STPM Mathematics T Past Year Question

STPM Mathematics T Past Year Questions

Compiled by: Lee Kian Keong

October 27, 2010

Abstract

This is a document which shows all the questions from year 2002 to year 2009 using LATEX.Students should use this document as reference and try all the questions if possible. Studentsare encourage to contact me via email1 or message on facebook2 if there are problems or typingerrors.

Contents

1 Numbers and Sets 2

2 Polynomials 3

3 Sequences and series 4

4 Matrices 6

5 Coordinate geometry 8

6 Functions 10

7 Differentiation 11

8 Integration 13

9 Differential Equations 15

10 Trigonometry 18

11 Geometry Deduction 19

12 Vectors 23

13 Data Description 25

14 Probability 28

15 Discrete Probability Distributions 30

16 Continuous Probability Distributions 32

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1

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Numbers and Sets Lee Kian Keong

1 Numbers and Sets

1. If loga

( xa2

)= 3 loga 2− loga(x− 2a), express x in terms of a.

[Answer : x = 4a ]

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

log2 a+ loga(1− 2a), where 0 < a <

1

2, find x.

[Answer :2

3]

3. Find the values of x if y = |3− x| and 4y − (x2 − 9) = −24.

[Answer : x = 7, x = −9 ]

4. Using the laws of the algebra of sets, show that

(A ∩B)′ − (A′ ∩B) = B′

5. Using definitions, show that, for any sets A, B and C,

A ∩ (B ∪ C) ⊂ (A ∩B) ∪ (A ∩ C)

6. Using the laws of the algebra of sets, show that, for any sets A and B,

(A−B) ∪ (B −A) = (A ∪B)− (A ∩B)

7. If A, B and C are arbitrary sets, show that

[(A ∪B)− (B ∪ C)] ∩ (A ∪ C)′ = ∅

8. If z is a complex number such that |z| = 1, find the real part of1

1− z.

[Answer :1

2]

9. Express

√59− 24

√6 as p

√2 + q

√3 where p and q are integers.

[Answer : 4√

2− 3√

3 ]

10. Simplify

(a)(√

7−√

3)2

2(√

7 +√

3),

(b)2(1 + 3i)

(1− 3i)2, where i =

√−1.

[Answer : (a) 2√

7− 3√

3 ; (b) −13

25−

9

25i ]

11. The complex numbers z1 and z2 satisfy the equation z2 = 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 =√

3− i, z2 = −√

3 + i ; (c) |z1| = 2, |z2| = 2 , arg(z1)=−π

6, arg(z2)=

5π

6]

12. If (x+ iy)2 = i, find all the real values of x and y.

[Answer : x = ±1√

2, y = ±

1√

2]

2

Polynomials Lee Kian Keong

2 Polynomials

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

2x− k has factor (x+ 1).

(a) Find the value of k.

(b) Factorise p(x) completely.

[Answer : (a) k =3

2; (b)

1

2(x+ 1)(2x− 1)(2x+ 3) ]

2. Show that −1 is the only one real root of the equation x3 + 3x2 + 5x+ 3 = 0.

3. 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.

Using the substitution y =1

x, solve the equation 12y4 − 8y3 − 7y2 + 2y + 1 = 0.

[Answer : a = 2, b = 12; y = −1

3,−

1

2, 1,

1

2]

4. Using the substitution y = x+1

x, express f(x) = x3 − 4x− 6− 4

x+

1

x3as a polynomial in y.

[Answer : y3 − 7y − 6 ]

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

[Answer : −1,−1,3 +√

5

2,

3−√

5

2]

5. The polynomial p(x) = 6x4 − ax3 − bx2 + 28x + 12, where a and b are real constants, hasfactors (x+ 2) and (x− 2).

(a) Find the values of a and b, and hence, factorise p(x) completely.

(b) Give that p(x) = (2x − 3)[q(x) − 41 + 3x3], find q(x), and determine its range whenx ∈ [−2, 10].

[Answer : (a) a = 7, b = 27, (x− 2)(2x− 3)(3x+ 1)(x+ 2); (b) q(x) = x2 − 12x+ 37, {y : 1 ≤ y ≤ 65} ]

6. Show that polynomial 2x3 − 9x2 + 3x+ 4 has x− 1 as factor.Hence,

(a) find all the real roots of 2x6 − 9x4 + 3x2 + 4 = 0.

(b) determine the set of values of x so that 2x3 − 9x2 + 3x+ 4 < 12− 12x.

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

7. Find the set of values of x such that −1 < x3 − 2x2 + x− 2 < 0.

[Answer : {x : 0 < x < 1, 1 < x < 2} ]

8. Find the solution set of inequality |x− 2| < 1

xwhere x 6= 0.

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

2} ]

3

Sequences and series Lee Kian Keong

9. Determine the set of values of x satisfying the inequality

x

x+ 1≥ 1

x+ 1

[Answer : {x : x < −1, x ≥ 1} ]

10. Find the solution set of the inequality ∣∣∣∣ 4

x− 1

∣∣∣∣ > 3− 3

x.

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

11. Sketch, on the same coordinate axes, the graph of y = 2− x and y = |2 +1

x|.

Hence, solve the inequality 2− x > |2 +1

x|

[Answer : {x : x < 2−√

5} ]

12. Find the constants A, B, C and D such that

3x2 + 5x

(1− x2)(1 + x)2=

A

1− x+

B

1 + x+

C

(1 + x)2+

D

(1 + x)3

.

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

3 Sequences and series

1. For the geometric series 7 + 3.5 + 1.75 + 0.875 + ..., find the smallest value of n for which thedifferent between the sum of the first n terms and the sum to infinity is less than 0.01.

[Answer : 11 ]

2. For geometric series 6 + 3 +3

2+ . . ., obtain the smallest value of n if the difference between

the sum of the first n+ 4 terms and the sum of first n terms is less than45

64.

[Answer : 5 ]

3. Express the infinite recurring decimal 0.72̇5̇ (= 0.7252525 . . . ) as a fraction in its lowest terms.

[Answer :359

495]

4. Determine the set of x such that the geometric series 1 + ex + e2x + . . . converges. Find theexact value of x so that the series converges to 2.

[Answer : {x : x < 0} ; x = − ln 2 ]

5. Prove that the sum of the first n terms of a geometric series a+ ar + ar2 + . . . is

a(1− rn)

1− r

4

Sequences and series Lee Kian Keong

(a) The sum of the first five terms of a geometric series is 33 and the sum of the first tenterms of the geometric series is -1023. Find the common ratio and the first term of thegeometric series.

(b) The sum of the first n terms and the sum to infinity of the geometric series 6−3+3

2− . . .

are Sn and S∞ respectively. Determine the smallest value of n such that |Sn−S∞| < 0.001

[Answer : (a) r = −2, a = 3 ; (b) n = 12 ]

6. The nth term of an arithmetic progression is Tn, show that Un =5

2(−2)2( 10−Tn

17 ) is the nth

term of a geometric progression.

If Tn =1

2(17n− 14), evaluate

∞∑n=1

Un.

[Answer : −10

3]

7. At the beginning of this year, Mr. Liu and Miss Dora deposited RM10 000 and RM2000respectively in a bank. They receive an interest of 4% per annum. Mr Liu does not make anyadditional deposit nor withdrawal, whereas, Miss Dora continues to deposit RM2000 at thebeginning 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.

(c) Determine in which year the total savings of Miss Dora exceeds the total savings of Mr.Liu.

[Answer : (a) 10000(1.04)n

; (b) 52000[1.04n − 1]; (c) 6 ]

8. Express1

4k2 − 1as partial fraction.

Hence, find a simple expression for Sn =

n∑k=1

1

4k2 − 1and find lim

n→∞Sn

[Answer :1

2(2k − 1)−

1

2(2k + 1); Sn =

1

2

(1−

1

2n+ 1

);

1

2]

9. If x is so small that x2 and higher powers of x may be neglected, show that

(1− x)(

2 +x

2

)10

≈ 29(2− 7x)

10. Express

(1 + x

1 + 2x

) 12

as a series of ascending powers of x up to the term in x3.

By taking x =1

30, find

√62 correct to four decimal places.

[Answer : 1−1

2x+

7

8x2 −

25

16x3

+ . . . ;√

62 = 7.8740 ]

11. Express ur =2

r2 + 2rin partial fractions.

[Answer :1

r−

1

r + 2]

5

Matrices Lee Kian Keong

Using the result obtained,

(a) show that u2r = −1

r+

1

r2+

1

r + 2+

1

(r + 2)2,

(b) show that

∞∑r=1

ur =3

2− 1

n+ 1− 1

n+ 2and determine the values of

∞∑r=1

ur and

∞∑r=1

(ur+1 +

1

3r

).

[Answer : (b)3

2,

4

3]

12. Expand (1− x)12 in ascending powers of x up to the term in x3. Hence, find the value of

√7

correct to five decimal places.

[Answer : 1−1

2x−

1

8x2 −

1

16x3

;√

7 = 2.64609 ]

4 Matrices

1. Matrix M and N is given by M =

−10 4 915 −4 −14−5 1 6

, and N =

2 3 44 3 11 2 4

Find MN and deduce N−1.Product X, Y , Z are assembled from three components A, B, C according to different pro-portions. Each product X consists of two components of A, four components of B, and onecomponent 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 componentof B, and four components of C. A total of 750 components of A, 1000 components of B, and500 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.Hence, find the number of products of X, Y , and Z assembled.

[Answer : x=200, y=50, z=50. ]

2. A, B, C are square matrices such that BA = B−1 and ABC = (AB)−1. Show that A−1 =B2 = C.

If B =

1 2 00 −1 01 0 1

, find C and A.

[Answer : A =

1 0 00 1 0−2 −2 1

, C =

1 0 00 1 02 2 1

]

3. (a) The matrix P, Q and R are given by

P =

1 5 62 −2 41 −3 2

,Q =

−13 −50 −33−1 −6 −57 20 15

,R =

4 7 −131 −5 −1−2 1 11

Find matrices PQ and PQR and hence, deduce (PQ)−1.

(b) Using the result in (a), solve the system of linear equations

6x + 10y + 8z = 4500x − 2y + z = 0x + 2y + 3z = 1080

.

6

Matrices Lee Kian Keong

[Answer : (a)

24 40 324 −8 44 8 12

,

72 0 00 72 00 0 72

,

1/18 7/72 −13/721/72 −5/72 −1/72−1/36 1/72 11/72

; (b) x = 220, y = 190, z = 160 ]

4. Determine the values of a, b, c so that the matrix

2b− 1 a2 b2

2a− 1 a bcb b+ c 2c− 1

is a symmetric

matrix.

[Answer : a = 1, b = 0, c = 0 ]

5. The matrices A and B are given by

A =

−1 2 1−3 1 40 1 2

, B =

−35 19 18−27 −13 45−3 12 5

.

Find the matrix A2B and deduce the inverse of A.Hence, solve the system of linear equations

x− 2y − z = −8,3x− y − 4z = −15,

y + 2z = 4.

[Answer :

121 0 00 121 00 0 121

,

−2/11 −3/11 7/116/11 −2/11 1/11−3/11 1/11 5/11

; x = −3, y = 2, z = 1 ]

6. The matrix A is given by A =

1 2 −33 1 10 1 −2

(a) Find the matrix B such that B = A2 − 10I, where I is the 3× 3 identity matrix.

(b) Find (A+ I)B, and hence find (A+ I)21B.

[Answer : (a)

−3 1 56 −2 −103 −1 −5

; (b)

−3 1 56 −2 −103 −1 −5

,

−3 1 56 −2 −103 −1 −5

]

7. The matrices P and Q, where PQ = QP , are given by

P =

2 −2 00 0 2a b c

and Q =

−1 1 00 0 −10 −2 2

Determine the values of a, b and c.Find the real numbers m and n for which P = mQ+ nI, where I is the 3× 3 identity matrix.

[Answer : a = 0, b = 4, c = −4 ; m = −2, n = 0 ]

8. Matrix A is given by A =

3 3 45 4 11 2 3

.

Find the adjoint of A. Hence, find A−1.

[Answer :

10 −1 −13−14 5 17

6 −3 −3

;

5/6 −1/12 −13/12−7/6 5/12 17/121/2 −1/4 −1/4

]

7

Coordinate geometry Lee Kian Keong

9. If P =

5 2 31 −4 33 1 2

, Q =

a 1 −18b −1 12−13 −1 c

and PQ = 2I, where I is the 3×3 identity

matrix, determine the values of a, b and c. Hence find P−1.Two groups of workers have their drinks at a stall. The first group comprising ten workers havefive 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 fruitjuice at a total cost of RM7.10. The cost of a cup of tea and three glasses of fruit juice is thesame as the cost of four cups of coffee. If the costs of a cup of tea, a cup of coffee and a glassof fruit juice are RM x, RM y and RM z respectively, obtain a matrix equation to representthe above information. Hence determine the cost of each drink.

[Answer : a = 11, b = −7, c = 22 ;

5/2 1 3/21/2 −2 3/23/2 1/2 1

; x=RM 1, y=RM 1.30, z=RM 1.40 ]

10. Determine the values of k such that the determinant of the matrix

k 1 32k + 1 −3 2

0 k 2

is 0.

[Answer : k = −1

4, k = 2 ]

11. Matrix A is given by A =

1 0 01 −1 01 −2 1

.

(a) Show that A2 = I, where I is the 3× 3 identity matrix, and deduce A−1.

(b) Find the matrix B which satisfies BA =

1 4 30 2 1−1 0 2

.

[Answer : (a)

1 0 01 −1 01 −2 1

; (b)

8 −10 33 −4 11 −4 2

]

5 Coordinate geometry

1. Given that PQRS is a parallelogram where P (0, 9), Q(2,−5), R(7, 0) and S(a, b) are pointson the plane. Find a and b.Find the shortest distance from P to QR and the area of the parallelogram.

[Answer : a = 5, b = 14; shortest distance=8√

2 units; Area=80 unit2

]

2. The straight line l1 which passes through the points A(4, 0) and B(2, 4) intersects the y-axisat point P . The straight line l2 is perpendicular to l1 and passes through B. If l2 intersectsthe x-axis and y-axis at points Q and R respectively, show that PR : QR =

√5 : 3.

3. The sum of distance of the point P from the point (4,0) and the distance of P from the origin

is 8 units. Show that the locus of P is the ellipse(x− 2)2

16+y2

12= 1 and sketch the ellipse.

4. The point R divides the line joining the points P (3, 2) and Q(5, 8) in the ratio 3 : 4. Find theequation of the line passing through R and perpendicular to PQ.

8

Coordinate geometry Lee Kian Keong

[Answer : 7x+ 21y − 123 = 0 ]

5. Find the perpendicular distance from the centre of the circle x2 + y2 − 8x + 2y + 8 = 0 tothe straight line 3x + 4y = 28. Hence, find the shortest distance between the circle and thestraight line.

[Answer : 4 ; 1 ]

6. The lines y = 2x and y = x intersect the curve y2 + 7xy = 18 at points A and B respectively,where A and B lie in the first quadrant.

(a) Find the coordinates of A and B.

(b) Calculate the perpendicular distance of A to OB, where O is the origin.

(c) Find the area of the triangle OAB.

[Answer : (a) A(1, 2) and B

(3

2,

3

2

); (b)

√2

2; (c)

3

4]

7. The parametric equations of a straight line l are given by x = 4t− 2 and y = 3− 3t.

(a) Show that the point A(1,3

4) lies on line l,

(b) Find the Cartesian equation of line l,

(c) Given that line l cuts the x and y-axes at P and Q respectively, find the ratio PA : AQ.

[Answer : (b) 3x+ 4y − 6 = 0 ; (c) 1:1 ]

8. The coordinates of the points P and Q are (x, y) and

(x

x2 + y2,

y

x2 + y2

)respectively, where

x 6= 0 and y 6= 0. If Q moves on a circle with centre (1, 1) and radius 3, show that the locusof P is also a circle. Find the coordinates of the centre and radius of the circle.

[Answer : centre = (−1

7,−

1

7) ; radius =

3

7]

9. Show that x2 +y2−2ax−2by+ c = 0 is the equation of the circle with centre (a, b) and radius√a2 + b2 − c.

The above figure shows three circles C1, C2 and C3 touching one another, where their centreslies on a straight line. If C1 and C2 have equations x2 + y2 − 10x− 4y+ 28 = 0 and x2 + y2 −16x+ 4y + 52 = 0 respectively. Find the equation of C3.

[Answer : 5x2

+ 5y2 − 74x+ 12y + 156 = 0 ]

9

Functions Lee Kian Keong

6 Functions

1. The function f is defined by f : x→√

3x+ 1, x ∈ R, x ≥ −1

3Find f−1 and state its domain and range.

[Answer : f−1: x→

x2 − 1

3, Df−1 = {x : x ≥ 0}, Rf−1 = {x : x ≥ −

1

3} ]

2. The function f is defined by

f(x) =

1 + ex, x < 1

3, x = 1

2 + e− x, x > 1

(a) Find limx→1−

f(x) and limx→1+

f(x). Hence, determine whether f is continuous at x = 1.

(b) Sketch the graph of f .

[Answer : (a) 1 + e , 1 + e ; not continuous ]


f(x) =

x− 1

x+ 2, 0 ≤ x < 2

ax2 + 1, x ≥ 2

where a ∈ R. Find the value of a if limx→2

f(x) exists. With this value of a, determine whether

f is continuous at x = 2.

[Answer : a =5

16; continuous at x = 2 ]

4. The functions f and g are given by

f(x) =ex − e−x

ex + e−xandg(x) =

2

ex + e−x

(a) State the domains of f and g,

(b) Without using differentiation, find the range of f ,

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

[Answer : (a) Df = {x : x ∈ R} , Dg = {x : x ∈ R} ; (b) {y : −1 < y < 1} ; (c) {y : 0 < y ≤ 1} ]

5. Given x > 0 and f(x) =√x, find lim

h→0

f(x)− f(x+ h)

h.

[Answer : −1

2√x

]


f(x) =

{√x+ 1, −1 ≤ x < 1,

|x| − 1, otherwise.

(a) Find limx→−1−

f(x), limx→−1+

f(x), limx→1−

f(x) and limx→1+

f(x).

(b) Determine whether f is continuous at x = −1 and x = 1.

10

Differentiation Lee Kian Keong

[Answer : (a) 0 , 0 ,√

2 , 0 ; (b) continuous at x = −1 , discontinuous at x = 1 ]

7. Functions f , g and h are defined by

f : x→ x

x+ 1; g : x→ x+ 2

x; h : x→ 3 +

2

x

(a) State the domains of f and g.

(b) Find the composite functions 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

[Answer : (a) Df = {x : x ∈ R, x 6= −1} , Dg = {x : x ∈ R, x 6= 0} ;

(b) 3 +2

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

(c) D = {x : x ∈ R, x 6= 0} , R = {y : y ∈ R, y 6= 3} ; (d) No. Different domain ]

8. The function f and g are defined by

f : x→ 1

x, x ∈ R \ {0};

g : x→ 2x− 1, x ∈ R

Find f ◦ g and its domain.

[Answer :1

2x− 1, D = {x : x ∈ R, x 6=

1

2} ]

7 Differentiation

1. Given that y = e−x cosx, finddy

dxand

d2y

dx2when x = 0.

[Answer :dy

dx= 1,

d2y

dx2= 0 ]

2. If y = ln√xy, find the value of

dy

dxwhen y = 1.

[Answer :1

e2]

3. Function f if defined by f(x) =2x

(x+ 1)(x− 2).

Show that f ′(x) < 0 for all values of x in the domain of f .Sketch the graph of y = f(x). Determine if f is a one to one function. Give reasons to youranswer.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.

[Answer : f is not one to one function. If k ≥ 0, 1 root. If k < 0, 3 roots. ]

4. If y =cosx

x, where x 6= 0, show that x

d2y

dx2+ 2

dy

dx+ xy = 0.

5. If y =x

1 + x2, show that x2 dy

dx= (1− x2)y2.

11

Differentiation Lee Kian Keong

6. If y =sinx− cosx

sinx+ cosx, show that

d2y

dx2= 2y

dy

dx.

7. A curve is defined by the parametric equations x = 1− 2t, y = −2 +2

t.

Find the equation of the normal to the curve at the point A(3,−4).The normal of the curve at the point A cuts the curve again at point B.Find the coordinates of B.

[Answer : x+ y + 1 = 0 ; B(−1, 0) ]

8. Using the sketch of y = x3 and x+ y = 1, show that the equation x3 + x− 1 = 0 has only onereal root and state the successive integers a and b such that the real root lies in the interval(a, b).Using the Newton-Raphson method to find the real root correct to three decimal places.

[Answer : a = 0 , b = 1 ; 0.683 ]

9. Sketch, on the same coordinate axes, the graphs y = ex and y =2

1 + x. Show that the equation

(1 + x)ex − 2 = 0 has a root in the interval [0, 1].Use the Newton-Raphson method with the initial estimate x0 = 0.5 to estimate the root correctto three decimal places.

[Answer : 0.375 ]

10. Find the coordinate of the stationary point on the curve y = x2 +1

xwhere x > 0; give the x-

coordinate and y-coordinate correct to three decimal places. Determine whether the stationarypoint is a minimum point or a maximum point.

The x-coordinate of the point of intersection of the curves y = x2 +1

xand y =

1

x2, where

x > 0, is p. Show that 0.5 < p < 1.Using the Newton-Raphson method to determine the value of p correct to three decimal placesand, hence, find the point of intersection.

[Answer : (0.794 , 1.890) , minimum ; p = 0.724 , (0.724 , 1.908) ]

11. A curve is defined by the parametric equations

x = t− 2

tand y = 2t+

1

t

where t 6= 0.

(a) Show thatdy

dx= 2− 5

t2 + 2, and hence, deduce that −1

2<dy

dx< 2.

(b) Find the coordinates of points whendy

dx=

1

3.

[Answer : (b) (-1 , 3) and (1 , -3) ]

12. Find the coordinates of the stationary points on the curve y =x3

x2 − 1abd determine their

nature.Sketch the curve.Determine the number of real roots of the equation x3 = k(x2 − 1), where k ∈ R, when kvaries.

12

Integration Lee Kian Keong

[Answer : (0, 0) is inflexion point , (√

3,3√

3

2) is local min. , (−

√3,−

3√

3

2) is local max.

1 real root for −3√

3

2< k <

3√

3

2, 2 real root for k = ±

3√

3

2, 3 real roots for k < −

3√

3

2or k >

3√

3

2]

13. 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.

[Answer :[

x

x+ 1+ ln(x+ 1)

]δx ; 0.698 ]

14. The function f is defined by f(t) =4ekt − 1

4ekt + 1where k is a positive constant,

(a) Find the value of f(0)

(3

5

)(b) Show that f ′(t) > 0

(c) Show that k[1− f(t)2] = 2f ′(t) and hence show that f ′′(t) < 0

(d) Find limt→∞

f(t) (1)

(e) Sketch the graph of f .

15. Show that the gradient of the curve y =x

x2 − 1is always decreasing.

Determine the coordinates of the point of inflexion of the curve, and state the intervals forwhich the curve is concave upwards.Sketch the curve.

[Answer : (0, 0) ; (−1, 0) ∪ (1,∞) ]

16. 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.

8 Integration

1. Show that

∫ e

1

lnx dx = 1.

2. Show that

∫ 3

2

(x− 2)2

x2dx =

5

3+ 4 ln

(2

3

)

3. By using suitable substitution, find

∫3x− 1√x+ 1

dx

[Answer : 2(x+ 1)32 − 8(x+ 1)

12 + C ]

4. Using an appropriate substitution, evaluate

∫ 1

0

x2(1− x)13 dx.

[Answer :27

140]

13

Integration Lee Kian Keong

5. Using the substitution u = 3 + 2 sin θ, evaluate

∫ π6

0

cos θ

(3 + 2 sin θ)2dθ.

[Answer :1

24]

6. Express2x+ 1

(x2 + 1)(2− x)in the form

Ax+B

x2 + 1+

C

2− xwhere A, B and C are constants.

Hence, evaluate

∫ 1

0

2x+ 1

(x2 + 1)(2− x)dx

[Answer :x

x2 + 1+

1

2− x;

3

2ln 2 ]

7. The gradient of the tangent to a curve at any point (x, y) is given bydy

dx=

3x− 5

2√x

, where

x > 0. If the curve passes through the point (1,−4).

(a) find the equation of the curve,

(b) sketch the curve,

(c) calculate the area of the region bounded by the curve and the x-axis.

[Answer : (a) y = x32 − 5x

12 ; (c)

20

5

√5 ]

8. 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,

(c) calculate the area of the region R bounded by the curve and the straight line,

(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)9

2; (d)

108

5π ]

9. Sketch, on the same coordinate axes, the curves y = 6 − ex and y = 5e−x, and find thecoordinates of the points of intersection.Calculate the area of the region bounded by the curves.Calculate the volume of the solid formed when the region is rotated through 2π radians aboutthe x-axis.

[Answer : (ln 5, 1) ; 6 ln 5− 8 ; π(36 ln 5− 48) ]

10. Find

(a)

∫x2 + x+ 2

x2 + 2dx,

(b)

∫x

ex+1dx.

[Answer : (a) x+1

2ln(x

2+ 2) + C ; (b) −

x

ex+1+

1

ex+1]

11. 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.

[Answer : 2 ln 3 ]

14

Differential Equations Lee Kian Keong

12. Sketch on the same coordinates axis y =1

2x and the curve y2 = x. Find the coordinate of the

points of intersection.

Find the area of region bounded by the line y =1

2x and the curve y2 = x.

Find the volume of the solid formed when the region is rotated through 2π radians about they-axis.

[Answer : (0, 0) , (4, 2) ;4

3;

64

15π ]

13. The curve y =a

2x(b − x), where a 6= 0, has a turning point at point (2, 1). Determine the

values of a and b.Calculate the area of the region bounded bt the x-axis and the curve.Calculate the volume of the solid formed by revolving the region about the x-axis.

[Answer : a =1

2, b = 4 ;

8

3;

32

15π ]

14. 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.Calculate the area of the region bounded by the curves y = −x2 + 3x and y = 2x3 − x2 − 5x.

[Answer : Point of intersection=(0,0), (2,2), (-2,-10) ; Area=16 units2. ]

15. Using trapezium rule, with five ordinates, evaluate

∫ 1

0

√4− x2 dx.

[Answer : 1.910 ]

9 Differential Equations

1. Find the particular solution of the differential equation

exdy

dx− y2(x+ 1) = 0

for which y = 1 when x = 0. Hence, express y in terms of x.

[Answer : y =ex

2 + x− ex]

2. Find the general solution of the differential equation

xdy

dx= y2 − y − 2.

[Answer : y =2 + Ax3

1− Ax3]

3. Show that the substitution u = x2 + y transforms the differential equation

(1− x)dy

dx+ 2y + 2x = 0

into the differential equation

(1− x)du

dx= −2u

15


4. The variables t and x are connected by

dx

dt= 2t(x− 1),

where x 6= 1. Find x in terms of t if x = 2 when t = 1.

[Answer : x = et2−1

+ 1 ]

5. The variables x and y, where x > 0, satisfy the differential equation

xdu

dx= u2 − 2u.

Hence, show that the general solution of the given differential equation maybe expressed in

the form y =2x

1 +Ax2, where A is an arbitrary constant.

Find the equation of the solution curve which passes through the point (1,4) and sketch thissolution curve.

[Answer : y =4x

2− x2]

6. Using the substitution y =v

x2, show that the differential equation

dy

dx+ y2 = −2y

x

may be reduced todv

dx= − v

2

x2.

Hence, find the general solution of the original differential equation.

[Answer : y =1

Ax2 − x]

7. Show thatd

dx(ln tanx) =

2

sin 2x,

Hence, find the solution of the differential equation

(sin 2x)dy

dx= 2y(1− y)

for which y =1

3when x =

1

4π. Express y explicity in terms of x in your answer.

[Answer : y =tan x

2 + tan x]

8. One of the rules at a training camp of 1000 occupants states that camp activities are to besuspended if 10% of the occupants are infected with a virus. A trainee infected with a flu virusenrolls in the camp causing an outbreak of flu. The rate of increase of the number of infectedoccupants x at t days is given by differential equation

dx

dt= kx(1000− x)

where k is a constant.Assume that the outbreak of flu begins at the time the infected trainee enrolls and no oneleaves the camp during the outbreak,

16


(a) Show that x =1000e1000kt

999 + e1000kt,

(b) Determine the value of k if it is found that, after one day, there are five infected occupants

(c) Determine the number of days before the camp activities will be suspended.

[Answer : (b) k =1

1000ln

(999

195

); (c) 5 ]

9. A 50 litre tank is initially filled with 10 litres of brine solution containing 20 kg of salt. Startingfrom 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 changeof Q is given by

dQ

dt= − Q

√k

10 + (4−√k)t

.

(b) Show that k = 4, and calculate the amount of salt in the tank at the instant overflowoccurs.

(c) Sketch the graph of Q against t for 0 ≤ t ≤ 20.

[Answer : (b) Q = 4 ]

10. The rate of change of water temperature is described by the differential equation

dθ

dt= −k(θ − θs)

where θ is the water temperature at time t, θs is the surrounding temperature, and k is apositive constant.A boiling water at 100◦C is left to cool in kitchen that has a surrounding temperature of 25◦C.

The water takes 1 hour to decrease to the temperature of 75◦C. Show that k = ln3

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 iceat 0◦C.

[Answer : Time = 4 hours 25 minutes ]

11. The rate of increase in the number of a species of fish in a lake is described by the differentialequation

dP

dt= (a− b)P

where P is the number of fish at time t weeks, a is the rate of reproduction, and b is themortality rate, with a and b as constants.

(a) Assuming that P = P0 at time t = 0 and a > b, solve the differential equation and sketchits solution curve.

(b) At a certain instant, there is an outbreak of an epidemic of a disease. The epidemicresults in no more offspring of the fish being produced and the fish die at a rate directly

proportional to

√1

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 theepidemic until all the fish of that species die.

17

Trigonometry Lee Kian Keong

[Answer : (a) P = P0e(a−b)t

; (b) 18 weeks ]

12. A particle moves from rest along a horizontal straight line. At time t s, the displacement andvelocity of the particle are x m and v ms−1 respectively and its acceleration, in ms−2, is givenby

dv

dt= sin(πt)−

√3 cos(πt)

Express v and x in terms of t.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 acceler-ation is zero.

[Answer : v = −1

π[cos(πt) +

√3 sin(πt)] , x = −

1

π2[sin(πt)−

√3 cos(πt)]

v = ±2

πms

−1; Distance =

4

π2m ]

10 Trigonometry

1. 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◦.

[Answer : 5 sin(θ − 36.9◦) ; 73.8

◦, 180

◦]

2. Find the values of x, where 0 ≤ x ≤ π, which satisfy the equation sin3 x secx = 2 tanx

[Answer : 0,π

4,

3π

4, π ]

3. If t = tanθ

2, show that sin θ =

2t

1 + t2and cos θ =

1− t2

1 + t2.

Hence, find the values θ between 0◦ and 360◦ that satisfy the equation

10 sin θ − 5 cos θ = 2

[Answer : 36.9◦

, 196.3◦

]

4. Express cosx+√

3 sinx in the form r cos(x− α), r > 0 and 0 < α <π

2.

Hence, find the values of x with 0 ≤ x ≤ 2π, which satisfies the inequality

0 < cosx+√

3 sinx < 1

.

[Answer : 2 cos

(x−

π

3

);

{x :

2π

3< x <

5π

6,

11π

6< x < 2π

}]

5. Starting from the formulae for sin(A+B) and cos(A+B), prove that

tan(A+B) =tanA+ tanB

1− tanA tanB

If 2x+ y =π

4, show that

tan y =1− 2 tanx− tan2 x

1 + 2 tanx− tan2 x

By substituting x =π

8, show that tan

π

8=√

2− 1.

18

Geometry Deduction Lee Kian Keong

6. Express cos θ + 3 sin θ in the form r cos(θ − α), where r > 0 and 0◦ < α < 90◦

[Answer :√

10 cos(θ − 71.6◦) ]

7. Find all values of x, where 0◦ < x < 360◦, which satisfy the equation tanx+ 4 cotx = 4 secx.

[Answer : 41.8◦

, 138.2◦

]

8. Find, in terms of π, all the values of x between 0 and π which satisfies the equation

tanx+ cotx = 8 cos 2x

[Answer :π

24,

5π

24,

13π

24,

17π

24]

9. The triangle PQR lies in a horizontal plance, with Q due west of R. The bearings of P fromQ and R are θ and φ respectively, where θ and φ are acute. The top A of a tower PA is atheight h above the plane and the angle of elevation of A from R is α. The height of a verticalpole QB is k ang the angle of elevation of B from R is β. Show that

h =k tanα cos θ

tanβ sin(θ − φ)

10. In the tetrahedronABCD, AB = BC = 10 cm, AC = 8√

2 cm, AD = CD = 8 cm andBD = 6cm. Show that the line from C perpendicular to AB and the line from D perpendicular to ABmeet at a point on AB. Hence, calculate the angle between the face ABC and the face ABD.

[Answer : 59.0◦

]

11 Geometry Deduction

1. Points A and B are in the side XY of triangle XY Z with XA = AB = BY . Points C andD are on the sides Y Z and XZ respectively such that ABCD is a rhombus. Prove that∠XZY = 90◦

2. The points P , Q, R, S are on the circumference of a circle, such that ∠PQR = 80◦ and∠RPS = 30◦ as shown in the diagram below. The tangent to the circle at P and the chordRS which is produced, meet at T .

(a) Show that PR = PT

19


(b) Show that the length of the chord RS is the same as the radius of the circle.

3. Vertices B and C of the triangle ABC lie on the circumference of a circle. AB and AC cutthe circumference of the circle at X and Y respectively. Show that ∠CBX + ∠CY X = 180◦

If AB = AC, show that BC is parallel to XY .

4. The diagram below shows two circles ABRP and ABQS which intersect at A and B. PAQand RAS are straight lines. Prove that the triangles RPB and SQB are similar.

5. The diagram below shows two isosceles triangles ABC and ADE which have bases AB andAD respectively. Each triangle has base angles measuring 75◦, with BC and DE parallel andequal in length. Show that

(a) ∠DBC = ∠BDE = 90◦,

(b) the triangle ACE is an equilateral triangle,

(c) the quadrilateral BCED is a square.

6. The diagram below shows two intersecting circles AXY B and CBOX, where O is the centreof the circle AXY B. AXC and BY C are straight lines. Show that ∠ABC = ∠BAC.

20


7. In the triangle ABC, the point P lies on the side AC such that ∠BPC = ∠ABC. Show thatthe triangles BPC and ABC are similar.

If AB = 4 cm, AC = 8 cm and BP = 3 cm, find the area of the triangle BPC.

[Answer : 6.54 cm2

]

8. Prove that an exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.

In the diagram, ABCD is a cyclic quadrilateral. The lines AB and DC extended meet at thepoint E and the lines AD and BC extended meet at the point F . Show that triangles ADEand CBE are similar.If DA = DE, ∠CFD = α and ∠BEC = 3α, determine the value of α.

[Answer : α = 18◦

]

9. The diagram above shows two intersecting circles APQ and BPQ, where APB is a straightline. The tangents at the points A and B meet at a point C. Show that ACBQ is a cyclicquadrilateral.

If the lines AQ and CB are parallel and T is the point of intersection of AB and CQ, showthat the triangles ATQ and BTC are isosceles triangles. Hence, show that the areas of thetriangles ATQ and BTC are in the ratio AT 2 : BT 2.

10. The diagram below shows the circumscribed circle of he triangle ABC.

21


The tangent to the circle at A meets the line BC extended to T . The angle bisector of theangle ATB at Pm AB at Q and the circle at R. Show that

(a) triangles APT and BQT are similar,

(b) PT ·BT = QT ·AT ,

(c) AP = AQ.

11. The circumscribed circle of the triangle JKL is shown in the diagram below.

The tangent to the circle at J meets the line KL extended to T . The angle bisector of theangle JTK cuts JL and JK at U and V respectively. Show that JV = JU .

12. A parallelogram ABCD with its diagonals meeting at the point O is shown in the diagrambelow.

AB is extended to P such that BP = AB. The line that passes through D and is parallel toAC meets PC produced at point R amd ∠CRD = 90◦.

(a) Show that the triangles ABD and BPC are congruent.

(b) Show that ABCD is a rhombus.

(c) Find the ratio CR : PC.

[Answer : (c) 1:2 ]

22

Vectors Lee Kian Keong

12 Vectors

1. If the angle between the vectors a =

(4

8

)and b =

(1

p

)is 135◦, find the value of p.

[Answer : −1

3]

2. A boat is travelling at a speed of 30 knots. A yacht is sailing northwards at a speed of 10knots. At 1300 hours, the boat is 14 nautical miles to the north-east of the yacht.

(a) Determine the direction in which the boat should be travelling in order to intercept theyacht.

(b) At what time does the interception occur?

[Answer : (a) S 58.6◦

W ; (b) 1341 hours ]

3. The position vectors of the points A, B and C, with respect to the origin O, are a, b andc respectively. The points L, M , P and Q are the midpoints of OA, BC, OB, and ACrespectively.

(a) Show that the position vector of any point on the line LM is1

2a +

1

2λ(b + c − a) some

scalar λ, and express the position vector of any point on the line PQ in terms of a, b andc.

(b) Find the position vector of the point of intersection of the line LM and the line PQ.

[Answer : (a)1

2b +

1

2µ(a + c− b) ; (b)

1

4a +

1

4b +

1

4c ]

4. In triangle ABC, the point X divides BC internally in the ratio m : n, where m+ n = 1.

Express AX2 in terms of AB, BC, CA, m and n.

[Answer : nAB2 −mnBC2+mCA

2]

5. Wind is blowing with a speed of w from the direction of N θ◦W. When a ship is cruisingeastwards with a speed of u, the captain of ship found that the wind seem like blowing witha speed of v1, from the direction N α◦W. When the ship is cruising north with a speed of u,the captain of the ship, however found that the wind seemed to be blowing with a speed of v2

from the direction N β◦W.

(a) Draw the triangles of velocity of both situations

(b) Show that tan θ =tanα− 1

1− cotβ

(c) Express v22 − v2

1 in terms of u, w and θ.

[Answer : −2uw(sin θ + cos θ) ]

6. Position vectors of the points P and Q relative to the origin O are 2 i∼

and 3 i∼

+4j∼

respectively.

Find the angle between vector−−→OP and vector

−−→OQ

[Answer : 53.1◦

]

7. The points P , Q, and R are the midpoints of the sides BC, CA and AB respectively of thetriangle ABC. The lines AP and BQ meet at the point G, where AG = mAP and BG = nBQ.

23

Vectors Lee Kian Keong

(a) Show that−→AG =

1

2m−−→AB +

1

2m−→AC and

−→AG = (1− n)

−−→AB +

1

2

−→AC.

Deduce that AG =2

3AP and CG =

2

3CR.

(b) Show that CR meets AP and BQ at G, where CG =2

3CR.

8. A force of magnitude 2p N acts along the line OA abd a force of magnitude 10 N acts alongthe line OB. The angle between OA and OB is 120◦. The resultant force has magnitude

√3p

N. Calculate the value of p and determine the angle between the resultant force and OA.

[Answer : 30◦

]

9. Let u = cosφ i + sinφ j and v = cos θ i + sin θ j, where i and j are perpendicular unit vectors.Show that

1

2|u− v| = sin

1

2(φ− θ)

10. A canal of width 2a has parallel straight banks and the water flows due north. The points Aand 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 theorigin. The speed of the current in the canal, vc, is given by

vc = v0

(1− x2

a2

),

where v0 is the speed of the current in the middle of the canal and x is the distance eastwardsfrom the middle of the canal. A swimmer swims from A towards the east at speed vr relative tothe current in the canal. Taking y to denote the distance northwards travelled by the swimmer,show that

dy

dx=v0

vr

(1− x2

a2

).

If the width of the canal is 12 m, the speed of the current in the middle of the canal is 10 ms−1

and the speed of the swimmer is 2 ms−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 hisdistance from B when he reaches the east bank of the canal,

(b) sketch the actual path taken by the swimmer.

[Answer : (a) 20 m north of O , 40 m north of B ]

11. 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.

[Answer : (b) x = 1 ]

24

Data Description Lee Kian Keong

12. The points P and Q lie on the diagonals BD and DF respectively of a regular hexagonABCDEF such that

BP

BD=DQ

DF= k.

Express−−→CP and

−−→CQ in terms of k, a and b, where

−−→AB = a and

−−→BC = b.

If the points C, P and Q lie on a straight line, determine the value of k. Hence, find CP : CQ.

[Answer : −−→CP = (2k − 1)b− ka ,−−→CQ = (1− k)b− (1 + k)a , k =

1√

3, CP : CQ = 1 :

√3 ]

13. The diagram above shows non-linear points O, A and B, with P on the line OA such thatOP : PA = 2 : 1 and Q on the line AB such that AQ : QB = 2 : 3. The lines PQ and OB

produced meet at the point R. If−→OA = a and

−−→OB = b,

(a) show that−−→PQ = − 1

15a +

2

5b,

(b) find the position vector of R, relative to O, in terms of b

[Answer : (b) 4b ]

13 Data Description

1. The mean and standard deviation of Physics marks for 25 school candidates and 5 privatecandidates are shown in the table below.

School candidates Private candidatesNumber of candidates 25 5Mean 55 40Standard deviation 4 5

Calculate the overall mean and standard deviation of the Physics marks.

[Answer : 52.5 ; 6.98 ]

2. A sample of 100 fuses, nominally rated at 13 amperes, are tested by passing increasing elec-tric current through them. The current at which they blow are recorded and the followingcumulative frequency table is obtained.

25


Currents (amperes) Cumulative frequency<10 0<11 8<12 30<13 63<14 88<15 97<16 99<17 100

Calculate the estimates of the mean, median and mode. Comment on the distribution.

[Answer : 12.65 , 12.61 , 12.58 ; positively skewed ]

3. The number of teenagers, according to age, that patronize a recreation centre for a certainperiod of time is indicated in the following table.

Age in Years Number of teenagers12 - 413 - 1014 - 2715 - 11016 - 21217 - 23818 - 149

[ Age 12 - means age 12 and more but less than 13 years ]

(a) Display the above data using histogram

(b) Find the median and semi-interquartile range for the age of teenagers who patronize therecreation centre. Give your answer to the nearest months.

[Answer : (b) 17 years 1 month , 10 months ]

4. The table below shows the number of defective electronic components per lot for 500 lots thathave been tested.

Numbers of defectivecomponents per lot

0 1 2 3 4 5 6 or more

Relative frequency 0.042 0.054 0.392 0.318 0.148 0.014 0.032

(a) State the mode and the median number of defective electronic components per lot.

(b) For the lots with defective components of more than 5, the mean number of defectivecomponents per lot is 6.4. Find the mean number of defective electronic components perlot for the given 500 lots.

[Answer : (a) 2 , 3 ; (b) 2.7 ]

5. The number of ships which anchor at a port every week for 26 particular weeks are as follows

32 28 43 21 35 19 25 45 35 32 18 26 3026 27 38 42 18 37 50 46 23 40 20 29 46

(a) Display the data in a stemplot

26


(b) Find the median and interquartile range

(c) Draw a boxplot to represent the data

(d) State the shape of the frequency distribution. Give a reason for your answer

[Answer : (b) 31 , 15 ; (d) positively skewed ]

6. Show that, for the numbers x1, x2, x3, . . . , xn with mean x̄,∑(x− x̄)2 =

∑x2 − nx̄2

The numbers 4, 6, 12, 5, 7, 9, 5, 11, p, q, where p < q, have mean x̄ = 6.9 and∑

(x− x̄)2 =

102.9. Calculate the of p and q.

[Answer : 1, 9 ]

7. The following data show the masses, in kg, of fish caught by 22 fishermen on a particular day.

23 48 51 25 39 37 41 38 37 20 8869 22 42 46 23 52 41 40 59 68 59

(a) Display the above data in an ordered stemplot.

(b) Find the mean and standard deviation.

(c) Find the median and interquartile range.

(d) Draw a boxplot to represent the above data.

(e) State whether the mean or the median is more suitable as a representative value of theabove data. Justify your answer.

[Answer : (b) 44 , 16.8 ; (c) 41 , 15 ; (e) median ]

8. The times taken by 22 students to breakfast are shown in the following table.

Time (x minutes) 2 ≤ x < 5 5 ≤ x < 8 8 ≤ x < 11 11 ≤ x < 14 14 ≤ x < 17 17 ≤ x < 20Number of students 1 2 4 8 5 2

(a) Draw a histogram of the grouped data. Comment on the shape of frequency distribution.

(b) Calculate estimates of the mean, median, and mode of the breakfast times. Use yourcalculations to justify your statement about the shape of the frequency distribution.

[Answer : (b) 12.23 , 12.50 , 12.71 ]

9. The mean mark for a group of students taking a statistics test is 70.6. The mean marks formale and female students are 68.5 and 72.0 respectively. Find the ratio of the number of maleto female students.

[Answer : 2:3 ]

10. The masses (in thousands of kg) of solid waste collected from a town for 25 consecutive daysare as follows:

41 53 44 55 48 57 50 38 53 50 43 56 5148 33 46 55 49 50 52 47 39 51 49 52

27

Probability Lee Kian Keong

(a) Construct a stemplot to represent the data.

(b) Find the median and interquartile range.

(c) Calculate the mean and standard deviation.

(d) Draw a boxplot to represent the data.

(e) Comment on the shape of the distribution and give a reason for your answer.

[Answer : (b) 50 , 6 ; (c) 48.4 , 5.97 ]

11. Overexposure to a certain metal dust at the workplace of a factory is detrimental to the healthof its workers. The workplace is considered safe if the level of the metal dust is less than 198µ g m−3. The level of the metal dust at the workplace is recorded at a particular time of dayfor a period of 90 consecutive working days. The results are summarised in the table below.

Metal dust level (µ g m−3) Number of days170 - 174 8175 - 179 11180 - 184 25185 - 189 22190 - 194 15195 - 199 7200 - 201 2

(a) State what the number 11 in the table means.

(b) Calculate estimates of the mean and standard deviation of the levels of the metal dust.

(c) Plot a cumulative frequency curve of the above data. Hence, estimate the median andthe interquartile range.

(d) Find the percentage of days for which the workplace is considered unsafe.

[Answer : (b) 185 , 7.22 ; (c) 184.7 , 9.8 ; (d) 4.44% ]

14 Probability

1. There are 20 doctors and 15 engineers attending a conderence. The number of women doctorsand that of women engineers are 12 and 5 respectively. Four participants from this group areselected randomly to chair some sessions of panel discussion.

(a) Find the probability that three doctors are selected.

(855

2618

)(b) Given that two women are selected, find the probability that both of them are doctors.

(33

68

)

2. In a basket of mangoes and papayas, 70% of mangoes and 60% of papayas are ripe. If 40% ofthe fruits in the basket are mangoes,

(a) find the percentage of the fruits which are ripe, (0.64)

(b) find the percentage of the ripe fruits which are mangoes. (0.4375)

28

Probability Lee Kian Keong

3. Three balls are selected at random from one blue ball, three red balls and six white balls. Find

the probability that all the three balls selected are of the same color.

(7

40

)

4. There are 12 towels, two of which are red. If five towels are chosen at random, find the

probability that at least one is red.

(15

22

)

5. A factory has 36 male workers and 64 female workers, with 10 male workers earning less thanRM1000.00 a month and 17 female workers earning at least RM1000.00 a month. At the endof the year, workers earning less than RM1000.00 are given a bonus of RM1000.00 whereas theothers receive a month’s salary.

(a) If two workers are randomly chosen, find the probability that exacly one worker receivesa bonus of one month’s salary. (0.495)

(b) If a male worker and a demale worker are randomly chosen, find the probability thatexactly one worker receives a bonus of one month’s salary. (0.604)

6. Two transistors are chosen at random from a batch of transistors containing ninety good andten defective ones.

(a) Find the probability that at least one out of the two transistors chosen is defective.(0.1909)

(b) If at least one out of the two transistors chosen is defective, find the probability that bothtransistors are defective. (0.0476)

7. Two archers A and B take turns to shot, with archer A taking the first shot. The probabilities

of A and B hitting the bull’s eye in each shot are1

6and

1

5respectively. Show that the

probability of archer A hitting the bull-eye first is1

2.

8. The probability that it rains in a certain area is1

5. The probability that an accident occurs

at a particular corner of a road in that area is1

20if it rains and

1

50if it does not rain. Find

the probability that it rains if an accident occurs at the corner.

(5

13

)

9. There are eight parking bays in a row at a taxi stand. If one blue taxi, two red taxis and fiveyellow taxis are parked there, find the probability that two red taxis are parked next to eachother.

[Assume that a taxi may be parked at any of the parking bays.]

(1

4

)

29

Discrete Probability Distributions Lee Kian Keong

10. Two events A and B are such that P (A) =3

8, P (B) =

1

4and P (A|B) =

1

6.

(a) Show that the events A and B are neither independent nor mutually exclusive.

(b) Find the probability that at least one of the events A and B occurs.

(7

12

)(c) Find the probability that either one of the events A and B occurs.

(13

24

)

11. A four-digit number, in the range 0000 to 9999 inclusive, is formed. Find the probability that

(a) the number begins or ends with 0, (0.19)

(b) the number contains exactly two non-zero, digits. (0.0486)

15 Discrete Probability Distributions

1. The independent random variable Yi, where i = 1, 2, . . . , n, takes the values of 0 and 1 withthe probabilities of q and p respectively, where q = 1− p.

(a) Show that E(Yi) = p and Var(Yi) = pq.

(b) If X = Y1 + Y2 + . . .+ Yn, determine E(X) and Var(X). Comment on the distribution ofX.

[Answer : (b) np, npq ; binomial distribution ]

2. A discrete random variable X takes the values of 0, 1 and 2 with the probabilities of a, b and

c respectively. Given that E(X) =4

3and Var(X) =

5

9, find the values of a, b and c.

[Answer : a =1

6, b =

1

3,

1

2]

3. The discrete random variable X has the probability function

P (X = x) =

{k(4− x)2, x = 1, 2, 3,

0, otherwise

where k is a constant.

(a) Determine the value of k and tabulate the probability distribution of X.

(b) Find E(7X − 1) and Var(7X − 1).

[Answer : (a)1

14; (b) 9 , 19 ]

4. The probability distribution function of the discrete random variable Y is

P (Y = y) =y

5050, y = 1, 2, 3, . . . , 100

(a) Show that E(Y ) = 67 and find Var(Y ).

(b) Find P (|Y − E(Y )| ≤ 30).

30

Discrete Probability Distributions Lee Kian Keong

[Answer : (a) 561 ; (b)4087

5050]

5. A car rental shop has four cars to be rented out on a daily basis at RM50.00 per car. Theaverage daily demand for cars is four.

(a) Find the probability that, on a particular day,

i. no cars are requested, (0.0183)

ii. at least four requests for cars are received. (0.5665)

(b) Calculate the expected daily income received from the rentals. (160.93)

(c) If the shop wishes to have one more car, the additional cost incurred is RM20.00 per day.Determine whether the shop should buy another car for rental. (YES)

6. Two percent of the bulb produced by a factory are not usable. Find the smallest number ofbulbs that must be examined so that the probability of obtaining at least one non-usable bulbexceeds 0.5.

[Answer : 35 ]

7. The independent Poisson random variables X and Y have parameters 0.5 and 3.5 respectively.The random variable W is defined by W = X − Y .

(a) Find E(W ) and Var(W )

(b) Give one reason why W is not a Poisson random variable.

[Answer : (a) -2 , 4 ]

8. The probability of a person allergic to a type of anaesthetic is 0.002. A total of 2000 personsare injected with the anaesthetic. Using a suitable approximate distribution, calculate theprobability that more than two persons are allergic to the anaesthetic.

[Answer : 0.7619 ]

9. A type of seed is sold in packets which contain ten seeds each. On the average, it is found thata seed per packet does not germinate. Find the probability that a packet chosen at randomcontains less than two seeds which do not germinate.

[Answer : 0.7361 ]

10. The probability that a heart patient survives after surgery in a country is 0.85.

(a) Find the probability that, out of five randomly chosen heard patients undergoing surgery,four survive.

(b) Using a suitable approximate distribution, find the probability that more than 160 surviveafter surgery in a random sample of 200 heart patients.

[Answer : (a) 0.3915 ; (b) 0.97 ]

11. The probability that a lemon sold in a fruit store is rotten is 0.02.

(a) If the lemons in the fruit store are packed in packets, determine the maximum numberof lemons per packet so that the probability that a packet chosen at random does notcontain rotten lemons is more than 0.85.

31

Continuous Probability Distributions Lee Kian Keong

(b) If the lemons in the fruit store are packed in boxed each containing 60 lemons, find usinga suitable approximation, the probability that a box chosen at random contains less thanthree rotten lemons.

[Answer : (a) 8 ; (b) 0.8795 ]

12. A computer accessories distributor obtains its supply of diskettes from manufacturers A and B,with 60% of the diskettes from manufacturer A. The diskettes are packed by the manufacturersin packets of tens. The probability that a diskette produced by manufacturer A is defective is0.05 whereas the probability that a diskette produced by manufacturer B is defective is 0.02.Find the probability that a randomly chosen packet contains exactly one defective diskette.

[Answer : 0.2558 ]

16 Continuous Probability Distributions

1. The time to repair a certain type of machine is a random variable X (in hours). The probabilitydensity function is given by

f(x) =

0.01x− p, 10 ≤ x < 20,

q − 0.01x, 20 ≤ x ≤ 30,

0, otherwise,

where p and q are constants.

(a) Show that p = 0.1 and q = 0.3.

(b) Find the probability that the repair work takes at least 15 hours.

(c) Determine the expected value of X.

(d) If the total cost of repair of the machine comprises a surcharge of RM500 and an hourlyrate of RM100, express the total cost of repair in terms of X, and determine the expectedtotal cost of repair.

[Answer : (b)7

8; (c) 20 hours ; (d) RM 2500 ]

2. The continuous random variable X has probability density function

f(x) =

0, x < 0,5

4− x, 0 ≤ x < 1,

1

4x2, x ≥ 1.

(a) Find the cumulative distribution function of X.

(b) Calculate the probability that at least one of two independent observed values of X isgreater than three.

[Answer : (b)23

144]

3. Continuous random variable X is defined in the interval 0 to 4, with

P (X > x) =

1− ax, 0 ≤ x ≤ 3

b− 1

2x, 3 < x ≤ 4

with a and b as constants,

32


(a) Show that a =1

6and b = 2,

(b) Find the cumulative distribution function of X and sketch its graph

(c) Find the probability density function of X

(d) Calculate the mean and standard deviation of X.

[Answer : (d)5

2, 1.190 ]

33


4. The lifespan of a species of plant is a random variable T (tens of days). The probability densityfunction is given by

f(t) =

1

8e−

18 t, t > 0

0, otherwise

(a) Find the cumulative distribution function of T and sketch its graph.

(b) Find the probability, to three decimal places, that a plant of that species randomly chosenhas a lifespan of more than 20 days.

(c) Calculate the expected lifespan of that species of plant.

[Answer : (b) 0.779 ; (c) 80 days ]

5. The continuous random variable X has the probability density function

f(x) =

4

27x2(3− x), 0 < x < 3,

0, otherwise.

(a) Calculate P

(X <

3

2

).

(b) Find the cumulative distribution function of X.

[Answer : (a)5

16]


f(x) =

√x− 1

12, 1 ≤ x ≤ b

0, otherwise

where b is a constant.

(a) Determine the value of b.

(b) Find the cumulative distribution function of X and sketch its graph.

(c) Calculate E(X).

[Answer : (a) b = 4 ; (c)14

5]


f(x) =

1

25(1− 2x), −2 ≤ x ≤ 1

23

25(2x− 1),

1

2≤ x ≤ 3

0, otherwise.

(a) Sketch the graph of y = f(x)

(b) Given that P (0 ≤ X ≤ k) =13

100, determine the value of k.

[Answer : (b) k =3

2]

34


8. The number of hours spent in a library per week by arts and science students in a college isnormally distributed with mean 12 hours and standard deviation 5 hours for arts students,and mean 15 hours and standard deviation 4 hours for science students.A random sample of four arts students and six science students is chosen. Assuming that Xis the mean number of hours spent by these 10 students in a week,

(a) calculate E(X) and Var(X),

(b) find the probability that in a given week, the mean number of hours spent by this sampleof students is between 11 hours and 15 hours.

[Answer : (a) 13.8 , 1.96; (b) 0.0.7814 ]

9. The random variable X is normally distributed with mean µ and standard deviation 100. Itis known that P (X > 1169) ≤ 0.117 and P (X > 879) ≥ 0.877. Determine the range of thevalues of µ.

[Answer : {µ : 995 ≤ µ ≤ 1050} ]

10. The mass of a small loaf of bread produced in a bakery may be modelled by a normal randomvariable with mean 303 g and standard deviation 4 g. Find the probability that a randomlychosen loaf has a mass between 295 g and 305 g.

[Answer : 0.6687 ]

11. The random variable X has a binomial distribution with parameters n = 500 and p =1

2.

Using a suitable approximate distribution, find P (|X − E(X)| ≤ 25).

[Answer : 0.9774 ]

12. The mass of yellow water melon produced by a farmer is normally distributed with a mean of4 kg and a standard deviation of 800 g. The mass of red water melon produced by the farmeris normally distributed with a mean of 6 kg and a standard deviation of 1 kg.

(a) Find the probability that the mass of a red water melon, selected at random, is less than5 kg. Hence, find the probability that a red water melon with mass less than 5 kg hasmass less than 4 kg.

(b) If Y = M − 2K, where M represents the mass of a red water melon and K the mass ofa yellow water melon, determine the mean and variance of Y .

Assuming that Y is normally distributed, find the probability that the mass of a red watermelon selected at random is more than twice the mass of yellow water melon selected atrandom.

[Answer : (a) 0.1587 , 0.144 ; (b) -2 , 3.56 ; 0.1446 ]

13. Tea bags are labelled as containing 2 g of tea powder. In actual face, the mass of tea powderper bag has mean 2.05 g and standard deviation 0.05 g. Assuming that the mass of tea powderof each bag is normally distributed, calculate the expected number of tea bags which contain1.95 g to 2.10 g of tea powder in a box of 100 tea bags.

[Answer : 82 ]

14. The lifespan of an electrical instrument produced by a manufacturer is normally distributedwith a mean of 72 months and a standard deviation of 15 months.

35


(a) If the manufacturer guarantees that the lifespan of an electrical instrument is at least 36months, calculate the percentage of the electrical instruments which have to be replaced-free of charge.

(b) If the manufacturer specifies that less than 0.1% of the electrical instruments have to bereplacedfree of charge, determine the greatest length of the gurantee period correct to thenearest month.

[Answer : (a) 0.82% ; (b) 25 months ]

36

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STPM Mathematics T Past Year Question