, the mean yield of rice is 996 kg per hectare. Test, at the 5% significance level, the hypothesis that the mean yield of rice in Kedah is more than the mean yield of rice in Malaysia. Give any assumptions that need to be made in the test of this hypothesis. [8 marks] [Answer : Test statistic=1.027
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