Stpm Mathematics S Past Year Questions By Akeong 1r6835

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STPM Mathematics S Past Year Questions Lee Kian Keong & LATEX [email protected] http://www.facebook.com/akeong Last Edited by November 14, 2011 Abstract

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This is a document which shows all the STPM questions from year 2002 to year 2010 using LATEX. Students should use this document as reference and try all the questions if possible. Students are encourage to me via email1 or facebook2 . Students also encourage to send me your collection of 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 STPM 2002 . . . . . . . . . STPM 2003 . . . . . . . . . STPM 2004 . . . . . . . . . STPM 2005 . . . . . . . . . STPM 2006 . . . . . . . . . STPM 2007 . . . . . . . . . STPM 2008 . . . . . . . . . STPM 2009 . . . . . . . . . STPM 2010 . . . . . . . . .

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2 2 3 4 6 7 9 10 11 13

2 PAPER 2 QUESTIONS STPM 2002 . . . . . . . . . STPM 2003 . . . . . . . . . STPM 2004 . . . . . . . . . STPM 2005 . . . . . . . . . STPM 2006 . . . . . . . . . STPM 2007 . . . . . . . . . STPM 2008 . . . . . . . . . STPM 2009 . . . . . . . . . STPM 2010 . . . . . . . . .

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15 15 17 20 23 26 29 32 35 38

1 2

[email protected] http://www.facebook.com/akeong

1

PAPER 1 QUESTIONS

1

Lee Kian Keong

PAPER 1 QUESTIONS 1. The function f is defined by f :x→

√

1 3x + 1, x ∈ R, x ≥ − . 3

Find f −1 and state its domain and range.

2. Given that y = e−x cos x, find

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STPM 2002

[4 marks]

d2 y dy when x = 0. and dx dx2

[4 marks]

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3. Determine the values of a, b, and c so that the matrix   2b − 1 a2 b2  2a − 1 a bc  b b + c 2c − 1 is a symmetric matrix.

Z

4. By using suitable substitution, find

[5 marks]

3x − 1 √ dx. x+1

[5 marks]

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

6. Express

7. Express

√ √ √ 59 − 24 6 as p 2 + q 3 where p, q are integers.

[7 marks]

1 as partial fraction. −1

[4 marks]

4k 2

Hence, find a simple expression for Sn =

n X

k=1

1 and find lim Sn . n→∞ −1

[4 marks]

4k 2

8. Given that P QRS is a parallelogram where P (0, 9), Q(2, −5), R(7, 0) and S(a, b) are points on the plane. Find a and b. [4 marks] Find the shortest distance from P to QR and the area of the parallelogram P QRS. [6 marks] 9. Find the point of intersection of the curves y = −x2 + 3x and y = 2x3 − x2 − 5x. Sketch on the same coordinate system these two curves. [5 marks] 2 3 2 Calculate the area of the region bounded by the curve y = −x + 3x and y = 2x − x − 5x.[6 marks] 

−10 4 10. Matrices M and N are given as M =  15 −4 −5 1

  9 2 −14 , and N =  4 6 1

3 3 2

 4 1  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 different proportions. Each product X consists of two components of A, four components of B, and one component of C; each product of Y consists of three components of A, three components of B, and two components of C; each product of Z consists of four components of A, one component of B, and 2

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2003

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four components of C. A total of 750 components of A, 1000 components of B, and 500 components of C are used. With x, y 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. Hence, (a) find all the real roots of 2x6 − 9x4 + 3x2 + 4 = 0. 3

2

(b) determine the set of values of x so that 2x − 9x + 3x + 4 < 12 − 12x. 12. Function f is defined by f (x) =

[2 marks]

[5 marks]

[6 marks]

2x . (x + 1)(x − 2)

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Show that f 0 (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]

STPM 2003

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

[5 marks]

dy √ 2. If y = ln xy, find the value of when y = 1. dx

[5 marks]

Z

π 6

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

0

cos θ dθ. (3 + 2 sin θ)2

[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]

6. The function f is defined by

 x  x<1 1 + e , f (x) = 3, x=1   2 + e − x, x > 1

(a) Find lim− f (x) and lim+ f (x). Hence, determine whether f is continuous at x = 1.

[4 marks]

(b) Sketch the graph of f .

[3 marks]

x→1

x→1

7. The straight line l1 which es through the points A(4, 0) and B(2, 4) intersects the y-axis at the point P . The straight line l2 is perpendicular to l1 and es through√B. If l2 intersects the x-axis [8 marks] and y-axis at the points Q and R respectively, show that P R : QR = 5 : 3. 8. Express

1+x 1 + 2x

12

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

3

[6 marks]

PAPER 1 QUESTIONS

STPM 2004

√ 1 , find 62 correct to four decimal places. 30 

1 9. The matrix A is given by A =  3 0

2 1 1

 −3 1  −2

[3 marks]

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By taking x =

Lee Kian Keong

(a) Find the matrix B such that B = A2 − 10I, where I is the 3 × 3 identity matrix. 21

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

[3 marks] [6 marks]

a 10. The curve y = x(b − x), where a 6= 0, has a turning point at point (2, 1). Determine the values of 2 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] 2 . Show that the equation 1+x

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

(1 + x)ex − 2 = 0 has a root in the interval [0, 1]. [7 marks] Use the Newton-Raphson method with the initial estimate x0 = 0.5 to estimate the root correct to three decimal places. [6 marks] 2 in partial fractions. r2 + 2r Using the result obtained,

12. Express ur =

[3 marks]

1 1 1 1 (a) show that u2r = − + 2 + + , [2 marks] r r r + 2 (r + 2)2 ∞ ∞ n X X X 1 3 1 1 ur+1 + r . ur and (b) show that ur = − − and determine the values of 2 n+1 n+2 3 r=1 r=1 r=1 [9 marks]

STPM 2004

Z

1. Show that

e

ln x dx = 1.

[4 marks]

1

1

2. Expand (1 − x) 2 in ascending powers of x up to the term in x3 . Hence, find the value of to five decimal places.

√

7 correct [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]



3 4. Matrix A is given by A =  5 1

3 4 2

 4 1 . 3

Find the adt of A. Hence, find A−1 .

[6 marks]

4

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2004

5. The function f is defined by

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 x − 1, 0≤x<2 f (x) = x + 2 ax2 − 1, x ≥ 2

where a ∈ R. Find the value of a if lim f (x) exists. With this value of a, determine whether f is x→2 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 y2 (x − 2)2 + = 1 and sketch the ellipse. [7 marks] 8 units. Show that the locus of P is the ellipse 16 12

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1 7. Sketch, on the same coordinate axes, the graphs of y = 2 − x and y = 2 + . x 1 Hence, solve the inequality 2 − x > 2 + . x

[4 marks]

[4 marks]

8. Using the sketch graphs of y = x3 and x + y = 1, show that the equation x3 + x − 1 = 0 has only one real root and state the successive integers a and b such that the real root lies in the interval (a, b). [4 marks]

Use the Newton-Raphson method to find the real root correct to three decimal places. 9. The matrices P and Q, where PQ = QP, are given by    −1 2 −2 0 P =  0 0 2  and Q =  0 0 a b c

1 0 −2

[5 marks]

 0 −1  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] 2 10. A curve is defined by the parametric equations x = 1 − 2t, y = −2 + . Find the equation of the t normal to the curve at the point A(3, −4). [7 marks] The normal to the curve at the point A cuts the curve again at point B. Find the coordinates of B. [4 marks]

1 11. Sketch on the same coordinates axes, the line y = x and the curve y 2 = x. Find the coordinates of 2 the points of intersection. [5 marks] 1 Find the area of region bounded by the line y = x and the curve y 2 = x. [4 marks] 2 Find the volume of the solid formed when the region is rotated through 2π radians about the y-axis. [4 marks]

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

a(1 − rn ) [3 marks] 1−r

(a) The sum of the first five of a geometric series is 33 and the sum of the first ten of the geometric series is -1023. Find the common ratio and the first term of the geometric series. [5 marks]

5

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2005

3 − . . . are 2 respectively. Determine the smallest value of n such that |Sn − S∞ | < 0.001.[7 marks]

Sn and S∞

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

STPM 2005 1. Using the laws of the algebra of sets, show that

(A ∩ B)0 − (A0 ∩ B) = B 0

[4 marks]

2. If y =

dy cos x d2 y , where x 6= 0, show that x 2 + 2 + xy = 0. x dx dx

[4 marks]

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3. The point R divides the line ing the points P (3, 2) and Q(5, 8) in the ratio 3 : 4. Find the equation of the line ing through R and perpendicular to P Q. [5 marks] 4. For the geometric series 7 + 3.5 + 1.75 + 0.875 + ..., find the smallest value of n for which the difference between the sum of the first n and the sum to infinity is less than 0.01. [6 marks]

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

1 where x 6= 0. x

[7 marks]

6. Find the perpendicular distance from the centre of the circle x2 + y 2 − 8x + 2y + 8 = 0 to the straight line 3x + 4y = 28. Hence, find the shortest distance between the circle and the straight line. [7 marks] 7. Sketch, on the same coordinate axes, the curves y = ex and y = 2 + 3e−x . Calculate the area of the region bounded by the y-axis and the curves. 8. A, B and C B2 = C.  1 If B =  0 1

[2 marks] [6 marks]

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

2 −1 0

[3 marks]



0 0 , find C and A. 1

[7 marks]

√ 9. The complex numbers z1 and z2 satisfy the equation z 2 = 2 − 2 3i.

(a) Express z1 and z2 in the form a + bi, where a and b are real numbers.

[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−x

and 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]

2

2

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

6

[6 marks]

PAPER 1 QUESTIONS

11. Express f (x) =

Lee Kian Keong

STPM 2006

x2 − x − 1 in partial fractions. (x + 2)(x − 3)

[5 marks]

1 1 up to the term in 3 . x x Determine the set of values of x for which this expansion is valid.

[6 marks]

ng

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

[2 marks]

1 12. Find the coordinates of the stationary point on the curve y = x2 + where x > 0; give the xx coordinate and y-coordinate correct to three decimal places. Determine whether the stationary point is a minimum point or a maximum point. [5 marks] 1 1 The x-coordinate of the point of intersection of the curves y = x2 + and y = 2 , where x > 0, is x x p. Show that 0.5 < p < 1. Use the Newton-Raphson method to determine the value of p correct to three decimal places and, hence, find the point of intersection. [9 marks]

STPM 2006

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1. If A, B and C are arbitrary sets, show that [(A ∪ B) − (B ∪ C)] ∩ (A ∪ C)0 = ∅.

[4 marks]

2. If x is so small that x2 and higher powers of x may be neglected, show that x 10 ≈ 29 (2 − 7x). (1 − x) 2 + 2



k 3. Determine the values of k such that the determinant of the matrix  2k + 1 0 Z

4. Using trapezium rule, with five ordinates, evaluate

1

p

[4 marks]

1 −3 k

 3 2  is 0.[4 marks] 2

4 − x2 dx.

[4 marks]

0

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. Ax + B C 2x + 1 in the form 2 + where A, B and C are constants. (x2 + 1)(2 − x) x +1 2−x Z 1 2x + 1 Hence, evaluate dx. 2 + 1)(2 − x) (x 0

6. Express

[6 marks]

[3 marks]

[4 marks]

10−Tn 5 7. The nth term of an arithmetic progression is Tn , show that Un = (−2)2( 17 ) is the nth term of 2 a geometric progression. [4 marks] ∞ X 1 If Tn = (17n − 14), evaluate Un . [4 marks] 2 n=1

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

7

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2006

C3

ng

C1

C2

The above figure shows three circles C1 , C2 and C3 touching one another, where their centres lie on a straight line. If C1 and C2 have equations x2 + y 2 − 10x − 4y + 28 = 0 and x2 + y 2 − 16x + 4y + 52 = 0 respectively. Find the equation of C3 . [7 marks]

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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, has remainder 60. Find the values of a and b and factorise p(x) completely. [9 marks] 1 4 3 2 Using the substitution y = , solve the equation 12y − 8y − 7y + 2y + 1 = 0. [3 marks] x    a 1 −18 5 2 3 −1 12  and PQ = 2I, where I is the 3 × 3 identity 11. If P =  1 −4 3 , Q =  b 3 1 2 −13 −1 c 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 five cups of tea, two cups of coffee and three glasses of fruit juice at a total cost of RM11.80. The second group of six workers have three cups of tea, a cup of coffee and two glasses of fruit juice at a total cost of RM7.10. The cost of a cup of tea and three glasses of fruit juice is the same as the cost of four cups of coffee. If the costs of a cup of tea, a cup of coffee and a glass of fruit juice are RM x, RM y and RM z respectively, obtain a matrix equation to represent the above information. Hence determine the cost of each drink. [6 marks] 

12. The function f is defined by f (t) =

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

(a) Find the value of f (0)

[1 marks]

0

(b) Show that f (t) > 0

[5 marks] 2

0

00

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

[6 marks]

(d) Find lim f (t).

[2 marks]

(e) Sketch the graph of f .

[2 marks]

t→∞

8

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2007

STPM 2007

ng

1. Express the infinite recurring decimal 0.72˙ 5˙ (= 0.7252525 . . . ) as a fraction in its lowest .[4 marks]

2. If y =

x dy , show that x2 = (1 − x2 )y 2 . 2 1+x dx

3. If loga

x = 3 loga 2 − loga (x − 2a), express x in of a. a2

[[ marks]4

4. Simplify √ √ ( 7 − 3)2 √ √ , (a) 2( 7 + 3)

[3 marks]

√ 2(1 + 3i) −1. , where i = (1 − 3i)2

[3 marks]

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(b)

[6 marks]



x y 5. The coordinates of the points P and Q are (x, y) and , respectively, where x 6= 0 x2 + y 2 x2 + y 2 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 a circle. Find the coordinates of the centre and radius of the circle. [6 marks] 6. Find

x2 + x + 2 dx, x2 + 2 Z x (b) dx. ex+1 Z

(a)

[3 marks]

[4 marks]

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

3x2 + 5x A B C D = + + + . (1 − x2 )(1 + x)2 1 − x 1 + x (1 + x)2 (1 + x)3

8. The function f is defined by

(√

f (x) =

(a) Find

lim f (x),

x→−1−

x + 1, −1 ≤ x < 1, |x| − 1, otherwise.

lim f (x), lim− f (x) and lim+ f (x).

x→−1+

[8 marks]

x→1

(b) Determine whether f is continuous at x = −1 and x = 1. 9. The matrices A and B are given by  −1 A =  −3 0

[4 marks]

x→1

  2 1 −35 19 1 4  , B =  −27 −13 1 2 −3 12

Find the matrix A2 B and deduce the inverse of A.

9

[4 marks]

 18 45  . 5 [5 marks]

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2008

Hence, solve the system of linear equations − 2y − y y

− z − 4z + 2z

= −8, = −15, = 4.

ng

x 3x

[5 marks]

10. The gradient of the tangent to a curve at any point (x, y) is given by the curve es through the point (1, −4). (a) find the equation of the curve,

dy 3x − 5 = √ , where x > 0. If dx 2 x [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]

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1 4 1 11. Using the substitution y = x + , express f (x) = x3 − 4x − 6 − + 3 as a polynomial in y. [3 marks] x x x Hence, find all the real roots of the equation f (x) = 0. [10 marks]

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

x3 and determine their nature. −1

x2

[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]

STPM 2008

1. The function f and g are defined by

f :x→

1 , x ∈ R \ {0}; x

g : x → 2x − 1, x ∈ R

Find f ◦ g and its domain. Z

2. Show that

2

3

(x − 2)2 5 dx = + 4 ln x2 3

2 . 3

[4 marks]

[4 marks]

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

A ∩ (B ∪ C) ⊂ (A ∩ B) ∪ (A ∩ C)

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

1 . 1−z

[5 marks]

[6 marks]

1 5. The polynomial p(x) = 2x3 + 4x2 + x − k has factor (x + 1). 2 (a) Find the value of k.

[2 marks]

(b) Factorise p(x) completely.

[4 marks]

10

PAPER 1 QUESTIONS

STPM 2009

sin x − cos x dy d2 y = 2y . , show that sin x + cos x dx2 dx 

1 7. Matrix A is given by A =  1 1

[6 marks]

 0 0 −1 0 . −2 1

(a) Show that A2 = I, where I is the 3 × 3 identity  1 4 (b) Find matrix B which satisfies BA =  0 2 −1 0

ng

6. If y =

Lee Kian Keong

matrix, and deduce A−1 .  3 1 . 2

[4 marks]

[4 marks]

8. The lines y = 2x and y = x intersect the curve y 2 + 7xy = 18 at points A and B respectively, where A and B lie in the first quadrant. (a) Find the coordinates of A and B.

[4 marks] [2 marks]

(c) Find the area of the OAB triangle.

[3 marks]

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

4 > 3 − 3. 9. Find the solution set of the inequality x − 1 x

[10 marks]

x is always decreasing. [3 marks] x2 − 1 Determine the coordinates of the point of inflexion of the curve, and state the intervals for which the curve is concave upwards. [5 marks] Sketch the curve. [3 marks]

10. Show that the gradient of the curve y =

11. Sketch, on the same coordinate axes, the curves y = 6 − ex and y = 5e−x , and find the coordinates of the points of intersection. [7 marks] Calculate the area of the region bounded by the curves. [4 marks] Calculate the volume of the solid formed when the region is rotated through 2π radians about the x-axis. [5 marks] 12. At the beginning of this year, Mr. Liu and Miss Dora deposited RM10 000 and RM2000 respectively in a bank. They receive an interest of 4% per annum. Mr Liu does not make any additional deposit nor withdrawal, whereas, Miss Dora continues to deposit RM2000 at the beginning of each of the subsequent years without any withdrawal. (a) Calculate the total savings of Mr. Liu at the end of n-th year.

[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]

STPM 2009

11

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2009

2. Given x > 0 and f (x) =

√

x, find lim

h→0

f (x) − f (x + h) . h

x 1 ≥ . x+1 x+1

[4 marks]

ng

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

[4 marks]

3 + . . ., obtain the smallest value of n if the difference between the 2 45 sum of the first n + 4 and the sum of the first n is less than . [6 marks] 64

3. For the geometric series 6 + 3 +

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. Find the values of p and q. [6 marks] 5. Given that

2 + loga (1 − 2a), log2 a

ak eo

loga (3x − 4a) + loga 3x =

where 0 < a <

1 , find x. 2

1

Z

[7 marks]

1

x2 (1 − x) 3 dx.

6. Using an appropriate substitution, evaluate

[7 marks]

0

7. The parametric equations of a straight line l are given by x = 4t − 2 and y = 3 − 3t. 3 (a) Show that the point A(1, ) lies on line l, 4 (b) Find the Cartesian equation of line l,

[2 marks] [2 marks]

(c) Given that line l cuts the x and y-axes at P and Q respectively, find the ratio P A : AQ.[4 marks] 8. Find the values of x if y = |3 − x| and 4y − (x2 − 9) = −24. 9. (a) The matrices P, Q and  1 5 P =  2 −2 1 −3

R are given by     −13 −50 −33 4 6 4  , Q =  −1 −6 −5  , R =  1 2 7 20 15 −2

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

[9 marks]

7 −5 1

 −13 −1  . 11 [5 marks]

(b) Using the result in (a), solve the system of linear equations 6x + x − x +

10y 2y 2y

+ 8z + z + 3z

= 4500 = 0 = 1080

.

[5 marks]

10. A curve is defined by the parametric equations x=t−

2 t

and y = 2t +

where t 6= 0. 12

1 t

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2010

dy 5 1 dy =2− 2 , and hence, deduce that − < < 2. dx t +2 2 dx dy 1 (b) Find the coordinates of points when = . dx 3 (a) Show that

[3 marks]

ng

11. Given a curve y = x2 − 4 and straight line y = x − 2,

[8 marks]

(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).

ak eo

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

[7 marks]

3

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

STPM 2010

1. Solve the following simultaneous equations:



log3 (xy) = 5

2. Given that u = show that

and

log9

x2 y



= 2.

[4 marks]

dy 1 1 x −x (e +e ), where x > 0 and y = f (u) is a differentiable function f . If =√ , 2 du u2 − 1

dy = 1. dx

[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 ln x, where x > 0. Find f 0 (x), and hence, determine the value of

Z

2e

ln x dx. e

[6 marks]

5. Let A − B denotes a set of elements which belongs to set A, but does not belong to set B. Without using Venn diagram, show that A − B = A ∩ B 0 . [3 marks] 0 0 Hence, prove that (A ∪ B ) − (B ∩ C) = B ∪ (A − C). [4 marks] 6. The graph of a function f is as follows:

13

Lee Kian Keong

STPM 2010

ng

PAPER 1 QUESTIONS

(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]

ak eo

(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 by 2x2 + 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)

B Cx + D + 2 , [5 marks] x−1 x +1 (b) when x is sufficiently small such that x4 and higher powers can be neglected, show that f (x) ≈ 4 + 7x + 3x2 − x3 . [4 marks] (a) find the constants A, B, C and D such that f (x) = A +

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

4 . Show that the equation 2−x

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 initial estimate x0 = −0.4. [5 marks] 10. A circle C1 es 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, ii. find the coordinates of the point of intersection of C1 with C2 .

[2 marks] [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. 14

[5 marks]

PAPER 2 QUESTIONS

Lee Kian Keong

(b) Sketch the graph of y = h(x).

[2 marks]

1 . h(x)

[3 marks]

ng

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

Hence, determine the set of values of k such that the equation i. one root, ii. two roots, iii. three roots.

1 = k has h(x)

[1 marks]

[1 marks] [1 marks]

 1 12. Matrix P is given by P = 2 2

2 1 −1

 1 3 . −1

(a) Find the determinant and adt of P. Hence, find P−1 .

[6 marks]

ak eo

(b) A factory assembles three types of toys Q, R and S. The total time taken to assemble one unit of R and one unit of S exceeds the time taken to assemble two units of Q by 8 minutes. One unit of Q, two units of R and one unit of S take 31 minutes to be assembled. The time taken to assemble two units of Q, one unit of R and three units of S is 48 minutes. If x, y and z represent the time, in minutes, taken to assemble each unit of toys Q, R and S respectively, i. write a system of linear equations to represent the above information, ii. using the results in (a), determine the time taken to assemble each type of toy.

2

[2 marks] [5 marks]

PAPER 2 QUESTIONS

STPM 2002

1. The discrete random variable X 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 business programs. 4.7% of the university students are female students who study business programs. A student 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 90 cm2 . A random sample of size 10 is taken. (a) State the distribution of the sample mean with its mean and variance. 15

[2 marks]

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2002

(b) Find the probability that the sample mean is located between 18 cm and 24 cm.

[3 marks]

Drink Tea Coffee Chocolate

Price (sen) 80 150 220

Year 1997 Quantity (thousand cans) 15 3 1

Price (sen) 120 170 230

ng

5. The following table indicates the price and the quantity sold in a year for three types of drink in a district. Year 1998 Quantity (thousand cans) 12 3 3

Using 1997 as the base year, calculate

Price (sen) 160 180 240

Year 1999 Quantity (thousand cans) 10 4 5

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

ak eo

(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 I.Q. level of eldest child

90 100

98 95

102 114

103 116

104 98

105 99

110 112

114 106

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

7. The table below shows the number of students for a certain program according to in a college from 1998 to 2001. Year

1998 1999 2000 2001

Number of students First term Second term Third term 20 32 62 21 42 75 23 39 77 27 39 92

(a) Calculate the three- 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 the speed limit. (a) Obtain a 95% confidence interval for the proportion of motorists who have exceeded the speed limit on the highway. [5 marks] (b) Determine the smallest sample size which should be surveyed so that the error of estimation is not more than 0.04 at the 90% confidence level. [5 marks] 9. The following table indicates the age, x years and the price, RMy × 103 , for eight cars of the same model in a second-hand car shop. x y

2 100

8 15

3 72 16

9 16

6 36

5 34

6 30

3 68

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2003

(a) Find the equation of the regression line y on x in the form y = a + bx, where a and b is accurate to two decimal places. [7 marks] [2 marks]

ng

(b) Explain the value a and b that you obtained.

(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. The activities and their duration times (in days) are listed as follows: Preceding Activities A B C D, E

Duration (in days) 4 2 3 8 6 3 4

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Activity A B C D E F G

(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 of higher learning is as follows. Marks 10-29 30-39 40-49 50-59 60-69 70-79 80-99

Number of students 6 7 12 19 15 13 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 the percentage 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 hours and 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 the factory each day. The factory also decided that the number of units of product B produced each day should not be more than 60% of the total daily production of both products A and B. The profit for each unit of A is RM120 and each unit of B is RM200. The factory intends to maximize the total profit 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 should be produced daily in order to maximize the total profit, and find the maximum total daily profit. [9 marks]

STPM 2003

17

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2003

1. The mean and standard deviation of the sleeping period of a sample of 100 students chosen at random in a school are 7.15 hours and 1.10 hours respectively.

(b) Estimate the standard error of the mean.

ng

(a) Estimate the mean and standard deviation of the sleeping period of all the students in the school. [3 marks] [1 marks]

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

3 25

4 32

4 26

7 38

7 34

(a) Plot a scatter diagram for the above data.

9 39

11 46

12 44

12 51

ak eo

(b) Comment on the relationship between x and y.

8 41

[2 marks]

[2 marks]

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

(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]

(b) State the range of the possible values of r.

[1 marks]

18

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2003

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

ng

A sample of ten data points may be summarised as follows:

X X X (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 prefer either documentary or drama programmes on television.

Male Female

Documentary 96 45

Drama 24 85

ak eo

A television viewer involved in the survey is selected at random. A is the event that a female viewer is 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 market in the first week of January and September of 2000.

Fish

Parang Tenggiri Bawal Putih Kembung Selar Kuning

January Price Quantity (RM per kg) (kg) 11.00 2 12.00 2 10.00 2 8.00 3 4.00 5

September Price Quantity (RM per kg) (kg) 12.00 1 13.00 2 a 1 10.00 2 5.00 6

(a) If the simple aggregate price index increases by 20% from January to September, determine the value of a. [3 marks] (b) Calculate the Laspeyres price index, and comment on the housewife’s change in expenditure on fish. [3 marks] (c) Calculate the Paasche quantity index, and comment on the housewife’s change in expenditure on fish. [3 marks] 8. Three companies X, Y , and Z offer taxi services in a town. The percentages of residents in the town using the taxi services from companies X, Y and Z are 40%, 50%, and 10% respectively. The probabilities of taxis from companies X, Y , and Z being late are 0.09, 0.06, and 0.20 respectively. A taxi 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 duration for each activity are shown in the table below.

19

Lee Kian Keong Activity A B C D E F G

Preceding activities B A, D C E, F

STPM 2004 Duration (days) 2 5 1 10 3 6 8

ng

PAPER 2 QUESTIONS

(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 training programme. [8 marks]

ak eo

10. The time taken by the customers of a company to settle invoices is normally distributed with mean 20 days and standard deviation 5 days. A discount is given for every invoice which is settled in less than 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. (d) Find the probability that at most 2 out of 10 invoices are given discounts.

[2 marks] [4 marks]

11. A manufacturer of wooden furniture produces two types of furniture: chairs and tables. Two machines are used in the production: a jigsaw and a lathe. Each chair requires 1 hour on the jigsaw and 1 hour on the lathe, whereas each table requires 1 hour on the jigsaw and 2 hours on the lathe. The jigsaw and lathe can operate 10 hours and 12 hours per day respectively. The profit made is RM27.00 on a chair 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 tables made 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 the decorative lamps at prices less than the recommended prices. (a) Find the 90% symmetric confidence interval for the proportion of shops selling the decorative lamps at prices less than the recommended prices. [5 marks] (b) Determine the smallest sample size required so that the estimated proportion of shops selling the decorative lamps at prices less than the recommended prices is within 2% of the actual proportion at the 90% confidence level. [5 marks] (c) Calculate the probability that more than 60% of a random sample of 500 shops sell the decorative lamps at prices less than the recommended prices. [3 marks]

STPM 2004

20

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2004

ng

1. According to a survey conducted in a company on job satisfaction, salary and pension benefits are two important issues. It is found that 74% of the employees are of the opinion that salary is important whereas 65% think that pension benefits are important. Among those who think that pension benefits are important, 60% think that salary is also important. Determine the percentage of employees who are 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 normally distributed with mean 300 kg and standard deviation 5 kg. A random sample of four packages is selected. Find the probability that the mean mass of the sample lies between 292 kg and 296 kg. [4 marks]

3. The table below shows the marks for the mid-semester examination, x, and the marks for the final semester examination, y, of 10 students. 70 85

58 66

81 78

66 76

85 93

92 81

54 65

93 95

75 80

65 70

ak eo

x y

(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 1 2 3 4 5 6

3 2 5 8 1

6 8 1

9 5 6

1

0

5

2

Key: 6k2 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 destinations within 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 y

50 185

64 231

35 165

58 193

45 213

34 160

74 198

Calculate the Pearson correlation coefficient of the above data. 7. The following table shows the activities involved in a particular project.

21

[6 marks]

Activity A B C D E F

Lee Kian Keong Preceding activities B A, C A D, E

STPM 2004

Duration (days) 5 1 2 4 6 3

Earliest start 0 0 1 5 5 11

Latest start 5 5 7 11 11 14

ng

PAPER 2 QUESTIONS

(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 the project. [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) Number of persons

0-9 3

10 - 14 6

15 - 19 15

20 - 24 25

25 - 29 16

30 - 49 10

ak eo

(a) Calculate the mode and the median. (b) Calculate the percentage of persons whose ages are between 18 and 35 years.

[4 marks] [6 marks]

9. The letters in the word BANANA are to be rearranged. A word can be considered formed without being 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 ). (b) Find P(R ∩ S), P(R ∪ S), P(R ∩ T ) and P(R ∪ T ).

[5 marks] [5 marks]

10. A telecommunications company wants to estimate the proportion of customers who require an additional line. A random sample of 500 customers is taken and it is found that 135 customers require an additional line. (a) Obtain the 99% symmetric confidence interval for the proportion of customers who require an additional line. Interpret the confidence interval obtained. [6 marks] (b) If the company wants to estimate the proportion of customers who require an additional line at a different location, determine the smallest sample size required so that the error of estimation does not exceed 0.03 at the 95% confidence level. [5 marks] 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, whereas it costs RM0.40 to produce a 15 g strawberry-flavoured chocolate bar which is sold at RM0.55. The manufacturer has 426 kg of chocolate in stock. A minimum of 10 000 orange-flavoured chocolate bars and 12 000 strawberry-flavoured chocolate bars have to be produced. The number of strawberryflavoured 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 bars of each flavour that should be produced in order to maximise total profit. (b) Show the feasible region and hence solve the linear programming problem using the method.

chocolate [6 marks]

graphical [9 marks]

12. The table below shows the quarterly gas consumption (in thousand cubic metres) of a factory from the year 1999 to 2002. 22

Lee Kian Keong

Year

Quarter 1 87 86 83 90

1999 2000 2001 2002

STPM 2005

Gas consumption Quarter 2 Quarter 3 68 62 70 60 72 59 70 64

Quarter 4 82 81 82 84

ng

PAPER 2 QUESTIONS

(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]

STPM 2005

1. Find the median and the interquartile range of the following numbers. 8

17

22

19

10

29

6

6

21

20

12

18

ak eo

21

25

[3 marks]

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

21.8 16.6

22.2 16.7

22.3 16.4

22.3 16.5

22.5 16.8

22.6 17.0

23.3 16.8

23.6 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 µ and standard deviation 5.1 minutes. (a) Given that 97.5% of the pupils require less than 40 minutes to complete the task, find the value of µ. [3 marks] (b) Find the probability that a pupil chosen at random takes more than 35 minutes to complete the task. [2 marks] 4. A study is conducted to assess the impact of the size of a store, x (in m2 ) on daily sales, y (in RM). A random sample of six stores is taken from several shopping centres. The data obtained are summarised as follows. X X X x = 24400, y = 28368, (x − x ¯)(y − y¯) = 6780, X X 2 (x − x ¯) = 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 the coefficient of determination is 0.674. State whether the size of a store or the size of a shopping centre is 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 a competition held by the magazine. A preliminary survey of 100 subscribers is conducted to find out their participation in the competition. (a) Determine the sampling distribution of the proportion of the subscribers who will participate in the competition, stating its mean and variance. [3 marks] 23

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2005

(b) Find the probability that at least 30% of the subscribers will participate in the competition.

ng

[3 marks]

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

Type of vegetables Spinach Water spinach Cabbage Red chilli

November 2003 Price Quantity (RM per kg) (kg) 3.00 250 2.00 130 6.00 50 10.00 20

December 2003 Price Quantity (RM per kg) (kg) 4.00 200 3.00 150 5.00 100 8.00 20

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

ak eo

i. a simple aggregate price index for December 2003, ii. a weighted price index for December 2003.

[2 marks] [2 marks]

(b) Which of the two price indices in (a) is more suitable to measure the changes in prices of the vegetables? Based on your answer, comment on the changes in prices of the vegetables from November 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 random variable having probability distribution function  2  2k , x = 0, 3, P(X = x) = 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 0.35 if 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: iron frame, electrical component and plastic component. Each lamp A requires 1 unit of iron frame, 2 units of electrical components and 3 units of plastic components, whereas each lamp B requires 3 units of iron frames, 2 units of electrical components and 1 unit of plastic component. The company has 300 000 units of iron frames, 300 000 units of electrical components and 400 000 units of plastic components in stock. The profits made from each lamp A and lamp B are RM15.00 and RM20.00 respectively. (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 which give the maximum profit and find this maximum profit. [8 marks]

24

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2005

35 34 29 26 47

Urban 35 36 34 45 39 40 40 31 43 36

38 30 40 35 35

45 47 40 40 25

ng

10. An advertising firm conducted a survey on television viewing habits in urban and rural areas. The table below shows the number of hours per week spent watching television by 20 persons in urban areas and 18 persons in rural areas. Rural 16 48 24 50 48 34 34 40 8

31 32 42 44

(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]

(c) Calculate the mean and the standard deviation of the number of hours spent watching television for each area. [6 marks] [1 marks]

ak eo

(d) Compare the dispersion of the two distributions.

11. The following table shows the activities, their preceding activities and their durations for a project. Activity A B C D E F G

Preceding activities A A B, C B A D, E, F

Durations (weeks) 7 3 3 4 5 3 6

(a) Draw an activity network for the project.

[3 marks]

(b) Construct a table which shows the earliest start time, earliest finish time, latest start time, latest finish 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 the project will be delayed. [3 marks] 12. The following table shows the quarterly amounts of mileage claims (in thousand RM) made by employees of a company from year 2001 to year 2003. Year

2001 2002 2003

Amount of mileage claims (in thousand RM) Quarter 1 Quarter 2 Quarter 3 Quarter 4 300 455 560 590 314 470 570 610 420 480 600 620

(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 quarter of year 2004. [3 marks]

25

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2006

STPM 2006

18 19 21 19 21

18 20 51 20 62

20 8 20 19 19

30 19 19 64 20

26 21 20 19 22

ng

1. The ages of patients visiting a clinic on a particular day are as follows. 28 43 22 18 19

18 19 20 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.

ak eo

(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 in a factory for the years 2000 and 2003.

Year

2000 2003

Operator Wage Number of (RM) workers 15.50 65 19.00 95

Wage (RM) 17.00 22.50

Clerk Number of workers 24 28

Supervisor Wage Number of (RM) workers 21.50 10 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 with mean 45 minutes and standard deviation 3 minutes. Determine the time when the manager has to leave 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

P(X = x)

p

1 5 21

2 10 21

3 5 21

4

p

(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 first quarter of year 2000 to the fourth quarter of year 2003 is shown below.

26

Lee Kian Keong

STPM 2006

ak eo

ng

PAPER 2 QUESTIONS

(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 moving averages. Year

2000

2001

2002

2003

Quarter 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Number of houses 112 118 155 95 119 126 168 110 129 126 182 135 136 145 195 158

Centred moving average

120.875 122.750 125.375 128.875 132.000 133.250 135.000 139.875 143.875 147.125 151.125 155.625

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 shown in the following table. Amount of purchase Less than RM50 RM50 or more

Mode of payment Cash Credit card 50 25 75 150

A customer is selected at random from this group of customers. 27

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2006

(a) Find the probability that the payment is made by cash.

[1 marks]

ng

(b) Find the probability that the amount of purchase is less than RM50 and the payment is made by cash. [1 marks] (c) If the amount of purchase is at least RM50, find the probability that the payment is made by cash. [1 marks]

(d) Find the probability that the amount of purchase is less than RM50 given that the payment is made by cash. [1 marks] (e) State, with a reason, whether the events “the amount of purchase is less than RM50” and “the payment 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 “the payment 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 deviation 10 000 km.

ak eo

(a) Determine the probability that a randomly chosen tyre has a lifespan of less than 80 000 km. [2 marks]

(b) Find the probability that the mean lifespan of 10 randomly chosen tyres is more than 68 000 km but less than 75 000 km. [4 marks] (c) Determine the minimum number of tyres to be chosen so that the standard error does not exceed 3500 km at the symmetric 99% confidence interval. [4 marks] 9. The following table shows the activities, their durations and their preceding activities for a project. Activity A B C D E F G

Duration (weeks) 2 1 3 2 3 2 1

(a) Draw an activity network for the project.

Preceding activities A B C, D E F

[2 marks]

(b) Construct a table showing the earliest start time, earliest finish time, latest start time and latest finish time for each activity. Hence, determine the critical activities and find the minimum time needed to complete the project. [8 marks] (c) If the durations for activities C and D are each reduced by a week, determine whether the project 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 correlation. [2 marks] (c) Obtain the equation of the regression line of y on x in the form y = a + bx, where a and b are given to three decimal places. [6 marks] (d) Estimate the value of y corresponding to x = 4.5.

28

[2 marks]

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2007

Process Assembling Testing Packaging

ng

11. A computer manufacturing company produces three types of computer: home desktop, business desktop and notebook. Each type of computer needs to 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 a home desktop, a business desktop and a notebook and the number of man-hours available per week. Number of hours required Home desktop Business desktop Notebook 5 6 8 10 12 12 2 4 2

Number of man-hours available per week 400 648 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 maximise the weekly profit and find this maximum profit. [9 marks]

ak eo

12. The table below shows the duration, in seconds, taken by 100 workers to finish a task. Duration (x seconds) 0 < x ≤ 100 100 < x ≤ 200 200 < x ≤ 250 250 < x ≤ 300 300 < x ≤ 350 350 < x ≤ 400 400 < x ≤ 500

Number of workers 2 10 20 26 24 10 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 median and the percentage of workers who finish the task in more than 270 seconds. [7 marks]

STPM 2007

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

(a) Name the above diagrams.

[1 marks]

(b) Comment on the relationship shown in each of the diagrams. Between quality and price, which has a stronger relationship to customers preference? [3 marks] 29

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2007

ng

2. A study on 100 visitors to a book fair shows that 60 visitors have seen the ment about the fair. Out of 40 visitors who make purchases, 30 have seen the ment. Find the probability that a visitor who has not seen the ment 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 20 persons, 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 of µ 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 are given in the following table. Type of chalet

Number of chalets rented 2004 2006 400 200 320 380 80 200 50 100

ak eo

A B C D

Tariff (RM per day) 2004 2006 150 160 180 200 200 220 250 260

(a) Taking 2004 as the base year, calculate the price index for the year 2006 with the quantity for the 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 for the 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? Give a reason. [2 marks] 6. The table below shows the number of new current holders in certain branch of a bank from year 1991 to 2006. Year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Number of holders 21 23 17 20 15 18 15 10 15 11 16 11 16 15 20 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 of new current holders. Give a reason. [2 marks]

30

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2007

ng

7. The following table shows the age distribution of drivers who are at fault in accidents during a one-month period in a country. Age (years) 18-25 26-35 36-45 46-55 56-65 66-80

Number of drivers 20 24 14 18 18 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 shoppers are asked whether they consume vitamins and 83% of them say “Yes”.

ak eo

(a) Obtain a symmetric 95% confidence interval for the proportion of shoppers who say “Yes” and interpret this confidence interval. [6 marks] (b) Explain why an interval estimate is more informative than a point estimate.

[2 marks]

9. The following table shows the activities for a project and their preceding activities and duration. Activity A B C D E F G H I

Preceding activities A B B E C, D, F E, G H

Duration (weeks) 11 5 9 6 8 6 8 7 9

(a) Draw an activity network for the project.

[3 marks]

(b) Construct a table showing the duration, earliest start time, latest finish time and total float for each 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 22

90 51

13 83

43 11

59 32

52 43

(a) Construct a stemplot to represent the data.

32 34

40 73

58 81

68 65

76 62

53 38

37 45 [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.

31

PAPER 2 QUESTIONS

Lee Kian Keong x y

2 1.8

4 1.8

6 1.4

STPM 2008 8 1.1

10 0.9

ng

(a) Find the equation of the least squares regression line of Y on X. Interpret the regression coefficient obtained. [7 marks] (b) Determine the proportion of the change in the amount of chlorine that is explained by the time after 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 hours after treatment. Give a reason. [2 marks]

ak eo

12. A factory produces two types of batteries A and B. Every unit of battery A requires 2 hours of assembly and 1 hour of testing, whereas every unit of battery B requires 2.5 hours of assembly and 1.5 hours of testing. The factory has at most 500 hours of assembly per week and at most 300 hours of testing per week. It is specified that the number of battery B produced per week exceeds the number of battery A produced per week and that the number of battery A produced per week exceeds 50 units. 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 B that should be produced per week and find the maximum profit per week. [10 marks]

STPM 2008

1. Eight persons are invited to be seated in the front row of eight seats to watch a concert. If two of them 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) 40-44 45-49 50-54 55-59 60-64 65-74

Number of works 6 14 23 42 30 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 obtained from random observations on the variables are summarised as follows: X X X X 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 by a location. (a) According to a previous study, the standard deviation of the number of vehicles ing by the location per day is 245. Calculate the number of days required so that he is 99% confident that the estimate is within 100 vehicles of the true mean. [3 marks] 32

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2008

ng

(b) The standard deviation of the number of vehicles is actually 356. Based on the sample size obtained in (a), determine the confidence level for the estimate to be within 100 vehicles of the true mean. [3 marks] 5. A random variable X has a binomial distribution with parameters n and p. (a) Given that E(X) = 6 and Var(X) =

12 , determine the values of n and p. 5

(b) Calculate P(X = 5).

[4 marks]

[3 marks]

6. In a country, 78% of consumers are in favour of government control over prices. A random sample of 400 consumers is selected.

ak eo

(a) Find the mean and standard deviation of the distribution of the sample proportion. [3 marks] (b) Find the probability that the sample proportion is at least 5% lower than the population proportion. [4 marks]

7. The table below shows the prices and the total sales of three grades of eggs for the month of January in 2000 and 2005. Grade A B C

January 2000 Price Total sale (RM per dozen) (RM ’000) 1.80 180 1.50 300 1.20 240

January 2005 Price Total sale (RM per dozen) (RM ’000) 2.60 780 2.00 600 1.20 120

Taking 2000 as the base year, calculate the Laspeyres and the Paasche price indices for the year 2005. Explain why one index is higher than the other.

[6 marks] [2 marks]

8. The distribution of the selling prices of 80 houses in a city is shown below. Selling price (RM ’000) 120150180210240270300330360-

Number of houses 8 23 17 18 8 4 1 1 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 offered scholarship S is 0.4. If he is offered scholarship S, the probability of him being offered scholarship T is 0.2. If he is not offered scholarship S, the probability of him being offered scholarship T is 0.7. 33

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2008

(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]

ng

(c) State, with a reason, whether the events ‘he is offered scholarship S’ and ‘he is offered scholarship T ’ are mutually exclusive. [2 marks] 10. The following table shows the activities for a project and their preceding activities and duration. Activity A B C D E F

Preceding activities A B, C B D, E

Durations (weeks) 3 4 2 5 2 3

ak eo

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

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

(c) If the duration of activity E has to be extended for a week, determine whether the project will be delayed. [3 marks] 11. A factory assembles three models of chairs using six components. The following table shows the number of units of components needed for each model and the total units of components available per week. Component

Type 1 seat Type 2 seat Chair frame Chair leg Type 1 backrest Type 2 backrest

Number of units of components Model P Model Q Model R 1 0 0 0 1 1 1 1 1 4 4 4 1 1 0 0 0 1

Total units of components available per week 500 1000 1000 4000 1000 500

The profits for models P , Q and R are RM35, RM40 and RM50 per unit respectively. The manager of the factory wishes to determine the number of chairs of each model to be produced per week in order to maximise the total profit. (a) If x1 , x2 , and x3 represent the numbers of chairs of models P , Q and R respectively, formulate a linear programming model to determine the number of chairs of each model that should be produced 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 should be produced per week in order to maximise the total profit, and calculate the maximum total profit. [4 marks] Basic s1 x2 x1 s4 s5 x3

x1 0 0 1 0 0 0

x2 0 1 0 0 0 0

x3 0 0 0 0 0 1

s1 1 0 0 0 0 0

s2 1 1 -1 0 0 0 34

s3 -1 0 1 -4 -1 0

s4 0 0 0 1 0 0

s5 0 0 0 0 1 0

s6 0 -1 0 0 1 1

Solution 500 500 0 0 500 500

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2009

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

2005 2006 2007

Quarter 1 23 29 31

Profit (RM ’000) Quarter 2 Quarter 3 45 69 44 81 49 97

ng

Year

Quarter 4 52 51 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 for quarter 1 of the year 2005. Using an additive model, calculate the adjusted quarterly seasonal variations. [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]

STPM 2009

ak eo

1. The duration of telephone calls made by 175 individuals at a call center on a particular day is shown in the table below: Duration (minutes) 1-7 8-14 15-21 22-28 29-35 36-42 43-49 50-56

Frequency 15 32 34 22 16 12 9 5

(a) Construct a histogram to display the data. (b) i. Draw a frequency polygon on the histogram in (a). ii. Comment on the skewness of the distribution obtained.

[2 marks] [1 marks] [1 marks]

2. On the average, a button making machine is known to produce 6% defective buttons. A random sample of 100 buttons is inspected and if eight or more buttons are found to be defective, the operation of the machine will be stopped. (a) State the sampling distribution for the sample proportion of defective buttons. (b) Find the probability that the operation of the machine will be stopped.

[1 marks] [3 marks]

3. A rice dispenser is capable of filling up a cup of rice with an average weight of 130 g. Given that the weight 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 are given in the table below. x y z

1 5 13

3 10 18

5 15 13

5 20 15

7 15 19 35

7 25 14

8 20 17

10 25 19

10 35 15

12 35 17

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2009

(a) Plot scatter diagrams to show the relationship between i. travelling time and distance, ii. travelling time and age.

[2 marks]

ng

[2 marks]

(b) Based on the scatter diagrams in (a), which variable has a stronger linear relationship with travelling 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 should be allowed to bring handphones to school. The data obtained are shown in the table below.

Town A Town B

Agree 100 20

Neutral 50 60

Disagree 50 120

(a) Find the probability that a mother agrees if she is from Town A. (b) Determine whether

ak eo

i. the events ‘Town B’ and ‘disagree’ are independent, ii. the events ‘Town A’ and ‘disagree’ are mutually exclusive.

[2 marks]

[4 marks]

[1 marks]

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

Tea Coffee Chocolate

1998 Price Number of (RM per cup) cups 0.60 10 0.70 13 0.80 3

2008 Price Number of (RM per cup) cups 1.00 18 1.20 7 1.50 10

(a) Taking 1998 as the base year,

i. calculate the simple aggregate price index for the year 2008, ii. comment on the changes in the prices of beverages from 1998 to 2008.

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

[2 marks] [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 P(X = x)

0 0.02

1 m

2 n

3 0.07

Given that the mean number of cars owned is 1.38.

(a) Determine the values of m and n. (b) Calculate the variance of X. (c) Find the probability that a randomly chosen household owns at least two cars.

[4 marks] [3 marks] [2 marks]

8. The number of pens x (in thousand) produced by a factory and the total cost of production y (in thousand RM) for 20 consecutive days are recorded. The results obtained ate summarised as follows: X X X X X x = 176.00, y = 400.00, x2 = 1780.00, y 2 = 8150.00, xy = 3700.20.

36

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2009

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

ng

(b) Find the equation of the least squares regression line of the total cost on the number of pens produced. 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 project are listed in the table below. Activity Project application and approval Project design Project design approval Land clearing Machinery and equipment purchase Building construction Landscaping Park construction Testing Opening ceremony

Duration (month) 10 6 3 2 6 8 6 10 3 1

Preceding activities A B A C C, D C, D C, D E, F , G, H I

ak eo

A B C D E F G H I J

(a) Draw an activity network for the project.

[4 marks]

(b)

[4 marks]

i. List all the possible paths of the project and their corresponding total duration. ii. Determine the critical path. iii. Find the minimum time required to complete the project.

[1 marks] [1 marks]

10. A random sample of 200 students of a university is selected and it is found that 120 of them stay in university hostels. (a) Estimate the proportion of the students who stay in the hostels and determine the standard error. [3 marks] (b) Construct a 95% confidence interval for the proportion of the students who stay in hostels, and interpret your answer. [4 marks] (c) What is the effect on the confidence interval if the confidence level is increased from 95% to 99%? [3 marks] 11. The number of luxury cars sold for each quarter y by a company and their coded quarters x for a duration of four years are shown in the table below.

37

Lee Kian Keong Year 2005

2006

2007

Coded quarter (x) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Number of cars (y) 10 14 11 21 11 16 10 22 14 18 13 22 13 16 9 25

ak eo

2008

Quarter 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

STPM 2010

ng

PAPER 2 QUESTIONS

(a) Plot the data as a time series.

[3 marks]

(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, find the adjusted seasonal variation for each of the four quarters. Write down your answers correct to three decimal places. [5 marks] (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 P requires 3.5 hours of production time, 1 hour of assembly time and 1 hour of packaging time. Each unit of model Q requires 8 hours of production time, 1.5 hours of assembly time and 1 hour of packaging time. The maximum available resources for each process in a day is as follows: Process Production Assembly Packaging

Resource available per day (hours) 280 60 50

The manager of the company wishes to maximise profit. Each unit of P yields a profit of RM400 while each unit of Q yields a profit of RM800. Due to high demand, the company has to produce at least 10 units of each model per day. (a) If x and y represent the quantities of models P and Q produced each day respectively, formulate the 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 maximum profit, [1 marks] ii. find the daily maximum profit.

STPM 2010

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PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2010

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

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(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 the employee is a female. [2 marks] (c) Determine whether the events ‘an employee has a degree qualification’ and ‘an employee is a female’ 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 the number of bags of rice sold in the years 2007 and 2008 are given as follows.

Type of rice

Number of bags of rice (million) 2007 2008 11 13 17 20 14 12

ak eo

A B C

Price (RM per bag) 2007 2008 19.50 25.00 23.00 32.50 30.50 40.00

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 random sample 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 a reason for your answer. [2 marks] 4. A census conducted in a school shows that the total hours per week pupils spent watching television has a mean of 16.87 hours and a standard deviation of 5 hours. If a random sample of 100 pupils is taken, find 3 hour of the population mean, 4 (b) the probability that the sample mean is more than 17 hours. (a) the probability that the sample mean is within

[4 marks] [2 marks]

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

i. What does r measure? ii. State the range of the possible values of r and what it means when r = 0.

[1 marks] [2 marks]

(b) A sample of 10 data points may be summarised as follows: X X X (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 period from 2006 to 2009 is shown below.

39

Lee Kian Keong

(a) Comment on the pattern of the time series.

STPM 2010

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PAPER 2 QUESTIONS

[1 marks]

ak eo

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

2006

2007

2008

2009

4-month period 1 2 3 1 2 3 1 2 3 1 2 3

Number of holiday packages sold 103 221 522 114 322 654 135 293 676 170 294 828

Moving average 282.000 285.667 319.333 363.333 370.333 360.667 368.000 379.667 380.000 430.667

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

7. A chemical company produces two types of organic fertilisers at its factory. Three types of raw materials P , Q and R are mixed to produce Type X and Type Y fertilisers. Each ton of Type X fertiliser is a mixture of 0.4 ton of P and 0.6 ton of R, while each ton of Type Y fertiliser is a mixture of 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 Y fertilisers are RM400 and RM300 respectively. The quantities of raw materials useable per week are shown in the following table. Raw material P Q R

Quantity of material useable per week (tons) 20 5 21

(a) If x and y represent the quantities, in tons, of Type X and Type Y fertilisers produced each week, formulate a linear programming model that can be used in order to maximise total profit per 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 be produced per week in order to maximise the total profit, and calculate the total profit.

40

Lee Kian Keong Basic

x

y

y

0

1

s2

0

0

x

1

0

s1 10 3 2 − 3 5 − 3

s2 0 1 0

STPM 2010 s3 20 − 9 4 9 25 9

Solution 20

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PAPER 2 QUESTIONS

1

25

[2 marks]

8. The prices of houses sold, x, in hundred thousand RM, in a municipality in the year 2007 is shown below. Frequency 15 10 14 8 10 7 5 3

ak eo

Price (hundred thousand RM) 2.5 < x ≤ 3.0 3.0 < x ≤ 3.4 3.5 < x ≤ 4.0 4.0 < x ≤ 4.5 4.5 < x ≤ 5.0 5.0 < x ≤ 5.5 5.5 < x ≤ 6.0 6.0 < x ≤ 6.5

(a) Plot a cumulative frequency curve for the above data. Hence, estimate the median and the quartiles. [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 proportion of a random sample of 100 adults is found to be in favour of banning cigarettes. (a) State the sampling distribution.

[2 marks]

(b) Find the probability that the sample proportion in favour of banning cigarettes is i. at least 6% lower than the population proportion, ii. within one standard deviation of the population proportion.

[3 marks] [3 marks]

10. Mr. Tan works five days a week and he takes either a bus, a commuter or a taxi to his office. The probability that he takes a taxi is 0.20 and the probability that he takes a commuter is 0.50. The probabilities 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 is likely 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 alternate days. [4 marks] 11. A group of students are involved in an orientation programme for new form six students. Nine activities are required in order to organise the programme. The activities and the duration for each activity are shown in the table below.

41

Lee Kian Keong Activity A B C D E F G H I

Preceding activities A A, B A C C E, F D, G

(a) Draw an activity network for the programme.

STPM 2010 Duration (days) 3 3 4 2 5 2 6 4 2

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PAPER 2 QUESTIONS

[3 marks]

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

ak eo

(c) If the duration of activity B has to be extended to four days, determine whether the programme will be extended or not. Give a reason for your answer. [2 marks] 12. The istration department of a private hospital conducted a study to determine whether a relationship exists between the number of days a patient is warded, X, and the total number of times the patient calls for a nurse, Y . The data collected is given below. Patient 1 2 3 4 5 6 7 8 9 10

Number of days in ward (X) 2 4 5 6 3 8 12 7 11 2

The number of calls for a nurse ((Y )) 2 3 3 4 2 6 10 5 8 1

(a) Plot a scatter diagram for the above data, and comment on the relationship between the two variables. [4 marks] (b) Find the equation of the least square regression line of Y on X, giving your answer correct to three 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]

(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 nurse made by the patient? Comment on your answer. [2 marks]

42