PYQ SM025

MATHEMATICS SM025

MATHEMATICS 2
SM025

Past Year Collections

Name : __________________________

2018/2019

JOHORE MATRICULATION COLLEGE
84900 TANGKAK JOHOR

BKS2018/19 Page 1

MATHEMATICS SM025 Pages
1-12
CONTENTS 13-19
20-24
Topic 25-31
1 INTEGRATION 32-40
2 FIRST ORDER DIFFERENTIAL EQUATIONS 41-53
3 NUMERICAL METHODS 54-60
4 CONICS 61-71
5 VECTORS 72-88
6 DATA DESCRIPTION 89-98
7 PERMUTATIONS AND COMBINATIONS
8 PROBABILITY
9 RANDOM VARIABLES
10 SPECIAL PROBABILITY DISTRIBUTIONS

BKS2018/19 Page 2

MATHEMATICS SM025

TOPIC 1: INTEGRATION

1. Evaluate each of the following integrals :



2

(a)  sin 5x cos x dx

0

(b)  cos 3  d
sin 2 

 ( 2003/04 )

3

(c)  sin 2 cos2  d

0

2. Show that 1  1  2sec2  Hence, evaluate
1 sin  1 sin 



4 1 1 3x  1 1 3x dx ( 2004/05 )
sin sin


0

3. Express 5x3  x2  x 1 in the form of partial fractions. Hence,
 x2 x2 1

 evaluate e 5x3  x 2  x 1 dx . Give the answer correct to three significant
1 x2 x2 1
figures.

( 2005/06 )

2 ( 2005/06 )

4. Evaluate  x3 x 4  5dx

1

5. Let f (x)  ln x , 1  x  e .
x

(a) Find the area of the region bounded by f (x) and the x-axis.

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MATHEMATICS SM025

(b) Hence, find the volume of the solid generated by revolving the region
2 radians about the x axis. Give the answer in terms of e and .
( 2005/06 )

6. Let R be the region bounded by y  x ln x, y  0, x  1 and x  4 . Find
(a) The area of R,
(b) The volume of revolution when R is rotated through 360o about the x-
axis ( 2006/07 )

7. Express 2x 4  4x 2 1 as partial fractions. Hence,
x3  x

evaluate 2 2x 4  4x2 1 dx .

1 x3  x
( 2006/07 )

8. Given 2x3  9x 2  4x  7  gx A  B . Determine the function
2x2 9x  4 2x 1 x  4

gx and find the values of A and B. Hence, find  2x 3 9x2  4x  7 dx .
2x2 9x 4

( 2007/08 )

9.

y

y 3 x y  4x
R 1 x

Ox

In the figure above, R is the region bounded by the line y  3  x , the curve
y  4x and the y-axis. Find

1 x

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MATHEMATICS SM025

(a) The area of R ,

(b) The volume of solid obtained when R is rotated through 3600 about the x-

axis. Give your answer in terms of  . ( 2007/08 )

10. Evaluate

(a) 1 dx (b)  lnxx dx ( 2008/09 )
1 ex

11. Given a function g is defined by

xe x2 , x 1
 , x 1
g x   ln x 2


x

Evaluate 3 gx dx ( 2009/10 )
1

12. A region R is bounded by the curve y  xx  2 and line y  x .

(a) Sketch the graphs and shade the region R.

(b) Find the area of R. ( 2009/10 )

2 ( 2010/11 )

13. Find the exact value of  t3 t 2  1 dt .

1

14. Express 8x2  15 as partial fraction. Hence evaluate  8x2  15 dx .( 2010/11 )
2x3  3x 2x3  3x

15. Consider the curve given by the equation f (x)  2  x2 .

(a) Sketch the region bounded by the curves f (x) , g(x)  x2 , the lines x  0
and x  2. Hence, find the area of the region.

(b) Find the volume of solid generated when the region bounded by the curve

f (x) , lines x  1 and x  2 is rotated completely about the x-axis.

( 2010/11 )

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MATHEMATICS SM025

16. 9 x 1x dx

By using the substitution t = x , find the exact value of the integral

4

( 2011/ 12 )

17. Find the values of A, B, C and D if x2  2x 1  A  B  C  D .
x2 (x2 1) x x2 x 1 x 1

Hence, 4  x2 2 x 1 dx . ( 2011/12 )
evaluate 2  x2 (x2  1)

18. Find the area of the region bounded by y  sin x, x = 0, x = π2 and the x-

axis. If the region rotated 360 about the x-axis, find the volume of the solid

generated. ( 2011/12 )

19. Find the values of A and B if 2x2  4x  3  A(x2  x  1)  B(2x  1) .
x2  x 1 x2  x 1

Hence, find 2x2  4x  3dx. . ( 2012/13 )

x2  x 1

20. (a) Find sin 3 x cos 4 xdx. by using the substitution u  cos x .

e ( 2012/13 )

(b) Evaluate x ln xdx..
1

21. (a) Sketch and shade the region R bounded by the curves y  x2  2 , the line

2y  x  2, x  0 and x  2.Hence, find the area of R.

(b) If the region R in part (a) is rotated through 2 radian about the x-

axis, find the volume of the solid generated. ( 2012/13 )

22. Determine cot 2 2 sin3 2d . ( 2013/14 )

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MATHEMATICS SM025

23. Express 7x2  3x  2 as partial fractions. Hence evaluate

x  12 x  2

4 7x2  3x  22dx . ( 2013/14 )

 x  12 x

3

24. Given f (x)  ln(x) .

(a) Sketch the graph of f. shade the region R which is bounded by f(x), x-

axis, x = 1 and x = 2.

(b) Find the area of R.

(c) Find the volume of the solid generated when the region R is rotated

3600 about the x-axis. ( 2013/14 )

25. Evaluate the following integrals: ( 2014/15 )

(a)  sin 6x cos 4x dx
(b) (3tan x  4)5 sec2 x dx

26. Given that f (x)  1 and g(x)  x
x3 4

(a) On the same axes, sketch the graphs of f and g for the values of

x between x  0 and x  2 . Shade the region R bounded by f , g , x  0 and

x 2.

(b) Find the area of region R

(c) Find the volume of the solid generated when the region R is rotated

through 2 radian about the and x  axis ( 2014/15 )

27. (a) Find the values of A,B and C if
x2 9  A  B  C

x2 (x  3) x x2 x  3

(b) 1 x2  9 dx
Hence, evaluate 2 x 2 (x  3) ( 2014/15 )

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MATHEMATICS SM025 ( 2015/16 )

 28. Show that e x ln x dx  1 1 e2 .
14

29. Express 1 4x in partial fractions and hence, find the exact value of
3 x  2x2

1 1 4x ( 2015/16 )

0 3  x  2x2 dx.

30. (a) Sketch and shade the region R bounded by the curve y  x , line
y  2  x and y  axis . Hence, find the area of the region R .

(b) If R1 is a region bounded by the curve y  x , line y  2  x and

x  axis , deduce the ratio of R : R1.

(c) Find the volume of the solid generated when the region R is rotated

through 360 about the x  axis . ( 2015/16 )

31. e2x dx
Solve 1 e2x ( 2016/17 )

32. (a) Show that the expression 4x4  2x2 1 can be written as
(2x  3)2 (x 1)

x2 A  B  C.
2x 3 (2x  3)2 x 1

(b) From part (a), determine the values of A,B and C. Hence, solve

 4x4  2x2 1 dx ( 2016/17 )
(2x  3)2 (x 1)

33. Given the curve y  4x2 and the line y  6x

(a) Find the intersection points.

(b) Sketch the region enclosed by the curve and the line.

(c) Calculate the area of the region enclosed by the curve and the line.

(d) Calculate the volume of the solid generated when the region is

revolved completely about the y-axis. ( 2016/17 )

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MATHEMATICS SM025 ( 2017/18 )


6

34. Evaluate  tan 2 cos2 2 d

0

35. Express 6x2  x  7 in partial fractions. Hence, show that
(4  3x)(1 x)2

1 6x2  x  7 dx  1 ln 2 ( 2017/18 )

0 (4  3x)(1 x)2

36. Given the curve y2  x and the line y  2x 1.

(a) Determine the points of intersection between the curve and the line.

(b) Sketch the curve and the line on the same axes. Shade the region R

bounded by the curve and the line. Label the points of intersection.

(c) Find the area of the region R.

(d) Calculate the volume of the solid generated when the region R is

rotated 2 radians about the y-axis. ( 2017/18 )

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MATHEMATICS SM025

Suggested Answers

1. (a) 1 (b)  1  sin   C (c) 0.1580
6 sin 

2.  2
3

3. 1  1  4x , 3.24
x x2 x2 1

4. 13.590

5. (a) 1 (b) 2  5
2 e 

6. a) 4.282 b) 8.03
7. 2x  1  x , 4.151 b) 4.514 

x x2 1

8. x 2  ln 2x 1  ln x  4  C
2

9. a) 1.273

 10. a) ln 1 ex  c b) x2 ln x  1  c
2 2 

11. ln 33

3

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MATHEMATICS SM025

12 .a) b) 4.5

13. 8

15

 14. 8x2 15  5  2x
8x2 15 dx  5ln x  1 ln 2x2  3  C
2x3  3x x 2x2  3 , 2x3  3x 2

15. (a)

-1 1 2

Area  4
(b) V  13  or 0.867

15

16. 2 + ln( 3 )
2

17. A  2 , B  1 , C = 2, D = 0 , 0.561
18. Area = 2π unit2 , Volume = 1 π3 unit3

2

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MATHEMATICS SM025

19. A  2 , B  1 ; 2x  ln x2  x 1  C

20. (a) cos7 x  cos5 x  C  (b) 1 e2 1 or 2.09726
75 4

21. (a) 11 (b) 102 
3 5

 22.  1 cos3 2  c
6

23. 7x2  3x  2  3  2  4

x 12 x  2 x 1 (x 1)2 x  2

24. (a) (b) 0.386 (c) 0.5916

25. (a) 1  cos10x  cos 2x   C (b) 1 3tan x  46  C
2 10 2 
18

26. (b) 0.3145unit 2 (c) 19  unit 3
27. (a) A  1, B  3 , C=2 120
(b) 3.361

28. DIY

29. ln 2 
3

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MATHEMATICS SM025

30. (a)

Area  5 unit 2 (c) 11 unit3
6 6

(b) 5 : 7

31.  1 ln 1 e2x  C
2

32. (a) x  2  101  19  1
10(2x  3) 2(2x  3)2 5(x 1)

(b) x2  2x  101 ln 2x  3  19  1 ln x 1  C
2 20 4(2x  3) 5

33. (a) 0,0 and  3 ,9

2 

(b) y  4x2 y y  6x

9

0 3 x
2
(c) Area  9 unit 2
4

(d) V  27  unit 3
8

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MATHEMATICS SM025

34. 3
16

35. 3  1  2
4  3x 1  x (1  x)2

36. (a)  1 , 1  and 1,1 x

4 2
(b)

y
y2  x

 1 , 1 
4 2

1,1 y  2x 1

(c) 9 unit 2
16

(d) 9  unit 3
20

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MATHEMATICS SM025

TOPIC 2: FIRST ORDER DIFFERENTIAL EQUATIONS

1. Determine the general solution of differential equation x dy  3y  x3 . Hence,
dx

find the particular solution of the equation if it has a stationary point

corresponding to x  1. ( 2003/04 )

2. Solve the following differential equation, ( 2004/05 )

dy  xe x2y ; y0  1 ;

dx

3. The motion of an object is governed by equation

dv  g  kv
dt

where v is the velocity at time t, g is the gravity and k is a constant.

(a) Find the velocity v by assuming that the object start from rest.

(b) Deduce that after a long period of time, the object will move with a

constant velocity g ( 2005/06 )
k

4. Consider a simple electric circuit with the resistance of 3 and inductance of

2H. If a battery gives a constant voltage of 24 V and the switch is closed when
t  0 , the current, I (t) , after t seconds is given by

dI  3 I  12 ; I (0)  0
dt 2

(a) Obtain I (t)

(b) Determine the difference in the amount of current flowing through the

circuit from fourth to eighth seconds. Give your answer correct to 3
decimal places.

(c) If current is allowed to flow through the circuit for a very long period of

time, estimate I (t) ( 2006/07 )

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MATHEMATICS SM025

5. A parachutist jumps off an aeroplane on a regular training session. When his

parachute open, he travels vertically downward with a velocity vo. The

velocity of the parachutist at time t minutes is v and his acceleration is given

by dv  g  v , where g is acceleration due to gravity and  is a constant.
dt

(a) Show that v  g   g  v0 e t
   

(b) Determine the difference in the velocities of the parachutist from the

fifth to the tenth minutes.

(c) Find the velocity of the parachutist after a very long period. ( 2007/08 )

6. A total of 30 rats are randomly captured from a plantation and kept to breed in

an experimental laboratory. After a month under observation, the number of

rats has increased by 10. The rate of increase per month of population is given

by dp  kp(50  p) where p is the current population and k is a constant.
dt

(a) Solve the differential equation. Give your answer for p in terms of t.

(b) Compute the number of rats after a period of one year. ( 2008/09 )

7. Solve the differential equation dy  1 with initial condition y(0) = 4 .
dx xy

Express y in terms of x . ( 2009/10 )

8. A model for the concentration of glucose solution in the bloodstream,

C  C(t) , is given by the differential equation dC  r  kC , where r is the
dt

constant rate at which glucose solution enters the bloodstream and k is a

positive constant. If C(0)  C0 , show that the concentration at any time t is

C(t)   C0  r e  kt  r
 k  k

After a very long period of time, the concentration of glucose is found to be 1

unit. If C0 = 9 , what is the concentration of glucose at t  2 ? ( 2009/10 )
k

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MATHEMATICS SM025 ( 2010/11 )

9. Obtain the general solution for dy  2y  4e2x .
dx

10. Newton’s law of cooling states that hot liquid at temperature H cools at a rate

dH proportional to the difference between its temperature and temperature of
dt

the surrounding environment H0 . Show that H  Aekt  H0 , where k is the

cooling rate constant and A is an integral constant. A hot tea at 76C is left in

a room of 22C .
(a) Find the Newton’s cooling equation.

(b) Using a container X, it is found that after 10 minutes in the room, the

temperature of tea has decreased by 10 C . Determine the temperature

of tea after 15 minutes in the room.

(c) Using a different container Y, whose k  0.10 , determine the time

taken for the tea to cool down to room temperature.

(d) In which of two container, X or Y, does the tea cools down to room

temperature faster? ( 2010/11 )

11. Solve the differential equation ex dy  2ex y  1, given that y  2 when x  0 .
dx
( 2011/ 12 )

12. A radioactive substance of mass N gram decays at the rate of dN  kN,
dt

where k is a constant. Initially the amount of the substance was 80 gram.
After 100 years it decayed to 20 gram.
(a) Express N in terms of the elapsed time t.
(b) Calculate the amount of the substance remains after 120 years.

( 2011/12 )

13. Solve the differential equation x dy  y  x sin x, y    1. ( 2012/13 )

dx

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MATHEMATICS SM025

14. (a) Solve the differential equation dy  4x3 , given that y2 when x  0.
dx 3y2

(b) Assume that P t  represents the size of a population at any time t and the

increment in population size at time t can be modeled by the differential

equation dP t   0.005P t  with an initial condition P0 1500.

dt

Determine the size of this population after 10 years. ( 2012/13 )

15. According to the Newton’s law of cooling, the rate change of temperature of

an object is proportional to the difference in temperature between the object

and the surrounding temperature, M. The law is given by the following

differential equation dT  k(T  M ) where T(t) is the temperature of the
dt

object at time t and k is a constant.

(a) Express T in terms of time t.

(b) A bowl of soup is removed from an oven with temperature at 600 .

What is the temperature after 5 minutes if k = 0.04 assuming that the

temperature of that surrounding is 26.90 . ( 2013/ 14 )

16. The amount and Q(t) of radioactive substance present at time and t in

reaction is given by the differential equation

dQ  kQ
dt

where k is a positive constant. If the initial amount of the substance is 100mg

and is decreased to 97mg in 6 days, determine

(i) the half-life of the substance

(ii) the amount of radioactive substance present after 30 days

(b) Find the general solution to the differential equation

(1 x) dy  y  1 x ( 2014/15 )
dx

 17. Find y in terms of x given that x dy  1 2x2 y where x  0 and y  1 when
dx

x 1. ( 2015/16 )

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MATHEMATICS SM025

18. Find the general solution of the differential equation dy  y cot x  2sin x .
dx
( 2015/16 )

19. In a Chemistry experiment, sodium hydroxide, NaOH, reacts with

hydrochloric acid, HCL, to form sodium chloride salt, NaCl, and water. Before

the reaction starts, no NaCl salt is formed. At time t (minute), the mass of

NaCl salt formed x grams and the rate of change of x is given by

dx   50  x, where  is a positive constant.

dt

(a) Find the general solution for the above equation.

(b) Find the particular solution if 35 grams of NaCl salt has formed in the first

30 minutes.

(c) Hence, find

(i) The mass of NaCl salt formed in 60 minutes.

(ii) The time taken to form 40 grams of NaCl salt. ( 2016/17 )

20. (a) Find the general solution of the differential equation dy  y2xe2x . Give
dx

your answer in the form y  f (x) .

(b) Find the particular solution of the differential equation

dy  xy  1 x2 ,given that y  1 when x  0 . ( 2017/18 )
dx 1 x2

BKS2018/19 Page 19

MATHEMATICS SM025

Suggested Answers

1. y  x3ln x  1 
 3

2. y  1 ln 2 xe x  ex  1 e2  1
2 2

 3. (a) V  g 1 ekt (b) V  g
kk

 81   3 t 
2 
4. (a) I (t ) e (b) 0.0198 (c) 8

 5.  g  c) v  g
b) v10  v5     v0  e5  e10 

6. a) P  150e0.981t b) P  50
2  3e0.981t

2

 1  3
 2 
7. y  3x  8

8. C  2.083

9. y  4xe2x  ce2x

10. (a) H  54ekt  22 (b) H  61.7 C
(c) t   (d) Y is faster than X

11. y  ex  e2x

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MATHEMATICS SM025

12. (a) N  80e0.0139t (b) N  15.09

13. y   cos x  sin x
x

14. (a) y  3 x4  8 (b) P10 1577

15. (a) T  Aekt  M (b) T= 540

16. (a) ( i) t  136.45 (ii) Q  85.87mg

(b) y  (1 x) ln(1 x)  C(1 x)

17. y  xe1 x2

18. y  x  cos x  C
sin x sin x

 19. (a) x  501 et  (b) x  501 e0.0401t

(c) ( i) x  45.5 (ii) t  40.1

20. (a) y  2xe2x 4
 e2x
C

(b) y  1   x3 
x  1
1 x2  3 

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MATHEMATICS SM025

TOPIC 3 : NUMERICAL METHODS

1. Using the Newton-Raphson method and by taking x  0.6 , estimate the root
of the equation x3  3x2 1  0 correct to three decimal place. ( 2003/04 )

1

2. Use the trapezoidal rule to approximate  1  x2 dx with 6 subintervals,

-1

giving your answer correct to three decimal place. ( 2004/05 )



4

3. Use the trapezoidal rule with four subintervals to approximate  cos 4 x dx .

0

Give your answer correct to three decimal places. ( 2005/06 )

4. (a) Use the trapezoidal rule with n4 to approximate 1 dx . Using definite
1 x


0

integration, find the value of 1 dx . Compare two answers and give a reason
1 x


0

for the difference.

(b) Approximate 3 7 by using Newton-Raphson method and initial value 2, up

to second iteration.

5. By sketching the graph of y  x3 and y  4x  2 , show that the equation
x3  4x  2  0 has three real roots.
a) Show that one of the real roots of the equation lies between 1 and 2.
b) By using the Newton- Raphson method and initial value 2, determine the
real root that lies between 1 and 2, correct to two decimal places. ( 2007/08 )

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1

6. (a) Approximate  x 1  x2 dx by using Trapezoidal Rule and n = 4. Give your

0

answer correct to four decimal places.

1 1  x2 dx  1 .

Using the substitution u  1  x2 , show that x
( b) 03

Give a reason for the difference of the values obtained in a) and b). ( 2008/09 )

7. (a) Given g(x)  (x 1) x  2. By using the Newton-Raphson method starting

at x0  1.1, find the root of g(x).

1

(b) By using trapezoidal method, obtain the approximate value of  xe x2dx

0

based on four subintervals, correct to four decimal places. ( 2009/10 )

8. By using the substitution u  ln x, evaluate 3 ln xdx up to five decimal places.
x


2

If 3 ln x dx is approximated using the trapezoidal method based on five equal
x


2

subintervals, compute the error. ( 2010/ 11 )

9. Show that the equation 5  3x13  x has a root in the interval 1,2 . By using

the Newton-Raphson method with the first approximation x1 =1, find an

approximate root of the equation correct to three decimal places. ( 2011/ 12 )

10. (a) Show that x4  3x2 1 has a solution on the interval (1,2)

(b) Use Newton Raphson’s method with x0 = 1 to estimate the solution for part

correct to four decimal places. ( 2012/ 13 )

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2

11. (a) Use trapezoidal rule with four subintervals to estimate x 2e x dx , correct to

0

four decimal places.

(b) Given f (x)  x3  5x  3 . Show that f (x)  0 can be written as

x  g(x)   3 . With the initial value x1  0.5 , find the roots of f (x) by
x2 5

using iteration method. Hence, calculate the root of f (x) accurate to three

decimal places. ( 2013/14 )

1

12. Use the trapezoidal rule to estimate  f (x)dx from the data given below:

0

x 0.00 0.25 0.5 0.75 1.00

f (x) 2.4 2.6 2.9 3.2 3.6

( 2014/ 15 )

13. Use the Newton-Raphson method with initial approximation x1  1 to find

6 2 on 0,2 correct to three decimal places. ( 2014/ 15)

14. (a) Given f1  x  2x and f2  x   ln x.

(i) Without using curve sketching, show that y  f1  x and y  f2  x

intersect on the interval of 0.1,1.

(ii) Use Newton-Raphson’s method to estimate the intersection point of

y  f1  x and y  f2  x with the initial value x1  1 . Iterate until

f  xn   0.005 . Give your answer correct to three decimal places.

(b) By using the trapezoidal rule, find the approximate value for

1

 x x 1 dx when n  4 , correct to four decimal places. ( 2015/16 )

0

15. (a) Show that the equation  4x2  5x  7  0 has a root on the interval  2,0.

Use the Newton-Raphson method to find the root of the equation correct to
four decimal places.

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0

(b) Estimate the value of  x cos x dx using trapezoidal rule with subinterval


 . Give your answer correct to four decimal places. ( 2016/17 )
4

16. Show that the equation ln x  x  4  0 has a root between 1 and 3. From the

Newton-Raphson formula, show that iterative equation of the root is

xn1  xn 5  ln xn  . Hence, if the initial value is x1  2 , calculate the root

1 xn

correct to three decimal places. ( 2017/18 )

Suggested Answers

1. x  0.653

2. x 1.459

3. x  0.541

4. (a) 0.697 , 0.693 Reason : value are different , trapezoidal rule only find the
approximate value, define integral find the exact value.
(b) x  1.9129

5. 1.68

6. (a) 0.2928
(b) Using the trapezoidal rule, we’ll get only the approximate value.
Definite integral, we’ll get the exact value

7. (a) 1.00 (b) 0.8959

8. Integration: 0.36325 Trapezoidal: 0.36296
The error = 0.36296  0.36325 = 0.00029

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9. 1.154

10. x = 0.6181

11. a) 13.9962 b) 0.657

12. 2.925

13. 1.123 (b) 0.6478

14. (a)(ii) 0.426,0.853

15. (a)  0.8381 (b) 2.1061

16. 2.926

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MATHEMATICS SM025

TOPIC 4: CONICS

1. A straight line x  2y  0 intersects a circle x2  y2  8x  6y 15  0 at the

points P and Q. Find the coordinate of P and Q. Hence, find the equation of

the circle passing through P, Q and the point 1,1. ( 2003/04 )

2. (a) Find the foci of 9x2  4y2  36 and sketch its graph.

(b) By using the implicit differentiation, find the gradient of the tangent to the

curve 9x2  4y2  36 . Hence, find the coordinates on the curve with

gradient 9 ( 2004/05 )
2

3. Find an equation of the circle passing through the origin and its center is the

focus of the parabola x2  8y 16 ( 2005/06 )

4. Let L be a line passing through the centre of the circle x2  y2  2x  2y  7
and perpendicular to the line 3x  4y  7 . Find
(a) the coordinates of the point of intersections of L and the circle.
(b) the equations of the tangents to the circle parallel to 3x  4y  7
( 2006/07 )

5. Find the values of p, q and r which make the ellipse
4x2  y2  px  qy  r  0

Touches the x-axis at the origin and passes through the point (1,2) .Express the
equation obtained in the standard form and hence find its foci. ( 2007/08 )

6. The end point of the diameter of a circle are A(2,0) and B(10,4).
(a) Determine the equation of the circle
( b) The equation of the tangent line to the circle at the point B. ( 2008/09 )

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7. An ellipse with centre at the origin passes through the points (2,0) and

1, 3 3  . Find the equation and the foci of the ellipse ( 2009//10 )
2 

8. Given the circles
C1 : x2  y2  2x  2y 1  0
C2 : x2  y2  1

Find

(a) the center and the radius of the circle C1 .

(b) the equations of the tangent from the point ( 0,3) to the circle C2 .
(c) the equation of the circle that passes through the point (-5,0) and the

points of intersection of the circles C1 and C2 . ( 2009/10 )

9. Find an equation of the circle that passes through the points 1,4 , 2,2 and

1,3. Hence, find the radius of the circle. ( 2010/11 )

10. A circle C passes through the origin and has its centre at point 3,3.

(a) Obtain the equation of the circle C.

(b) If the line y  x  6 meets the circle C at points P and Q, determine the

coordinate of P and Q.

(c) Find the coordinates of the points on the circle C where the tangents

are parallel to the line PQ. ( 2010/ 11)

11. Show that 4x2 16x  y2  2y  8  0 is an equation of an ellipse with vertical

major axis. Hence, find its centre and foci. ( 2011/12 )

12. A line segment joining 1,0 and 3,4 is a diameter of a circle.

(a) Find an equation of the circle.

(b) Find an equation of the tangent to the circle at the point 3,4

(c) Find the points of intersection of the circle with its chord of which the

midpoint is the origin. ( 2011/12 )

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13. Find the center and radius of the circle x2  y2  2x  4 . Obtain the equation of

The tangent to the circle at the point (0,2) ( 2012/13 )

14. An equation x2  4x  4y  8  0 represents a parabola

(a) Determine the vertex, focus and directrix of the parabola.
(b) Show that the tangent lines to the parabola at the points A(2,5) and

B(3, 5) intersect at the right angle. ( 2012/13 )
4

15. Two circles of radius 5 units pass through the origin with their centres lie on
the line x  y 1 . Show that the equations of the circles are

x2  y 2  6x  8y  0 and x2  y 2  8x  6y  0 ( 2013/14 )

16. Given 9x2  72x  16y2  32y  16 is the equation of an ellipse.

(a) Write the equation in standard form.

(b) Find the foci, vertices, lengths of the major and minor axes.

(c) Sketch the graph. ( 2013/14 )

17. Given a parabola with vertex (-2,1), opening to the right and passes through
the point (3,6). Find the equation of the parabola and determine its focus.
( 2014/15 )

18. Find the equation of a circle x2  y2  2gx  2 fy  c  0 which passes through

the points A(0,1), B(3,2) and C(-1,-4) . Hence, determine its center and radius.

Find the points of intersection of the circle with the y axis. ( 2014/15 )

19. (a) Find the equation in standard form of an ellipse which passes through

the point 1,6 and having foci at 5, 2 and 3, 2 .

(b) From the result obtained in part (a), sketch the graph of the ellipse.
( 2015/16 )

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20. Find the vertex, focus and directrix for parabola y2  64  8y 16x. Hence,

sketch and label the vertex, focus and directrix for the curve. ( 2016/17 )

21. The end points of the diameter of a circle are P(0,1) and Q(3,3) .
(a) Determine an equation of the circle.
(b) Find an equation of the tangent line to the circle at the point P(0,1) .
( 2016/17 )

22. Determine the vertices and foci of the ellipse

25x2  4y2  250x 16y  541  0 . Sketch the ellipse and label the foci,

center and vertices. ( 2017/18 )

23. Show that the line 2y  5x  4  0 does not intersect the circle
x2  y2  3x  2y  2  0 . Find the center and radius of the circle. Hence,
determine the shortest distance between the line and the circle. ( 2017/18 )

Suggested Answers

1. x2  y2  23x  36y 15  0

   2. (a) 0, 5 and 0, 5

y

F 0, 5

0,0 x

 F 0, 5

(b) dy   9x ,   6, 3  and  6 , 3 
dx 4 y 10 10 10 10

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MATHEMATICS SM025

3. x2  y2  8y  0

4. (a)   4 , 7  and 14 ,17  (b) y   3 x  11
 5 5  5 5  42

p  0 , q  4 , x2  ( y  2)2  1,
4
   5.
F1 0,2  3 , F2 0,2  3

6. (a) (x  6)2  ( y  2)2  20 or x2  y2 12x  4y  20  0
(b) 2x  y  24  0

x2  y2 1 ,
49
   7.
F1 0, 5 , F2 0, 5

8. (a) Center C = 1,1 Radius r  1

(b) y  8x  3 or y   8x  3
(c) x2  y2  4x  4y  5  0

9. x2  y2  x  5y  4  0 ; center  1 , 5  ; radius= 5
2 2 2

10. (a) x2  y2  6x  6y  0
(b)
(c) P0,6 and Q6,0
0,0 and 6,6

11. x  22  y 12 1 , center 2,1 Foci  2 ,1 5 3  ,  2 ,1  5 3 
2 2
25 25

4

12. (a) x2  y2 2x 4y 3  0 (b) y  7  x
(c)
 2 3 , 3  and   2 3, 3 
5 5 5 5

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13. Center (1, 0) , r  5 . Equation of tangent 2y  x  4  0

14. (a) V 2,1, F(2,2) , directrix y  0

15 . DIY

16. (a) x  42  ( y  1)2  1

16 9

(b) Foci : F1(4  7,1) and F2 (4  7,1) , length of major = 8 , length

of minor = 6 , Vertices : 0,1 and 8,1 .

(c)

17. y 12  5(x  2) ; F  3 ,1

 4

18. 3x2  3y2  2x 10y 13  0 ; C 1 , 5  ; r  65 ; 0,1,0, 13 
3 3 3  3

19. (a) x 12  y  22  1

32 16

(b)

y

F(5, 2) C(1, 2) F(3, 2)

x

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20. V   3,4 , F   7,4 , directrix x  1

y

F 7,4 V  3,4

x

x 1

21. (a)  x  3 2  y 12  25 (b) y  3 x 1
 2 4 4

   22. V  5,7 and V  5,3 ; F  5,2  21 and 5,2  21

y V 5,7

 F 5,2  21

C5,2 x

 F 5,2  21

V 5,3

23. r  5 , center   3 ,1 shortest distance = 1.39
2  2

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MATHEMATICS SM025

TOPIC 5 : VECTORS

1. Given two vector a  2 i  j q k and b  q i  2 j 2q k .

~ ~~ ~ ~ ~~ ~

(a) Determine the value of q such that a and b has the same magnitude.

(b) If q  4 , find the angle between the vector b and 1  a 1 b .
~ 2~ 2 ~

(c) (i) Find the value of q if a  b  8 i  4 j 2 k

~~ ~~~

(ii) Determine the Cartesian equation of the plane passing through

point (1,0,2) and perpendicular to a and b . ( 2003/04 )

~~

2. A(6,3,3), B(3,5,1) and C(-1,3,5) are the point in a three-dimensional space.

Find



(a) the vector BA and BC in terms of unit vector i , j and k . Hence ,

~~ ~



show that BA is perpendicular to BC ,

(b) a unit vector that is perpendicular to plane containing the point A, B

and C.

(c) a Cartesian equation of the plane described in (b). ( 2004/05 )

3. If u, v and w are three nonzero vectors such that u  v  w  0 show that

w2  u2  v2 ( 2005/06 )
uv 

2

4. The position vector of point P, Q and R are given respectively as

p  4 i  3 j11k , q  2 i  8k , r  i  2 j

~ ~ ~ ~~ ~ ~~~ ~

(a) Show that PQR is a right angle triangle and calculate its area.

(b) Find an equation of the plane containing P, Q and R

(c) Find parametric equation of the straight line passing through the point

3,5,2 and perpendicular to the plane containing P, Q and R ( 2005/06 )

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MATHEMATICS SM025

5. The vector a, b and c are such that b  c  3 i and c  a  2 i  k , where i, j

~ ~~

and k are unit vector. Express a  b a  b  4c in terms of i, j and k.

( 2006/07 )

6. The point A(2,1, 2 ) , B(5,7,3) and C(3,3,1) lie on the plane 1 . The
equation of second plane 2 , is given as 2x  y  2z  5 .



(a) Find the vector AB and AC

(b) Determine the Cartesian equation of 1 .
(c) find the acute angle between 1 and 2 , giving your answer in degrees.

( 2006/07 )

7. The position vectors p,q,r and s are given such that s  p q  r  0 and
s  q r  p  0
(a) Show that s  r p  q  0 .

(b) If p  4i  5j, q  3i  2j, r  4i  j and s  xi  yj, find the values of x

and y. ( 2007/08 )

8. Given that u  3i  3j  ak and v  bi  2k . If u  v  6i  2j 12k ,

determine the values of a and b.

Hence, determine

(a) the direction angles of u .

(b) the area of parallelogram with sides u and v .

(c) the angle between u and v . ( 2007/08 )

9. If u  v  5 and u  v  1, find u  v by using the property a  a  a 2 .
( 2008/09 )

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10. Let P (1, 3, 2) , Q (3, 1, 6) and R (5, 2, 0) be points in three dimensional

space.

Determine :

a) the direction cosines for the vector PQ.

b) whether PQ and PR are perpendicular vectors.

c) an equation of the plane containing P, Q, and R.

d) the parametric equations of the line passing through the point

B (0,1, 2) and perpendicular to the plane in part (c). ( 2008/09 )

11. Given u  2i  2 j  k . Find the vectors which have magnitude 6 and parallel

to u . ( 2009/10 )

12. The points A (1, 3, 2) , B (3, 1, 6) and C (5, 2, 0) lie on the plane  . A line L
Passes through the points P (1, 2, 2) and Q (0,1, 4) . Find

a) AB  AC and hence, obtain an equation of the plane  in Cartesian form
b) the parametric equations of the line L.the point of intersection of L and  .

( 2009/ 10 )

13. Given three vectors a  2i  j  4k , b  j  3k and c  5i  6 j  2k . Find

the value of  such that

(a) a is perpendicular to b.

(b) ab  c . ( 2010/11 )

14. Given the points A1,3,1, B4,1,2 , C 12,0,1 and D0,2,0 . Find

(a) A vector equation of the line AB.
(b) An equation of plane ABC in the Cartesian form.
(c) The acute angle between the plane ABC and plane ABD.

( 2010/ 11 )

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15. Given the point P4,2,3 , the straight line L : x  2  y  z 1 and the

4 3 5

plane  : 2x  y  2z  9. Find

(a) an acute angle between the straight line L and the plane Π.
(b) an intersection point between the straight line L and the plane Π.

(c) a Cartesian equation of the plane containing the point P and the

straight line L. ( 2011/12 )



16. P, Q and R are three points in space where PQ  a and PR  b . Given



a  2 i  2 j k and b  i  2 j 2 k

~ ~ ~~ ~~ ~ ~

(a) Find the area of triangle PQR.

(b) Find the parametric equations of the line L passing through the point

R  2, 0, 3 and parallel to vector a .

(c) If u   b a a b and v   a b b a  , evaluate u  v . Hence,
~  ~ ~ ~ ~  ~ ~ ~ ~
~~

interpret the geometrical relationship between u ( 2011/12 )

~

17. The plane 1 contains a line L with vector equation r  t j and a point

P3,1,2 .

(a) Find a Cartesian equation of 1 .
(b) Given a second plane 2 with equation x  2y  3z  4 , calculate the

angle between 1 and 2 .
(c) Find a vector equation for the line of intersection of 1 and 2

( 2012/13 )

18. Given nonzero vectors p and q are perpendicular. Prove that ( 2013/14 )

2 22

(a) p  q  p  q .
(b) p  q  p  q .

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MATHEMATICS SM025

19. Given the points A(1,2,-2) , B(2,4,6) and C(-4,3,-1). Find the area of the

triangle ABC. ( 2013/14 )

20. Given two planes 1 : x  2y  z 1 ,  2 : 2x  y  4z 1and the straight line

L: x  2  y  3  z 1.
2 45

(a) Find an acute angle between the planes  1 and  2 .

(b) Write the equation of L in parametric form. Hence, find the

intersection point between the straight line L and the plane  2

(c) Find a Cartesian equation of the plane which is orthogonal to the

straight line L and passes through the point (1,2,-3). ( 2013/14 )

21. Given two straight lines,

L1 : t  x 1  y2  z and L2 :t  x2 y  z4
3 8 3 10 10 7

(a) Show that L1 and L2 are not parallel and find the acute angle between

the two straight lines.

(b) Determine intersection point between L1 and plane
 : 2x  y  5z  25  0

(c) Find an equation of the plane containing L1 and L2 . ( 2014/15 )

22. Given P,Q and R are three points in a space where

PQ  a  3i  j  k , PR  b  2i  j  3k

and the coordinates of R is (3, 0, 1)
(a) Hence, show that

(i) a and b are not perpendicular

(ii) a  b 2  a 2 b 2  a  b2

(b) Find the area of triangle PQR

(c) Find the Cartesian equation for the

(i) plane that passes through the points P, Q and R

(ii) line that passes through the point R and perpendicular to the

plane in part (i). ( 2014/15 )

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23. (a) If p  3i  j  2k and q  2i  2 j  k , show that

 2 2 2 2
pq  p q  p q .

(b) Given a triangle ABC with AB  2a and AC  3b . Use the result in

part (a), show that the area of the triangle is 3 a 2 b 2  a b2 . Hence,

deduce the area of the triangle if a  p and b  q . ( 2015/16 )

24. Given a line l : x  2  t, y  3 4t, z  5  3t and two planes

1 : 2x  y  7z  53 and 2 : 3x  y  z 1. Find

(a) the point of intersection between the line l and the plane 1 .

(b) the acute angle between the line l and the plane 1 .

(c) the acute angle between planes 1 and  2 . ( 2015/16 )

25. Given four points A   2,8,4 , B  2,,1 , C  0,9,0 and

D   4,3,7. Determine the value of  if       64 . ( 2016/17 )

AB AC  AD

26. Find the angle between the line  : x, y, z  1, 3, 1  t 2,1, 0 and the plane

 : 3x  2y  z  5. ( 2016/17 )

27. Given vectors p  3 i  6 j k and q   i  4 j 5k where  and  are

~ ~~ ~ ~ ~~~

constants.

(a) Find the values of  and  if p and q are parallel.

~~

(b) Given  1, find  if p and q are perpendicular. ( 2017/18 )

~~

28. Given three point P 3,2,1 , Q 2,4,5 and R1,2,4. Calculate the area of

triangle PQR. ( 2017/18 )

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MATHEMATICS SM025

29. Given the line L: x 1  y3  z2 and the plane 1 : 2x  y  2z 17 and
2 1 3

 2 : 4x  3y  5z 10 .

(a) The intersection point between L and 1.

(b) The acute angle between 1 and  2 .

(c) The parametric equations of the line that passes through the point

2,1,3 and perpendicular to the plane  2 . ( 2017/18 )

Suggested Answers

1. (a) q   1 (b) 102.6
2 (ii) 4x  2y  z  6

(c) (i) q  2



2. (a) BA  3 i  2 j 2 k , BC  4 i  2 j 4k

~~~ ~~~

(b) 1  or 1  20 j14 
612  4 i  20 j14 k 612 4 i  k
 ~ ~ ~ ~ ~ ~

(c) 2x 10y  7z  63

3. DIY

4. (a) Area = 32.24 (b) 10x 19y  z  28

(c) x  3  30t , y  5  57t , z  2  3t

5. 12 i  8 j 4k

~~ ~



6. (a) AB  7 i  8 j 5k , AC  5 i  2 j k

~~~ ~ ~~

(b) x  y  3z  3 (c) 72.5

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7. (a) DIY MATHEMATICS SM025

(b) x  5, y  2

8. a  1 ; b  4

(a)   0.81rad ,   0.81rad  1.8 rad

(b) 13.56 unit2 (c)   2.37 rad or 135.9

9. 6

10. a) 1 ,  2 , 2 (b) not perpendicular
3 33 (d) x 12t , y 1 20t , z  2 14t

(c) 6x + 10y + 7z = 50

11. 4i – 4j + 2k or – 4i + 4j – 2k

12. (a) 12i + 20j + 14k , 6x + 10y + 7z = 50
(b) x 1 t , y  2  t , z  2  2t
(c) ( 6, 7, – 8)

13. (a)   12 (b)   3

14. (a) r  i  3 j  k  t 3i  4 j  k  (b) 3x  11y  35z  71

(c) 50.22

15. (a) 45   (b) 2,3,4 (c) 2x  y  z  3  0
4

16. (a) 65  4.03 unit 2 (b) x  2  2t , y  2t , z  3  t
2

(c) 0 ; u and v are perpendicular

~~

17. (a)  2x  3z  0 (b) 58.74
(c) r  2 j  t(6i  9 j  4k)

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MATHEMATICS SM025

18. DIY

19. 21.4360

20. (a)   69.10
(b) x  2  2t, y  3  4t, z 1  5t ,(1,-5,  3 )
2
(c) 2x  4y  5z  5

21. (a)   1.05rad / 60.21

(b)  2,6,3

(c) 26x  51y 110z  76  0

22. (b) 6.12 ii. x  3  y  z 1
(c) i. 2x 11y  5z 11 2 11 5

23. (b) 3 122 (b) 46.1 (c) 60.5

24. (a) 5,15,4

25. w  10
26. 28.6

27. (a)   15 ,   2 (b)    29
2 3

28. 20.4 unit 2 (b) 45 (c) x  2  4t ; y  1 3t ; z  3  5t

29. (a) 5,1,  4

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MATHEMATICS SM025

TOPIC 6: DATA DESCRIPTION

1. The number of accidents recorded yearly in a certain district for eight

consecutive years are 96, 82, 80, x, 94, 82, 96, and (x + 6). If the sample mean

is 88, find the value of x and the sample median. ( 2003/04 )

2. The frequency distribution table for ages (in years) of a sample of 50
participants in a motivation program is as follows :

Age Class Limit Number of Participants

7–9 4
10
10 – 12 12
13 – 15 18
6
16 – 18
19 – 21

Find the mean, median, mode and standard deviation of the above sample.
(2003/04 )

3. (a) The following data set represents the number of candies that was obtained

by 8 children in a game.

30212023

Determine the median.

(b) The sample variance of a set of data with 209 10
 10 is 90 . If is 65,
xi 2

i 1

find the sample mean. ( 2004/05 )

4. The following table shows the distribution for the weight of 50 parcels in a post

office.

Weights (kg) Number of Parcels

0.5-5.5 4

5.5-10.5 6

10.5-15.5 12

15.5-20.5 16

20.5-25.5 10

25.5-30.5 2

Calculate the mean and the first quartile of the above sample. ( 2004/05 )

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MATHEMATICS SM025

5. The cumulative distribution for the ages of 50 patients at a clinic on a

particular day is shown in the table below.

Age ( Years) Number of Patients
<2 0
<12 5
<22 12
<32 29
<42 38
<52 45
<62 50

Calculate the

(a) median

(b) mode ( 2005/06 )

6. Consider a sample consisting of the following observations :

24 12 22 12 20 32 4 10

(a) Calculate the mean, median and mode.

(b) A new sample is formed using the above observation with three values in

part (a) included. For this new sample, calculate the

(i) first quartile

(ii) standard deviation ( 2005/06 )

7. The mean sample
w, w, 7, 9, 11, 12, 3w, 18

is 10.25 with w constant.
(a) Find w
(b) Calculate the coefficient of variation and interpret your answer ( 2006/07 )

8. The following table show that the distribution for the number of medical

leaves ( in days) taken within a certain period by 65 employees of a company.

Number of Medical Leaves (days) Number of Employees
1-3 4
4-6 6
7-9 8
10-12 12
13-15 18
16-18 11
19-21 6

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MATHEMATICS SM025

Calculate

(a) the mean, mode and median. Hence describe the distribution of the data.

(b) the variance. ( 2006/07 )

9. Given a sample of heights ( in cm) of seedlings in an experiment as follows:
76.0, 68.1, 73.4, 80.2, 75.4, 78.3

a) Find the mean and standard deviation of the above data
b) If it was later discovered that the measuring scales were reading cm

below the correct height, state the changes ( if any) on the mean and the

variance ( 2007/08 )

10. The consultation times ( in minutes) for 100 patients at a private clinic is given
in the table below.

Time interval (minutes), Frequency,

0–9 9
10 – 19 34
20 – 29 20
30 – 39 18
40 – 49 9
50 – 59 7
60 – 69 3

Given that  xf  2620 and  x2 f  91858

a) Find
i. The mode and median

ii. The mean, standard deviation and Pearson skewness coefficient

b) Hence, state the reason, which of the above measures of central tendency

better describes the distribution of the data. ( 2007/08 )

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MATHEMATICS SM025

11. The following table shows the frequencies of daily income of 42 fruit sellers

Income, x (RM) Number of fruit sellers
100  x  120 4
120  x  140 8
140  x  160 14
160  x  180 12
180  x  200 4

a) Calculate the mean and the standard deviation of the daily income

b) What is the daily income earned by the most fruit sellers?
c) State the skewness of the daily income distribution using the Pearson’s

coefficient of skewness.

d) If the average income of fishmongers per day is RM180 with standard

deviation of RM20, determine whose income is more stable between the

fishmongers and fruits sellers? ( 2008/09 )

12. The relative frequency distribution of the marks for a Statistics test obtained
by a group of 100 students in the last semester is shown in the following table.

Marks Relative frequency
0 19 0.05
20  39 0.15
40  59 0.38
60  79 0.32
80  99 0.10

Determine the mean and median for the distribution. ( 2008/09 )

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MATHEMATICS SM025

13. The waiting time for 50 customers to have their food served at a restaurant on
a particular day is shown in the following table

Time (minutes) Number of Customers
1–5 4
6 – 10 9
11 – 15 15
16 – 20 11
21 – 25 6
26 – 30 3
31 – 35 2

Calculate the mean, median and mode of the waiting time

14. The summary statistics of the length (in cm) of a sample of 50 adult insects of

a certain species is as follows

 x  45 ,  x2  81

Calculate the mean and variance. ( 2010/11 )

15. The time (in minute) used by 120 students surfing the internet to perform a

certain project is given in the following relative cumulative frequency table.

Time (x), in minutes Relative cumulative frequency
0
x0
x  20 3
40
x  40 19

x  60 60
x  80 2
3
53

x 100 60

1

Find

(a) The median and mean.
(b) Pearson’s skewness coefficient and comment on the value obtained.

( 2010/11 )

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MATHEMATICS SM025

16. The frequency distribution of the age (in years) of 80 patients in a clinic is
given in the table below

Age 10-15 15-20 20-25 25-30 30-35 35-40

Number of 5 15 24 18 10 8
Patients

Find the mean and mode. Hence, calculate and interpret Pearson’s coefficient of

the skewness given that the standard deviation is 6.798 years. ( 2011/12 )

17. The following is the stem-and-leaf diagram for a sample of heights (in cm) of
a type of herbal plant. All observations are integers.
13 0 6
14 5 8 8
15 0 2 5 8
16 0 6 8 8
17 1 5
19 2
23 0

(a) Calculate the mean.
(b) Find the values of the median, first and third quartiles.
(c) Construct the box-and-whiskers plot and comment on the data distribution.

( 2011/12 )

18. The mean and median of the ordered sample data 1, 2, 4, 7, x, y, 11, 12, 15, 2y

are 8.7 and 8.5 respectively. Determine the values of x and y. Hence, find the

variance. ( 2012/13 )

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