PYQ SM015

MATHEMATICS SM015

TOPIC 1: NUMBER SYSTEM

1. Given that the complex number u, v and w such that 1  1  1 . If v 1 3i and
u vw

w  2  i , express u in the form a  bi , where a and b are real numbers. ( 2003/04 )

2. Given a complex number z  i ( 2003/04 )
2i

(a) State z in the form of a  bi
(b) Find the modulus and argument of z

3. Given z1 1  3i and z2  2  5i ( 2004/05 )

(a) Express 1  1 in the form of a  bi
z1 z2



(b) Find the argument of z 2 in radian

4. Given z  x  yi , where x and y are real numbers. If z  i  2 , show that
z 1i

3x2  3y2  8x  6y  7  0 ( 2004/05 )

5. Given two complex number z1  1 3i and z2  2  i , express z1  z2 in the form of
z1z2

a  bi , where a and b are real numbers.. Hence determine z1  z2 . ( 2005/06 )

z1z2



6. Given the complex number z and its conjugate z satisfy the equation z z 2z  12  6i .

Find the possible value of z. ( 2006/07 )

BKS 2018/19 Page 1

MATHEMATICS SM015

7. An equation in a complex number system is given by z 1  1 where
(z1  z2 )


z1

z1  1 2i and z2  2  i . Find

(a) the value of z in the Cartesian form a  bi

(b) the modulus and argument of z. ( 2006/07 )

8. If z1  4  i and z2 1  2i , find z1  5 . Express the answer in polar form. ( 2007/08 )
z2

9. Given z1  1  i and z2  4  2i . Express z12 in the form of a  bi , where a and b
z1  z2

are real numbers. Hence, determine z12 . ( 2008/09 )
z1  z2



10. Given that z  x  i y , where x and y are real numbers and z is the complex conjugate of

z. Find the positive values of x and y so that 1  2 3i ( 2008/09 )
z


z

11. Given a complex number z  a  bi which satisfy the equation z 2  8  6i ( 2009/10 )
(a) Find all the possible values of z.
(b) Hence, express z in polar form.

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MATHEMATICS SM015

12. (a) Given two complex number z1  2  i and z2  1 2i .Express z12  1 in the



z2



form x  yi , where x and y are real numbers and z2 is conjugate of z2

(b) Hence, find the modulus of z12  1 (2010/11)



z2

13. Given two complex numbers z1  5  3i and z2  2  i .

(a) State z1 and z2 .

(b) Determine the value of k if 1  k z1 .
z1

(c) Find z1 z2 . Hence, show that z1 z2  z1z2 . (2011/12)

14. Given a complex number z 1  3i . Determine the value of k if z 2  k 1 . (2012/13)
z

15. Find the values of p and q if p  q 1 5i
4  2i 4  2i 2

  (b) Given log10 2  m and log10 7  n. Express x in term of m and n if 143x1 82x3  7

(2013/14)

16. Let z  a  bi be a nonzero complex number .



(a) Show that 1  z
z z2

(b) Show that if z  z , then z is a complex number with only an imaginary part.

 (2014/15)

(c) Find the value of a and b if z (2  i)  (z1) (1 i)

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MATHEMATICS SM015

17.Given z1  3  3i and z2  3 2i .

(a) Write z1 in polar form

(b) Express z1 z2   i 3  in the form a  bi, a, b  ( 2015/16 )
13 
z 2

 18. Solve for p and q where p  q , such that ( p  qi)  3  16  i3 ( 2016/17 )
3i

19. Given a complex number z  2  i

(a) Express  1 in the form a  bi , where a and b are real numbers.

z 

z

(b) Obtain  1 . Hence, determine the values of real numbers  and  if

z 

z

  i   1   1  2 ( 2017/18 )
 
z  z 

z z

SUGGESTED ANSWERS
1. 25  5 i

13 13

2. (a)  1  2 i (b) z  5  0.447 , Arg(z)  2.03 rad
55 5

3. (a) 49  37 i 
290 290
(b) Arg(z 2 )  1.19 rad

4. Shown

5. (a) 1  1 i (b) 26  0.510
2 10 10

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MATHEMATICS SM015

6. z  3  3i or z 1  3i

7. (a) 1  1 i (b) z  2  0.141, Arg(z)  0.785 rad
10 10 10

8. 3  3i , 3 2cos(0.785)  i sin(0.785)

9. 1  1 i , 2  0.471
33 3

10. x  y  1
2

11. (a) z  3  i , z  3  i

(b) z  10cos(0.322)  i sin(0.322) , z  10cos(2.82)  i sin(2.82)

12. (a) 16  18 i (b) 4.816
55

13. (a) z1  5  3i and z2  2  i (b) k  1 (c) z1z2 13  i
34

14. k  8 x   10m
3n  9m
15. (a) p 15, q  10 (b)

16. (a) Shown (b) shown (c) a 1,b 1

17. (a) z1  3 2cos   i sin  
4 4

(b) 5  12i
13 13

18. p  15 and q  9

19 (a) 8  6 i ( b) 2 ,   56 ,    192
55 25 25

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MATHEMATICS SM015

TOPIC 2: EQUATIONS, INEQUALITIES AND ABSOLUTE VALUES

 1. By substituting a  3x , solve the equation 9x  3  28 3x1 ( 2003/04 )

2. Solve the following inequalities ( 2003/04 )
(a) x2  x 12  0
(b) 2x 1  1
x2

3. Solve 3ln 2x  3  ln 27 ( 2003/04 )

4. Solve x5e3ln x  4x  21 ( 2004/05 )

5. Solve the following inequalities : ( 2004/05 )
(a) 4x  9  12
x
(b) 1 log2 x  6log x 2  0
(c) x  5  1
2x  4

   6. Find the values of x satisfying the equation log4 x4  4  1 log4 x2  4 ( 2005/06 )

7. Solve the following inequalities : ( 2005/06 )
(a) 7x2  x  6  x2  4
(b) x  3  3
x 1

8. By substituting a  2x , solve the equation 4x  3  2x2 ( 2006/07 )

BKS 2018/19 Page 6

MATHEMATICS SM015 ( 2006/07 )

9. Obtain the solution set for 2x 1  x2  4

10. (a) Find the solution set of the inequality 1  1
3 2x x  4

(b) Solve the following inequality equation for all x is real numbers. Write your answer in

set form 4  3  2x  1 ( 2006/07 )
1 x

11. Given that 81y  3(2y3)x and 218y6x  64xy . Find the values of x and y . ( 2007/08 )

12. Solve the following inequalities : ( 2007/08 )
(a) x  1
x  4 2x 1
(b) x  2
x4

13. Solve the equation 3log x 3  log3 3 x  10 ( 2008/09 )
3 ( 2008/09 )
( 2009/10 )
14. Determine the interval of x satisfying the inequality x  2  10  x2

 15. Solve the equation 32x 10 3x1 1  0

16. Determine the solution set for 2x  3  5 . ( 2009/10 )
x ( 2009/10 )

17. Solve 2 5  x  x

18. Solve the equation ln x  3  2 . ( 2010/11 )
ln x
Page 7
BKS 2018/19

MATHEMATICS SM015 ( 2010/11 )
( 2011/12 )
19. Solve the following inequalities.
(a) 3x2  x  4  0
2x2  3x  2
(b) x  1  2
x3

 20. 32x1  28 3x  9  0

21. (a) Solve the equation logx  4  2log 3  1 log x  ( 2011/12 )

2
(b) Find the solution set of the inequality

x3 2
x 1

22. Find the value of x which satisfies the equation ( 2012/13 )
log2 (5  x)  log2 (x  2)  3  log2 (1 x)

23. Determine the solution set of the inequality ( 2012/13 )
1 1

2x 1 x  2

24. (a) Solve x2  x  3  3 ( 2012/13 )
(b) Find the solution set of the inequality 2x2  9x  4  4
x2

25. Find the value of x which satisfies the equation log9 x  log3 x2 , x  1 ( 2013/14 )

26. Solve the equation 22x2  2x1  2x  23 ( 2013/14 )

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MATHEMATICS SM015 ( 2013/14 )
( 2014/15 )
27. Find the solution set of 2  3x  x  3

28. Solve the equation 3x  33x  12.

29. Solve the inequality 1  1 . ( 2014/15 )
6 x x 1

30. (a) Solve the following equation 6x2  x 11  4 .

(b) Find the solution set for the inequality 2   x  2  < 5 ( 2014/15 )
 x  4 

31. Evaluate the solution of 4 y2  1 up to three decimal places. ( 2015/16 )
3 y ( 2015/16 )

32. Solve the equation 2  log2x  15logx 2

33. (a) Solve the inequality x 1  2
x3

(b) Show that 2x  42x  22x. Hence , find theinterval for x so that ( 2015/16 )
8x ( 2016/17 )

2x  42x 13(2x )  36  0.
8x

 34. Determine the values of x which satisfy the equation 32x1  4 3x  9

35. (a) If 7  3 5  x  y . Determine the values of x and y. ( 2016/17 )
(b) Solve the equation log2 x  log4 (3x  4)  0
( 2016/17 )
36. (a) Solve the following equation 3  7, x  4
x4 Page 9

(b) Find the solution set for the inequality  4  x  x  4, x  3
x3

BKS 2018/19

MATHEMATICS SM015 ( 2017/18 )
( 2017/18 )
37. Solve the equation 32x1  (16)3x  5  0
( 2017/18 )
38. Solve the equation 3log9 x  log3 x2

39. Find the interval of x for which the following inequalities are true.
(a) 5 1  0
x3
(b) 3x  2  2
2x  3

SUGGESTED ANSWERS
1. x  1, x  2

2. (a)  ,4 3, (b)  ,2  2, 1   3,

 3

3. x  3 e  4.08
2

4. x  3

5. (a) 0, (b)  1 ,1  4,  (c) [ 1 ,2)  (2,9]
3
8 

6. x   6

7. (a)  2 , 1  (b) 0,1 1,3
3 2 

8. x  0 , x  1.59

9. x : x  1.45 or x  1

10. (a) x : 4  x   1 or x 3 (b) x : x  6 or x0 
 3 
2 

11. x  0, y  0 and x  5, y  5
2

12. (a) (4,1]  (1 ,2] (b)  ,8   8 , 

2 3 

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MATHEMATICS SM015

13. x  3 , x  39  19683

14.  ,3 2.37,

15. x  1 or x  1

16. x : x  0 or 1 x  3
 2 

17.  ,10   10 ,

3 

18. x  e or x  e3

19. (a)   , 4     1 ,1  2, (b)  ,7  5 , 
3
 3  2 
20. x  1, x  2

21. (a) x  9 (b) x : x  5  x  1
22. x  3  3

23. x : 2  x  1 or x  3
 2 

24. (a) x  2,0,1,3 (b) x : x  4 or 2  x  3
 
2 

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MATHEMATICS SM015

25. x  3

26. x  2 or x  3

27. (a) x :  1  x  5
 4 2 

28. x  1or x  2

29. x : 1  x  7 or x  6
 2 

30. (a) x   5 ,  7 , 1, 3 (b) {x : x  5 or x  4}
36 2 2

31. y  9.638 (b) (,2] or [3.17, )

32. x  8 or x  1
32

33. (a) (7,3)  (3, 5)
3

34. x  1 or x  2

35 (a) x  9 , y  5 ( b) x  4
22

36. (a) x  25 , 31 ( b) x : x  4 or 2  x  3 
77

37. x  1 , x  1.46

3

38. x  1, x  32  5.20

39. (a)  ,3[ 2,) (b)   8, 3     3 , 4 
 2  2 7

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MATHEMATICS SM015

TOPIC 3 : SEQUENCE AND SERIES

1. Find the sum of even numbers between 199 and 1999. ( 2003/04 )

2. The sum of the first four terms of a geometric series with common ratio  1 is 30.
2

Determine the tenth term and the infinite sum , S . ( 2003/04 )

3. The sum of the first n terms of an arithmetic sequence is n 4n  20

2

(a) Write down the expression for the sum of the first n 1terms.

(b) Find the first term and the common difference of the above sequence. ( 2003/04 )

4. Express 5.555… in the form of geometric series. Hence, find the ( 2004/05 )
(a) sum of the first n terms.
(b) infinite sum of the series.

5. The third and the sixth terms of a geometric series are 1 and 1 . Determine the values
2 16

of the first term and the common ratio. Hence, find the sum of the first nine terms of the

series. ( 2005/06 )

6. Expand 1 up to the term x3 and determine the interval of x for which the
(3  x)3
1

expansion is valid. Hence, approximate (2.9)3 correct to four decimal places. ( 2005/06 )

7. The sum of the first k terms of an arithmetic series is 777. The first term is  3 and the
k-th term is 77. Obtain the value of k and the eleventh term of the series. ( 2006/07 )

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MATHEMATICS SM015

8. (a) Find the first four terms in the binomial expansion of the following functions:

(i) 1 2x (ii) 1
(1  x)2

(b) Hence, expand 1  2x in ascending power of x up to the term containing x3 . By
(1  x)4

putting x  1 , show that 12000 is approximately 10947 . ( 2006/07 )
10 100

9. The sum of the first n terms of an arithmetic series is n (3n  5) . If the second and fourth
2

terms of the arithmetic series are the second and the third terms of a geometric series

respectively, find the sum of the first eleven terms of this geometric series. ( 2007/08 )

10. The fifth term and the tenth term of a geometric series are 3125 and 243 respectively.

(a) Find the value of common ratio, r of the series.

(b) Determine the smallest value of n such that S  Sn  0.02 , where Sn is the sum of
S

the first n term and S is the sum to infinity of the geometric series. ( 2008/09 )

11. (a) The rth term of an arithmetic progression is 1 6r. Find in terms of n, the sum of

the first n terms of the progression.

1 1 1  x   1
2
(b) (i) Show that 
9 x 3 9

(ii) Find the first three terms in the binomial expansion 1  x   1 in ascending power
2

 9

of x and state the range of the values of x for which this expansion is valid.
(iii) Find the first three terms in the expansion of 3(1  x) in ascending powers of x

9x

( 2008/09 )

BKS 2018/19 Page 14

MATHEMATICS SM015

12. The first term of common difference of an arithmetic progression are a and  2 ,

respectively. The sum of first n terms is equal to the sum of first 3n terms. Express a in

terms of n. Hence, show that n  7 if a  27 . ( 2009/10 )

13. (a) Expand 1 and 1 x2 .

(4  x) 2 (1  3x) 2 in ascending powers of x up to the terms

1 1 x2 and determine the

(b) Find the expansion of (4  x) 2 (1  3x) 2 up to the terms

range of x such that this expansion is valid. Hence, by substituting x  1 ,
13

approximate the value of 51 correct to four significant figures. ( 2009/10 )

14. The sum Sn of the first n terms of an arithmetic progression is given by Sn  pn  qn2 .

The sum of the first five and ten terms are 40 and 155 respectively.

(a) Find the values of p and q.

(b) Hence, find the nth term of arithmetic progression and the values of the first term, a

and the common difference, d. ( 2010/11 )

15. (a) Given that 1  0.015151515..  p  q  s  ..., where p, q and s are the first three
u

terms of geometric progression. If p  0.015, state the value of q and s in decimal

form. Hence, find the value of u.

1

(b) Find the expansion for 1  x 3 up to the term x2 . State the range of x for which

 16 

1

the expansion is valid. Show that 3 8  x  21  x 3 . Hence, by substituting
2  16 

x  2, approximate 3 7 correct to four significant figures. ( 2010/11 )

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MATHEMATICS SM015

16. The ninth term and the sum of first fifteen terms of an arithmetic progression are 24
and 330 respectively. Find the first term, a and the common difference, d. Hence, find
the least possible value n, such that the sum of the first n terms is greater than 500.
( 2011/12 )

17. (a) Given that the sum of the first n terms, Sn of a series as Sn  1   1 n . Find an
3

expression for the nterm. Show that the series is a geometric series and find the sum

to infinity, S .

1

(b) Expand 1  2  2 in the descending power of x up to the term in x3 . Hence, by
 x

substituting x  3 , evaluate 5 correct to three decimal places. ( 2011/12 )
3

18. Given k  2, k  4, k  7 are the first three terms of geometric series. Determine the value

of k. Hence, find the sum to infinity of the series. ( 2012/13 )

19. The first four terms of a binomial expansion (1 ax)n is 1 x  1 x2  px3  ... . Find
2

(a) The values of a and n where n  0 .

(b) The values of p. Hence, by substituting x  1 , show that 3 is approximately equal
42

to 157 . ( 2012/13 )
128

20. (a) In an arithmetic progression, the sum of first four terms is 46 and the seventh term
exceeds twice of the second term by 5. Obtain the first term and the common
difference for the progression. Hence, calculate the sum of the first ten even terms of
the progression .

BKS 2018/19 Page 16

MATHEMATICS SM015

(b) A ball is dropped from a height of 2 m. Each time the ball hits the floor, it bounces
vertically to a height that is 3 of its previous height.
4
(i) Find the height of the ball at tenth bounce.
(ii) Find the total distance that the ball will travel before eleventh bounce.
( 2013/14 )

21. Using algebraic method, find the least value of n for which the sum of the first n terms of a

geometric series

0.88  (0.88)2  (0.88)3  (0.88)4 ...

is greater than half of its sum to infinity. ( 2014/15 )

3

22. (a) State the interval for x such that the expansion for (4  3x) 2 is valid.

3

(b) Expand (4  3x)2 in ascending power of x up to the term in x3

3

(c) Hence, by substituting an appropriate value of x, evaluate (5) 2 correct to three decimal

p laces. ( 2014/15 )

23. The first three terms of a geometric series are 4 m  2, (2m 1) and 12. Determine the value
3 

of m. Hence, find the sixth terms for this sequence. ( 2015/16 )

24. (a) Expand (2  1 in ascending power of x, up to the term x3

x) 2

(b) Use the expansion in (a) to approximate 2 ( 2015/16 )
3

25. The seventh term of a geometric series is 16, the fifth term is 8 and the sum of first ten

terms is positive. Find the first term and the common ratio. Hence, show that

 S12  126 2 1 ( 2016/17 )

BKS 2018/19 Page 17

MATHEMATICS SM015

1

26. (a) Obtain the expansion for (1  x ) 4 in ascending powers of x up to the term x3 . State
4

1

the interval for x such that the expansion (1  x ) 4 is valid. Hence, obtain the simplest
4

1

form of the expansion (16  4x) 4

 1
(1 x) 
(b) Write 4 12 in the form of K  4 4  . Hence, approximate 4 12 correct to three



decimal places. ( 2016/17 )

27. The first and three more successive terms in geometric progression are given as follows:
7, …, 189, y , 1701, …

Obtain the common ratio r. Hence, find the smallest integer n such that the n-th term

exceeds 10 000. ( 2017/18 )

1

28. (a) Expand (1  x ) 2 in ascending powers of x up to the term x3 and state the interval of x
3

for which the expansion is valid.

1

(b) From part (a), express 9  3x in the form of a (1  x ) 2 , where a is an integer.
3

(c) Hence, by substituting the suitable value of x, approximate 8.70 correct to two
decimal places.

BKS 2018/19 Page 18

MATHEMATICS SM015

SUGGESTEDANSWERS

1. 989100

2. T10   3 , S  32
32

3. (a) 2(n 1)(n  4) (b) T1  12 , d  4

4. 5.555...  5  0.5  0.05  0.005  ...

 (a)
Sn  501 0.1n (b) S  50
9
9

5. 511
128

6. 1 1  x  2 x2  10 x3  ... , the expansion is valid when 3 x3 , 0.0410
27 3 27

7. k  21, T11  37

8. (a) (i) 1  x  x2  x3  ... (ii) 1 2x  3x2  4x3  ...
22

(b) 1 3x  9 x2  13 x3  ...
22

9. 699050.5

10. (a) r  3 (b) n  8
5

BKS 2018/19 Page 19

MATHEMATICS SM015

11. (a) Sn  n(4  3n)

(b) (i) Shown (ii) 1 x  x2  ... expansion is valid when  9  x  9
18 216

(iii) 1 19 x  13 x2  ...
18 216

12. a  4n 1

13. (a) (4  1  2 x  x2  ... , 1 1 3 x  27 x2  ...

x) 2 4 64 (1  3x) 2 28

(b) 2  13 x  455 x2  ...., expansion is valid when  1  x  1 , 51  7.168
4 64 33

14. (a) p  1 , q  3
22

(b) Tn  3n 1 , a  2, d  3

15. (a) q  0.00015 , s  0.0000015 , u  66

1
(b) 1  x  3  1  x  x2  ... ;
 16  48 2304  x 1 16  x 16 , 3 7  1.913

16

16. a  8, d  2, n  20

1 n1, 1
3
17. (a) Tn  2  S 1 (b) 1  2  2 1 1  1  1 , 5  1.296
3   x x 2x2 2x3 3

18. k  10 , S  24 (b) p  1
19. (a) n  1 , a  2 2

2

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MATHEMATICS SM015

20. (a) a  7 d  3 ; S10  370

(b) (i) T10  2 3 10  0.1126 (ii) S11  13.3242
 4 

21. n  6

22. (a) 4  x 4 (b) 8  9x  27 x2  27 x3  ...
33 16 128

3

(c) 52  11.180

23. , m  5 T6  324
2

24. (a) 1  x  3 x2  5 x3  ...
2 4 2 32 2 128 2

(b) 2  0.816
3

25. r  2 , a  2

1

26. (a) (1 x) 4  1 x  3 x2  7 x3  ... the expansion is valid when  4  x  4
4 16 512 8192

11

(16  4x) 4  2(1  x ) 4
4

1

(b) 4 12  2(1  1) 4 , 4 12  1.862
4

27. r  3 , n  8

1 1 x  x2  x3  ... the expansion is valid when 3 x3

28. (a) (1  x ) 2
3 6 72 432

1

(b) 9  3x  3(1  x ) 2
3

(c) 8.70  2.95

BKS 2018/19 Page 21

MATHEMATICS SM015

TOPIC 4 : MATRICES AND SYSTEM OF LINEAR EQUATIONS

1. (a) Let matrix A  6  4 . If A2  pA  qI 0 where p and q are real numbers, I is a
1 
0 

2  2 identity matrix and 0 is a 2  2 null matrix. Find p and q.

(b) Given a matrix equation Ax  B as

1 1  3 x 2
2 4  y 3
1   

1 1 1  z  1

(i) Find the determinant of matrix A.

5 p 3
(ii) Given the cofactor matrix of A   4 2  2 , find p and q.

 q  2 1 

(iii)Determine the adjoint matrix of A and hence find the inverse of A. ( 2003/04 )

2. (a) Matrices A and B are given as

 1 2 3  4 1  4
A  1 0 4 , B  1 1 3.5
 0 2 2  1 1 1

Find AB and hence find A1.

(b) A company produces three grades of mangoes: X, Y and Z. The total profit from 1kg

grade X, 2kg grade Y and 3kg grade Z mangoes is RM20. The profit from 4kg grade Z

is equal to the profit from 1kg grade X. The total profit from 2kg grade Y and 2kg

grade Z mangoes is RM10.

(i) Obtain a system of linear equations to represent the given information.

(ii) Write down the system in (i) as a matrix equation.

(iii) Use the inverse method to solve the system of linear equation. Hence, state the

profit per kg for each grade. ( 2003/04 )

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MATHEMATICS SM015

3. (a) Show that the determinant of the matrix A   x2 x 1
 y2 

 y 1 is ( y  x)(z  x)(y  z) for
 z 2 z 1

real x, y and z

1 1 1
(b) By substituting x  1, y  2 and z  3 , the matrix A becomes A  4 2 1 . Find

9 3 1

the adjoint and inverse of the matrix A.

(c) The graph of a quadratic equation y  ax2  bx  c passes through the points whose

coordinates are 1,2 , (2,3) and (3,6) , and

(i) obtain a system of linear equations to represent the given information.
(ii) Write down the system in (i) as a matrix equation in the form of AX  B where

a
X  b

 c 

(iii) Use the inverse of matrix to solve the system of linear equation in (ii). Hence,

find the quadratic equation of the graph. ( 2004/05 )

 1 0  2  1  1 2 1 0
 1 ,  1  3 0  1
4. Given A   2 1 0  B  0  and C 
0
 1  2  1

(a) Find matrix D  A  (BC)T

(b) Show that |AD| = |DA| ( 2005/06 )

1  2 1 0 1
If P  1  0
5. 0 1  and Q  1 0  , find matrix R such that

 1

 0 2 2 ( 2006/ 07 )
R  2(PQ)   2 4 3
Page 23
 4 5 3

BKS 2018/19

MATHEMATICS SM015

1 a 2
6. Given that A  2 1 2 , where a and b are constants.

2 2 b

4 1 2
a) If A  13, evaluate the determinant of matrix 2 a 2 using determinant

4 2 b

properties.

b) Given that A2  4A  5I , where I is a 3 3 identity matrix. Show that a  2

and b  1. Hence find A1 . ( 2007/08 )

1 2 1  2 2  3
(a) Given the matrices P  2 2  
7. 1 and Q   2 1 0  . Find PQ and hence

1 2 2  3 0 3 

determine P1 .

(b) The following tables shows the quantities (kg) and the amount paid (RM) for the
three types of items bought by three housewives in a supermarket.

Housewives Sugar (kg) Flour (kg) Rice (kg) Amount paid (RM)

Aminah 3 6 3 16.50

Malini 6 3 6 21.30

Swee Lan 3 6 6 21.00

The price in (RM) per kilogram (kg) of sugar, flour and rice are x, y and z

respectively.

i) Form a system of linear equations from the above information and write

the system of linear equations in the form of matrix equation AX  B .

ii) Rewrite AX  B above in the form kPX  B , where A  kP, P is the

matrix in (a) and k is a constant. Determine the value of k and hence find

the values of x, y and z. ( 2008/ 09 )

BKS 2018/19 Page 24

MATHEMATICS SM015

3 x 2x 
Matrix A is given as A  0 x 
8. 4  and A  75 . Find

0 0 x 10

a) the value of x.

b) the cofactor and the adjoint matrix of A. Hence, determine the inverse of A.

( 2009/10 )

9. The following table shows the quantities in kilogram (kg) and the amount paid (RM) for
three types of fruits bought from three stalls at a night market.
Fruit Mango (kg) Durian (kg) Rambutan Amount paid (RM)
Stall (kg)
P 5 3 2 34.00
Q 3 4 4 37.00
R 2 3 4 29.00

The price in RM per kilogram (kg) for mango, durian and rambutan are x, y and z
respectively.
a) Form a system of linear equations which represent the total expenditure per stall

calculated based on the weight bought and price per kiligram. Hence, write the
system in the form of a matrix equation AX  B .
b) Find the determinant, minor and adjoint of matrix A.
c) Based on part (b) above, find A1 . Hence, solve the matrix equation. ( 2009/10 )

BKS 2018/19 Page 25

MATHEMATICS SM015

10. The following table shows the price(RM) per type of 0.5kg cakes old at the shops P, Q
and R together with the total expenditure if a customer buys a number of each type of
cake from the listed shop.

Cake types Banana Chocolate Vanilla Total Expenditure
Shops (RM)
P 5 8 5 36
Q 4 6 6 30
R 5 9 7 40

Let the number of banana, chocolate and vanilla cakes bought from each shop be x, y
and z respectively.

(a) Write the matrix equation AX  B using above information.

(b) Obtain the adjoint matrix of A. Hence, find the inverse of matrix A.
(c) Determine the value of x, y and z using the inverse matrix of A obtained in (b).

( 2010/11 )

1 2 1
11. Matrix A is given as 2 3  3.

2 2 1

 3 x  y  2
 
(a) Given the cofactor of A is  0 1 2  where x  0. Determine the values of x

 3 x 2 1

and y.

(b) Given A2  4A  I  0, show that A3  15A  4I where I is the 3×3 identity matrix.

Hence, find A3. ( 2011/12 )

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MATHEMATICS SM015

12. The following table shows the quantities (unit) and the amount paid (RM) for pens

bought from three shops.

Pilot Kilometrico Papermate Amount paid

(unit) (unit) (unit) (RM)

S1 p 2p 18.00

T1 q 3q 31.00

U 1 r 4r 37.00

Given the price in RM per unit of pilot, kilometrico and papermate pens by x, y and z

respectively.

(a) Obtain a system of linear equations to represent the given information.

 x 
(b) Write the system in the form of matrix equation AX  B where X   y .

 z 

(c) Given the minor a11, a21 and a22 of matrix A is 9, 12 and 8 respectively. Find the
values of p, q and r.

(d) Find the determinant, cofactor, adjoint and A1 of matrix A. Hence, find the values of

x, y and z. ( 2011/12 )

13. (a) Matrix M is given as 3 1 Show that M2  7M  8I, where I is the 2×2 identity
 4 .
4 

matrix. Deduce that M 1  7 I  1 M.
88

p 1 1 1 
 
(b) Given matrix A   3 2 4  and A  27. Find the value of p, where p is an

 1 0 p  2

integer. ( 2012/13 )

BKS 2018/19 Page 27

MATHEMATICS SM015

2 2 3
14. Given A  1 5 4

3 1 4
(a) Find the determinant of matrix A.
(b) Find the minor, cofactor and adjoint of matrix A.
(c) Given A(Adjoint(A))  A I, where I is 3×3 identity matrix, show that

A1  1 adjoint A. Hence, find A1 .

A

(d) By using A1 in part (c), solve the following simultaneous equations. ( 2012/13 )

2x  2y  3z  49
x  5y  4z  74
3x  y  4z  49

5 3 0 c 3 0
15. Given matrices A  3 1 0 and B  3 d 0. Find the values of c, d and e such that
0 0 2 0 0 e

AB  14I, where I is the identity matrix. Hence, determine A1. ( 2013/14 )

16. An osteoporosis patient was advised by a doctor to take enough magnesium, vitamin D and
calcium to improve bone density. In a week, the patient has to take 8 units magnesium, 11
units vitamin D and 17 units calcium. The following are three types of capsule that contains
the three essential nutrients for the bone:
Capsule of type P: 2 units magnesium, 1 unit vitamin D and 1 unit calcium.
Capsule of type Q: 1 unit magnesium, 2 units vitamin D and 3 units calcium.
Capsule of type R: 4 units magnesium, 6 units vitamin D and 10 units calcium.

Let x, y and z represent the number of capsule of types P, Q and R respectively that the

patient has to take in a week.

(a) Obtain a system of linear equation to represent the given information and write the

x
 y.
system in the form of matrix equation AX  B, where X  

 z 

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MATHEMATICS SM015

(b) Find the inverse of matrix A from part (a) by using the adjoint method. Hence, find

the values of x, y and z.

(c) The cost for each capsule of type P, Q and R are RM10, RM15 and RM17

respectively. How much will the expenses be for 4 weeks if the patient follows the

doctor’s advice? ( 2013/14 )

1 0 0 1 0 0
17. Given matrices A  4 1 0 and B  z 1 0 where B is the inverse of A. find x, y and
a b 1 x y 1

z in terms of a and b. ( 2014/15 )

18. Two companies P and Q decided to award prizes to their employees for three work ethical

values, namely punctuality (x), creativity (y) and efficiency (z). Company P decided to

award a total RM3850 for the three values to 6, 2 and 3 employees respectively, while
company Q decided to award RM3200 for three values to 4, 1 and 5 employees respectively.

The total amount for all the three prizes is RM1000.

(a) Construct a system of linear equations to represent the above situations.
(b) By forming a matrix equation, solve this equation system using the elimination

method.
(c) With the same total amount of the money spent by company P and Q, is it possible for

company P to award 15 employees for their creativity instead of 2 employees? Give

your reason. ( 2014/15 )

1 x 1
 
19. (a) Determine the values of x so that  x 0 1  is singular.

1 3 1

(b) If A  3 5 and B  1 4 A  BCB 1. ( 2015/16 )
1 2 0 1, find C when

BKS 2018/19 Page 29

MATHEMATICS SM015

20. A curve y  ax2  bx  c where a, b and c are constants, passes through the points (2,11),

(-1,-16) and (3,28).

(a) By using the above information, construct a system containing three linear equations.

(b) Express the above system as a matrix equation AX  B.

(c) Find the inverse of matrix A by using the adjoint matrix method. Hence, obtain the

values of a, b and c.

( 2015/16 )

2 0  4 1 3 1 2 1
 1 6 2   2 R  3 2 2 . Find
21. (a) Given P  , Q   0 5 and 4 1 R1by using
3
 6

elementary row operation method. Hence , if RX  3Q  PT , determine the matrix X.

(b) Ahmad bought an examination pad, 2 pens and a tube of liquid paper for RM 18. Ali

spent RM 24 for 3 examination pads, 2 pens and 2 tubes of liquid paper. In the meantime

Abu spent RM 36 at the same store for 3 examination pads, 4 pens and a tube of liquid

paper. Let x , y and z represent the price per unit for examination pad, pen and liquid paper

respectively.

(i) Obtain the system of linear equations from the above information.
(ii) Write the system in the form of matrix equation AX  B .

(iii) State the price of each unit of examination pad, pen and liquid paper.

(iv) Aminah bought 4 examination pads, 5 pens and 1 tube of liquid paper. What is the

total price paid ? ( 2016/17 )

22. Given matrix A 2 3 A2  A  I  0,  and  are constants, where I and
 2 5 such that

0 are identity matrix and zero matrix of 2  2 respectively. Determine the values of  and

 . ( 2017/18 )

BKS 2018/19 Page 30

MATHEMATICS SM015

23. Given the system of linear equations as follow:
2x  4 y  z  77
4x  3y  7z  114
2x  y  3z  48

(a) Express the system of equations in the form of matrix equation AX  B where
 x 

X   y  . Hence, determine matrix A and matrix B .
 z 

(b) Based on part 10 (a) , obtain A . Hence, find
(i) P if PA  I where I is an identity matrix 3 3.
(ii) Q if Q  (2A)T
(iii) find adjoint A . Hence, obtain A1 and find the values of x, y and z. ( 2017/18 )

SUGGESTEDANSWERS

1. (a) p  6, q  4

(b) (i) A  2 (ii) p  2, q  1

 5 4 1  1  5 4 1 
(iii) adjA   2  2 , 2  2  2
(c)  3 3 1  A1   3 3 1 
2 2

5 0 0 1  4 1  4
AB  0 0 5  1 3.5
2. (a) 0 5 5 , A1   1 1  1
0 1

(b) (i) x  2y  3z  20 , 4z  x , 2y  2z  10

 1 2 3x 20
(ii) 1 4 y  
0   0 

 0 2 2 z  10

(iii) x  8 , y  3 , z  2

BKS 2018/19 Page 31

MATHEMATICS SM015

3. (a) Shown

1 2 1 1 1 2 1
  2  
(b) adjA   5 8 3  , A1   5 8 3 
6 6
 6  2  6  2

(c) (i) a  b  c  2 , 4a  2b  c  3 , 9a  3b  c  6

1 1 1a 2
(ii) 4 2 1b  3

9 3 1c 6

(iii) a  1, b  2 , c  3, Quadratic equation y  x2  2x  3

 0 2  3
 
4. (a) D   3 0 2  (b) shown

 2 1 1

 2  2 4
5. R   4 6 5

 4 3 3

412  3 2 2 
1
6. (a) 2 a 2  26 (b) A1   2 3 2 
42b 5  2 
2  3

2 2  1
 3 
 3

3 0 0  
 
7. (a) PQ  0 3 0 , P 1   2 1 0 
0 0 3  3 3

 
1 0 1 
 

(b) (i) 3x  6y  3z  16.50 , 6x  3y  6z  21.30 , 3x  6y  3z  21.00

3 6 3x 16.50
6 3 6 y  21.30
3 6 6 z  21.00

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1 2 1x 16.50
(ii) 32 1 2 y  21.30 , k  3 , x  1.4 , y  1.3 , z  1.5

1 2 2 z  21.00

8. (a) x  5

1 1 2
 3 3 
 25 0 0   25 25  30 A1  0 1 5 
   15 12 4 
(b) C   25  15 0  , Adj A   0
 5 25 
 30 12 15  0 0 15  0 1
0  5 

5x  3y  2z  34 5 3 2 x 34
3x  4 y  4z  37 , 3 4  y 37
9. (a) 2x  3y  4z  29 4 

2 3 4  z  29

4 4 1   4 6 4 
A  10, Minor A  6 16  , Adj A   4  14
(b) 9  16

4 14 11  1  9 11 

2 3 2
A1  52 5 
(c) 8 5  , x  3, y  5, z  2
7 
 5 5 5
1 9 11 
 10 10 10 

5 8 5 x 36
4 6  y 30
10. (a) 6  

5 9 7  z  40

6 11 9 
A1  71 7 
 12  11 18  14 5 
 10 5 
(b) Adjoint of A   2 5 10 , 7 7
7
 6  2   3 5 1
7
14 7 

(c) x  3, y  2, z  1

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11. (a) x  1, y  5 ; 11 30 15
(b) A3  30 41  45

30 30 19

12. (a) x  py  2 pz  18 , x  qy  3qz  31, x  ry  4 r z  37

1 p 2 p x 18
(b) 1   y 31
q 3q   

1 r 4r   z  37

(c) p  2, q  3, r  3

 9 3 0   9 12 6 
(d) A  3, cofactor  12 8 1, adjo int   3 8  5 ;
 6  5 1   0 1 1 


 25 ;
 3 4
8
A1  1 x  4, y  3, z  2

 3 3
 1 1
 0 3 3 

13. (a) Shown (b) p  1

 16  5  7
 1  5
14 (a) A  6 ; (b) adjA   8

14 4 8 

8 5  7 
 6  
(c) A 1  3 6  (d) x  71, y  73, z  1
 4 1 5  6 63

 3 6 6
 7 2 4
3 3 3 

1 3 0 
A1  134 
14 
c 1, d  5, e  7 ; 5 0
15. 14 14 

 0 0 1
 2 

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MATHEMATICS SM015

2 1 4 x  8  1 1 1 
(a) 1  y 11 (b) A1  121 2 
16. 2 6   ; 2  ; x  1, y  2, z 1;
4 2 
1 3 10 z   7  5 3 

 4 4 4 

(c) RM228

17. x  4b  a, y  b, z  4

x  y  z  1000 (b) x  RM 400, y  RM350, z  RM 250;

18 (a) 6x  2y  3z  3850;

4x  y  5z  3200

(c) need to show determinant of new matrix equals 0.

19. (a) x  1,3; (b) C  7 15 
1  2

4a  2b  c  11 4 2 1a  11 
20. (a) a  b  c  16 (b) 1 1 1b  16

9a  3b  c  28 9 3 1c  28 

(c) a  2,b  7,c  11

 1 1 1  19 5
  
21. (a) R 1   1 3 3  X   3 3 
 1 1  25 17 

2 3 6  6 6
 1 2  41 2
 1 3  3   3 3 

(b) (i) x  2y  z  18 , 3x  2y  2z  24 , 3x  4y  z  36

1 2 1x 18 (iv) RM 45
(ii) 3 2 2 y  24

3 4 1 z  36
(iii) x  2, y  7, z  2

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22.   7 ,   16

2 4 1x  77 
23. (a) 4 3 7 y  114

2 1 3 z   48 

(b) A  10 (i) P  1 (ii) Q  80
10

 2 11 25  1  2 11 25 
  10 10  10 , x  10, y  13, z  5
(iii) adjA   2 4  10 , A1   2 4  10
6 6
 2  2

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TOPIC 5 : FUNCTIONS AND GRAPHS

1. Given that f (x)  2x 1 and h(x)  2x2  4x 1, find a function g such that

 f  g(x)  h(x) . Write g(x) in the form a(x  b)2  c , where a ,b and c are constants.

( 2003/04 )

2. Given g(x)  3 x and h(x)  1 . Find f (x) such that  f  g  h(x)  x ( 2004/05 )

x3 x 1

3. Function f and g are defined as f (x)  e2x , g(x)  1 x , x   . Find f 1(x) and

 hence obtain g  f 1 (x) ( 2005/06 )

4. A function f is defined by f (x)  x2  2x  3 for 0  x  5 . State the range of f and

determine whether f is one to one. ( 2007/08 )

5. Given h(x)  3x . Defining h2 (x)  h  h(x) , determine the function h2 (x) and hence

x3

deduce the inverse of h(x) . Evaluate h13(9) . ( 2007/08 )

6. Given f (x)  2x2 1, x  0 and g(x)  x  3 , find

(a) The inverses of f and g and verify that (g  f )1  f 1  g 1

(b) The function of h if (g  f )1  h(x)  1 ( 2007/ 08 )
x

(c) The value of x for which f  g  g  f

7. Given that f (x)  10  2x and g(x)  5  2x2 . Find the value of k so that
k

 f1  x 
( x2 )  g  2  . Hence, find f 1 o g (0). ( 2008/09 )

8. Let f (x)  4x 1 and g(x)  x  2 ( 2008/09 )
(a) Find the interval of x which f (x)  g(x) .
(b) If h(x)  f (x)  2g(x), express h(x) as a piecewise functions.

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9. Let f (ax)  a3x2  a2x  3a where a is non-zero.

(a) Find a if f (0)  6 .

(b) Determine f (x) .

(c) Determine the domain and range of f (x) . Hence, state the interval in which f is

one to one ( 2008/09 )

10. A function g is defined by g(x)  1 , x  1.Find g 1(x) and state its domain and
x 1

range , ( 2009/10 )

11. (a) Show that y  y2 1  0 for all real values of y.

(b) Let f be a function defined by f (x)  ex  ex . Find f 1(x) . ( 2009/10 )
2 ( 2010/11 )

12. Given the function f and g as follows:

f (x)  2  x2 , g(x)  x  2

(a) Find f  g and g  f

(b) Find g  f 1

(c) Determine the value of x such that f  g(x)  g  f (x)

13. The functions f and g are defined as:

f (x)  x 1, x 1
g(x)  x2, x0

Find the inverse function, f 1(x) and determine its range. Then, evaluate  f  g(2)

14. (a) Given f (x)  ex and g(x)  x2 . ( 2011/12 )
( 2011/12 )
(i) Find the domain and range of f and g.

(ii)Show that g  f (x)  e2x

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MATHEMATICS SM015

(b) Given h(x)  e2x ,    x  0

 x 1, x0

(i) Find h1(x) .

(ii) Sketch the graph for h(x) and h1(x) ( 2011/12 )

15. (a) A function f(x) is defined by f (x)  3x for x  6 . Show that f(x) is a one–to–one
x6

function. Find the values of x such that  f  f (x)  0

(b) Given that f (x)  1 3x and g(x)  x 1. Find f  g 1   7  . ( 2011/12 )
2  2 

16. The functions f and g are defined as f (x)  3x  4 , x  2 and g(x)  3  x .
x2

(a) Find f 1(x) and g 1(x)

 (b) Evaluate f  g 1 (3).

 (c) If g  f 1 (k)  2 , find the value of k. ( 2012/13 )
3

17. Given f (x)  ln(2x  3) and g(x)  ex  3 .
2

(a) Show that f(x) is a one-to-one function algebraically.

(b) Find  f  g(x) and g  f (x) . Hence, state the conclusion about the results.

(c) Sketch the graph of f(x) and g(x) on the same axes. Hence, state the domain and range

of f(x). ( 2012/13 )

18. Consider the function f (x)  1 ln(x), x  1. Determine f 1(x) and state its range. Hence,

evaluate f 1(3). ( 2013/14 )

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19. Given g(x)  kx  8 , x  5 where k is a constant.
4x  5 4

(a) Find the value of k if g  g(x)  x.

(b) Find the value of k so that g(x) is not a one-to-one function. ( 2013/14 )
20. Given f (x)  e3x  4 , x  R.

(a) Find f 1(x).

(b) On the same axes, sketch the graphs of f (x) and f 1(x) . State the domain of

f (x) and f 1(x) . ( 2013/14 )

21. (a) Given f (x)  2x 1and g(x)  x2  2x 1.

(i) Find  f  g(x).
(ii) Evaluate 3g  2 f (1).

(b) Given f (x)  2x  1 . State the domain and range of f (x) . Hence, on the same
2

axes, sketch the graph of f (x) and f 1(x) . ( 2014/15 )

22. (a) Determine whether f (x)  1 and g(x)  4x 1 are inverse function of each
x4 x

other by computing their composite functions.
(b) Given f (x)  ln(1 3x).

(i) Determine the domain and range of f (x) . Then sketch the graph of f (x) .

(ii) Find f 1(x) , if it exists. Hence, state the domain and range of f 1(x) .

( 2014/15 )

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23. Given a function f (x)  3  2x.
(a) Show that f is a one to one function.
(b) Find the domain and range of f
(c) Determine the inverse functions of f and state its domain and range.
(d) Sketch the graphs of f and f 1 on the same axis.
( 2015/16 )

24. (a) The function f is given as f (x)  ax  2 , x  4 . If  f  f (x)  x , find the value

3x  4 3
of a.

(b) Let f (x)  ln3x  2 and g(x)  ex  2 be two functions. Evaluate g  f 1(3).

( 2015/16 )

25. Given f (x)  x2 1, x  0
5

 (a) Determine f 1(x) . Hence, if f g(x)  1 e2(3x1) 1 , show that g(x)  e3x1
5

(b) Evaluate g f (2) correct to three decimal places.

(c) Assume that the domain for g(x) is x  0 , determine g 1(x) and state its domain

and range. ( 2016/17 )

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MATHEMATICS SM015

26. Given f (x)  e x and g(x)  x  3

(a) Show that  f  g(x)  ex3 x  3
e(x3) x  3

(b) Sketch the graph of y   f  g(x) . Hence, state the interval in which  f  g1(x)

exists.

(c) Determine  f  g1(x) , for x  3

(d) Find the function h(x) for x  1 , given that h  f (x)  2e x . Hence show that
3 1  3e x

h(x) is one to one function. ( 2016/17 )

27. Consider functions of f (x)  (x  2)2 1, x  2 and g(x)  ln(x 1) , x  0

(a) Find f 1(x) and g 1(x) , and state the domain and range for each of the inverse
function.

(b) Obtain g  f (x) . Hence, evaluate g  f (2) . ( 2017/18 )

28. Given the function g(x)  1 ( 2017/18 )
2x  5

(a) Find the domain and range of g(x) .
(b) Show that g(x) is one –to-one function. Hence, find g 1(x) .
(c) On the same axis, sketch the graph of g(x) and g 1(x)
(d) Show that g  g 1(x)  x .

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SUGGESTEDANSWERS

1. g(x)  (x 1)2 1

2. f (x)  1
1 x

 3. f 1(x)  1 ln x , g  f 1 (x)  1 1 ln x
22

4. Rf   4,12, f(x) not one to one function

5. h2 (x)  x , h1 (x)  3x , h13(9)  9
x3 2

6. (a) f 1(x)  x 1, g 1(x)  x  3
2

(b) h(x)  2  2x2 (c) 7
x2 4

7. k  1, ( f 1  g)(0)  5
2

8. (a)   1 ,1  6x  3 if x 1
5  if 4
(b) h(x)  
 2x  5 x 1
4

9. (a) a = 2 (b) f (x)  2x2  2x  6

(c) Df  (,) , Rf  11 ,   ,   , 1  or  1 , 
 2   2  2

10. g 1(x)  1  1 , D 1  (0, ) , R 1  (1, )
x2
g g

11. (b) f 1(x)  ln(x  x2 1)

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MATHEMATICS SM015

 12. (a) f  g(x)   x2  4x  2 , g  f (x)  4  x2

(b) g  f 1(x)  4  x
(c) x   3

2

13. f 1(x)  x2 1 R f 1(x)  [1, )  f  g(2)  3

14. (a) (i) Df  , Rf  0,  Dg   , Rg  0, 

(b) (i) h 1 ( x)   1 ln x, x 1
 2
 x 1, x  1

(ii)

15. (a) x  0

(b) f  g 1   7    4
 2 

16. (a) f 1(x)  4  2x , g 1(x)  3  x (c) k = 33
x3

 (b) Evaluate f  g 1 (3)  2

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MATHEMATICS SM015

17. (b)  f  g(x)  x g  f (x)  x . Since  f  g(x)  g  f (x) , f and g are inverse to

each other.
(c)

Df :   3 ,  , R f :  ,
 2 

 18. f 1(x)  e x1 R f 1 : 1,  f 1(3)  e2

19. (a) k = 5 (b) k =  32
5

20. (a) f 1(x)  1 ln(x  4)
3

Df :  ,  Df 1 : 4,  Page 45

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MATHEMATICS SM015

21 (a) (i)  f  g(x)  x2  2 (ii) 3g  2 f (1)  0

(b) Df :  1 ,   Rf : 0,
4 

(0, 1 ) yx
2 f (x)

1 f 1(x)
4 ( 1 ,0)
1
4 2

22. (a) f(x) and g(x) are inverse to each other.

(b) (i) Df  (, 1) Rf :  ,
3

f(x)

01 x
3

(ii) f 1(x)  1 (1 ex )
3

D f 1 :  , R f 1 :   , 1 
 3

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MATHEMATICS SM015

23. (b) Df    , 3  , Rf  0,
 2 

(c) f 1(x)  3  x2  D f 1  0,  R f 1    , 3
2  2 

(d)

f (x)
3
3

2

33
2

f 1(x)

24. (a) a = 4

(b) g  f 1(3)   1

3

25. (a) f 1(x)  5x 1

(b) g f (2)  e2

(c) g 1(x)  1 ln(x) 1 , D 1  Rg  1 ,   and R 1  Dg  0,
 e 
3 g g

26. (a) Shown
(b)

y y  ex3

y  e(x3)

X

 f  g1(x) exists at (,3] or [3,) Page 47

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MATHEMATICS SM015

(c) ( f  g)1(x)  3  ln x
(d) h(x)  2x

1 3x

27 (a) f 1(x)  2  x 1 D f 1  R f  (1, ) R f 1  D f  (2, )
g1(x)  ex 1
D 1  Rg  (0,) R 1  Dg  (0, )

g g

 (b) g  f (x)  ln (x  2)2  2  (g  f )(2)  ln 2

28. (a) Dg  (, 5)  (5 , ) Rg  (,0)  (0, )
22

(b) g 1(x)  1  5x
2x

(c)

(d) Shown Page 48

BKS 2018/19

MATHEMATICS SM015 ( 2003/04 )

TOPIC 6 : POLYNOMIALS
5x2  17x  17
1. Express (x  2)(x  1)2 as a sum of partial fraction.

2. If (x 1) and (x  2) are factors of the expression 4x4  6x3  ax2  bx 12 , determine

a and b. Hence, factorise the expression completely. ( 2003/04 )

3. Given (x  3) is one factor of P(x)  9 12x 11x2  2x3 . Factorise completely P(x) ,

and express 13x  18 as a sum of partial fraction. ( 2004/ 05 )
P(x)

4. A polynomial has the form P(x)  2x3  3x2  px  q , with x real and p, q constants.

When P(x) is divided by (x 1) the remainder is (2  4x) . Find the values of p and q,

and factorise P(x) completely if 2 is one of the roots. ( 2004/05 )

5. Polynomial P(x)  2x3  ax2  x  b has (x 1) as a factor and leaves a remainder 12

when divided by (x  3) . Determine the values of a and b . ( 2005/06 )

6. Two factor of the polynomial P(x)  x3  ax2  bx  6 are (x 1) and (x  2) . Determine

the values of a and b and find the third factor of the polynomial. Hence express

2x2  5x 13 as a sum of partial fraction. ( 2005/06 )
P(x)

7. Find the values of A, B, C and D for the expression 4x3  3x2  6x  27 in the form of
x4  9x2

the partial fractions A  B  Cx  D where A, B, C and D are constants. ( 2006/07 )
x x2 x2  9

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MATHEMATICS SM015

8. (a) Show that x  3 is a factor of the polynomial P(x)  x3  2x2  5x  6 . Hence,

factorize P(x) completely.

(b) If f (x)  ax2  bx  c leaves remainder 1, 25 and 1 on division by x 1, (x 1) and

(x  2) respectively, find the values of a, b and c. Hence, show that f (x) has two equal

real roots. ( 2006/07 )

9. Express 2x 1 in partial fractions ( 2007/08 )

(x  2)(x2  2x  4)

10. (a) Find a cubic polynomial Q(x)  (x  a)(x  b)(x  c) satisfying the following
conditions : the coefficient of x3 is 1, Q(-1) = 0 , Q(2) = 0 and Q(3) = -8

(b) A polynomial P(x)  ax3  4x2  bx 18 has a factor (x  2) and a remainder

(2x 18) when divided by (x 1) . Find the value of a and b. Hence, factorise

P(x) completely ( 2007/08 )

11. Express 5x2  3x  8 in partial fractions ( 2008/09 )
(1 x2 )(1 x)

12. Polynomial P(x)  mx3  8x2  nx  6 can be divided exactly by (x2  2x  3) . Find the
values of m and n. Using these values of m and n, factorise the polynomial completely.
Hence, solve the equation 3x4 14x3 11x2 16x 12  0 using the polynomial P(x)
( 2008/09 )

Express 4x  3 ( 2009/10 )

x  2 x2  2x  2
 13. in partial fractions.

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