stpm mathematics m

STPM Mathematics S Past Year Questions

Lee Kian Keong & [email protected]

http://www.facebook.com/akeong

Last Edited by December 23, 2012

Abstract

This is a document which shows all the STPM questions from year 2002 to year 2012 using LATEX.Students should use this document as reference and try all the questions if possible. Students areencourage to contact me via email1 or facebook2. Students also encourage to send me your collectionof papers or questions by email because i am collecting various type of papers. All papers are welcomed.Special thanks to Zhu Ming for helping me to check the questions.

Contents

1 PAPER 1 QUESTIONS 2STPM 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3STPM 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5STPM 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7STPM 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9STPM 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11STPM 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13STPM 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15STPM 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17STPM 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19STPM 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21STPM 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 PAPER 2 QUESTIONS 25STPM 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26STPM 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29STPM 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32STPM 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35STPM 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38STPM 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42STPM 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45STPM 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48STPM 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52STPM 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57STPM 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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1

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PAPER 1 QUESTIONS Lee Kian Keong

1 PAPER 1 QUESTIONS

2

PAPER 1 QUESTIONS Lee Kian Keong STPM 2002

STPM 2002

1. The function f is defined by

f : x→√

3x+ 1, x ∈ R, x ≥ −1

3.

Find f−1 and state its domain and range. [4 marks]

2. Given that y = e−x cosx, finddy

dxand

d2y

dx2when x = 0. [4 marks]

3. Determine the values of a, b, and c so that the matrix 2b− 1 a2 b2

2a− 1 a bcb b+ c 2c− 1

is a symmetric matrix. [5 marks]

4. By using suitable substitution, find

∫3x− 1√x+ 1

dx. [5 marks]

5. Determine the set of x such that the geometric series 1 + ex + e2x + . . . converges. Find the exactvalue of x so that the series converges to 2. [6 marks]

6. Express

√59− 24

√6 as p

√2 + q

√3 where p, q are integers. [7 marks]

7. Express1

4k2 − 1as partial fraction. [4 marks]

Hence, find a simple expression for Sn =

n∑k=1

1

4k2 − 1and find lim

n→∞Sn. [4 marks]

8. Given that PQRS is a parallelogram where P (0, 9), Q(2,−5), R(7, 0) and S(a, b) are points on theplane. Find a and b. [4 marks]

Find the shortest distance from P to QR and the area of the parallelogram PQRS. [6 marks]

9. Find the point of intersection of the curves y = −x2 + 3x and y = 2x3−x2− 5x. Sketch on the samecoordinate system these two curves. [5 marks]

Calculate the area of the region bounded by the curve y = −x2 + 3x and y = 2x3 − x2 − 5x.[6 marks]

10. Matrices M and N are given as M =

−10 4 915 −4 −14−5 1 6

, and N =

2 3 44 3 11 2 4

Find MN and deduce N−1. [4 marks]

Products X, Y and Z are assembled from three components A, B and C according to differentproportions. 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, andtwo components of C; each product of Z consists of four components of A, one component of B, andfour components of C. A total of 750 components of A, 1000 components of B, and 500 components

3


of C are used. With x, y and 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]

11. Show that polynomial 2x3 − 9x2 + 3x+ 4 has x− 1 as factor. [2 marks]

Hence,

(a) find all the real roots of 2x6 − 9x4 + 3x2 + 4 = 0. [5 marks]

(b) determine the set of values of x so that 2x3 − 9x2 + 3x+ 4 < 12− 12x. [6 marks]

12. Function f is defined by

f(x) =2x

(x+ 1)(x− 2).

Show that f ′(x) < 0 for all values of x in the domain of f . [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]

4


STPM 2003

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

2. If y = ln√xy, find the value of

dy

dxwhen y = 1. [5 marks]

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

∫ π6

0

cos θ

(3 + 2 sin θ)2dθ. [5 marks]

4. If (x+ iy)2 = i, find all the real values of x and y. [6 marks]

5. Find the set of values of x such that −2 < x3 − 2x2 + x− 2 < 0. [7 marks]


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. [4 marks]

(b) Sketch the graph of f . [3 marks]

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

√5 : 3. [8 marks]

8. Express

(1 + x

1 + 2x

) 12

as a series of ascending powers of x up to the term in x3. [6 marks]

By taking x =1

30, find

√62 correct to four decimal places. [3 marks]

9. 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. [3 marks]

(b) Find (A + I)B, and hence find (A + I)21B. [6 marks]

10. 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. [4 marks]

Calculate the area of the region bounded by 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]

11. 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]. [7 marks]

Use the Newton-Raphson method with the initial estimate x0 = 0.5 to estimate the root correct tothree decimal places. [6 marks]

5


12. Express ur =2

r2 + 2rin partial fractions. [3 marks]

Using the result obtained,

(a) show that u2r = −1

r+

1

r2+

1

r + 2+

1

(r + 2)2, [2 marks]

(b) show that

n∑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

).

[9 marks]

6


STPM 2004

1. Show that

∫ e

1

lnx dx = 1. [4 marks]

2. 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. [5 marks]

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

(A−B) ∪ (B −A) = (A ∪B)− (A ∩B)

[6 marks]

4. Matrix A is given by A =

3 3 45 4 11 2 3

.

Find the adjoint of A. Hence, find A−1. [6 marks]


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. [6 marks]

6. The sum of the 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. [7 marks]

7. Sketch, on the same coordinate axes, the graphs of y = 2− x and y =

∣∣∣∣2 +1

x

∣∣∣∣. [4 marks]

Hence, solve the inequality 2− x >∣∣∣∣2 +

1

x

∣∣∣∣. [4 marks]

8. Using the sketch graphs 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).

[4 marks]

Use the Newton-Raphson method to find the real root correct to three decimal places. [5 marks]

9. 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. [5 marks]

Find the real numbers m and n for which P = mQ+nI, where I is the 3× 3 identity matrix.[5 marks]

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

t. Find the equation of the

7


normal to the curve at the point A(3,−4). [7 marks]

The normal to the curve at the point A cuts the curve again at point B. Find the coordinates of B.[4 marks]

11. Sketch on the same coordinates axes, the line y =1

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

the points of intersection. [5 marks]

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

2x and the curve y2 = x. [4 marks]

Find the volume of the solid formed when the region is rotated through 2π radians about the y-axis.[4 marks]

12. Prove that the sum of the first n terms of a geometric series a+ ar + ar2 + . . . isa(1− rn)

1− r[3 marks]

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

[5 marks]

(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.[7 marks]

8


STPM 2005

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

(A ∩B)′ − (A′ ∩B) = B′

[4 marks]

2. If y =cosx

x, where x 6= 0, show that x

d2y

dx2+ 2

dy

dx+ xy = 0. [4 marks]

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

4. For the geometric series 7+3.5+1.75+0.875+ ..., find the smallest value of n for which the differencebetween the sum of the first n terms and the sum to infinity is less than 0.01. [6 marks]

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

xwhere x 6= 0. [7 marks]

6. Find the perpendicular distance from the centre of the circle x2 + y2−8x+ 2y+ 8 = 0 to the straightline 3x+ 4y = 28. Hence, find the shortest distance between the circle and the straight line. [7 marks]

7. Sketch, on the same coordinate axes, the curves y = ex and y = 2 + 3e−x. [2 marks]

Calculate the area of the region bounded by the y-axis and the curves. [6 marks]

8. A, B and C are square matrices such that BA = B−1 and ABC = (AB)−1. Show that A−1 =B2 = C. [3 marks]

If B =

1 2 00 −1 01 0 1

, find C and A. [7 marks]

9. 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. [6 marks]

(b) Represent z1 and z2 in an Argand diagram. [1 marks]

(c) For each of z1 and z2, find the modulus, and the argument in radians. [4 marks]

10. The functions f and g are given by

f(x) =ex − e−x

ex + e−xand g(x) =

2

ex + e−x

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

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

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

11. Express f(x) =x2 − x− 1

(x+ 2)(x− 3)in partial fractions. [5 marks]

Hence, obtain an expansion of f(x) in ascending powers of1

xup to the term in

1

x3. [6 marks]

9


Determine the set of values of x for which this expansion is valid. [2 marks]

12. Find the coordinates 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 stationary pointis a minimum point or a maximum point. [5 marks]

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. Use the Newton-Raphson method to determine the value of p correct tothree decimal places and, hence, find the point of intersection. [9 marks]

10


STPM 2006

1. If A, B and C are arbitrary sets, show that [(A ∪B)− (B ∪ C)] ∩ (A ∪ C)′ = ∅. [4 marks]

2. 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).

[4 marks]

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

k 1 32k + 1 −3 2

0 k 2

is 0.[4 marks]

4. Using trapezium rule, with five ordinates, evaluate

∫ 1

0

√4− x2 dx. [4 marks]

5. 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. [6 marks]

6. Express2x+ 1

(x2 + 1)(2− x)in the form

Ax+B

x2 + 1+

C

2− xwhere A, B and C are constants. [3 marks]

Hence, evaluate

∫ 1

0

2x+ 1

(x2 + 1)(2− x)dx. [4 marks]

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

2(−2)2(

10−Tn17 ) is the nth term of

a geometric progression. [4 marks]

If Tn =1

2(17n− 14), evaluate

∞∑n=1

Un. [4 marks]

8. Show that x2 + y2 − 2ax − 2by + c = 0 is the equation of the circle with centre (a, b) and radius√a2 + b2 − c. [3 marks]

C1

C2

C3

The above figure shows three circles C1, C2 and C3 touching one another, where their centres lie on astraight line. If C1 and C2 have equations x2 +y2−10x−4y+28 = 0 and x2 +y2−16x+4y+52 = 0respectively. Find the equation of C3. [7 marks]

11


9. 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. [2 marks]

(b) Find the composite function g ◦ f and state its domain and range. [5 marks]

(c) State the domain and range of h. [2 marks]

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

10. The polynomial p(x) = x4 + ax3 − 7x2 − 4ax+ b has a factor x+ 3 and when divided by x− 3, hasremainder 60. Find the values of a and b and factorise p(x) completely. [9 marks]

Using the substitution y =1

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

11. 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. [8 marks]

Two groups of workers have their drinks at a stall. The first group comprising ten workers have fivecups of tea, two cups of coffee and three glasses of fruit juice at a total cost of RM11.80. The secondgroup of six workers have three cups of tea, a cup of coffee and two glasses of fruit juice at a totalcost of RM7.10. The cost of a cup of tea and three glasses of fruit juice is the same as the cost offour 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. Hencedetermine the cost of each drink. [6 marks]

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

4ekt + 1where k is a positive constant, t > 0,

(a) Find the value of f(0) [1 marks]

(b) Show that f ′(t) > 0 [5 marks]

(c) Show that k[1− f(t)2] = 2f ′(t) and, hence, show that f ′′(t) < 0. [6 marks]

(d) Find limt→∞

f(t). [2 marks]

(e) Sketch the graph of f . [2 marks]

12


STPM 2007

1. Express the infinite recurring decimal 0.725 (= 0.7252525 . . . ) as a fraction in its lowest terms.[4 marks]

2. If y =x

1 + x2, show that x2

dy

dx= (1− x2)y2. [[ marks]4

3. If loga

( xa2

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

4. Simplify

(a)(√

7−√

3)2

2(√

7 +√

3), [3 marks]

(b)2(1 + 3i)

(1− 3i)2, where i =

√−1. [3 marks]

5. 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 locus of P is also acircle. Find the coordinates of the centre and radius of the circle. [6 marks]

6. Find

(a)

∫x2 + x+ 2

x2 + 2dx, [3 marks]

(b)

∫x

ex+1dx. [4 marks]

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

3x2 + 5x

(1− x2)(1 + x)2=

A

1− x+

B

1 + x+

C

(1 + x)2+

D

(1 + x)3.

[8 marks]


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). [4 marks]

(b) Determine whether f is continuous at x = −1 and x = 1. [4 marks]

9. 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. [5 marks]

13


Hence, solve the system of linear equations

x − 2y − z = −8,3x − y − 4z = −15,

y + 2z = 4.

[5 marks]

10. 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, [4 marks]

(b) sketch the curve, [2 marks]

(c) calculate the area of the region bounded by the curve and the x-axis. [5 marks]

11. Using the substitution y = x+1

x, express f(x) = x3−4x−6− 4

x+

1

x3as a polynomial in y. [3 marks]

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

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

x2 − 1and determine their nature.

[10 marks]

Sketch the curve. [4 marks]

Determine the number of real roots of the equation x3 = k(x2 − 1), where k ∈ R, when k varies.[3 marks]

14


STPM 2008

1. 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. [4 marks]

2. Show that

∫ 3

2

(x− 2)2

x2dx =

5

3+ 4 ln

(2

3

). [4 marks]

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

A ∩ (B ∪ C) ⊂ (A ∩B) ∪ (A ∩ C)

[5 marks]

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

1− z. [6 marks]

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

2x− k has factor (x+ 1).

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

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

6. If y =sinx− cosx

sinx+ cosx, show that

d2y

dx2= 2y

dy

dx. [6 marks]

7. 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. [4 marks]

(b) Find matrix B which satisfies BA =

1 4 30 2 1−1 0 2

. [4 marks]

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

(a) Find the coordinates of A and B. [4 marks]

(b) Calculate the perpendicular distance of A to OB, where O is the origin. [2 marks]

(c) Find the area of the OAB triangle. [3 marks]

9. Find the solution set of the inequality

∣∣∣∣ 4

x− 1

∣∣∣∣ > 3− 3

x. [10 marks]

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

x2 − 1is always decreasing. [3 marks]

15


Determine the coordinates of the point of inflexion of the curve, and state the intervals for which thecurve is concave upwards. [5 marks]

Sketch the curve. [3 marks]

11. Sketch, on the same coordinate axes, the curves y = 6 − ex and y = 5e−x, and find the coordinatesof 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 thex-axis. [5 marks]

12. At the beginning of this year, Mr. Liu and Miss Dora deposited RM10 000 and RM2000 respectivelyin a bank. They receive an interest of 4% per annum. Mr Liu does not make any additional depositnor withdrawal, whereas, Miss Dora continues to deposit RM2000 at the beginning of each of thesubsequent years without any withdrawal.

(a) Calculate the total savings of Mr. Liu at the end of n-th year. [3 marks]

(b) Calculate the total savings of Miss Dora at the end of n-th year. [7 marks]

(c) Determine in which year the total savings of Miss Dora exceeds the total savings of Mr. Liu.[5 marks]

16


STPM 2009

1. Determine the set of values of x satisfying the inequalityx

x+ 1≥ 1

x+ 1. [4 marks]

2. Given x > 0 and f(x) =√x, find lim

h→0

f(x)− f(x+ h)

h. [4 marks]

3. For the 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 the first n terms is less than45

64. [6 marks]

4. The line y+x+ 3 = 0 is a tangent to the curve y = px2 + qx, where p 6= 0 at the point x = −1. Findthe values of p and q. [6 marks]

5. Given that

loga(3x− 4a) + loga 3x =2

log2 a+ loga(1− 2a),

where 0 < a <1

2, find x. [7 marks]

6. Using an appropriate substitution, evaluate

∫ 1

0

x2(1− x)13 dx. [7 marks]

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, [2 marks]

(b) Find the Cartesian equation of line l, [2 marks]

(c) Given that line l cuts the x and y-axes at P and Q respectively, find the ratio PA : AQ.[4 marks]

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

9. (a) The matrices 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 the matrices PQ and PQR and hence, deduce (PQ)−1. [5 marks]

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

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

. [5 marks]

10. A curve is defined by the parametric equations

x = t− 2

tand y = 2t+

1

t

where t 6= 0.

17


(a) Show thatdy

dx= 2− 5

t2 + 2, and hence, deduce that −1

2<dy

dx< 2. [8 marks]

(b) Find the coordinates of points whendy

dx=

1

3. [3 marks]

11. 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, [2 marks]

(b) determine the coordinate of their points of intersection, [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. [5 marks]

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

(a) Find the values of a and b, and hence, factorise p(x) completely. [7 marks]

(b) Give that p(x) = (2x−3)[q(x)−41 + 3x3], find q(x), and determine its range when x ∈ [−2, 10].[8 marks]

18


STPM 2010

1. Solve the following simultaneous equations:

log3(xy) = 5 and log9

(x2

y

)= 2.

[4 marks]

2. Given that u =1

2(ex+e−x), where x > 0 and y = f(u) is a differentiable function f . If

dy

du=

1√u2 − 1

,

show thatdy

dx= 1. [5 marks]

3. 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. [6 marks]

4. Given that f(x) = x lnx, where x > 0. Find f ′(x), and hence, determine the value of

∫ 2e

e

lnx dx.

[6 marks]

5. Let A−B denotes a set of elements which belongs to set A, but does not belong to set B. Withoutusing Venn diagram, show that A−B = A ∩B′. [3 marks]

Hence, prove that (A ∪B′)− (B ∩ C) = B′ ∪ (A− C). [4 marks]

6. The graph of a function f is as follows:

(a) State the domain and range of f . [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]

7. The polynomial p(x) = 2x4 − 7x3 + 5x2 + ax + b, where a and b are real constants, is divisible by2x2 + x− 1.

(a) Find a and b. [4 marks]

(b) For these values of a and b, determine the set of values of x such that p(x) ≤ 0. [4 marks]

8. Given f(x) =x3 − 3x− 4

(x− 1)(x2 + 1),

19


(a) find the constants A, B, C and D such that f(x) = A+B

x− 1+Cx+D

x2 + 1, [5 marks]

(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]

9. Sketch, on the same coordinate axes, the graphs y = e−x and y =4

2− x. Show that the equation

x+ 4ex = 2 has a root in the interval [-1,0]. [6 marks]

Estimate the root correct to three decimal places by using Newton-Raphson method with initialestimate x0 = −0.4. [5 marks]

10. A circle C1 passes through the points (-6, 0), (2, 0) and (-2, 8).

(a) Find the equation of C1. [4 marks]

(b) Determine the coordinates of the centre and the radius of C1. [2 marks]

(c) If C2 is the circle (x− 4)2 + (y − 11)2 = 25,

i. find the distance between the centres of the two circles, [2 marks]

ii. find the coordinates of the point of intersection of C1 with C2. [3 marks]

11. 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]

(c) On a separate diagram, sketch the graph of y =1

h(x). [3 marks]

Hence, determine the set of values of k such that the equation1

h(x)= k has

i. one root, [1 marks]

ii. two roots, [1 marks]

iii. three roots. [1 marks]

12. Matrix P is given by P =

1 2 12 1 32 −1 −1

.

(a) Find the determinant and adjoint of P. Hence, find P−1. [6 marks]

(b) A factory assembles three types of toys Q, R and S. The total time taken to assemble one unitof R and one unit of S exceeds the time taken to assemble two units of Q by 8 minutes. Oneunit of Q, two units of R and one unit of S take 31 minutes to be assembled. The time takento 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 Srespectively,

i. write a system of linear equations to represent the above information, [2 marks]

ii. using the results in (a), determine the time taken to assemble each type of toy. [5 marks]

20


STPM 2011

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

2. Show that

n∑r=1

r2 + r − 1

r2 + r=

n2

n+ 1. [4 marks]

3. Use the substitution u = lnx, evaluate

∫ e

1

(x+ 1) lnx

x2dx. [6 marks]

4. Find the set of values of x satisfying the inequality 2x− 1 ≤ |x+ 1|. [6 marks]

5. Given that y is differentiable and y√x = sinx, where x 6= 0. Using implicit differentiation, show that

x2d2y

dx2+ x

dy

dx+

(x2 − 1

4

)y = 0.

[6 marks]

6. The lines l1 : y = mx+ a and l2 : y = − 1

mx+ b, where m 6= O and b > a > 0, intersect at R.

(a) Find the coordinates of R in terms of a, b and m. [2 marks]

(b) The line l1 cuts the y-axis at P and the line l2 cuts the x-axis at Q. If m = 1, find, in termsof a and b, the perpendicular distance from R to line PQ, and determine the area of trianglePQR. [5 marks]

7. 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. [3 marks]

(b) Find the modulus and argument of z. [3 marks]

(c) Represent z and its conjugate in an Argand diagram. [3 marks]

8. Differentiate ex2

with respect to x.

Hence, determine integers a, b and c for which∫ 2

1

x3ex2

dx =a

bec.

[9 marks]

21


9. Functions f and g are defined by

f : x→ x

2x− 1for x 6= 1

2;

g : x→ ax2 + bx+ c, where a, b and c are constants.

(a) Find f ◦ f , and hence, determine the inverse function of f . [4 marks]

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

(2x− 1)2. [4 marks]

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

2x2 − 5in terms of f and p. [2 marks]

10. A and B are two matrices such that

A =

−4 −3 6−2 −2 42 2 −3

and A2B =

−2 6 02 0 40 4 2

.

(a) Find the determinant and adjoint of A. Hence, determine A−1. [6 marks]

(b) Using A−1 obtained in (a), find B. [4 marks]

11. The polynomial p(x) = ax3 + bx2 − 4x+ 3, where a and b are constants, has a factor (x+ 1). Whenp(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]

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

x− 3≥ 0. [4 marks]

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

x− 3, x 6= 3, and the value of x at which it

occurs. [4 marks]


f(x) =ln 2x

x2, where x > 0.

(a) State all asymptotes of f . [2 marks]

(b) Find the stationary point of f , and determine its nature. [6 marks]

(c) Obtain the intervals, where

i. f is concave upwards, and

ii. f is concave downwards.

Hence, determine the coordinates of the point of inflexion. [6 marks]

(d) Sketch the graph y = f(x). [2 marks]

22


STPM 2012

1. The sum of the first n terms of a progression 3n2. Determine the n-th term of the progression, andhence, deduce thetype of progression. [4 marks]

2. Given that y = (2x)2x, finddy

dxin terms of x. [4 marks]

3. Differentiate tanx with respect to x, and hence, show that∫ π3

0

x sec2 xdx =π√3− ln 2.

[6 marks]

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

5. Matrix A is given by

A =

1 x 1−1 −1 01 0 0

and A2 = A−1. Determine the value of x. [7 marks]

6. Functions f and g ◦ f are defined by f(x) = ex+2 and (g ◦ f)(x) =√x, for all x ≥ 0.

(a) Find the function g, and state its domain. [5 marks]

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

7. Solve the simultaneous equations log9

(x

y

)=

3

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

8. Express in partial fractions3

(3r − 1)(3r + 2).

[4 marks]

Show thatn∑

r=1

3

(3r − 1)(3r + 2)=

1

2− 1

(3n+ 2),

[2 marks]

and hence, find∞∑r=1

1

(3r − 1)(3r + 2).

[2 marks]


f(x) =e−x√1 + x2

, where x ∈ R,

23


(a) Show that

f ′(x) =−e−x(x2 + x+ 1)

(1 + x)32

.

[3 marks]

(b) Show that f is a decreasing function. [4 marks]

(c) Sketch the graph of f . [2 marks]


f : x→ x2 − x, for x ≥ 1

2.

(a) Find f−1, and state its domain. [4 marks]

(b) Find the coordinates of the point of intersection of graph f and f−1. [3 marks]

(c) Sketch, on the same coordinates axes, the graph of f and f−1. [3 marks]

11. A straight line 2x+ y = 1 intersects an ellipse 4x2 + y2 = 5 at points A and B.

(a) Find the coordinates of points A and B. [4 marks]

(b) The tangent to the ellipse at points A and B intersect at a point C. Find the coordinates ofpoint C. [7 marks]

(c) Find the shortest distance from point C to the line AB. [4 marks]

12. Given that z2 =2i

(1 + 3i)2.

(a) Find the real and imaginary parts of z2. Hence, obtain z1 and z2 which satisfy the aboveequation. [10 marks]

(b) Given that z1 and z3 are roots of 5x2 + ax+ b = 0, where a and b are integers.

i. Find the values of a and b. [3 marks]

ii. Determine z3 and deduce the relationship between z1 and z3. [3 marks]

24

PAPER 2 QUESTIONS Lee Kian Keong

2 PAPER 2 QUESTIONS

25


STPM 2002

1. The discrete random variableX can only take the values 1, 3, 5 and 9, with probabilities: P(X=1)=0.2,P(X=3)=0.3, P(X=5)=0.4, and P(X=9)=0.1. Find E(X) and Var(X). [4 marks]

2. The number of hand phones that are sold in a week by 15 representatives in a town is as follows:

5 10 8 7 25 12 5 14 11 10 21 9 8 11 18

(a) Find the median, lower quartile, and upper quartile for this distribution. [2 marks]

(b) Draw a box plot to represent the data. [3 marks]

3. In a university, 48% of the students are females and 17.5% of the students are taking businessprograms. 4.7% of the university students are female students who study business programs. Astudent is selected randomly. Events A and B are defined as follows:

A: A female student of the university is selected.B: A student of the university, who studies the business program is selected.

(a) Determine whether A and B are mutually exclusive and whether A and B are independent.[3 marks]

(b) Find P(A|B). [2 marks]

4. The height of a certain type of mustard is distributed normally with mean 21.5 cm and variance 90cm2. 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]

5. The following table indicates the price and the quantity sold in a year for three types of drink in adistrict.

DrinkYear 1997 Year 1998 Year 1999

Price Quantity Price Quantity Price Quantity(sen) (thousand cans) (sen) (thousand cans) (sen) (thousand cans)

Tea 80 15 120 12 160 10Coffee 150 3 170 3 180 4Chocolate 220 1 230 3 240 5

Using 1997 as the base year, calculate

(a) Weighted price indices for years 1998 and 1999 with base year quantities as the weights.[3 marks]

(b) Weighted quantity indices for years 1998 and 1999 with base year prices as the weights.[3 marks]

6. The following table indicates the I.Q. levels of eight pairs of fathers and eldest children, in an I.Q.test.

I.Q. level of father 90 98 102 103 104 105 110 114I.Q. level of eldest child 100 95 114 116 98 99 112 106

Find the Pearson’s correlation coefficient for the above data. Explain the result that you obtained.[7 marks]

26


7. The table below shows the number of students for a certain program according to terms in a collegefrom 1998 to 2001.

YearNumber of students

First term Second term Third term1998 20 32 621999 21 42 752000 23 39 772001 27 39 92

(a) Calculate the three-terms moving averages for the data above. [2 marks]

(b) Plot the actual data and the moving averages on the same axes. [4 marks]

(c) Give a summary regarding the basic trend and the seasonal variation. [2 marks]

8. In a survey of 500 motorists on a certain highway, it is found that 120 of them have exceeded thespeed limit.

(a) Obtain a 95% confidence interval for the proportion of motorists who have exceeded the speedlimit on the highway. [5 marks]

(b) Determine the smallest sample size which should be surveyed so that the error of estimation isnot more than 0.04 at the 90% confidence level. [5 marks]

9. The following table indicates the age, x years and the price, RMy × 103, for eight cars of the samemodel in a second-hand car shop.

x 2 8 3 9 6 5 6 3y 100 15 72 16 36 34 30 68

(a) Find the equation of the regression line y on x in the form y = a+ bx, where a and b is accurateto two decimal places. [7 marks]

(b) Explain the value a and b that you obtained. [2 marks]

(c) Estimate the price of a car of the same model of age 7 years. [2 marks]

10. A project on setting up a student-registration system of a college involves seven activities. Theactivities and their duration times (in days) are listed as follows:

Activity Preceding Activities Duration (in days)A - 4B - 2C - 3D A 8E B 6F C 3G D, E 4

(a) Draw a network diagram for the project. [3 marks]

(b) Determine the minimum duration for the project to be completed. [5 marks]

(c) Calculate the total float for each activity and state the critical path of the project. [3 marks]

11. The frequency distribution of the final examination marks for statistics course at an institute ofhigher learning is as follows.

27


Marks Number of students10-29 630-39 740-49 1250-59 1960-69 1570-79 1380-99 8

(a) Plot a histogram for the above data. [3 marks]

(b) Plot a cumulative frequency curve. Hence, estimate the median, semi-quartile range, and thepercentage of students who obtained 45 to 70 marks. [10 marks]

12. A factory produces two types of products, A and B. Each unit of product A requires 2 labour hoursand 1 machine hour, whereas each unit of product B requires 2 labour hours and 4 machine hours.There are not more than 120 labour hours and not more than 96 machine hours available in thefactory each day. The factory also decided that the number of units of product B produced each dayshould not be more than 60% of the total daily production of both products A and B. The profit foreach unit of A is RM120 and each unit of B is RM200. The factory intends to maximize the totalprofit each day.

Formulate the problem as a linear programming problem. [6 marks]

By using graphical method, determine the number of units of product A and product B that shouldbe produced daily in order to maximize the total profit, and find the maximum total daily profit.

[9 marks]

28


STPM 2003

1. The mean and standard deviation of the sleeping period of a sample of 100 students chosen at randomin 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 theschool. [3 marks]

(b) Estimate the standard error of the mean. [1 marks]

2. The marketing manager of a car trading company wishes to predict the delivery period, y months, ofa car model based on the number of accessories, x, chosen by customers. The following table showsthe results obtained from a random sample of 10 cars booked by customers.

x 3 4 4 7 7 8 9 11 12 12y 25 32 26 38 34 41 39 46 44 51

(a) Plot a scatter diagram for the above data. [2 marks]

(b) Comment on the relationship between x and y. [2 marks]

3. The time taken by 50 customers to browse through books in a bookshop is shown in the histogrambelow.

(a) State the modal class. [1 marks]

(b) Calculate the mean time taken by the customers to browse through books in the bookshop.[2 marks]

(c) If 25% of the customers take more than x minutes to browse through books in the bookshop,determine the value of x. [3 marks]

4. The probability distribution of a random variable X is given by P(X = 0) = P(X = 2) = 3k, P(X=l)=P(X=3)=2k,and P(X ≥ 4)=0.

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

(b) If Y = 2X + 3, find the probability distribution of Y , and hence find the expected value of Y .[4 marks]

5. Let Pearson’s correlation coefficient between variables x and y for a random sample be r.

(a) What does r measure? [1 marks]

29


(b) State the range of the possible values of r. [1 marks]

(c) What is the effect of change in the unit of measurement of either variable on the value of r?[1 marks]

A sample of ten data points may be summarised as follows:∑(x− x)2 = 600.1,

∑(y − y)2 = 444.4,

∑(x− x)(y − y) = 466.2.

Calculate Pearson’s correlation coefficient between x and y. Comment on your answer. [3 marks]

6. The following table, based on a survey, shows the numbers of male and female viewers who prefereither documentary or drama programmes on television.

Documentary DramaMale 96 24Female 45 85

A television viewer involved in the survey is selected at random. A is the event that a female vieweris selected, and B is the event that a viewer prefers documentary programmes.

(a) Find P(A ∩B) and P(A ∪B). [4 marks]

(b) Determine whether A and B are independent and whether A and B are mutually exclusive.[3 marks]

7. The table below shows the prices of fish and the quantities of it bought by a housewife at a marketin the first week of January and September of 2000.

FishJanuary September

Price Quantity Price Quantity(RM per kg) (kg) (RM per kg) (kg)

Parang 11.00 2 12.00 1Tenggiri 12.00 2 13.00 2Bawal Putih 10.00 2 a 1Kembung 8.00 3 10.00 2Selar Kuning 4.00 5 5.00 6

(a) If the simple aggregate price index increases by 20% from January to September, determine thevalue of a. [3 marks]

(b) Calculate the Laspeyres price index, and comment on the housewife’s change in expenditure onfish. [3 marks]

(c) Calculate the Paasche quantity index, and comment on the housewife’s change in expenditureon fish. [3 marks]

8. Three companies X, Y , and Z offer taxi services in a town. The percentages of residents in thetown using the taxi services from companies X, Y and Z are 40%, 50%, and 10% respectively. Theprobabilities of taxis from companies X, Y , and Z being late are 0.09, 0.06, and 0.20 respectively. Ataxi is booked at random. Find the probability that

(a) the taxi is from company X and is not late, [4 marks]

(b) the taxi is from company Y given that it is late. [6 marks]

9. A training programme for young managers involves seven activities. The activities and the durationfor each activity are shown in the table below.

30


Activity Preceding activities Duration (days)A - 2B - 5C - 1D B 10E A, D 3F C 6G E, F 8

(a) Draw the network diagram for the training programme. [3 marks]

(b) Determine the critical activities, and find the minimum time needed to complete the trainingprogramme. [8 marks]

10. The time taken by the customers of a company to settle invoices is normally distributed with mean20 days and standard deviation 5 days. A discount is given for every invoice which is settled in lessthan 12 days.

(a) Find the probability that an invoice is settled in less than 12 days. [2 marks]

(b) Find the probability that an invoice is settled in 18 to 26 days. [3 marks]

(c) Determine, out of 200 invoices, the expected number of invoices which are given discounts.[2 marks]

(d) Find the probability that at most 2 out of 10 invoices are given discounts. [4 marks]

11. A manufacturer of wooden furniture produces two types of furniture: chairs and tables. Two machinesare used in the production: a jigsaw and a lathe. Each chair requires 1 hour on the jigsaw and 1 houron the lathe, whereas each table requires 1 hour on the jigsaw and 2 hours on the lathe. The jigsawand lathe can operate 10 hours and 12 hours per day respectively. The profit made is RM27.00 on achair and RM48.00 on a table. The daily profit is to be maximised.

(a) Formulate the problem as a linear programming problem. [4 marks]

(b) Using the simplex method, find the maximum daily profit and the numbers of chairs and tablesmade which give this profit. [9 marks]

12. A survey carried out by a manufacturer of decorative lamps finds that 136 out of 400 shops sell thedecorative lamps at prices less than the recommended prices.

(a) Find the 90% symmetric confidence interval for the proportion of shops selling the decorativelamps at prices less than the recommended prices. [5 marks]

(b) Determine the smallest sample size required so that the estimated proportion of shops sellingthe decorative lamps at prices less than the recommended prices is within 2% of the actualproportion at the 90% confidence level. [5 marks]

(c) Calculate the probability that more than 60% of a random sample of 500 shops sell the decorativelamps at prices less than the recommended prices. [3 marks]

31


STPM 2004

1. According to a survey conducted in a company on job satisfaction, salary and pension benefits are twoimportant issues. It is found that 74% of the employees are of the opinion that salary is importantwhereas 65% think that pension benefits are important. Among those who think that pension benefitsare important, 60% think that salary is also important. Determine the percentage of employees whoare of the opinion that salary and pension benefits are important. [3 marks]

2. A factory receives its supply of raw materials in packages. The mass of each package is normallydistributed with mean 300 kg and standard deviation 5 kg. A random sample of four packages isselected. Find the probability that the mean mass of the sample lies between 292 kg and 296 kg.

[4 marks]

3. The table below shows the marks for the mid-semester examination, x, and the marks for the finalsemester examination, y, of 10 students.

x 70 58 81 66 85 92 54 93 75 65y 85 66 78 76 93 81 65 95 80 70

(a) Plot a scatter diagram of the above data. [2 marks]

(b) What conclusion can be made from your scatter diagram? [2 marks]

4. The stemplot below shows the driving experience (in thousand km) of 15 express bus drivers.

0 3 6 91 2 8 5 1 0 52 5 1 63 84 156 2

Key: 6‖2 means 62 000 km

(a) Find the median, the first quartile and the third quartile. [2 marks]

(b) Draw a boxplot to represent the data. [3 marks]

5. A courier service company claims that 95% of the letters sent using its service reach their destinationswithin a day. If six letters are randomly chosen, find

(a) the probability that at least two letters take more than a day to reach their destinations,[4 marks]

(b) the mean and variance of the number of letters that reach their destinations within a day.[2 marks]

6. The following table shows the annual incomes (RM x × 103) and the amounts of insurance purchased(RM y × 103) for seven persons.

x 50 64 35 58 45 34 74y 185 231 165 193 213 160 198

Calculate the Pearson correlation coefficient of the above data. [6 marks]

7. The following table shows the activities involved in a particular project.

32


Activity Preceding activities Duration (days) Earliest start Latest startA - 5 0 5B - 1 0 5C B 2 1 7D A, C 4 5 11E A 6 5 11F D, E 3 11 14

(a) Draw an activity network for the project. [3 marks]

(b) Calculate the total float and free float of each activity. Hence, determine the critical path of theproject. [7 marks]

8. The ages of 75 persons under the age of 50 years are shown in the table below.

Age at last birthday (in years) 0 - 9 10 - 14 15 - 19 20 - 24 25 - 29 30 - 49Number of persons 3 6 15 25 16 10

(a) Calculate the mode and the median. [4 marks]

(b) Calculate the percentage of persons whose ages are between 18 and 35 years. [6 marks]

9. The letters in the word BANANA are to be rearranged. A word can be considered formed withoutbeing meaningful. The events R, S and T are defined as follows.

R: The word starts and ends with an A.

S: All the N’s in the word are kept together.

T : All the A’s in the word are kept together.

(a) Find P(R), P(S) and P(T ). [5 marks]

(b) Find P(R ∩ S), P(R ∪ S), P(R ∩ T ) and P(R ∪ T ). [5 marks]

10. A telecommunications company wants to estimate the proportion of customers who require an addi-tional line. A random sample of 500 customers is taken and it is found that 135 customers requirean additional line.

(a) Obtain the 99% symmetric confidence interval for the proportion of customers who require anadditional line. Interpret the confidence interval obtained. [6 marks]

(b) If the company wants to estimate the proportion of customers who require an additional line ata different location, determine the smallest sample size required so that the error of estimationdoes not exceed 0.03 at the 95% confidence level. [5 marks]

11. A chocolate manufacturer produces two types of chocolate bars of orange and strawberry flavours.It costs RM0.22 to produce a 10 g orange-flavoured chocolate bar which is sold at RM0.35, whereasit costs RM0.40 to produce a 15 g strawberry-flavoured chocolate bar which is sold at RM0.55. Themanufacturer has 426 kg of chocolate in stock. A minimum of 10 000 orange-flavoured chocolate barsand 12 000 strawberry-flavoured chocolate bars have to be produced. The number of strawberry-flavoured chocolate bars produced has to be more than that of orange-flavoured chocolate bars.

(a) Formulate a linear programming model that can be used to determine the number of chocolatebars of each flavour that should be produced in order to maximise total profit. [6 marks]

(b) Show the feasible region and hence solve the linear programming problem using the graphicalmethod. [9 marks]

12. The table below shows the quarterly gas consumption (in thousand cubic metres) of a factory fromthe year 1999 to 2002.

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YearGas consumption

Quarter 1 Quarter 2 Quarter 3 Quarter 41999 87 68 62 822000 86 70 60 812001 83 72 59 822002 90 70 64 84

(a) Plot the time series and comment on the appropriateness of a linear trend. [4 marks]

(b) Calculate the centred four-quarter moving averages for this times series. [4 marks]

(c) Calculate the seasonal variations using an additive model. [4 marks]

(d) Forecast the gas consumption for the first quarter of the year 2003. [4 marks]

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STPM 2005

1. Find the median and the interquartile range of the following numbers.

21 8 17 22 19 10 29 6 6 21 20 12 18 25

[3 marks]

2. The values of x and y for a set of bivariate data are given in the table below.

x 21.8 22.2 22.3 22.3 22.5 22.6 23.3 23.6y 16.6 16.7 16.4 16.5 16.8 17.0 16.8 17.1

Plot the data on a scatter diagram and state the relationship between x and y. [3 marks]

3. The time taken by year 1 pupils of a school to complete a task is normally distributed with mean µand standard deviation 5.1 minutes.

(a) Given that 97.5% of the pupils require less than 40 minutes to complete the task, find the valueof µ. [3 marks]

(b) Find the probability that a pupil chosen at random takes more than 35 minutes to complete thetask. [2 marks]

4. A study is conducted to assess the impact of the size of a store, x (in m2) on daily sales, y (inRM). A random sample of six stores is taken from several shopping centres. The data obtained aresummarised as follows.∑

x = 24400,∑

y = 28368,∑

(x− x)(y − y) = 6780,∑(x− x)2 = 186333,

∑(y − y)2 = 130110.

Calculate the coefficient of determination. Comment on your answer. [3 marks]

The study also assesses the impact of the size of a shopping centre on daily sales and finds that thecoefficient of determination is 0.674. State whether the size of a store or the size of a shopping centreis more suitable to be used to predict daily sales. Give a reason for your answer. [2 marks]

5. A marketing research firm believes that 40% of the subscribers of a magazine will participate in acompetition held by the magazine. A preliminary survey of 100 subscribers is conducted to find outtheir participation in the competition.

(a) Determine the sampling distribution of the proportion of the subscribers who will participate inthe 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]

6. The table below shows the prices and quantities of vegetables sold in a day at a market in Novemberand December 2003.

Type of vegetablesNovember 2003 December 2003

Price Quantity Price Quantity(RM per kg) (kg) (RM per kg) (kg)

Spinach 3.00 250 4.00 200Water spinach 2.00 130 3.00 150Cabbage 6.00 50 5.00 100Red chilli 10.00 20 8.00 20

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(a) Taking November 2003 as the base period, calculate

i. a simple aggregate price index for December 2003, [2 marks]

ii. a weighted price index for December 2003. [2 marks]

(b) Which of the two price indices in (a) is more suitable to measure the changes in prices of thevegetables? Based on your answer, comment on the changes in prices of the vegetables fromNovember 2003 to December 2003. [2 marks]

7. The number of requests, X, received by a company to deliver pianos in a day is a discrete randomvariable having probability distribution function

P(X = x) =

2k2, x = 0, 3,

kx, x = 1, 2,

0, otherwise.

(a) Determine the value of the constant k and construct a probability distribution table for X.[4 marks]

(b) Find the probability that the company receives at least two requests in a day. [2 marks]

(c) Find the expected number of requests per day. [2 marks]

8. The probability that an employee of a company is late for work is 0.15 in any working day and 0.35if it rains. The probability that it rains is 0.24. Calculate

(a) the probability that it rains and the employee is late, [2 marks]

(b) the probability that it rains if the employee is late, [2 marks]

(c) the probability that the employee is late on at least 2 out of 5 consecutive working days.[4 marks]

9. A company produces two types of lamps, A and B, which are made of three types of materials: ironframe, electrical component and plastic component. Each lamp A requires 1 unit of iron frame, 2units of electrical components and 3 units of plastic components, whereas each lamp B requires 3units of iron frames, 2 units of electrical components and 1 unit of plastic component. The companyhas 300 000 units of iron frames, 300 000 units of electrical components and 400 000 units of plasticcomponents in stock. The profits made from each lamp A and lamp B are RM15.00 and RM20.00respectively.

(a) Formulate a linear programming problem to maximise profit within the constraints. [4 marks]

(b) Using the graphical method, determine the number of lamp A and the number of lamp B whichgive the maximum profit and find this maximum profit. [8 marks]

10. An advertising firm conducted a survey on television viewing habits in urban and rural areas. Thetable below shows the number of hours per week spent watching television by 20 persons in urbanareas and 18 persons in rural areas.

Urban Rural35 35 36 38 45 16 48 3134 34 45 30 47 24 50 3229 39 40 40 40 48 34 4226 40 31 35 40 34 40 4447 43 36 35 25 8

(a) Construct a suitable stemplot for each of the above data set. [3 marks]

(b) Comment on the skewness of the two distributions. [2 marks]

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(c) Calculate the mean and the standard deviation of the number of hours spent watching televisionfor each area. [6 marks]

(d) Compare the dispersion of the two distributions. [1 marks]

11. The following table shows the activities, their preceding activities and their durations for a project.

Activity Preceding activities Durations (weeks)A - 7B A 3C A 3D B, C 4E B 5F A 3G D, E, F 6


(b) Construct a table which shows the earliest start time, earliest finish time, latest start time, latestfinish time, total float, free float and independent float for each activity. [7 marks]

(c) Determine the critical path and the minimum time required to complete the project. [2 marks]

(d) If the duration of activity D has to be extended to 8 weeks, determine the number of weeks theproject will be delayed. [3 marks]

12. The following table shows the quarterly amounts of mileage claims (in thousand RM) made byemployees of a company from year 2001 to year 2003.

YearAmount of mileage claims (in thousand RM)

Quarter 1 Quarter 2 Quarter 3 Quarter 42001 300 455 560 5902002 314 470 570 6102003 420 480 600 620

(a) Plot the above data as a time series. [2 marks]

(b) Find the equation of the trend line using the method of least squares. [6 marks]

(c) Calculate the centered four-quarter moving averages. [3 marks]

(d) Calculate the average seasonal variation for each quarter using a multiplicative model. [3 marks]

(e) Forecast the amount of mileage claims made by employees of the company for the first quarterof year 2004. [3 marks]

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STPM 2006

1. The ages of patients visiting a clinic on a particular day are as follows.

18 18 20 30 26 28 1819 20 8 19 21 43 1921 51 20 19 20 22 2019 20 19 64 19 18 1921 62 19 20 22 19 23

(a) Construct an ordered stemplot to display the above data. [2 marks]

(b) State a measure of central location that best describes the data. Give a reason for your answer.[2 marks]

2. A teacher, 3 male students and 2 female students line up for a photograph.

(a) Find the number of different arrangements if the teacher stands at the end of the line. [2 marks]

(b) Find the number of different arrangements if all the male students stand together. [2 marks]

3. The table below shows the average daily wage and the number of workers for three job categories ina factory for the years 2000 and 2003.

YearOperator Clerk Supervisor

Wage Number of Wage Number of Wage Number of(RM) workers (RM) workers (RM) workers

2000 15.50 65 17.00 24 21.50 102003 19.00 95 22.50 28 28.50 15

Taking 2000 as the base year, calculate

(a) the Laspeyres index for the average wage in year 2003, [2 marks]

(b) the Paasche index for the number of workers in year 2003. [2 marks]

4. The time taken by a manager to travel from his home to his workplace is normally distributed withmean 45 minutes and standard deviation 3 minutes. Determine the time when the manager has toleave his house so that he is 95% confident of arriving at the workplace by 8:00 am. [5 marks]

5. The probability distribution of a random variable X is as shown in the table below.

x 0 1 2 3 4

P(X = x) p5

21

10

21

5

21p

(a) Determine the value of p. [2 marks]

(b) Calculate the mean and variance of X. [5 marks]

6. The time series plot of the number of houses sold through an entrepreneur scheme from the firstquarter of year 2000 to the fourth quarter of year 2003 is shown below.

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(a) Give comments on the time series plot. [2 marks]

(b) The table below shows the number of houses sold for each quarter and the centred movingaverages.

Year Quarter Number of houses Centred moving average

2000

1 1122 1183 155 120.8754 95 122.750

2001

1 119 125.3752 126 128.8753 168 132.0004 110 133.250

2002

1 129 135.0002 126 139.8753 182 143.8754 135 147.125

2003

1 136 151.1252 145 155.6253 1954 158

Using an additive model, calculate the adjusted seasonal variation for each of the four quarters.[2 marks]

7. The amounts of purchase and the modes of payment of 300 customers of a supermarket are shownin the following table.

Amount of Mode of paymentpurchase Cash Credit card

Less than RM50 50 25RM50 or more 75 150

A customer is selected at random from this group of customers.

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(a) Find the probability that the payment is made by cash. [1 marks]

(b) Find the probability that the amount of purchase is less than RM50 and the payment is madeby cash. [1 marks]

(c) If the amount of purchase is at least RM50, find the probability that the payment is made bycash. [1 marks]

(d) Find the probability that the amount of purchase is less than RM50 given that the payment ismade by cash. [1 marks]

(e) State, with a reason, whether the events “the amount of purchase is less than RM50” and “thepayment is made by cash” are mutually exclusive. [2 marks]

(f) State, with a reason, whether the events “the amount of purchase is less than RM50” and “thepayment is made by cash” are independent. [2 marks]

8. The lifespan of a type of tyre is normally distributed with mean 70000 km and standard deviation10 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 000km 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 exceed3500 km at the symmetric 99% confidence interval. [4 marks]

9. The following table shows the activities, their durations and their preceding activities for a project.

Activity Duration (weeks) Preceding activitiesA 2 -B 1 -C 3 AD 2 BE 3 C, DF 2 EG 1 F


(b) Construct a table showing the earliest start time, earliest finish time, latest start time and latestfinish time for each activity. Hence, determine the critical activities and find the minimum timeneeded to complete the project. [8 marks]

(c) If the durations for activities C and D are each reduced by a week, determine whether theproject can be completed within 10 weeks. [2 marks]

10. Eight pairs of values obtained from random observations on two variables x and y are (4, 63), (2,89), (5, 58), (3, 73), (4, 72), (5, 48), (3, 75) and (2, 84).

(a) Plot these values on a scatter diagram. [2 marks]

(b) State, with a reason, whether the scatter diagram in (a) displays a positive or a negative corre-lation. [2 marks]

(c) Obtain the equation of the regression line of y on x in the form y = a + bx, where a and b aregiven to three decimal places. [6 marks]

(d) Estimate the value of y corresponding to x = 4.5. [2 marks]

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11. A computer manufacturing company produces three types of computer: home desktop, businessdesktop and notebook. Each type of computer needs to pass through three processes: assembling,testing and packaging. The profits for a home desktop, a business desktop and a notebook are RM200,RM350 and RM450 respectively. The table below shows the number of hours required to produce ahome desktop, a business desktop and a notebook and the number of man-hours available per week.

ProcessNumber of hours required Number of man-hours

Home desktop Business desktop Notebook available per weekAssembling 5 6 8 400Testing 10 12 12 648Packaging 2 4 2 60

(a) Formulate the problem as a linear programming problem. [4 marks]

(b) Using the simplex method, find the number of each type of computer to be produced to maximisethe weekly profit and find this maximum profit. [9 marks]

12. The table below shows the duration, in seconds, taken by 100 workers to finish a task.

Duration (x seconds) Number of workers0 < x ≤ 100 2

100 < x ≤ 200 10200 < x ≤ 250 20250 < x ≤ 300 26300 < x ≤ 350 24350 < x ≤ 400 10400 < x ≤ 500 8

(a) Calculate an estimate of the mean. [2 marks]

(b) Plot a histogram for the above data. Hence, estimate the mode. [5 marks]

(c) Plot a relative cumulative frequency curve for the above data. Hence, determine the medianand the percentage of workers who finish the task in more than 270 seconds. [7 marks]

41


STPM 2007

1. A random sample of 10 supermarkets is selected from a metropolitan area. Diagram (i) shows therelationship between the ratings of the supermarkets in the quality of merchandise and customerspreference. Diagram (ii) shows the relationship between the ratings of the supermarkets in the price ofmerchandise and customers preference. The ratings are based on a certain scale, with larger numbersindicating higher ratings.

(a) Name the above diagrams. [1 marks]

(b) Comment on the relationship shown in each of the diagrams. Between quality and price, whichhas a stronger relationship to customers preference? [3 marks]

2. A study on 100 visitors to a book fair shows that 60 visitors have seen the advertisement about thefair. Out of 40 visitors who make purchases, 30 have seen the advertisement. Find the probabilitythat a visitor who has not seen the advertisement makes a purchase. [4 marks]

3. In a country, one person in 20 is left-handed. Find the probability that, in a random sample of 20persons, at least three will be left-handed. [5 marks]

4. A random variable X is normally distributed with mean µ and variance 25. Find the least value ofµ for which P (X ≥ 500) > 0.9. [5 marks]

5. The tariffs and the numbers of chalets rented by tourists at a resort for the years 2004 and 2006 aregiven in the following table.

Type of chaletTariff (RM per day) Number of chalets rented2004 2006 2004 2006

A 150 160 400 200B 180 200 320 380C 200 220 80 200D 250 260 50 100

(a) Taking 2004 as the base year, calculate the price index for the year 2006 with the quantity forthe current year as the weight. [2 marks]

(b) Taking 2004 as the base year, calculate the quantity index for the year 2006 with the price forthe current year as the weight. [2 marks]

(c) Which index gives a better picture of the change in the number of tourists at the resort? Givea reason. [2 marks]

6. The table below shows the number of new current account holders in certain branch of a bank fromyear 1991 to 2006.

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Year Number of account holders1991 211992 231993 171994 201995 151996 181997 151998 101999 152000 112001 162002 112003 162004 152005 202006 21

(a) Plot the data as a time series. [3 marks]

(b) Comment on the trend of the time series. [1 marks]

(c) State whether it is appropriate to use the linear regression method to forecast the number ofnew current account holders. Give a reason. [2 marks]

7. The following table shows the age distribution of drivers who are at fault in accidents during aone-month period in a country.

Age (years) Number of drivers18-25 2026-35 2436-45 1446-55 1856-65 1866-80 26

(a) Draw a histogram for the data. [3 marks]

(b) Estimate the mean and standard deviation of the ages of the drivers. [5 marks]

8. A market survey is conducted at a number of shopping complexes. A random sample of 1250 shoppersare 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” andinterpret this confidence interval. [6 marks]

(b) Explain why an interval estimate is more informative than a point estimate. [2 marks]

9. The following table shows the activities for a project and their preceding activities and duration.

43


Activity Preceding activities Duration (weeks)A - 11B - 5C A 9D B 6E B 8F E 6G C, D, F 8H E, G 7I H 9


(b) Construct a table showing the duration, earliest start time, latest finish time and total float foreach activity. Hence, determine the critical path and the minimum duration of the project.

[9 marks]

10. The marks for 26 students in a test are as follows:

22 90 13 43 59 52 32 40 58 68 76 53 3722 51 83 11 32 43 34 73 81 65 62 38 45

(a) Construct a stemplot to represent the data. [2 marks]

(b) Find the probability that a randomly selected student has a mark between 53 and 65, inclusively.[3 marks]

(c) Determine the interquartile range. [4 marks]

(d) Draw a boxplot to represent the data. [3 marks]

11. The amount of chlorine Y , in parts per million, in the water in a swimming pool at time X, in hours,after treatment are given in the table below.

x 2 4 6 8 10y 1.8 1.8 1.4 1.1 0.9

(a) Find the equation of the least squares regression line of Y on X. Interpret the regression coefficientobtained. [7 marks]

(b) Determine the proportion of the change in the amount of chlorine that is explained by the timeafter treatment. [3 marks]

(c) Estimate the amount of chlorine in the water five hours after treatment. [2 marks]

(d) State whether it is appropriate to estimate the amount of chlorine in the water fifteen hoursafter treatment. Give a reason. [2 marks]

12. A factory produces two types of batteries A and B. Every unit of battery A requires 2 hours ofassembly and 1 hour of testing, whereas every unit of battery B requires 2.5 hours of assembly and1.5 hours of testing. The factory has at most 500 hours of assembly per week and at most 300 hours oftesting per week. It is specified that the number of battery B produced per week exceeds the numberof battery A produced per week and that the number of battery A produced per week exceeds 50units. The profits for battery A and battery B are RM80 and RM90 per unit respectively.

(a) Formulate the above problem as a linear programming problem to maximise the profit.[6 marks]

(b) Using the graphical method, determine the number of battery A and the number of battery Bthat should be produced per week and find the maximum profit per week. [10 marks]

44


STPM 2008

1. Eight persons are invited to be seated in the front row of eight seats to watch a concert. If two ofthem have to sit next to each other, find the number of different ways in which this can be done.

[3 marks]

2. The time taken by 128 workers to complete a task is summarised in the table below.

Time (minutes) Number of works40-44 645-49 1450-54 2355-59 4260-64 3065-74 13

Plot a cumulative frequency curve for the data. [3 marks]

Hence, estimate the number of workers who take more than 66 minutes to complete the task.[2 marks]

3. Two variables x and y are linearly related by the equation y = a + bx. Six pairs of values obtainedfrom random observations on the variables are summarised as follows:∑

x = 136,∑

y = 7650,∑

xy = 187350,∑

x2 = 3344.

Find the values of a and b correct to two decimal places. [5 marks]

4. 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 thelocation per day is 245. Calculate the number of days required so that he is 99% confident thatthe 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 sizeobtained in (a), determine the confidence level for the estimate to be within 100 vehicles of thetrue mean. [3 marks]

5. A random variable X has a binomial distribution with parameters n and p.

(a) Given that E(X) = 6 and Var(X) =12

5, determine the values of n and p. [4 marks]

(b) Calculate P(X = 5). [3 marks]

6. In a country, 78% of consumers are in favour of government control over prices. A random sample of400 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 pro-portion. [4 marks]

7. The table below shows the prices and the total sales of three grades of eggs for the month of Januaryin 2000 and 2005.

45


GradeJanuary 2000 January 2005

Price Total sale Price Total sale(RM per dozen) (RM ’000) (RM per dozen) (RM ’000)

A 1.80 180 2.60 780B 1.50 300 2.00 600C 1.20 240 1.20 120

Taking 2000 as the base year, calculate the Laspeyres and the Paasche price indices for the year 2005.[6 marks]

Explain why one index is higher than the other. [2 marks]

8. The distribution of the selling prices of 80 houses in a city is shown below.

Selling price (RM ’000) Number of houses120- 8150- 23180- 17210- 18240- 8270- 4300- 1330- 1360- 0

The interval ‘120-’, for example, means RM 120 000 ≤ selling price < RM 150 000.

(a) Construct a histogram for the data above. [3 marks]

(b) State the most appropriate measure of location to describe the selling prices. Give a reason.[2 marks]

(c) Calculate the measure that you have selected in (b), and interpret your answer. [4 marks]

9. A student applies for two scholarships S and T to further study. The probability that he is offeredscholarship S is 0.4. If he is offered scholarship S, the probability of him being offered scholarship Tis 0.2. If he is not offered scholarship S, the probability of him being offered scholarship T is 0.7.

(a) Find the probability that he is offered both scholarships. [2 marks]

(b) Find the probability that he is offered only one scholarship. [5 marks]

(c) State, with a reason, whether the events ‘he is offered scholarship S’ and ‘he is offered scholarshipT ’ are mutually exclusive. [2 marks]

10. The following table shows the activities for a project and their preceding activities and duration.

Activity Preceding activities Durations (weeks)A - 3B - 4C A 2D B, C 5E B 2F D, E 3

(a) Draw an activity network for the project showing the earliest start time and the latest starttime for each activity. [7 marks]

(b) State the critical activities of the project and the minimum time required to complete the project.[2 marks]

46


(c) If the duration of activity E has to be extended for a week, determine whether the project willbe delayed. [3 marks]

11. A factory assembles three models of chairs using six components. The following table shows thenumber of units of components needed for each model and the total units of components availableper week.

ComponentNumber of units of components Total units of componentsModel P Model Q Model R available per week

Type 1 seat 1 0 0 500Type 2 seat 0 1 1 1000Chair frame 1 1 1 1000

Chair leg 4 4 4 4000Type 1 backrest 1 1 0 1000Type 2 backrest 0 0 1 500

The profits for models P , Q and R are RM35, RM40 and RM50 per unit respectively. The managerof the factory wishes to determine the number of chairs of each model to be produced per week inorder to maximise the total profit.

(a) If x1, x2, and x3 represent the numbers of chairs of models P , Q and R respectively, formulatea linear programming model to determine the number of chairs of each model that should beproduced per week in order to maximise the total profit. [5 marks]

(b) Construct the initial tableau for the linear programming problem. [4 marks]

(c) Based on the final tableau given below, state the number of chairs of each model that shouldbe produced per week in order to maximise the total profit, and calculate the maximum totalprofit. [4 marks]

Basic x1 x2 x3 s1 s2 s3 s4 s5 s6 Solutions1 0 0 0 1 1 -1 0 0 0 500x2 0 1 0 0 1 0 0 0 -1 500x1 1 0 0 0 -1 1 0 0 0 0s4 0 0 0 0 0 -4 1 0 0 0s5 0 0 0 0 0 -1 0 1 1 500x3 0 0 1 0 0 0 0 0 1 500

12. The table below shows the quarterly profits of a company from year 2005 to year 2007.

YearProfit (RM ’000)

Quarter 1 Quarter 2 Quarter 3 Quarter 42005 23 45 69 522006 29 44 81 512007 31 49 97 63

(a) Plot the above data as a time series. [3 marks]

(b) Given that the trend line is Y = 35.42 + 2.68t, where Y is the profit in period t with t = 1 forquarter 1 of the year 2005. Using an additive model, calculate the adjusted quarterly seasonalvariations. [8 marks]

(c) Interpret the seasonal variations for quarters 1 and 3. [2 marks]

(d) Forecast the profit for quarter 1 of year 2008. [3 marks]

47


STPM 2009

1. The duration of telephone calls made by 175 individuals at a call center on a particular day is shownin the table below:

Duration (minutes) Frequency1-7 158-14 3215-21 3422-28 2229-35 1636-42 1243-49 950-56 5

(a) Construct a histogram to display the data. [2 marks]

(b) i. Draw a frequency polygon on the histogram in (a). [1 marks]

ii. Comment on the skewness of the distribution obtained. [1 marks]

2. On the average, a button making machine is known to produce 6% defective buttons. A randomsample of 100 buttons is inspected and if eight or more buttons are found to be defective, theoperation 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]

3. A rice dispenser is capable of filling up a cup of rice with an average weight of 130 g. Given that theweight of the rice is normally distributed with a standard deviation of 6 g.

(a) Find the percentage number of cups of rice that weigh more than 140g. [3 marks]

(b) If a cup can hold a maximum of 150 g of rice, find the probability that an overflow occurs.[3 marks]

4. A study is conducted to determine the correlation between the distance to school X (in kilometer),student’s travelling time Y (in minutes) and age of a student Z (in years). The data obtained aregiven in the table below.

x 1 3 5 5 7 7 8 10 10 12y 5 10 15 20 15 25 20 25 35 35z 13 18 13 15 19 14 17 19 15 17

(a) Plot scatter diagrams to show the relationship between

i. travelling time and distance, [2 marks]

ii. travelling time and age. [2 marks]

(b) Based on the scatter diagrams in (a), which variable has a stronger linear relationship withtravelling time? Give a reason for your answer. [2 marks]

5. A survey was carried out among mothers in Town A and Town B on the issue whether pupils shouldbe allowed to bring handphones to school. The data obtained are shown in the table below.

Agree Neutral DisagreeTown A 100 50 50Town B 20 60 120

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(a) Find the probability that a mother agrees if she is from Town A. [2 marks]

(b) Determine whether

i. the events ‘Town B’ and ‘disagree’ are independent, [4 marks]

ii. the events ‘Town A’ and ‘disagree’ are mutually exclusive. [1 marks]

6. The prices of three beverages and the average number of cups consumed by an individual per weekin a town for the years 1998 and 2008 are shown in the table below.

Beverage1998 2008

Price Number of Price Number of(RM per cup) cups (RM per cup) cups

Tea 0.60 10 1.00 18Coffee 0.70 13 1.20 7Chocolate 0.80 3 1.50 10

(a) Taking 1998 as the base year,

i. calculate the simple aggregate price index for the year 2008, [2 marks]

ii. comment on the changes in the prices of beverages from 1998 to 2008. [1 marks]

(b) Taking 1998 as the base year and the number of cup for the year 1998 as the weight,

i. calculate the weighted aggregate price index for the year 2008. [2 marks]

ii. comment on the changes in the prices of beverages from 1998 to 2008. [1 marks]

(c) State one advantage of the weighted aggregate price index over the simple aggregate price index.[1 marks]

7. The probability distribution of the number of cars owned by households in a city is as follows:

x 0 1 2 3P(X = x) 0.02 m n 0.07

Given that the mean number of cars owned is 1.38.

(a) Determine the values of m and n. [4 marks]

(b) Calculate the variance of X. [3 marks]

(c) Find the probability that a randomly chosen household owns at least two cars. [2 marks]

8. The number of pens x (in thousand) produced by a factory and the total cost of production y (inthousand RM) for 20 consecutive days are recorded. The results obtained ate summarised as follows:∑

x = 176.00,∑

y = 400.00,∑

x2 = 1780.00,∑

y2 = 8150.00,∑

xy = 3700.20.

(a) Calculate the Pearson correlation coefficient between the number of pens produced and the totalcost. Interpret your answer. [5 marks]

(b) Find the equation of the least squares regression line of the total cost on the number of pensproduced. Interpret the slope of the regression line. [4 marks]

9. A company wishes to develop a theme park on a 120-acre land. The major activities of the projectare listed in the table below.

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Activity Duration (month) Preceding activitiesA Project application and approval 10 -B Project design 6 AC Project design approval 3 BD Land clearing 2 AE Machinery and equipment purchase 6 CF Building construction 8 C, DG Landscaping 6 C, DH Park construction 10 C, DI Testing 3 E, F , G, HJ Opening ceremony 1 I


(b) i. List all the possible paths of the project and their corresponding total duration. [4 marks]

ii. Determine the critical path. [1 marks]

iii. Find the minimum time required to complete the project. [1 marks]

10. A random sample of 200 students of a university is selected and it is found that 120 of them stay inuniversity hostels.

(a) Estimate the proportion of the students who stay in the hostels and determine the standarderror. [3 marks]

(b) Construct a 95% confidence interval for the proportion of the students who stay in hostels, andinterpret your answer. [4 marks]

(c) What is the effect on the confidence interval if the confidence level is increased from 95% to99%? [3 marks]

11. The number of luxury cars sold for each quarter y by a company and their coded quarters x for aduration of four years are shown in the table below.

Year Quarter Coded quarter (x) Number of cars (y)

2005

1 1 102 2 143 3 114 4 21

2006

1 5 112 6 163 7 104 8 22

2007

1 9 142 10 183 11 134 12 22

2008

1 13 132 14 163 15 94 16 25


(b) Comment on the pattern of the time series that you have plotted in (a). [1 marks]

(c) Given that the equation of the trend line is T = 0.325x+ 12.550. Using an additive model, findthe adjusted seasonal variation for each of the four quarters. Write down your answers correctto three decimal places. [5 marks]

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(d) Hence, predict the number of luxury cars sold for the fourth quarter of 2009. [4 marks]

12. An electronics company manufactures LCD televisions of models P and Q. Each unit of model Prequires 3.5 hours of production time, 1 hour of assembly time and 1 hour of packaging time. Eachunit of model Q requires 8 hours of production time, 1.5 hours of assembly time and 1 hour ofpackaging time. The maximum available resources for each process in a day is as follows:

Process Resource available per day (hours)Production 280Assembly 60Packaging 50

The manager of the company wishes to maximise profit. Each unit of P yields a profit of RM400while each unit of Q yields a profit of RM800. Due to high demand, the company has to produce atleast 10 units of each model per day.

(a) If x and y represent the quantities of models P and Q produced each day respectively, formulatethe problem as a linear programming problem. [5 marks]

(b) Plot a graph for the above problem, and shade the feasible region. [7 marks]

(c) Using the graph that you have plotted in (b),

i. determine the quantity of the daily production for each model which gives the maximumprofit, [1 marks]

ii. find the daily maximum profit.

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STPM 2010

1. A company has 400 employees of whom 240 are females. There are 57 female employees with degreequalifications and 85 male employees with non-degree qualifications.

(a) Find the probability that a randomly chosen employee has a degree qualification. [1 marks]

(b) Find the probability that a randomly chosen employee has a degree qualification given that theemployee is a female. [2 marks]

(c) Determine whether the events ‘an employee has a degree qualification’ and ‘an employee is afemale’ are independent. [2 marks]

2. Three types of rice A, B and C are sold in a country. The selling price of a 10kg bag of rice and thenumber of bags of rice sold in the years 2007 and 2008 are given as follows.

Type of ricePrice Number of bags of rice

(RM per bag) (million)2007 2008 2007 2008

A 19.50 25.00 11 13B 23.00 32.50 17 20C 30.50 40.00 14 12

Using 2007 as the base year, calculate

(a) a simple aggregate quantity index for the year 2008, [2 marks]

(b) the Paasche price index for the year 2008. Interpret your answer. [3 marks]

3. Five per cent of credit card holders of a bank do not pay their monthly bills on time. A randomsample of 10 credit card holders is taken.

(a) Find the probability that at least one card holder do not pay his/her bills on time. [4 marks]

(b) State the modal number of card holders who do not pay their monthly bills on time. Give areason for your answer. [2 marks]

4. A census conducted in a school shows that the total hours per week pupils spent watching televisionhas a mean of 16.87 hours and a standard deviation of 5 hours. If a random sample of 100 pupils istaken, find

(a) the probability that the sample mean is within3

4hour of the population mean, [4 marks]

(b) the probability that the sample mean is more than 17 hours. [2 marks]

5. Let Pearsons correlation coefficient between variables x and y for a random sample be r.

(a) i. What does r measure? [1 marks]

ii. State the range of the possible values of r and what it means when r = 0. [2 marks]

(b) A sample of 10 data points may be summarised as follows:∑(x− x)2 = 2534,

∑(y − y)2 = 1497.6,

∑(x− x)(y − y) = −1382.

Calculate Pearsons correlation coefficient between x and y. Comment on your answer. [3 marks]

6. The time series plot of the number of holiday packages sold by a company for every 4-month periodfrom 2006 to 2009 is shown below.

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(a) Comment on the pattern of the time series. [1 marks]

(b) The number of holiday packages sold for each 4-month period and the moving averages areshown in the table below.

Year 4-month period Number of holiday packages sold Moving average

20061 1032 221 282.0003 522 285.667

20071 114 319.3332 322 363.3333 654 370.333

2008

1 135 360.6672 293 368.0003 676 379.667

2009

1 170 380.0002 294 430.6673 828

Using a multiplicative model, calculate the adjusted seasonal variation for each 4-month period.[5 marks]

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7. A chemical company produces two types of organic fertilisers at its factory. Three types of rawmaterials P , Q and R are mixed to produce Type X and Type Y fertilisers. Each ton of Type Xfertiliser is a mixture of 0.4 ton of P and 0.6 ton of R, while each ton of Type Y fertiliser is a mixtureof 0.5 ton of P , 0.2 ton of Q and 0.3 ton of R. The profit yields for each ton of Type X and Type Yfertilisers are RM400 and RM300 respectively. The quantities of raw materials useable per week areshown in the following table.

Raw material Quantity of material useable per week (tons)P 20Q 5R 21

(a) If x and y represent the quantities, in tons, of Type X and Type Y fertilisers produced eachweek, formulate a linear programming model that can be used in order to maximise total profitper week. [4 marks]

(b) Construct the initial tableau for the linear programming model. [2 marks]

(c) Based on the following final tableau, state the quantity of each type of fertiliser that should beproduced per week in order to maximise the total profit, and calculate the total profit.

Basic x y s1 s2 s3 Solution

y 0 110

30 −20

920

s2 0 0 −2

31

4

91

x 1 0 −5

30

25

925

[2 marks]

8. The prices of houses sold, x, in hundred thousand RM, in a municipality in the year 2007 is shownbelow.

Price (hundred thousand RM) Frequency2.5 < x ≤ 3.0 153.0 < x ≤ 3.4 103.5 < x ≤ 4.0 144.0 < x ≤ 4.5 84.5 < x ≤ 5.0 105.0 < x ≤ 5.5 75.5 < x ≤ 6.0 56.0 < x ≤ 6.5 3

(a) Plot a cumulative frequency curve for the above data. Hence, estimate the median and thequartiles. [5 marks]

(b) Draw a boxplot to represent the above data. Comment on the prices of the houses sold.[4 marks]

9. According to a report, 80% of the adult population is in favour of banning cigarettes. A proportionof 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, [3 marks]

ii. within one standard deviation of the population proportion. [3 marks]

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10. Mr. Tan works five days a week and he takes either a bus, a commuter or a taxi to his office. Theprobability that he takes a taxi is 0.20 and the probability that he takes a commuter is 0.50. Theprobabilities that he will arrive late at his office if he takes a bus, a commuter, or a taxi are 0.30,0.10 and 0.15 respectively.

(a) Find the probability that Mr. Tan will arrive late at his office on any given day. [3 marks]

(b) Mr. Tan arrives late at his office on a given day. Determine the mode of transport that he islikely to take. Give a reason for your answer. [4 marks]

(c) Determine the probability that in a given week, Mr. Tan arrives late at his office on alternatedays. [4 marks]

11. A group of students are involved in an orientation programme for new form six students. Nineactivities are required in order to organise the programme. The activities and the duration for eachactivity are shown in the table below.

Activity Preceding activities Duration (days)A - 3B - 3C A 4D A, B 2E A 5F C 2G C 6H E, F 4I D, G 2

(a) Draw an activity network for the programme. [3 marks]

(b) Construct a table which shows the earliest start time, latest finish time and the total float foreach activity. Hence, determine the critical path and the minimum number of days needed tocomplete the programme. [8 marks]

(c) If the duration of activity B has to be extended to four days, determine whether the programmewill be extended or not. Give a reason for your answer. [2 marks]

12. The administration department of a private hospital conducted a study to determine whether arelationship exists between the number of days a patient is warded, X, and the total number of timesthe patient calls for a nurse, Y . The data collected is given below.

Patient Number of days in ward (X) The number of calls for a nurse ((Y ))1 2 22 4 33 5 34 6 45 3 26 8 67 12 108 7 59 11 810 2 1

(a) Plot a scatter diagram for the above data, and comment on the relationship between the twovariables. [4 marks]

(b) Find the equation of the least square regression line of Y on X, giving your answer correct tothree decimal places. Hence, interpret the slope of the regression line. [7 marks]

(c) Plot a regression line obtained in (b) on the scatter diagram in (a). [2 marks]

55


(d) Estimate the total number of calls for a nurse made by a patient who is warded for a week.[2 marks]

(e) If a patient is warded for two months, could you estimate the total number of calls for a nursemade by the patient? Comment on your answer. [2 marks]

56


STPM 2011

1. Some summary statistics about the annual expenditure, in RM, of a sample of recreational golferson golf activities are shown in the table below.

Minimum Maximum Mean Median200.00 15500.00 6285.67 3500.00

(a) Comment on the shape of the distribution of expenditure. Give a reason for your answer.[2 marks]

(b) State, with a reason, the appropriate measure of central tendency to describe the expenditureof the golfers. [2 marks]

2. There are eight male and four female architects in a consultant company. Three architects arerandomly chosen to be posted to Johor Bahru, Kuala Lumpur and Miri. Find the probability that

(a) three male architects are posted to the three cities, [2 marks]

(b) one male architect is posted to Kuala Lumpur, one female architect is posted to Johor Bahruand one female architect is posted to Miri. [2 marks]

3. A discrete random variable X has probability distribution function

P(x) =

(

1

3

)x

, x = 1, 2, 3, 4,

k, x = 5, 6,

0, otherwise.

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

(b) State the mode, and calculate the mean of X. [3 marks]

4. The sale values (output quantity multiplied by price) for four types of product of a company atcurrent price and 1998 pricc arc as follows:

Type of productSale value at current price Sale value at 1998 price

(RM million) (RM million)1998 2009 2010 2009 2010

A 50 120 175 100 125B 10 30 24 25 20C 40 40 50 20 20D 20 50 90 40 60

Total 120 240 339 185 225

Calculate

(a) the Laspeyres quantity index for the year 2010 by using 1998 as the base year, [2 marks]

(b) the Laspeyres quantity index for the year 2010 by using 2009 as the base year, [3 marks]

(c) the percentage change in the output quantity of the company from 2009 to 2010. [1 marks]

5. In a preliminary sample of 40 postgraduate students in a university, 32 students are satisfied withthe services at the main library of the university.

(a) Determine the smallest sample size needed to estimate the population proportion with an errornot exceeding 0.05 at the 90% confidence level. State any assumption made. [5 marks]

(b) State the effect on the sample size

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i. if the error is larger than 0.05 with the confidence level unchanged. [1 marks]

ii. if the confidence level is higher than 90% with the error unchanged. [1 marks]

6. A sample of 14 pairs of observations of variables x and y is summarised as follows:∑x = 43,

∑x2 = 157.42,

∑y = 572,

∑y2 = 23530,

∑xy = 1697.80.

The equation of the regression line of y on x is y = a+ bx.

(a) Determine the values of a and b. [4 marks]

(b) Calculate the coefficient of determination and interpret your answer. [3 marks]

7. An opinion poll on a certain political party is conducted on 1000 voters, of whom 600 are males. Itis found that 250 voters are in favour of the party. It is also found that 450 male voters are not infavour of the party.

(a) Construct a two-way classification table based on the above information. [2 marks]

(b) Find the probability that a randomly chosen voter is in favour of the party if the voter is afemale. [1 marks]

(c) Find the probability that a randomly chosen voter is a male or not in favour of the party.[2 marks]

(d) Determine whether the events “a voter is a male” and “a voter is in favour of the party” areindependent. [3 marks]

8. Experience shows that 40% of the throws of a bowler result in strikes.

(a) Find the probability that, out of ten throws made by the bowler, at least two throws result instrikes. [4 marks]

(b) Find the probability that at most five throws need to be made by the bowler so that four throwsresult in strikes. [5 marks]

9. The networks of the activities on nodes of a project is shown below.

(a) Determine the values of r and s. [4 marks]

(b) State the critical path, and determine the time required to complete the project. [3 marks]

(c) Calculate the total floats for activities A and J . [2 marks]

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(d) If the duration of activity J is extended to four weeks, determine whether the project will bedelayed. [2 marks]

10. The age of women in country A suffering from kidney problems is found to be normally distributedwith mean 40 years and standard deviation 5 years.

(a) Find the probability that 10 randomly selected women who suffer from kidney problems havethe mean age less than 42 years. [3 marks]

(b) Find the probability that four randomly selected women who suffer from kidney problems havea total age of more than 145 years. [3 marks]

(c) The ages of eight randomly selected women from country B who suffer from kidney problemsare 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 whetherthe mean age differs from that of country A, and explain your answer. [6 marks]

11. A food company produces a cereal from several ingredients. The cereal is enriched with vitamins Aand B which are provided by two of the ingredients, oats and rice. A 1 g of oats contributes 0.32 mgof vitamin A and 0.08 mg of vitamin B, whereas 1 g of rice contributes 0.24 mg of vitamin A and0.12 mg of vitamin B. Each box of cereal produced has to meet the minimum requirements of 19.20mg of vitamin A and 7.20 mg of vitamin B. The cost of 1 kg of oats is RM5 and the cost of 1 kg ofrice is RM4. The company wants to determine how many grams of oats and rice are to be includedin each box of cereal in order to minimise cost.

(a) Formulate a linear programming model for the problem to minimise the cost. [5 marks]

(b) Using a graphical method, determine how many grams of oats and rice are included in each boxof cereal to minimise cost, and find this minimum cost. [8 marks]

12. The lengths (in minutes) of 200 telephone calls made by customers to a pizza customer call centre ina particular day is shown in the table below.

Length of call (minutes) Number of calls0.0 - 3.0 73.0 - 4.5 194.5 - 6.0 346.0 - 7.5 677.5 - 9.0 389.0 - 10.5 1810.5 - 12.0 1312.0 - 15.0 4

(a) Construct a histogram to display the data. Comment on the shape of the distribution obtained.[4 marks]

(b) Use your histogram to estimate the mode and the median. [5 marks]

(c) Calculate the mean of the length of calls made by the customers. [2 marks]

(d) Determine the percentage of calls whose lengths are at least 10 minutes. [3 marks]

59


STPM 2012

1. There are three male and two female workers who will be assigned to four different tasks Each workerhas an equal chance of being assigned to any task and may perform at most one task.

(a) Determine the number of different ways the workers can be assigned to the tasks, with two ofthe tasks allocated to the female workers. [2 marks]

(b) Find the probability that the female workers are assigned to two of the tasks. [2 marks]

2. The daily closing prices (in RM per share) of a company stock for the month of March in the year2010 are as follows:

Date Price (RM) Date Price (RM) Date Price (RM)1 8.13 11 8.10 23 8.032 8.11 12 8.10 24 8.103 8.04 15 8.08 25 8.094 8.13 16 8.14 26 8.065 8.05 17 8.15 29 8.008 8.06 18 8.08 30 8.109 7.95 19 8.05 31 8.0510 7.95 22 8.00


(b) Comment on the pattern of the time series that you have plotted in (a). [1 marks]

3. A random variable X is normally distributed with mean 20 and variance 6.25. The mean of a randomsample of size n is X.

(a) State the sampling distribution of X. [1 marks]

(b) If P(X < 18) = 0.0057, find the value of n. [4 marks]

4. At a cineplex 70% of the moviegoers buy popcorn at the snack counter. Of those who buy popcorn,80% of them buy drinks. Among those moviegoers who do not buy popcorn, 10% of them buy drinks.

(a) Find the probability that a moviegoer buys a drink. [4 marks]

(b) Find the probability that a moviegoer does not buy popcorn if he buys a drink. [2 marks]

5. The cumulative frequency distribution of monthly charges, to the nearest RM, of 200 randomly chosenpostpaid subscribers to a particular telecommunication company is shown below.

Monthly charge Cumulative frequency≤ 30 0≤ 50 10≤ 70 40≤ 90 76≤ 110 120≤ 150 170≤ 200 192≤ 250 200

(a) Calculate the mean of the monthly charges. [3 marks]

(b) Determine the percentage of subscribers whose month charges are more than the mean.[4 marks]

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6. There are two boxes which each contain three golf balls numbered 1, 2 and 3. A player draws oneball at random from each box, and the score X of the player is the sum of the numbers on the twoballs.

(a) Determine the probability distribution function of X. [3 marks]

(b) Find E(X) and Var(X). [4 marks]

7. The amount of time devoted to studying Economics each week by students who achieve a grade A inan examination is normally distributed with a mean of 8.0 hours.

(a) Given that 14% of the grade A students study more than 10.7 hours weekly, show that itsstandard deviation is 2.5 hours. [3 marks]

(b) Find the percentage of the grade A students who study less than 5 hours weekly. [3 marks]

(c) Find the probability that all three randomly selected grade A students study less than 5 hoursweekly. [2 marks]

8. According to a recent census, children under 18 years of age spend an average of 16.87 hours perweek 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 timespent 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]

9. A company is involved in construction projects. One of the projects awarded to the company containsseven activities. The activities, the preceding activities and the duration required for each activityare shown in table below.

Activity Preceding activity Duration (weeks)A - 0B - 10C A 40D A 10E B 2F B 15G C,D,E 8


(b) Determine the critical activities of the project, and find the minimum number of weeks requiredto complete the project. [4 marks]

(c) Shortly before the company starts to implement the project, a technical assistant points outthat the duration required to undertake activity F could be shortened to 11 weeks with a newinnovative approach. Determine whether the new approach adopted by the company for activityF would affect your answer in (b). [3 marks]

10. Three groups of sales trainees of a company undergo different sales training programmes which areProgramme A, Programme B and Programme C. The boxplots below represent the sales (RM ’000)earned by each group in the first month after the completion of the training programmes.

61


(a) Estimate the median of the three sales distributions. State the training programme which ismost effective. [3 marks]

(b) Which of the sales distributions has a mean closest to its median? Give a reason for your answer.[2 marks]

(c) Which of the sales distributions has the largest dispersion? [1 marks]

(d) Comment on the skewness of the three sales distributions. [3 marks]

(e) State the effect on the mean and median sales when the outlier of the sales distribution forprogramme A is removed. [2 marks]

11. A study was conducted to investigate the influence of the quality and fair price of products onpreference to shop at a hypermarket. A random sample of 14 customers were asked to rate thehypermarket in terms of preference to shop, Y , quality of product, X1 and fair price of product, X2.The ratings were based on an 11-point scale with higher numbers indicating higher ratings. The datacollected are given in the table below.

Customer Preference to shop (Y ) Quality of product (X1) Fair price of product (X2)1 6 5 42 9 6 103 8 6 54 3 2 25 10 6 116 4 3 17 5 4 78 2 1 49 11 9 910 9 5 1011 10 8 812 3 1 513 8 8 514 5 3 2

(a) Plot a scatter diagram of preference to shop against quality of product. Hence, interpret thisdiagram. [3 marks]

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(b) i. Calculate the Pearson correlation coefficient between preference to shop and fair pricing.[6 marks]

ii. Given that the value of Pearson correlation coefficient between preference to shop andquality of product is 0.924, state whether quality of product or fair price of product has agreater influence on preference to shop. [1 marks]

(c) Calculate the percentage of variation in preference to shop which is accounted for by fair priceof product. [2 marks]

12. An investor has RM5 million to invest in corporate bond, fixed deposit and unit trust. The interestrate and maximum investment allowed are as follows.

Type of investmentInterest rate Maximum investment allowed

(%) (RM million)Corporate bond 7 1.0Fixed deposit 3 2.5Unit trust 11 1.5

(a) Formulate a linear programming problem to maximise total interest earned within the constrains.[5 marks]

(b) Using the simplex method, find the optimal amount for each type of investment and the totalinterest earned. [9 marks]

(c) State whether the RM5 million is fully utilised. Explain your answer. [2 marks]

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stpm mathematics m

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