STPM Mathematics M

STPM Mathematics S Past Year Questions Lee Kian Keong & LATEX [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. Studen Students ts should should use this this documen documentt as reference reference and try all the questi questions ons if possible possible.. Studen Students ts are 1 2 encourage to contact me via email emai l or facebook . Students Students also encourage encourage to send me your collection of papers or questions questions by email because i am collecting various various type of papers. All papers are welcomed. Special thanks to Zhu Ming for helping me to check the questions.

Contents 1

PAPER APER 1 QUES QUESTIO TIONS NS

STPM 2002   . STPM 2003   . STPM 2004   . STPM 2005   . STPM 2006   . STPM 2007   . STPM 2008   . STPM 2009   . STPM 2010   . STPM 2011   . STPM 2012   . 2

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PAPER APER 2 QUES QUESTIO TIONS NS

STPM 2002   . STPM 2003   . STPM 2004   . STPM 2005   . STPM 2006   . STPM 2007   . STPM 2008   . STPM 2009   . STPM 2010   . STPM 2011   . STPM 2012   .

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3 5 7 9 11 13 15 17 19 21 23 25

1

[email protected]

2

http://www.facebook.com/akeong

1

26 29 32 35 38 42 45 48 52 57 60

PAPER 1 QUESTIONS

1

Lee Kian Keong

PAPER APER 1 QUES QUESTI TION ONS S

2

PAPER 1 QUESTIONS

1

Lee Kian Keong

PAPER APER 1 QUES QUESTI TION ONS S

2

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2002

STPM 2002 1. The function function f  is defined by f  :  x

 → √ 3x + 1, x ∈ R, x ≥ − 13 .

Find  f −1 and state its domain and range.

2. Given Given that y  =  e −x cos x, find

[4 marks]

dy d2 y and   when  x  = 0. dx dx2

[4 marks]

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

 

is a symmetric matrix.

4. By using suitable substitutio substitution, n, find

2b 2a

a2 a

−1 −1

b2 bc 2c 1

−

b+c

b

 

[5 marks]

3x

1 dx. x+1

  √  −

[5 marks]

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

6. Express Express

7. Express Express

  − 59

√ 

√ 

√ 

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

[7 marks]

1 4k

2

− 1   as partial fraction.

[4 marks] n

Hence, find a simple expression for S n  =



k=1

1 4k

2

and find find − 1   and

lim lim S n .

[4 marks]

n→∞

8. Given Given that  PQRS  is   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  PQRS . [6 marks]

−

9. Find the point of intersection intersection of the curves y  = x2 + 3x  and  y  = 2x3 x2 5x. Sketch on the same coordinate system these two curves. [5 marks] Calculate the area of the region bounded by the curve y  = x2 + 3x  and y  = 2x3 x2 5x.[6 marks]

 −

− −

 −

10. Matrices Matrices  M and  N  are given as  M =

− −

10 15 5

4 4 1

9 14 6

− −

 

, and  N =

− −

 

2 3 4 4 3 1 1 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 different proportions. proportions. Each Each product 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  C ; each product of  Z   Z  consists of four components of  A  A , one component of  B  B , and two components of  C   C . A total of 750 components of  A  A , 1000 components of  B  B , and 500 components four components of  C 

3

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2002

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 Hence,

2

− 9x

(a) find all the real roots of 2x6

+ 3x + 4 has  x 4

− 9x

− 1 as factor.

[2 marks]

+ 3x2 + 4 = 0.

(b) determine the set of values of  x  so that 2x3 12. Function f  is defined by f (x) =

− 9x

2

[5 marks]

+ 3x + 4  <  12

2x (x + 1)(x

− 12x.

[6 marks]

− 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

|

−

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2003

STPM 2003

 −1 is the only one real root of the equation  x

1. Show that

√ 

2. If  y  = ln xy , find the value of 

3

+ 3x2 + 5x + 3 = 0.

dy   when y  = 1. dx

[5 marks]

π

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

 

cos θ dθ. (3 + 2 sin θ)2

6

0

[5 marks]

4. If (x + iy )2 =  i , find all the real values of  x  and y. 5. Find the set of values of  x  such that

[5 marks]

3

[6 marks]

2

 −2 < x − 2x

+x

− 2 <  0.

[7 marks]

6. The function f  is defined by f (x) =

 

x <  1 1 + ex , 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 passes 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 passes through B . If  l 2  intersects the x-axis and y -axis at the points  Q  and R  respectively, show that P R  :  QR  = 5 : 3. [8 marks]

√ 





1

1+x 2 8. Express as a series of ascending powers of  x  up to the term in  x 3 . 1 + 2x 1 By taking  x  = , find 62 correct to four decimal places. 30

√ 

 

1 3 0

2 1 1

 

−3 1 9. The matrix A  is given by  A = −2 (a) Find the matrix B  such that B = A − 10I, where I  is the 3 × 3 identity matrix. 2

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

−

[6 marks] [3 marks]

[3 marks] [6 marks]

 

10. The curve y = x(b x), where a = 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] 11. Sketch, on the same coordinate axes, the graphs y = ex and y =

2 . Show that the equation 1+x

(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]

−

5

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2003

2   in partial fractions. r 2 + 2r Using the result obtained,

12. Express ur =

 −1r + r1

(a) show that u2r =

2

n

3 ur = (b) show that 2 r =1



−

+

[3 marks]

1 1 + , r + 2 (r + 2) 2

1 n+1

[2 marks] ∞

−

∞





1 1 ur and ur+1 + r .  and determine the values of  n+2 3 r=1 r=1



6

[9 marks]

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2004

STPM 2004 e

1. Show that

 

ln x dx  = 1.

[4 marks]

1

√ 

1

2. Expand (1 x) 2 in ascending powers of  x  up to the term in  x 3 . 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 4 5 4 1 1 2 3

 

.

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

[6 marks]

5. The function f  is defined by

 

f (x) =

 ∈

x 1 , x+2 ax2 1,

−

−

0

 ≤ x < 2 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 8 units. Show that the locus of  P  is the ellipse +  = 1 and sketch the ellipse. [7 marks] 16 12

−

7. Sketch, on the same coordinate axes, the graphs of  y  = 2 Hence, solve the inequality 2

−x>

 

2+

1 x

 

− x  and y =

.



1

2+

x



.

[4 marks] [4 marks]

8. Using the sketch graphs of  y  =  x 3 and  x + y  = 1, show that the equation  x 3 + 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.

[5 marks]

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

 

2 0

−2 0

0 2

a

b

c

 

and Q =

− 

1 0 0

1 0 2

−

0 1 2

−

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

10. A curve is defined by the parametric equations x = 1 7

 

[5 marks]

× 3 identity matrix.[5 marks]

− 2t, y = −2 + 2t .

Find the equation of the

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2004

−

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 y2 =  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 2 π  radians about the  y -axis. [4 marks]

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

−r 1−r

a(1

n

)

[3 marks]

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

− 3 + 23 − . . .  are  respectively. Determine the smallest value of  n  such that |S  − S  |  <  0 .001.[7 marks]

(b) The sum of the first n terms and the sum to infinity of the geometric series 6 S n and  S ∞

n

8

∞

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2005

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





∩ B) − (A ∩ B) = B



[4 marks]

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

 

x

dx

[4 marks]

dx

3. The point R  divides the line joining the points  P (3, 2) and Q (5, 8) in the ratio 3 : 4. Find the equation of the line passing 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  terms and the sum to infinity is less than 0 .01. [6 marks]

 | − 2| < x1   where  x  = 0.

5. Find the solution set of inequality x

[7 marks]

6. Find the perpendicular distance from the centre of the circle x 2 + 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  =  e x and  y  = 2 + 3e−x . Calculate the area of the region bounded by the y -axis and the curves. 8. A, B and C  are B2 = C. 1 2 0 1 If  B = 1 0

 

−

[2 marks] [6 marks]

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

0 0 1

 

, find C  and  A .

[7 marks]

9. The complex numbers z1 and  z 2   satisfy the equation  z 2 = 2

−2

√ 

3 i.

(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  z 1 and  z 2 , 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 .

11. Express f (x) =

[6 marks]

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

− − −

[5 marks]

1 1 Hence, obtain an expansion of  f (x) in ascending powers of   up to the term in 3 . x

9

x

[6 marks]

PAPER 1 QUESTIONS

Lee Kian Keong

Determine the set of values of  x  for which this expansion is valid.

STPM 2005 [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  =  x 2 + 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]

10

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2006

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)

x

10

  2+

9

≈ 2 (2 − 7x).

2

[4 marks]

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

 

k

2k + 1 0

1 3 3 2 k 2

−

 

is 0.[4 marks]

1

4. Using trapezium rule, with five ordinates, evaluate

    − 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.

6. Express

Ax + B C  2x + 1  in the form 2 +   where  A ,  B and C  are constants. (x + 1)(2 x) 2 x x +1 2

−

1

Hence, evaluate

  0

−

2x + 1  dx . (x + 1)(2 x) 2

[6 marks]

[3 marks]

[4 marks]

−

10− 5 7. The nth term of an arithmetic progression is T n , show that U n = ( 2)2( 17 ) is the nth term of  2 a geometric progression. [4 marks]

−

1 If  T n  = (17n 2

∞

− 14), evaluate

8. Show that x2 +  y 2 a2 + b2 c.

 

−

T n



U n .

[4 marks]

n=1

− 2ax − 2by  +  c   = 0 is the equation of the circle with centre (a, b) and radius

[3 marks]

C 3 C 1

C 2

The above figure shows three circles  C 1 ,  C 2  and  C 3  touching one another, where their centres lie on a straight line. If  C 1 and  C 2  have equations  x 2 + y2 10x 4y +28 = 0 and  x 2 + y 2 16x + 4y + 52 = 0 respectively. Find the equation of  C 3 . [7 marks]

−

11

−

−

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2006

9. Functions  f , g and  h  are defined by f  :  x

 → x +x 1 ;

g  :  x

 → x +x 2 ;

h  :  x

 → 3 + x2

(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

[2 marks]

◦ f . Give a reason for your answer.

10. The polynomial p(x) =  x 4 + ax3 7x2 4ax + b  has a factor x + 3 and when divided by  x 3, has remainder 60. Find the values of  a  and b  and factorise p(x) completely. [9 marks] 1 Using the substitution  y  = , solve the equation 12y 4 8y 3 7y 2 + 2y + 1 = 0. [3 marks]

−

−

−

−

x

 

 

 −

−

 

a 5 2 3 1 18 b 1 4 3 , Q = 1 12 11. If  P = and PQ = 2I, where I   is the 3  3 identity 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]

(b) Show that f  (t)  >  0

[5 marks]

(c) Show that k [1

2





− 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→∞

12

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2007

STPM 2007 1. Express the infinite recurring decimal 0.72˙ 5˙ (= 0.7252525 . . . ) as a fraction in its lowest terms.[4 marks]

2. If  y  =

3. If loga

x

1+x

2

x a2

 

, show that x2

= 3 loga 2

dy = (1 dx

2

2

− x )y .

[[ marks]4

− log (x − 2a), express x  in terms of  a .

[6 marks]

a

4. Simplify

√  − √  √  √ 

( 7 3)2 (a) , 2( 7 + 3) 2(1 + 3i) (b) , where i  = (1 3i)2

−

[3 marks]

√ −1.

[3 marks]

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

 



x

y

, x2 + y 2 x2 + y 2



 

respectively, where  x = 0

and y = 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 (a) (b)

   

x2 + x + 2 dx, x2 + 2 x dx. x+1 e

[3 marks] [4 marks]

7. Find the constants A,  B , C  and D  such that A B C  D 3x2 + 5x . = +  + + 2 2 2 (1 x )(1 + x) 1 x 1 + x (1 + x) (1 + x)3

−

−

[8 marks]

8. The function f  is defined by f (x) =

(a) Find

lim f (x),

x→−1−

√ 

x + 1,

|x| − 1,

−1 ≤ x < 1,

  otherwise.

lim f (x), lim f (x) and lim f (x).

x→−1+

[4 marks]

x→1+

x→1−

(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 4 0 1 2

 −   −− ,B =

Find the matrix A2B  and deduce the inverse of  A.

13

35 27 3

19 18 13 45 12 5

−

 

.

[5 marks]

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2007

Hence, solve the system of linear equations x 3x

− −

2y y y

− −

+

z 4z 2z

= = =

−8, −15, 4. [5 marks]

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

√ −

dy 3x 5 =   , where x >  0. If  dx 2 x

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]

 1 11. Using the substitution y  =  x + , express  f (x) =  x 3 x

− 4x − 6 − 4x + x1  as a polynomial in  y .   [3 marks] 3

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

12. Find the coordinates of the stationary points on the curve y = Sketch the curve. Determine the number of real roots of the equation x3 = k(x2

[10 marks]

x3 x2

− 1  and determine their[10nature. marks] [4 marks]

− 1), where k ∈ R, when k   varies.

[3 marks]

14

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2008

STPM 2008 1. The function f  and g  are defined by

 → x1 , x ∈ R \ {0}; g  :  x  → 2 x − 1, x ∈ R

f  :  x

Find  f  g  and its domain.

 ◦

3

2. Show that

  2

(x

2

− 2) 2

x

[4 marks]

5 dx  =  + 4ln 3



2 . 3

[4 marks]

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

∩ (B ∪ C ) ⊂ (A ∩ B) ∪ (A ∩ C )

A

[5 marks]

 | |

4. If  z  is a complex number such that z  = 1, find the real part of  1 5. The polynomial p (x) = 2x3 + 4x2 + x 2

1

1

− z.

[6 marks]

− k  has factor (x + 1).

(a) Find the value of  k .

[2 marks]

(b) Factorise p(x) completely.

[4 marks]

6. If  y  =

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

−

7. Matrix  A  is given by A =

 

1 1 1

0 0 1 0 2 1

− −

(a) Show that A2 = I, where  I  is the 3

[6 marks]

  × − .

3 identity matrix, and deduce  A −1 . 1 0 1

(b) Find matrix B  which satisfies BA =

4 3 2 1 0 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]

(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

10. Show that the gradient of the curve y  =

1

>  3

x

2

x

− x3 .

− 1  is always decreasing. 15

[10 marks]

[3 marks]

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2008

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] 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]

16

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2009

STPM 2009 1  ≥ . x+1 x+1 x

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

2. Given x > 0 and f (x) =

[4 marks]

√ x, find lim f (x) − f (x + h) .

[4 marks]

h

h→0

3 + . . ., obtain the smallest value of  n  if the difference between the 2 45 sum of the first n + 4 terms and the sum of the first n  terms 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  =  px 2 + qx , where  p = 0 at the point  x  = 1. Find the values of  p  and q . [6 marks]

 

5. Given that loga (3x

 −

− 4a) + log  3x = log2 a  + log (1 − 2a), a

a

2

where 0  < a <

1 , find  x . 2

[7 marks]

1

6. Using an appropriate substitution, evaluate

 

x2 (1

0

− x)

1 3

dx.

[7 marks]

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 , [2 marks] 4 (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  P A  :  AQ .[4 marks] 2

8. Find the values of  x  if  y  = 3

 | − x|  and 4y − (x − 9) = −24.

[9 marks]

9. (a) The matrices P,  Q  and R  are given by P=

 

1 2 1

5 6 2 4 3 2

− −

 −  −

13 1 7

,Q =

−50 −33 −6 −5 20

15

   − ,R =

Find the matrices  PQ  and  PQR  and hence, deduce ( PQ)−1 . (b) Using the result in (a), solve the system of linear equations 6x + 10y 2y x x + 2y

−

+ 8z   = + z = + 3z   =

4 1 2

−13 − −1 7 5 1

11

 

.

[5 marks]

4500 0 1080

.

[5 marks]

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

−  2t

and y  = 2t +

where  t = 0.

 

17

 1 t

PAPER 1 QUESTIONS

Lee Kian Keong

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

dy =2 dx

11. Given a curve y  =  x 2

−t

STPM 2009 dy  − 12 < dx < 2.

[8 marks] [3 marks]

− 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 p oints 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 (x + 2) and (x 2).

−

3

2

− ax − bx

+ 28 x  + 12, where a and b  are real constants, has factors

(a) Find the values of  a  and b, and hence, factorise  p (x) completely. (b) Give that p (x) = (2x

[7 marks]

3

− 3)[q (x) − 41+3x ], find q (x), and determine its range when  x ∈ [−2, 10].

[8 marks]

18

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2010

STPM 2010 1. Solve the following simultaneous equations: log3 (xy ) = 5 and

log9

x2 y

 

= 2. [4 marks]

2. Given that u  = show that

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

dy = 1. dx

√ u 1− 1 , 2

[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] 2e



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

 

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  . [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  = 7. The polynomial p(x) = 2x4 2x2 + x 1.

−

3

− 7x

 −1. Give a reason for your answer.

+ 5 x2 + ax  +  b , where a and b  are real constants, is divisible by

(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.

8. Given f (x) =

x3 3x 4 , (x 1)(x2 + 1)

−

[3 marks]

− −

19

[4 marks]

PAPER 1 QUESTIONS

Lee Kian Keong

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

STPM 2010 B x

−1

+

Cx + D , x2 + 1

[5 marks]

(b) when  x  is sufficiently small such that  x 4 and higher powers can be neglected, show that  f (x) 4 + 7 x + 3x2 x3 . [4 marks]

 ≈

−

9. Sketch, on the same coordinate axes, the graphs y = e−x and y = x + 4ex = 2 has a root in the interval [-1,0].

4

2

− x.

Show that the equation [6 marks]

Estimate the root correct to three decimal places by using Newton-Raphson method with initial estimate  x 0  = 0.4. [5 marks]

 −

10. A circle  C 1  passes through the points (-6, 0), (2, 0) and (-2, 8). (a) Find the equation of  C 1.

[4 marks]

(b) Determine the coordinates of the centre and the radius of  C 1 .

[2 marks]

(c) If  C 2  is the circle (x

2

− 4)

+ (y

2

− 11)

= 25,

i. find the distance between the centres of the two circles, ii. find the coordinates of the point of intersection of  C 1  with C 2 .

[2 marks] [3 marks]

11. The functions f  and g  are defined by 3

 → x − 3x + 2, x ∈ R. g  :  x  → x − 1, x ∈ R.

f  :  x

(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)

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

[3 marks]

1 =  k has h(x)

i. one root, ii. two roots, iii. three roots.

12. Matrix  P  is given by  P =

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

 

1 2 2

2 1 1

−

  − 1 3 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 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.

20

[2 marks] [5 marks]

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2011

STPM 2011 1. Solve the equation ln x + ln(x + 2) = 1. n

2. Show that

 r =1

[4 marks]

r2 + r 1 n2 = . r2 + r n+1

−

[4 marks]

e

3. Use the substitution  u  = ln x, evaluate

(x + 1)ln x

 

dx.

x2

1

4. Find the set of values of  x   satisfying the inequality 2x

√ 

[6 marks]

− 1 ≤ |x + 1|.

[6 marks]

 

5. Given that  y  is differentiable and  y x  = sin x, where  x = 0. Using implicit differentiation, show that d2 y dy x x + + dx2 dx 2

 − x2

 1 4

y  = 0. [6 marks]

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

 − 1m x + b, where m = 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 terms of  a and b, the perpendicular distance from R   to line P Q, and determine the area of triangle P QR. [5 marks] 7. The complex number z  is such that z

− 2z

∗

=

√ 3 − 3 , where i

 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]

2

8. Differentiate  e x with respect to x. Hence, determine integers  a ,  b  and c  for which 2

 

2

x3 ex dx  =

1

a c e . b [9 marks]

21

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2011

9. Functions  f  and  g  are defined by f  :  x g  :  x

 → ax

2

 → 2xx− 1 for x = 21 ;

+ bx + c,   where  a, b and  c  are constants.

 ◦

(a) Find  f  f , and hence, determine the inverse function of  f . 2

◦

(b) Find the values of  a,  b  and c  if  g f (x) =

 −3x

−

+ 4x 1 . (2x 1)2

2

(c) Given that p (x) =  x 2

[4 marks]

−

[4 marks]

− 2, express h(x) = 2xx −−25  in terms of  f  and  p .

[2 marks]

2

10. A  and B  are two matrices such that A=

− −

−3 −2

4 2 2

2

  − 6 4 3

and A2 B =

− 

 

2 6 0 2 0 4 0 4 2

(a) Find the determinant and adjoint of  A. Hence, determine A−1. (b) Using A −1 obtained in (a), find  B .

.

[6 marks] [4 marks]

11. The polynomial p (x) =  ax 3 + bx2 4x + 3, where a  and  b  are constants, has a factor (x + 1). When  p(x) is divided by (x 2), it leaves a remainder of  9.

−

−

 −

(a) Find the values of  a  and b, and hence, factorise  p (x) completely. [6 marks] p(x) (b) Find the set of values of  x  which satisfies  0. [4 marks] x 3 p(x) (c) By completing square, find the minimum value of  , x = 3, and the value of  x  at which it x 3 occurs. [4 marks]

−  ≥

−

12. The function f  is defined by 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. (d) Sketch the graph  y  =  f (x).

[6 marks] [2 marks]

22

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2012

STPM 2012 1. The sum of the first n  terms of a progression 3 n2 . Determine the n-th term of the progression, and hence, deduce thetype of progression. [4 marks] 2. Given that y  = (2x)2x , find

dy  in terms of  x . dx

[4 marks]

3. Differentiate tan x  with respect to x, and hence, show that π

 

3

x sec2 xdx  =

0

√ π3 − ln 2. [6 marks]

4. Given that 2 x x2 is a factor of  p (x) =  ax 3 x2 + bx the set of values of  x  for which  p (x) is negative.

− −

−

− 2. Find the values of  a  and  b . Hence, find

[6 marks]

5. Matrix  A  is given by A=

and A2 = A−1 . Determine the value of  x . 6. Functions  f  and  g

 −

1 1 1

◦ f  are defined by f (x) = e

x+2

 

1 0 0

x

−1 0

[7 marks]

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  and (log3 x)(log3 y) = 1. 4

=

8. Express in partial fractions (3r Show that

n

 r=1

(3r

−

−

[8 marks]

3 . 1)(3r + 2) [4 marks]

3 1 = 1)(3r + 2) 2

 − (3n 1+ 2) , [2 marks]

and hence, find ∞



r =1

9. The function f  is defined by

(3r

1 . 1)(3r + 2)

−

[2 marks]

−x

e f (x) = √  1+x

2

23

,   where  x

 ∈ R,

PAPER 1 QUESTIONS

Lee Kian Keong

STPM 2012

(a) Show that −x

f  (x) =

 −e

(x2 + x + 1) . 3 (1 + x) 2 [3 marks]

(b) Show that f  is a decreasing function.

[4 marks]

(c) Sketch the graph of  f .

[2 marks]

10. The function f  is defined by f  :  x

 → x − x, for x ≥ 21 . 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 + y 2 = 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 of  point  C . [7 marks] (c) Find the shortest distance from point C  to the line  AB .

12. Given that z 2 =

[4 marks]

2i . (1 + 3i)2

(a) Find the real and imaginary parts of  z 2 . Hence, obtain z1 and z2   which satisfy the above equation. [10 marks] (b) Given that z 1 and  z 3  are roots of 5x2 + ax + b  = 0, where a  and b  are integers. i. Find the values of  a  and b. ii. Determine z3   and deduce the relationship between z1 and z3 .

24

[3 marks] [3 marks]

PAPER 2 QUESTIONS

2

Lee Kian Keong

PAPER 2 QUESTIONS

25

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2002

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.

[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 a district. Drink  Tea Coffee Chocolate

Year 1997 Price Quantity (sen) (thousand cans) 80 15 150 3 220 1

Year 1998 Price Quantity (sen) (thousand cans) 120 12 170 3 230 3

Year 1999   Price Quantity   (sen) (thousand cans) 160 10 180 4 240 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 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]

26

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2002

7. The table below shows the number of students for a certain program according to terms 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-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 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, RM y model in a second-hand car shop. x   2 y   100

8 15

3 72

9 16

6 36

5 34

3

 × 10 , for eight cars of the same

6 30

3 68

(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] (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. The activities and their duration times (in days) are listed as follows: Activity A B C  D E F G

Preceding Activities A B C  D,  E 

Duration   (in days) 4 2 3 8 6 3 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 of  higher learning is as follows. 27

PAPER 2 QUESTIONS

Lee Kian Keong Marks 10-29 30-39 40-49 50-59 60-69 70-79 80-99

STPM 2002

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]

28

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2003

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. (a) Estimate the mean and standard deviation of the sleeping period of all the students in the school. [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, 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   3 y   25

4 32

4 26

7 38

7 34

8 41

9 39

11 46

12 44

12 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 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]

29

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2003

(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

2

− ¯y)

= 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

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. 30

PAPER 2 QUESTIONS

Lee Kian Keong Activity A B C  D E F G

Preceding activities B A, D C  E , F 

STPM 2003 Duration   (days) 2 5 1 10 3 6 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 training programme. [8 marks] 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. [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 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]

31

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2004

STPM 2004 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. x   70 y   85

58 66

81 78

66 76

85 93

92 81

54 65

93 95

75 80

65 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 1 2 3 4 5 6

3 2 5 8 1

6 8 1

9 5 6

1

0

5

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 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 (RM  y  10 3) for seven persons.

 ×

x   50 y   185

64 231

35 165

3

 × 10 ) and the amounts of insurance purchased

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. 32

[6 marks]

PAPER 2 QUESTIONS 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

(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

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

30 - 49 10 [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

∩

∪

[5 marks]

∩ T ) and P(R ∪ T ).

[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 chocolate bars 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 graphical method. [9 marks] 12. The table below shows the quarterly gas consumption (in thousand cubic metres) of a factory from the year 1999 to 2002. 33

PAPER 2 QUESTIONS

Lee Kian Keong

Year  1999 2000 2001 2002

Quarter 1 87 86 83 90

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

STPM 2004

Quarter 4 82 81 82 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]

34

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2005

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 y   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  = 24400,



(x



− ¯x)

2

− −

y  = 28368,

= 186333,

(x

(y

 ¯ x)(y

− ¯y) = 6780,

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] (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 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 35

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

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2005

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

P(X  =  x ) =

 

2k 2 ,

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 0.35 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] 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.

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

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]

36

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2005

(c) Calculate the mean and the standard deviation of the number of hours spent watching television for 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 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]

37

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2006

STPM 2006 1. The ages of patients visiting a clinic on a particular day are as follows. 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

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. (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 Clerk Supervisor   Wage Number of Wage Number of Wage Number of  (RM)   workers    (RM)   workers    (RM)   workers  15.50 65 17.00 24 21.50 10 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 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.

38

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2006

(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. 39

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2006

(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 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. (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.

40

[2 marks]

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2006

11. A computer manufacturing company produces three types of computer: home desktop, business desktop 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 a home desktop, a business desktop and a notebook and the number of man-hours available per week. Process  Assembling Testing Packaging

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] 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]

41

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2007

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] 2. A study on 100 visitors to a book fair shows that 60 visitors have seen the advertisement about the fair. Out of 40 visitors who make purchases, 30 have seen the advertisement. Find the probability that 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 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.   Tariff  (RM per day) 2004 2006   150 160 180 200   200 220   250 260

Type of chalet  A B C  D

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

(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 account holders in certain branch of a bank from year 1991 to 2006. 42

PAPER 2 QUESTIONS

Lee Kian Keong Year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

STPM 2007

Number of account 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 account holders. Give a reason. [2 marks] 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”. (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.

43

PAPER 2 QUESTIONS

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

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

STPM 2007 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

32 34

40 73

58 81

68 65

76 62

53 38

(a) Construct a stemplot to represent the data.

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. x 2 y   1.8

4 1.8

6 1.4

8 1.1

10 0.9

(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] 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]

44

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2008

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  = 136,



y  = 7650,



xy  = 187350,

Find the values of  a  and b  correct to two decimal places.



x2 = 3344. [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 the location per day is 245. Calculate the number of days required so that he is 99% confident that the estimate is within 100 vehicles of the true mean. [3 marks] (b) The standard deviation of the number of vehicles is actually 356. Based on the sample size obtained in (a), determine the confidence level for the estimate to be within 100 vehicles of the true mean. [3 marks] 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. (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.

45

PAPER 2 QUESTIONS

Grade  A B C 

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

STPM 2008 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. [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) 120150180210240270300330360The interval ‘120-’, for example, means RM 120 000

Number of houses  8 23 17 18 8 4 1 1 0

 ≤ 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. (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 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

(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]

46

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2008

(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 

x1

x2

x3

s1

s2

s3

s4

s5

s6   Solution 

s1 x2 x1 s4 s5 x3

0 0 1 0 0 0

0 1 0 0 0 0

0 0 0 0 0 1

1 0 0 0 0 0

1 1 -1 0 0 0

-1 0 1 -4 -1 0

0 0 0 1 0 0

0 0 0 0 1 0

0 -1 0 0 1 1

500 500 0 0 500 500

12. The table below shows the quarterly profits of a company from year 2005 to year 2007. Year  2005 2006 2007

  Quarter 1 23 29 31

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

(a) Plot the above data as a time series.

Quarter 4 52 51 63 [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]

47

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2009

STPM 2009 1. The duration duration of telephone telephone calls made by 175 individuals individuals at a call center center on a particular particular day is shown in the table below: Duration  (minutes) Duration  (minutes)   Frequency  1-7 15 8-14 32 15-21 34 22-28 22 29-35 16 36-42 12 43-49 9 50-56 5 (a) Construct Construct a histogram histogram to display the data.

[2 marks]

(b)

[1 marks]

i. Draw Draw a frequ frequenc ency y polygo polygon n on the the histo histogra gram m in (a). (a). ii. Comment Comment on the skewness skewness of the distribution distribution obtained. obtained.

[1 marks]

2. On the avera average, ge, a button button making making mac machin hinee is known known to produce produce 6% defectiv defectivee button buttons. s. A random random sample sample of 100 buttons buttons is inspect inspected ed and if eight eight or more more buttons buttons are found found to be defect defectiv ive, e, the operation of the machine will be stopped. (a) State the sampling distribution distribution for the sample sample proportion of defective defective buttons.

[1 marks]

(b) Find the probability probability that the operation of the machine will be stopped.

[3 marks]

3. A rice dispenser is capable capable of filling filling up a cup of rice with an average average weight weight of 130 g. Given Given that the weight of the rice is normally distributed with a standard deviation of 6 g. (a) Find the percentage percentage number 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 conducted to determine the correlation correlation between between the distance to school X   (in kilometer), student’s travelling time Y   (in minutes) and age of a student Z   (in years). years). The data obtained obtained are given in the table below. x   1 y 5 13 z   13

3 10 18

5 15 13

5 20 15

7 15 19

7 25 14

8 20 17

10 25 19

10 35 15

12 35 17

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

[2 marks] [2 marks]

(b) Based Based on the scatte scatterr diagra diagrams ms in (a), which which variab variable le has a strong stronger er linear linear relation relationshi ship p with with travelling time? Give a reason for your answer. [2 marks] 5. A survey was carried carried out among mothers in Town Town A and Town Town B on the issue whether whether pupils should be allowed to bring handphones to school. The data obtained are shown in the table below.

Town A Town B

Agre gree 100 20 48

Neut Neutrral 50 60

Disa Disagr greee  50 120

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2009

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

[2 marks]

(b) Determine Determine whether whether i. the events ‘Town ‘Town B ’ and ‘disagree’ are independent, ii. the events ‘Town ‘Town A’ and ‘disagree’ are mutually exclusive.

[4 marks] [1 marks]

6. The prices of three beverages beverages and the average average number number of cups consumed consumed by an individual 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 (RM  per cup )   cups  0.60 10 0.70 13 0.80 3

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

(a) Taking 1998 as the base year, i. calculate calculate the simple aggregate aggregate price index for the year 2008, ii. comment comment on the changes changes in the prices of beverages beverages from 1998 to 2008.

[2 marks] [1 marks]

(b) Taking 1998 as the base year and the number number of cup for the year 1998 as the weight, weight, i. calculate calculate the weighted weighted aggregate aggregate price index for the year 2008. ii. comment comment on the changes changes in the prices of beverages beverages from 1998 to 2008.

[2 marks] [1 marks]

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

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

P(X  =  x )

0 0.02

1

2

m

n   0.07

3

Given that the mean number of cars owned is 1.38. (a) Determine Determine the values values of  m and  n .

[4 marks]

 X . (b) Calculate Calculate the variance variance of  X 

[3 marks]

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

[2 marks]

8. The numbe numberr 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  = 176.00,



y  = 400.00,



x2 = 1780.00,



y 2 = 8150.00,



xy  = 3700.20.

(a) Calculate the Pearson correlation correlation coefficient between between the number of pens produced and the total cost. Interpret your answer. [5 marks] (b) Find Find the equatio equation n of the least least squares squares regressi regression on line of the total total cost on the numbe numberr of pens produced. Interpret the slope of the regression line. [4 marks] 9. A company company wishes to develop develop a theme park on a 120-acre land. The major activities activities of the project are listed in the table below.

49

PAPER 2 QUESTIONS

A B C  D E  F  G H  I  J 

Lee Kian Keong

Activity Pro ject application and approval Pro ject design Pro ject design approval Land clearing Machinery and equipment purchase Building construction Landscaping Park construction Testing Opening ceremony

STPM 2009

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

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

(a) Draw Draw an activity activity network network for the pro ject.

[4 marks]

(b)

[4 marks]

i. List List all the possible possible paths paths of the project project and their their corres correspond ponding ing total total duratio duration. n. ii. Determine Determine the critical critical path. iii. Find the minimum minimum time required to complete the project.

[1 marks] [1 marks]

10. A random sample of 200 students of a university university is selected and it is found that 120 of them stay in university hostels. (a) Estimate Estimate the proportion proportion of the students students who stay stay in the hostels and determine determine the standard standard error. [3 marks] (b) Construct Construct a 95% confidence confidence interval interval for the proportion of the students students who stay in hostels, and interpret your answer. [4 marks] (c) What What is the effect on the confidenc confidencee inter interv val if the confidenc confidencee level level is increa increased sed from 95% to 99%? [3 marks] 11. The number 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. Year 2005

2006

2007

2008

Quar Quarte terr 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Code Coded d quar quarte ter  r  (x) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1155 16

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

(a) Plot the data as a time series. series.

[3 marks]

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

[1 marks]

(c) Given Given that the equation equation of the trend line is  T  = 0.325x + 12.550. Using an additive model, find the adjusted seasonal variation variation for each each of the four quarters. quarters. Write down your answers answers correct to three decimal places. [5 marks] 50

PAPER 2 QUESTIONS

Lee Kian Keong

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

STPM 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.

51

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2010

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. (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  A B C 

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

Number of bags of rice  (million) 2007 2008 11 13 17 20 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 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 (a) the probability that the sample mean is within  hour of the population mean, 4 (b) the probability that the sample mean is more than 17 hours.

[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

2

− ¯x)

= 2534,



(y

2

− ¯y)

= 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. 52

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2010

(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 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]

53

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2010

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

Raw material P  Q R

(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. Basic 

x

y

y

0

1

s2

0

0

x

1

0

s1

10 3 2 3 5 3

− −

s2

0 1 0

s3   Solution 

− 209 4 9 25 9

20 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. Price  (hundred thousand RM)   Frequency  2.5  < x  3 .0 15 3.0  < x  3 .4 10 3.5  < x  4 .0 14 4.0  < x  4 .5 8 4.5  < x  5 .0 10 5.0  < x  5 .5 7 5.5  < x  6 .0 5 6.0  < x  6 .5 3

 ≤  ≤  ≤  ≤  ≤  ≤  ≤  ≤

(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.

54

[3 marks] [3 marks]

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2010

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. Activity A B C D E F G H I

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

Duration   (days) 3 3 4 2 5 2 6 4 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 for each activity. Hence, determine the critical path and the minimum number of days needed to complete the programme. [8 marks] (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 administration 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). 55

[2 marks]

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2010

(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]

56

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2011

STPM 2011 1. Some summary statistics about the annual expenditure, in RM, of a sample of recreational golfers on golf activities are shown in the table below. Minimum 200.00

Maximum 15500.00

Mean 6285.67

Median  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 expenditure of the golfers. [2 marks] 2. There are eight male and four female architects in a consultant company. Three architects are randomly 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 Bahru and one female architect is posted to Miri. [2 marks] 3. A discrete random variable X   has probability distribution function

P(x) =

x

    1 3

,

x  = 5, 6,

k,

0,

x  = 1, 2, 3, 4,

 

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 at current price and 1998 pricc arc as follows: Type of product  A B C D Total 

Sale value at current price (RM million) 1998 2009 2010 50 120 175 10 30 24 40 40 50 20 50 90 120 240 339

Sale value at 1998 price  (RM million) 2009 2010 100 125 25 20 20 20 40 60 185 225

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 with the services at the main library of the university. (a) Determine the smallest sample size needed to estimate the population proportion with an error not exceeding 0.05 at the 90% confidence level. State any assumption made. [5 marks] (b) State the effect on the sample size 57

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2011

i. if the error is larger than 0.05 with the confidence level unchanged. ii. if the confidence level is higher than 90% with the error unchanged.

[1 marks] [1 marks]

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



x  = 43,



x2 = 157.42,



y  = 572,



y 2 = 23530,

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



xy  = 1697.80.

(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. It is found that 250 voters are in favour of the party. It is also found that 450 male voters are not in favour 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 a female. [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” are independent. [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 in strikes. [4 marks] (b) Find the probability that at most five throws need to be made by the bowler so that four throws result 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]

58

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2011

(d) If the duration of activity J   is extended to four weeks, determine whether the project will be delayed. [2 marks] 10. The age of women in country A  suffering from kidney problems is found to be normally distributed with mean 40 years and standard deviation 5 years. (a) Find the probability that 10 randomly selected women who suffer from kidney problems have the mean age less than 42 years. [3 marks] (b) Find the probability that four randomly selected women who suffer from kidney problems have a total age of more than 145 years. [3 marks] (c) The ages of eight randomly selected women from country B  who suffer from kidney problems are as follows: 52, 68, 22, 35, 30, 56, 39, 48. Assuming that the ages of the women who suffer from kidney problems are normally distributed, determine the 95% confidence interval for the mean age of the women. Hence, conclude whether the mean age differs from that of country A, and explain your answer. [6 marks] 11. A food company produces a cereal from several ingredients. The cereal is enriched with vitamins A and  B  which are provided by two of the ingredients, oats and rice. A 1 g of oats contributes 0.32 mg of vitamin A  and 0.08 mg of vitamin B , whereas 1 g of rice contributes 0.24 mg of vitamin A and 0.12 mg of vitamin  B . Each box of cereal produced has to meet the minimum requirements of 19.20 mg 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 of  rice is RM4. The company wants to determine how many grams of oats and rice are to be included in 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 box of 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 in a particular day is shown in the table below. Length of call  (minutes) 0.0 - 3.0 3.0 - 4.5 4.5 - 6.0 6.0 - 7.5 7.5 - 9.0 9.0 - 10.5 10.5 - 12.0 12.0 - 15.0

Number of calls  7 19 34 67 38 18 13 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

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2012

STPM 2012 1. There are three male and two female workers who will be assigned to four different tasks Each worker has 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 of  the 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 year 2010 are as follows: Date 1 2 3 4 5 8 9 10

Price   (RM)   Date 8.13 11 8.11 12 8.04 15 8.13 16 8.05 17 8.06 18 7.95 19 7.95 22

Price   (RM)   Date 8.10 23 8.10 24 8.08 25 8.14 26 8.15 29 8.08 30 8.05 31 8.00

Price   (RM) 8.03 8.10 8.09 8.06 8.00 8.10 8.05

(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]

3. A random variable  X  is normally distributed with mean 20 and variance 6.25. The mean of a random ¯. sample of size n is X  ¯. (a) State the sampling distribution of  X   ¯ <  18) = 0.0057, find the value of  n. (b) If P(X

[1 marks] [4 marks]

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 chosen postpaid subscribers to a particular telecommunication company is shown below. Monthly charge  30 50 70 90 110 150 200 250

≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤

Cumulative frequency  0 10 40 76 120 170 192 200

(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]

60

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2012

6. There are two boxes which each contain three golf balls numbered 1, 2 and 3. A player draws one ball at random from each box, and the score X  of the player is the sum of the numbers on the two balls. (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 in an 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 its standard 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 hours weekly. [2 marks] 8. According to a recent census, children under 18 years of age spend an average of 16.87 hours per week surfing the Internet with a standard deviation of 5 hours per week. Find the probability that in a random sample of 100 children under 18 years of age, the mean time spent surfing the internet per week is (a) between 16.5 and 17.5 hours, inclusive.

[4 marks]

(b) within 0.75 hour of the population mean,

[3 marks]

(c) at least 0.75 hour lower than the population mean.

[3 marks]

9. A company is involved in construction projects. One of the projects awarded to the company contains seven activities. The activities, the preceding activities and the duration required for each activity are shown in table below. Activity A B C D E F G

Preceding activity A A B B C, D, E  

(a) Draw an activity network for the project.

Duration   (weeks) 0 10 40 10 2 15 8 [3 marks]

(b) Determine the critical activities of the project, and find the minimum number of weeks required to complete the project. [4 marks] (c) Shortly before the company starts to implement the project, a technical assistant points out that the duration required to undertake activity F  could be shortened to 11 weeks with a new innovative approach. Determine whether the new approach adopted by the company for activity F  would affect your answer in (b). [3 marks] 10. Three groups of sales trainees of a company undergo different sales training programmes which are Programme  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

PAPER 2 QUESTIONS

Lee Kian Keong

STPM 2012

(a) Estimate the median of the three sales distributions. State the training programme which is most 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 for programme A  is removed. [2 marks] 11. A study was conducted to investigate the influence of the quality and fair price of products on preference to shop at a hypermarket. A random sample of 14 customers were asked to rate the hypermarket in terms of preference to shop,  Y  , quality of product,  X 1  and fair price of product,  X 2 . The ratings were based on an 11-point scale with higher numbers indicating higher ratings. The data collected are given in the table below. Customer 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Preference to shop (Y  ) 6 9 8 3 10 4 5 2 11 9 10 3 8 5

Quality of product  (X 1 ) 5 6 6 2 6 3 4 1 9 5 8 1 8 3

Fair price of product  (X 2 ) 4 10 5 2 11 1 7 4 9 10 8 5 5 2

(a) Plot a scatter diagram of preference to shop against quality of product. Hence, interpret this diagram. [3 marks] 62