MATHEMATICS
AM015
PAST YEARS EXAMINATION QUESTIONS
Table of content
Chapter Title Page
1 | P a g e
CHAPTER 1: NUMBER SYSTEM & EQUATIONS
1. Find the value of x that satisfies the equation log (1 x ) log (5 x ) log (x 2) 3
2
2
2
2. Express 1 20x 2 x 1 3
1 9
3. Express log 12 3 log 2 in the form a b log 3where a and b are real numbers to be
2
2
3 16
found.
4. Solve x 22 x 6 8
1 2 1 2
5. Simplify
1 2 1 2
1
6. Solve the equation 2 2 x 1 3 2 x
7. Solvelog x 2 3log 2 4
x
1 3
8 Simplify in the form of a b c where
2 3
2
3
9. Solve3log x 2log 10 x 2log x 8
10
10
3
10. Solve 2log 3 log x .
3
x
2
11. Solve 9 4x 1 27 .
k
12. Given that log x log 8 log 2 k log 4 0. If y log x , show that
2 x 2 x 2
y ky 2k 3 0. Find the value of x when k .
2
6
13. Solve 2log x 3 log 2 log 3 3 -4x .
3
6
14 Solve 5 x 5 x 1 .
5
15. Find the possible values of x if log x 2 3log 2 4.
x
Given
16.
A : 2x 2, x R
x
B :x x 1 or x 1, x R
C : 3x 3, x R
x
Represent , and A B C on a number line. Hence, find
(a) A . B
2 | P a g e
(b) A . B
(c) A . C
17. Solve log x 4log 3 3.
3
x
Simplify the following expressions
18.
1
)
x 2 (x y 2 3
(a)
y 2
2 5
(b)
2 5 2
19. Find the values of x that satisfy the equation 7 3 2 9 x 3 x .
-1
x
20. By substituting a 3 , solve the equation 9 x 3 28(3 ).
x
21. Solve 3 ln 2x 3 ln 27
5
22. Solve x e 3ln x 4x 21
2
4
log (x 4) 1 log (x 4)
23. Find the values satisfying the equation 4 4
5(2 ) 4 x 16
x
1
24. Solve the equation
25. By substituting a 2 , solve the equation 4 x 3 2 x 2
x
26. Given that 81 3 (2y 3)x and 2 18y 6x 64 . Find the possible values of and .x y
y
xy
10
27. Solve the equation3log 3 log 3 x
3
x
3
3 10 3 x 1
1 0
2x
28. Solve the equation
29. Given P ,7 , Q 3,4 and R 2, . Represent P ,Q and R on a line number.
Hence, find the solution sets for P ,Q and P R
'
R
Q
.
3
30. Solve the equation ln x 2 .
ln x
1 1
31. Prove that log xy log x log y
16
2 4 2 4
2x
x
9 3
32. Solve the equation 22 3 15 0 .
33. Given p 7 5 2 and q 7 5 2.Find the value of y such that
1 1 y 1 50 .
p 1 q 1
3 | P a g e
3 2n 3 18 3 2 n 1
34. (a) Simplify 2 .
5 3 n
1
log 8 log 27 2 log 5
(b) Without using the calculator, evaluate 10 10 10 .
3 log 6 log 5
2 10 10
(c) Solve x 5 4x 13.
35. Solve the equation 2log 3 log x 3 3 0.
x
36. Given √ + √3 = 5 where p and q are integers. Find the value of p and q without using a
2+√3
calculator.
37. Solve the equation 9 + 4 = 5(3 ).
38. Solve the equation 4 − 16 = 6(2 )
2
2
39. Show that log ( + √ − 1) + log ( + √ − 1) = 0
2 10
1/3
6
64 p q r
10 2
2
40. (a) Simplify 8 2/3 p q r
4 x 2 y
(b) Solve 1 if xy
9
3 x
41. Find the values of x which satisfies the equation log (5 x ) log (x 2) 3 log (1 x
)
2
2
2
42. Solve the equation: 3 2x 1 3 7 3 x
2
1 1
11 11 2
2
43. a) Evaluate 3 3 without using calculator
2 2
1 1
b) Show that 1
log pq log pq
q
p
n
243 3 2n 1
5
n
44. a) Evaluate 9 3 n 1
4 | P a g e
b) If x 3 2 2 , find the value of x 1 2 2
x
2
45. a) Consider the equation 2log 3 logx y x log y . Express in terms of
b) Solve the equation log p log 3p 1
27
3
46. Find the values of x that satisfies the equation 3 2x 1 3 2 28 3 x
47. Solve the equation log 2x 2 log 2 log 2 .
6
4
x
5 | P a g e
ANSWERS
1.x 3
3
2.x
4
10 5
3.x log 3
3 6 2
4.x 3
5. 6
6.x 2
7.x 2, x 8
8.5 3 3
9.x 100
1
10.x , x 3
81
1
11.x
8
1
12.x
8
13.x 2, x 4
14.x 0
15.x 2, x 8
16.( )( 2,1] (1,2] ( )( , ) ( )( 2,2]
a
c
b
1
17.x 3, x
81
y 8 10 10
a
18.( ) ( )
b
x 9
19.x 0.631, x 1
20.x 1, x 2
3e
21.x / 4.077
2
22.x 3
23.x 6, x 6
24.x 3, x 1
25.x 0, x 1.585
6 | P a g e
5
26.x 0, y 0; x 5, y
2
27.x 3, x 19683
28.x 1, x 1
29.P Q ( ,7);Q R (2,4];P R ' ( ,2]
30.x , e x e 3
31.
32.x 1
1
33. y
7
34. 5a 1b c x 4
35.x 3, x 9
36. p 175,q 100
37.x 0, x 1.262
38.x 3
39.
pqr
40. a 8 b x 18, y 9
2
41.x 3
42.x 1
1
43.
2
44. 9a 2b
1
45. p
3
46.x 2, x 1
47.x 2
7 | P a g e
CHAPTER 2: INEQUALITIES AND ABSOLUTE VALUES
1. Find the solution set for the inequalities (x x 5) 2x 10.
2. Solve the inequality x 3 2x 5.
1 2 1
3. Find the set of values of x for x x .
2 4
. c
.
4. Define x Hence, solve 1 2x 2 x
5. Solve the following inequalities
2
(a) 3x 3x 4
x
1
x
(b) 2x 3
6. Solve the inequalities
x 12 0.
2
x
7. Solve
2
7x x 4 .
2
x
6
8. Solve the following.
2x 3 3x 2
2
(a)
(3x 5)(x 2) 4
(b)
9. If 2x 4 6, find the values of m and n such that m x n 3 .
10. Determine the solution set which satisfies the following inequalities:
x x 2 3x 2.
(a)
(b) 3 2x 5.
11. (a) Find the solution set of the inequality 3x 2 x .
2
(b) Determine the solution set for x 16 .
12. (a) Determine the solution set for ( + 6) < 3 + 4 .
(b) Solve | 5 +1 | > 3 and express your answer in interval notation.
2
8 | P a g e
2
x
13. (a) Determine the solution set which satisfies the inequality x 2x
4
2
(b) Determine the possible values of x if x 2x 3. Express your answer in
interval number form
x 2
14. (a) Find the value of x such that 3
x 1
(b) Determine the solution set of 2x 2 13x
6
15. (a) Determine the solution set which satisfies the inequality 1 x 4 x 11
x
2x
(b) Determine the possible values of x if 1 4 . Express your answer in interval
3
number form
x 1
16. Find the solution set for 2 x 1
2
2
17. Solve x 2 4 2x 1 . Write your answer in the form of solution set
3
x 2x 4
18. Determine the possible values of if 4 .
2 3
19. (a) Find the solution for x x 2
x
4
(b) Given 3 2 2 a b 2 find a and b where ,a b
9 | P a g e
ANSWERS
1. :x x 5or x 2
2
2. 3 ,
3. :x x 0or x 2
1
4. 3, 3
2 4
5. a , 2, b ,2
3 3
6. , 4 3,
2 1
7. 3 2 ,
1 7 1 73 1 73
8. a 1, b x , x , x , x 2
5 3 6 6
9.m 3,n 2
10. :a x x 1or x 2 :b x x 4or x 1
1
11. a : x x 1 :b x x 4or x 4
2
x
b
12. : 4a x 1 ,1
13. a , 1 4, b 1 ,
3
5 1 1
14. a x , x b : x x or x 6
2 4 2
15 9
15. :a x x 1or x 7 b , ,
2 2
16. x x 5
:
3
17. :1x x 3
18. :x x 16
19.a 1,b 1
10 | P a g e
CHAPTER 3: SEQUENCES
1. If a contractor delays his housing project by one week, he will be fined RM3000. For every
subsequent week it is delayed, he will be fined 10% more than the previous week.
(a) How much will he be fined (i) in the second week, (ii) in the fifth week.
(b) If the project is delayed by five weeks, calculate the total fine the contractor has to
pay.
2 8
2. Given that the second term of a geometric sequence is and the fourth term is . If
5 125
T , p find the possible values of p.
3
2
3. The sum of the first n terms of an arithmetic progression is S n 7n n . Find the first term
and the common difference.
4. The sum of the first n terms of an arithmetic sequence is S 5n 2 n . Find the first term
n
th
and the common difference. Find the 15 term.
5. The sum of the first 20 terms of an arithmetic sequence is 50, and the sum of the next 20
terms is –50. Find the first term and common difference of the sequence.
th
6. The sum of the first 20 terms of an arithmetic progression is 400. The 15 term is 110. Find
the common difference and the first term.
7. (a) The fifth term of an arithmetic sequence is 10 and the sum of the first five terms is
30. Find the tenth term.
1
(b) Given 4,2,1, ,..., find the sum of the first eight terms.
2
11 | P a g e
8. (a) The sum of the first six terms and the sixth term of an arithmetic sequence are S 6
and T respectively. If S 6 T 6 35, find the eleventh term of the sequence given
6
that the first term is 3.
1 n 1
(b) The nth term of a sequence is given by T 3 .
n
2
(i) Show that the sequence is a geometric sequence.
(ii) Find the sum of the first six terms of this sequence.
x
1
9. (a) For the sequence 2,2 ,2 2x 1 ,2 3x 1 ,..., show that it is a geometric sequence.
(b) The second and the fifth terms of an arithmetic sequence are –12 and 324
respectively. Find the first term, a and the common difference, d.
10. The geometric sequence is given by 5, 25, 125,… .
(a) Find the first term and the common ratio.
(b) Find the nth term.
(c) Find the least number of terms so that the sum of this geometric sequence is exceed
20000.
11. Find the sum of the even numbers between 199 and 1999.
1
12. The sum of the first four terms of a geometric series with common ratio is 30.
2
Determine the tenth term.
n
13. The sum of the first n terms of an arithmetic sequence is ( 4 20 ) .
n
2
1)
(a) Write down an expression for the sum of the first (n terms.
(b) Find the first term and the common difference of the above sequence.
1 1
14. The third and the sixth terms of a geometric series are and . Determine the values of
2 16
the first term and the common ratio. Hence, find the sum of the first nine terms of the series.
12 | P a g e
15. 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.
n
16. The sum of the first n terms of an arithmetic series is (3n 5). If the second and the
2
fourth terms of the arithmetic series are the second and the third term of a geometric
series respectively, find the sum of the first eleven terms of this geometric series.
17. The fifth term and the tenth term of a geometric series are 3125 and 243 respectively. Find
the value of common ratio, r of the series.
th
18. The r terms of an arithmetic progression is (1 + 6r). Find in term of n, the sum of the first
n terms of the progression.
19. The first term and common difference of an arithmetic progression are a and –2
respectively. The sum of the first n terms is equal to the sum of the first 3n terms.
Express a in terms of n. Hence, show that n = 7 if a = 27.
20. The eighth and sixteenth terms of an arithmetic progression are 100 and 508 respectively.
Find the first term, the common difference and the sum of the first twenty terms.
21. The third term of an arithmetic series is 15. If the sixth term is half of the fourth term,
(a) determine the first five terms of the series.
(b) determine the sum of all the terms if given the last term is –126.
27
22. (a) The nth term of a sequence is T = n 13.Show that the sequence is an
n
2
arithmetic sequence.
(b) The second term of a geometric sequence is 12 and the fifth term is -96.
Determine the first term, a, and common ratio, r.
13 | P a g e
23. Julie bought an electric piano through an installment plan with a down payment of
RM 1 000. She paid RM 100 for the first month with an increment of RM 25 every
month until the plan is settled. If the final monthly instalment was RM 675, find
(a) the number of months she took to settle the installment.
(b) the total amount she paid to buy the piano.
2
24. (a) The first, second and third terms of a geometric series are r, s and s respectively.
2
The first, second and third term of an arithmetic series are r, s and s respectively.
Determine the values of r and s with s .
0
29
(b) If the sequence 4, , , x y forms an arithmetic sequence, find the values of x and
2
y.
(c) The nth term of a certain sequence is n 2 n 3.
(i) Find the sum of the first three terms.
(ii) Which term is 243?
25. The third term of a geometric sequence exceeds the second term by 6 and the second
term exceeds the first term by 9. Find the sum of the first four terms.
26. There are 20 rows of seats in a concert hall with 25 seats in first row, 27 seats in the
second row, 29 seats in the third row and son on.
(a) Find the number of seats in the last row.
(b) Determine the total number of seats in the hall.
(c) If the price per ticket is RM 500 for the first three rows and RM 200 for the rest,
how much will be the total sales for a one-night concert if all seats are sold?
27. The first three terms of a geometric sequence are m , 1 6 and m 4 where m .
m
Show that m satisfies the equation m 2 3 40 0. Hence, compute the possible
values for m.
28. The first term and sum of the first n terms of an arithmetic sequence are -21 and 26499
rd
respectively. Find the value of the 23 term of the sequence if the last term of the
sequence is 459.
14 | P a g e
29. A 26 years old ladu gets a job at a company. Her starting salary is RM1400 and the
annual increment is RM90.
(a) What will her monthly salary be when she is 45 years old?
(b) What will her age be when she gets a monthly salary of RM2480?
30. Tower A has fifty floors. A cleaning company estimates that the charge of cleaning on
floor increases by 10% for each floor above the previous floor. If the charges for cleaning
the first floor is RM 100,
(a) List in the form of sequence, the first five charges of the first five floor
(b) Determine the type of sequence and T that can be used to estimate the cleaning
n
charges for any floor.
(c) Determine the charges to clean the top floor
(d) Find the expression of total charges to clean the first n floors. Hence calculate the
total charges to clean the first thirty floors.
31. An employee pays his debt monthly by paying RM 20 in the first month. For the
following months, he pays an additional RM4 of the previous month. How many months
will it take for him to pay a debt of RM3572? Hence calculate the amount he has to pay
in the last month.
32. Given a geometric series with common ratio more than one r 1 . The ratio of the sum
of the first four terms to the sum of its first two term is 10:1. Find the common ratio.
Hence if the third term of the series is 54, find the first term.
33. The first term and the sum of all terms for a geometric sequence is 4 and -59048
respectively. Given that the common ratio of the sequence is -3, find the number of terms
of this geometric sequence.
15 | P a g e
ANSWERS
1. (a) (i) RM 3300 (ii) RM 4392.30
(b) RM 18315.30
2 4
2. r , p
5 25
3. First term, a = 6 , common difference , d 2
th
4. First term, a = 6 , common difference , d 10 , The 15 term= 146
39 1
5. First term, a , common difference , d
8 4
6. d = 20, a = – 170
(a) T 20
10
7.
255
(b) S
8
32
8. (a) T 23
11
(b) (i) proven
189
(ii) S or 1.477
6
128
9. (a) - Refer to Lecturer - (b) a 124 ; d 112
n
10. (a) a 5 ; r (b) T 5 (c) n 7
5
n
11. 989100
3
12.
32
13. (a) 2n 2 6n 8 (b) a 12 , d 4
1
14. a 2 , r ,3.9922
2
16 | P a g e
15. k 21, T 11 37
16. 699050.5
3
17. r
5
2
18. S n 4n 3n
19. a 4n 1
,
20. a 257 , d 51 S 4550
20
21. (a) First 5 terms = 21, 18, 15, 12, 9
(b) S 2625
50
2
22. (a) Proven (b) a 6 , r
23. (a) 24 months (b) The total amount is RM9300
1 15
24. (a) r 1 , s (b) x , y 11
2 2
(c) (i) S 17 (ii) n 16
3
25. a 27,r 2 ,S 65
3 4
26. (a) 63 seats (b) 880 seats (c) RM 200 300
27. m 5 or m 8
28. 67
29. (a) RM3110 (b) 38 years old
3
2
4
30. (a) 100,100 1.1 ,100 1.1 ,100 1.1 ,100 1.1
(b) Geometric sequence, T 100 1.1 n 1
n
(c) RM10671.90
(d) S n 1000 1.1 n 1 ; RM 16449.40
17 | P a g e
31. It will take 38 months. He has to pay RM 168 in the last month
32. r 3;a 6
33. = 10
18 | P a g e
CHAPTER 4: MATRICES AND SYSTEMS OF LINEAR EQUATIONS
3 6 3 1 1 4
T
1. Given the matrices P 3 3 0 and Q 1 2 5 . Find PQ . Hence, find P 1
9 3 3 1 1 3
.
4 1 6 33 2 17
2. Given the matrix A y 9 3 and its cofactor matrix is 10 4 6 .
57
x 1 4 6 37
2
Show that x .
Hence, find
(a) the value of y.
(b) the determinant, |A|.
(c) the adjoint matrix, Adj (A).
1
(d) the inverse matrix, A .
3 1 1
3. Given A 2 0 4 .
1 6 5
T
2A
(a) Find . A
(b) Determine the minor m21 and cofactor c22 for matrix A.
1 2 0 2 2 2
4. Given A 3 2 1 and B 1 1 1 .
2 4 1 8 0 4
(a) Show that AB 4I .
(b) Hence,
(i) determine A .
1
(ii) solve the following system of linear equations:
19 | P a g e
x 2y 5
3x 2y z 10
2x 4y z 13
5 8 1 2
5. Given P and Q .
3 1 1 5
1 3
(a) Determine matrix S if S (P Q )R and R .
2 1
(b) Hence, find the inverse of matrix S.
2 2 2
6. If A 1 1 1 and XA = 4I where X is a 3 x 3 matrix and I is an identity matrix.
8 0 4
(a) Find the interest of matrix A using the adjoint method.
(b) Hence, find matrix X.
6 4
2
7. (a) The matrix A . If A – pA – qI = 0 where p and q are real
1 0
numbers, I is the 2 x 2 identity matrix and 0 is the null matrix 2 x 2, find p and q.
1 1 3 x 2
(b) Given that the matrix equation AX = B is 2 1 4 y 3 .
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.
20 | P a g e
8. (a) Matrices A and B are given as
1 2 3 4 1 4
A 1 0 4 , B 1 1 5 . 3
0 2 2 1 1 1
-1
Find AB and hence find A .
(b) A company produces three grades of mangoes: X, Y, and Z. The total profit from
1 kg of grade X, 2 kg of grade Y and 3 kg of grade Z mangoes is RM20. The profit
from 4 kg of grade Z mangoes is equal to the profit from 1 kg of grade X mangoes.
The total profit from 2 kg of grade Y and 2 kg of 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) By using the result from part (a), solve the system of linear equation. Hence,
state the profit per kg for each grade.
1 0 2 1 1 2
9. Given A 2 1 0 , B 1 0 and C 1 0 .
1 1 0 2 1 3 0 1
T
(a) Find matrix D = A – (BC) .
(b) Show that AD DA
1 2 1 0 1 0 2 2
10. If P = 1 1 and Q= , find matrix R such that R 2PQ 2 4 3 .
0 1 0
0 1 4 5 3
1 0 0
2 0 0
11. Given A = 4 6 2 and B = 1 1 1 . Show that AB = kI. Where k is a
5
6 4 2 5
1 2 3
5 5
1
constant and I is an identity matrix. Find the value of k and hence obtain A
21 | P a g e
1 2 1 2 2 3
12. (a) Given the matrices P 2 1 2 and Q 2 1 0 . Find PQ and
1 2 2 3 0 3
hence, determine P .
1
(b) The following table 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
SweeLan 3 6 6 21.00
The prices 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 , where A kP( P is the
B
matrix in (a) and k is a constant. Determine the value of k and hence
find the values of x, y and z.
0 1 1
13. Matrix A is given by A 5 1 1 .
2 3 3
(a) Find
(i) the determinant of A,
(ii) the minor of A and
(iii) the adjoint of A.
1
(b) Based on part (a) above, find A . Hence, solve the simultaneous equations
3
y z
2
z
y
5x 9
3
2x 3y 3z
2
22 | P a g e
3 x 2x
14. Matrix A is given as 0 x 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.
15. 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 Durian Rambutan Amount paid
Stall (kg) (kg) (kg) (RM)
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
respectively.
(a) Form a system of linear equations which represent the total expenditure per stall
calculated based on the weight bought and price per kilogram. Hence, write the
system in the form of a matrix equation AX=B
(b) Find the determinant, minor and adjoint of matrix A.
1
(c) Based on part (b) above, find A . Hence, solve the matrix equation.
(d) Suppose the price per kilogram for mango, durian and rambutan has increased by
RM 2, RM 2 and RM 1, respectively. Obtain a new matrix representing the amount
spent on each type of fruit to be bought.
1 2 1 3
A B
16. (a) Given 3 4 and 1 2 .
1
(i) Find B .
23 | P a g e
1
,
(ii) If C B AB findC .
2 4 3 1 5 m
(b) Given A x 2 1 and B 1 8 5 .
2 3 1 1 14 8
(i) If the minor for element b 32 is -3, find m.
(ii) If the cofactor for element a 13 is -1, find x.
T
(iii) Hence, find B 2A .
17. Syarikat Dari Mata Ke Hati produces three types of souvenirs; key-chain, fridge-magnet
and book-mark. In a day, three shifts of workers are used to produce the company’s
products. The number of each type of souvenirs produced can be summarized as follows.
Workers’ ratio Number of
Types of
souvenirs
souvenirs Shift I Shift II Shift III
produced
Key-chain 2 1 1 180
Fridge-magnet 1 3 2 300
Book-mark 2 1 2 240
(a) Write the system of linear equations in the form of matrix equation, AX B where
A is the coefficient matrix, X is the variable matrix and B is the constant matrix.
(b) Using Cramer’s Rule, find the number of workers needed for each shift to produce
the souvenirs.
(c) To fulfill the demand for peak season, the company increases the number of
workers up to 100 workers for each shift. Calculate the number of each souvenir
produced per day during the season.
24 | P a g e
1 2 3 2 0 1
18. Given A 0 4 1 and B 1 3 4 .
2 2 5 6 1 2
(a) Find AB
.
T
T
(b) Show that AB B A T .
1 5 10 9 15 10
19. (a) If P 0 1 4 and Q 4 5 4 , find PQ .
1 6 15 1 1 1
Hence, determine P 1 .
(b) Ahmad is a vegetable seller. On a certain day, his regular customer, Ah Chong paid
a sum of RM120 for 1kg potatoes, 5kg cucumbers and 10kg cabbages.
Another customer, Samy paid RM38 for 1kg cucumbers and 4kg cabbages. The
total price of 1kg potatoes, 6kg cucumbers and 15kg cabbages is RM166.
Let x , and z represents the price of 1kg potatoes, 1kg cucumbers and 1kg
y
cabbages respectively.
(i) Write a system of linear equations to represent the above information.
(ii) Write the system of linear equations in the form of matrix equation.
(iii) By using the result from part (a), solve the matrix equation and hence
determine the price of 1kg for each vegetable that is sold.
3 1 2
20. (a) Given matrix A= 1 x 4 . Find the value of x if the determinant of matrix A
2 1 1
is 14.
25 | P a g e
1 3
p 2 1 4 5
(b) Find the values of p and q, if 2 1 .
3 0 q 11 19
4 5
2
3
21. (a) Given matrix B B 6B where B is a 3 3 matrix and I is a 3 3 identity
I
matrix.
-1
2
1
B
(i) By using B onto the equation, show that B B 6 .
I
1 2 1
-1
(ii) Hence , if B 1 1 0 , determine B .
3 1
1
(b) A system linear equations is given as follows.
3x 6y 3z 1
3x 3y 2
9x 3y 3z 1
(i) Write the above system of linear equations in the form of 3BX C where B
is the matrix from (a) (ii) above, X is an unknown column matrix and C is a
constant column matrix.
(ii) Hence, by using the inverse matrix, solve the above system of linear
equations.
2 0 2 4
22. (a) Given B 0 4 , and C 1 3 , find Y such that YC=B.
5 1
0 2 T 6 1
T
(b) Find the transpose matrix, A if3A 2 5 2 2 .
T
1 3 2 2 6
26 | P a g e
1 3
T
Hence, find the inverse of A D if D 2 0 .
1 1
1 2 1 1 1 2 0
23. Given the matrices A 0 1 1 , B 1 0 and C 1 .
2 0 0 2 1 3 0 1
T
Find the matrix D such that D A BC .
1 3 x 0 x y
3 0 2
24. Given the matrices A , B 1 0 , C 0 1 2 and
2 1 0 0 2 2 3 0
7 3 18
D 3 2 2 .If BA 2C , D find the values of x and y .
0 8 0
25. (a) Consider a system of linear equations
2x y z 1
x 2y 3 1
z
3x 2y 4z 5.
(i) Write the above system as a matrix equation in the form of AX=B. Hence,
determine the determinant of A, the cofactor matrix of A and the inverse
matrix of A.
(ii) Hence, solve the above system of linear equations.
1 y
1 0 1 2 3 z
(b) Given the matrices A 2 1 0 and B 1 0 1 .
1 1 1 y x y
3
If AB=I, where I is a 33 identity matrix, find the values of , and .x y z
27 | P a g e
5 2 3 a 2 36
26. If P 1 4 3 , P b 2 24 and PQ 4I , where I is the 3x3 identity
3 1 2 26 2 c
matrix, determine the values of a, b and c.
1 2 1
27. A and B are square matrices such that BA = B . If B -1 1 0 1 , find A and
-1
1
1 2
hence, A.
2 3 1
28. Given the matrix A 0 1 4 .
5 6 1
(a) Determine the adjoint matrix of A.
-1
(b) By using the result in (a) above, find A . Hence, solve the system of linear equations
below
2x 3y z 5
y 4z
4
5x 6y z 9
2 3 4 9 27 3
1
29. Given C 5 0 1 and D 7 26 18 . Find CD and hence determine C .
8 9 3 45 6 15
7 2
2
I
30. (a) If Y , show that Y satisfies the equation Y 8Y O with I is an
3 1
identity matrix 2 2 . Hence, find Y 1 .
28 | P a g e
1 2 0 2 4
(b) Given P 5 4 and Q , determine matrix R such that
3 2 1 3 2
2 2 5
T
PQ R 1 4 1 .
0 3 4
31. Table below shows the floor area of the rooms in Amir’s house.
Room Floor Area (square metre)
Living room 25
Bedroom 1 15
Bedroom 2 10
Bedroom 3 10
Kitchen 10
Amir intends to replace his home floor to ceramic tiles, parquet or mosaic. If he uses
ceramic tiles for the living room, parquet for all the bedrooms and mosaic for the kitchen,
the total cost is RM 2625. If he uses ceramic tiles for the living room and bedroom 1, and
mosaic for the other bedrooms and kitchen, the total cost is RM 2450. However, if
bedrooms 2 and 3 use parquet, and other rooms and kitchen use ceramic tiles, the total cost
is RM 3200.
Let the prices per square metre of ceramic tiles, parquet and mosaic be x, y and z
respectively.
(a) Obtain a system of linear equations to represent the given information. Hence, state
B
the equations in the form of AX .
(b) Based on (a), find the inverse of matrix A. Hence, determine the price (RM) per
square meter for each ceramic tiles, parquet and mosaic.
29 | P a g e
1 3 2
32. Given P 2 0 m . Find m if P 1.
1 2 1
1 0 1 1
T
1
33. Given A and B .
2 1 4 2
T
Find (i) AB ,
3 8
(ii) matrix C such that CAB
1 2
3 2 5 3
34. Given 2P Q 2 1 and 3P Q 4 2 . Find matrices P and Q.
4 5 5 6
35. The table shows the weight (in kilograms) of three types of fruits namely mango, pineapple
and watermelon which were supplied by a supplier to three restaurants K, L and M on a
daily basis.
Fruits (kg)
Restaurants
Mango Pineapple Watermelon
K 5 5 5
L 10 15 10
M 10 15 15
The payment received by the supplier from the K, L and M restaurants are RM66, RM147
and RM143 respectively. If x, y and z are the prices for each kilograms of the fruits,
(a) state a system of linear equations for the above information given above. Hence,
write in the form of matrix equation, AX = B.
(b) Based on (a), find the inverse matrix of A. Hence, find the price per kilograms for
each of the fruits.
2 3 2 1 1 1
T
1
1
36. (a) Given A and C where A B C . Find .
2 4 1 1 0 3
30 | P a g e
3 4
(b) Given P . Show that P 7I and P 7P. Hence, find P and P .
2
3
5
4
4 3
37. Aini spent RM 11 tp buy 4 pens, 2 rulers and 1 erasers. Nora spent RM9 fot 2 pens, 3 rulers
and 2 erasers. While, Dila spent RM13 for 3 pens, 4 rulers and 3 erasers.
(a) (i) Write the information given above as a system of linear equations in the
form of matrix equation, AX
B
(ii) Find the cofactor matrix of A and hence determine the determinant of matrix
A
(iii) Determine the adjoint matrix of A and hence find the inverse of A
(b) What is the price of each pen, ruler and eraser?
(c) How much will each person spend if the price of each pen, ruler and eraser increased
by 20%?
1 1 1 2 3 0 0 1
1 ,V
38. Given A 0 1 0 ,S 0 1 0 and U p
p
1 0 0 0 0 2 1
2
(a) Compute W S V .Hence find the value of p such that WU p .
T
T
VA
(b) Compute .VA T Hence, show that A V .
T
T
39. A system of linear equations is given as follows:
2x 3y 4z 11
4x 3y z 10
x 2y 4z 8
(a) Write the above system of linear equation in the form of matrix equation AX B
where , and are the coefficient matrix, the variable matrix and the constant
matrix respectively. Hence, determine
.
(i) the determinant of A, A
(ii) the matrix od cofactor , and
(iii) the adjoint matrix A, ( ).
31 | P a g e
1
(b) Find the inverse matrix, A . Hence, solve the above system of linear equations.
1 4 5
40. (a) Given matrix A 4 0 2 , find the cofactor and adjoint for the matrix. Hence,
8 0 3
find A .
1
(b) Find the values of , , and in the following matrix equation.
3 7 x 2 4 2y
3
6 1 5 z 5w 7
32 | P a g e
ANSWERS
1 1 1
3 3 3
1
1. PQ 3I ; P 1 2 1
T
3 3 3
4 5 1
3 3
2. – Refer to Lecturer –
(a) y 1 (b) A 28
23 5 57
33 10 57 28 14 28
1
(c) Adj ( ) A 2 4 6 (d) A 1 1 3
14 7 14
17 6 37
17 3 37
28 14 28
3 3 3
3. (a) 0 0 8 (b) m 21 11, c 22 16
3 2 5
1 1 1
2 2 2
4. (a) –Refer to Lecturer– (b) A 1 1 1 1
4 4 4
2 0 1
(c) x 1, y 2, z 3
9 7
26 28 10 5
5. (a) S (b) S 1
16 18 4 13
5 10
33 | P a g e
1 1
4 2 0
1 2 0
1
6. (a) A 3 1 1 (b) X 3 2 1
4 2 4
2 4 1
1 1 1
2 4
7. (a) p = 6, q = –4
(b) (i) A 2
(ii) p = –2 ,q = 1
5 1
5 4 1 2 2 2
-1
(iii) adjA = 2 2 2 ; A = 1 1 1
3 2 1 3 1 1
2 2
4 1 4
1 0 0 5 5 5
-1
8. (a) AB= 5 0 1 0 ; A = 1 1 7
5 5 10
0 0 1
1 1 1
5 5 5
(b) (i) x + 2y +3z = 20
4z = x
2y + 2z = 10
1 2 3 x 20
(ii) 1 0 4 y = 0
0 2 2 z 10
(iii) x = 8, y = 3, z = 2
0 2 3
9. (a) 3 0 2 (b) –Refer to Lecturer–
2 1 1
34 | P a g e
2 2 4
10. R 4 6 5
4 3 3
1
2 0 0
1
11. k = 2; A 1 1 1
2 10 10
1 1 3
2 5 10
2 2
1
3 0 0 3 3
12. (a) PQ 0 3 0 , P 1 2 1 0
0 0 3 3 3
1 0 1
x
3x 6y 3z 16.5 3 6 3 16.5
y
(b) (i) 6x 3y 6z 21.3 , 6 3 6 21.3
3x 6y 6z 21.0 3 6 6 21.0
z
(ii) k 3, x 1.4, y 1.3 , z 1.5
13. (a) (i) – 4
M 6 M 12 13 M 17
13
11
(ii) M 21 0 M 22 2 M 23 2
M 2 M 5 M 5
31 32 33
6 0 2
(iii) Adj . A 13 2 5
17 2 5
3 1
2 0 2
9 15
1
(b) A 13 1 5 , x 3, y , z
4 2 4 4 4
17 1 5
4 2 4
35 | P a g e
14. (a) x=5
25 0 0 25 25 30
(b) cofactor of A 25 15 0 ; adjoint of A 0 15 12
30 12 15 0 0 15
1 1 2
3 3 5
1
A 0 1 4
5 25
0 0 1
5
5x 3y 2z 34 5 3 2 34
x
y
15. (a) 3x 4y 4z 37 3 4 4 37
2x 3y 4z 29 2 3 4 29
z
(b) A 10
4 4 1 4 6 4
Minor of A = 6 16 9 ; Adjoint of A = 4 16 14
4 14 11 1 9 11
2 3 2
5 5 5
2
1
(c) A 2 8 7 x 3, y 5, z
5 5 5
1 9 11
10 10 10
5 3 2 52
x
y
(d) 3 4 4 55
2 3 4 43
z
2 3 3 17
1
16. (a) (i) B (ii) C
1 1 2 8
36 | P a g e
5
(b) (i) m 2 (ii) x (iii) – 1299
3
2 1 1 180
x
y
17. (a) 1 3 2 300
2 1 2 240
z
(b) x 36, y 48,z 60
2 1 1 100 400
(c) 1 3 2 100 600
2 1 2 100 500
18 3 15
18. (a) AB 2 13 14 (b) – Refer to Lecturer –
24 11 4
9 15 10
1
19. (a) P Q 4 5 4
1 1 1
(b) (i) x 5y 10z 120
y 4z 38
x 6y 15z 166
x
1 5 10 120
y
(ii) 0 1 4 38
1 6 15 166
z
(iii) Price of 1 kg potatoes = RM10
Price of 1 kg cucumbers = RM6
Price of 1 kg cabbages = RM8
2
20. (a) x 3 (b) p 4, q
37 | P a g e
21. (a) (i) – Refer to Lecturer –
1 1 1
1
(ii) B 1 2 1
4 5 3
1 2 1 1
x
y
(b) (i) 3 1 1 0 2
13 1 1 1
z
2 4
3
(ii) x , y , z
3 3
3 4
22. (a) Y 2 4
7 9
1 3
2 3 0 1 10 50
T
(b) A ; A D T
2 2 4 1 1
10 25
0 4 2
23. 1 0 3
3 0 1
24. x 5 , y 3
2 1 1 1
x
1 , A
y
25. (a) (i) 1 2 3 9
5
z
3 2 4
38 | P a g e
2 2 1
2 13 8 9 9 9
1
C 2 5 1 , A 13 5 7
A
1 7 5 9 9 9
8 1 5
9 9 9
(ii) x 1 , y 3 , z 2
1 3 1
x , y , z
(b) 2 2 2
26. a 22 , b 14 , c 44
2 1 5 8 1 11 1
8
1
27. A 2 3 1 , A 8 1 3 8 1 2
1 5 1
2 4 2 8 8 2
23 3 11
28. (a) adj A 20 7 8
5 27 2
23 3 11
101 101 101
28 144
1
1
(b) A 20 7 8 , x , y , z
101 101 101 101 101
5 27 2
101 101 101
3 9 1
141 0 0 47 47 47
1
29. CD 0 141 0 ; C 7 26 6
0 0 141 141 141 47
15 2 5
47 47 47
39 | P a g e
1 2 0 6 7
1
30. (a) Y (b) R 5 2 1
3 7 8 25 20
31. (a) 5x 7y 2z 525
8x 6z 490
10x 4y 640
5 7 2 525
x
y
8 0 6 490
10 4 0 640
z
6 2 3
91 91 26
1
(b) A 15 5 1
91 91 26
8 25 2
91 182 13
x RM 50, y RM 35, z RM 15
32. m =3
3 2 35 51
33. (i) 1 1 (ii)
7 11
2 2
2 1 1 0
P 6 1 Q 14 1
1 1 2 3
34. ,
35. (a)
5 + 5 + 5 = 66
10 + 15 + 10 = 147
10 + 15 + 15 = 143
5 5 5 66
[10 15 10] [ ] = [147]
10 15 15 143
(b) Mango = RM 11 Pineapple = RM 3.00 Watermelon = RM 0.80
40 | P a g e
15 18
36. (a) B 9
6
2
49 0 147 196
(b) P 4 ,P 5
0 49 196 147
1 2 1
3 3 3
37. (a) A 1 0 3 2
1 10 8
3 3 3
(b) Pen RM 2,Ruler RM 1,Eraser RM 1
(c) Aini RM 13.20, Nora RM 10.80,Dila RM 15.60
38. (a) W 6 1 2p p 3,2
;
3 0 2
T
VA
(b) 3 1 0 ; proof
3 0 0
x
2 3 4 11
y
10
39. (a) 4 3 1
1 2 4 8
z
(i) A 5
10 15 5
(ii) 4 4 1
9 14 6
10 4 9
(iii) Adj A 15 4 14
5 1 6
41 | P a g e
4 9
2 5 5
3
1
(b) A 3 4 14 ; x 2 , y 13 , z
5 5 5 5 5
1 1 6
5 5
3 1
0 28 14
1
40. (a) A 1 37 11
4 112 56
0 2 1
7 7
9 1 1
2
(b) w , x , y , z
5 3 2
42 | P a g e
CHAPTER 5 : FUNCTIONS AND GRAPHS
1. Given ( ) 4f x x 2 , x R .
(a) Sketch the graph of ( )f x .
(b) Hence, state the domain and range of ( )f x .
2. A function f is defined by ( )f x x 2 1. Sketch the graph of ( )f x and state the domain and
range of ( )f x .
x
2
h
3. Given that f 2x x 1 and 2x 4x 1 , find the function g such that
f g x h x . Write g in the form of (a x b ) 2 c where a, b and c are constants.
2x
4. Functions f and g are defined as ( )f x e , ( ) 1g x x , x R .
x
Find f 1 ( ) and hence obtain (g f 1 )( )
x
2
3
5. A function f is defined by ( )f x x 2x for 0 x 5 . State the range of f and
determine whether f is one to one.
3x
6. Given ( )h x . Defining h 2 ( ) (h h )( ) , determine the function h 2 ( ) and hence
x
x
x
x 3
deduce the inverse of ( )h x . Evaluate h 13 (9) .
2
0
7. Given ( ) 2f x x 1 , x . Defining ( )g x x 3, find
1
1
(a) the inverse f and g and verify that (g f ) f 1 g
(b) the values of x for which graph of f g g f
10 2x x
2
2
8. Given that ( )f x and ( ) 5 2g x x . Fınd the value k so that f 1 ( ) g .
x
k 2
Hence, find f 1 g (0) .
9. Let ( )f x 4x 1 and ( )g x x 2 . If ( ) h x f ( ) 2 ( ) , express ( )h x as a piecewise
x
g
x
function.
43 | P a g e
2
3 2
10. Let ( )f ax a x a x 3a where a is non-zero.
(a) Find a if (0)f 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
1
x
11. A function g is defined by ( )g x , x 1.Find g 1 ( ) .
x 1
x
12. Given that ( ) lnf x 1 .
3
(a) Show that f is one to one function.
x
(b) Find f 1 ( ) and hence, find the value of x when f 1 ( ) 6.
x
13. The functions f and g are defined by
f x x 2 1 , x R
g x x 1 , x R
(a) Find all the roots of 2 g f f g x 7x .
x
(b) Given ( )h x .
g x
(i) find h h x .
x
(ii) Hence, determine h 1 ( ) .
14. A function f is defined as ( ) 3f x x 2
x
x
(a) Show that the function f 1 ( ) exists and hence, find f 1 ( ).
(b) State the domain and range of f 1 ( ).
x
(c) On the same axes, sketch the graphs of ( )f x and f 1 ( ).
x
State the relationship between the two graphs.
x
15. Given that f ( ) e 2x 1 and g ( ) x 2x 1.
x
x
(a) Given ( )p x 4x 7.Find the function q such that (g q 1 )( ) p ( ).
44 | P a g e
(b) Solve the equation ( f g )( ) 2.
x
(c) Sketch the graph of x and determine whether the function is one-to-one. Hence,
f
f
state the domain and range of x .
16. Given the function 3f x x 1.Sketch ,f x hence, determine the domain and range
of the function.
17. Given log and f x 3 x g 3 .x x
(a) Show that f and g are one-to-one functions. Without finding the inverse function,
show that the functions f and g are inverses of each other.
(b) On the same axes, sketch the graphs of f and g. Clearly label all intercepts and
lines of symmetry. Hence, state the domain and range of each function.
f
x
18. Given x 2x 1 and g x 4.
f
(a) Write x as a piecewise function.
Hence, find the function 2h x f 3x g .x
(b) Sketch the graph of h.
k
(c) Find the function x if g k x x 2 where x 2.Hence, determine the
value of x such that k 1 11.x
e 2 px
19. A function f and g are defined as f ( ) x 5 2 ln3x and ( )g x respectively,
3
(a) write down an expression for f g ( )
x
(b) Find the value of p such that f and g are inverse functions.
4 , x 1
2
20. Given ( )f x x 3 , x 1
x
2 x , 1
(a) Sketch the graph of ( )f x and hence, find the range of ( )f x
45 | P a g e
(b) Find the value of ( 3)f , (0)f and (1)f .
x 2
g
21. (a) Functions f and g are defined by ( )f x , x 1 and x ax bx c ,
x 1
,x where a, b and c are constants.
(i) Find f f , and hence , determine the inverse of f .
2x 2 5x 2
(ii) Find the values of a, b and c if g f x 2 . .
x 1
2
q
(b) Find the values of q if the function 4f x x 2 and f 1 5 3 ; x 0 .
2
2
f
22. Given the function x p x 3 If f 1 2 2 , find the value of p.
23. The functions f and g are given by 3f x e 3x and ( )g x ln(3x 5) .
(a) Find the value of x such that f g 4x .
(b) If f g 1 4b , find the value of .
2
f
24. Consider the function x x 4x 3.
(a) Express x in the form of x h 2 k , such that x h . Hence, find the value of
f
h.
f
(b) Sketch the graph of x for x h and explain why f is a one-to-one function. State
f
the range of x .
(c) Find f 1 x and hence, evaluate f 1 3 .
5 ax
25. Given ( )g x and ( ) 2h x x 2 5. Find
2
x
(a) g 1 ( )
2
(b) the value of a so that 2g 1 ( ) h x
)
(
x
(c)
2x 2 , x 0
26. A piecewise function f is given as ( )f x
4
x
x 2 , 0
46 | P a g e
(a) Sketch the graph of f
(b) Hence, find the domain and range of f
27. (a) Given ( )f x 4(3 ln2 )and ( )g x e 3 bx , where b is a constant
x
2
(i) Write down an expression for ( f g )( )
x
(ii) Find the value of b such that f and g are inverses of each other
2
4
(b) Find the value of (2)h given that (h g )( ) 4x 4x and ( )g x 2x 1
x
x
x
x
x
28. Given f ( ) 2 2x , find f 1 ( ) . Hence, sketch f ( ) and f 1 ( ) on the same graph.
29. (a) Given ( )f x 6x p , find the function g for each of the composite function below,
(i) f g ( ) 8x x 2 5
2
2
(ii) g f ( )x 36x 12px p
2
(b) Given ( )f x x 2x where x 1, find the value of f 1 (6) .
3
30. Given the functions as follows:
x
g ( ) e x 3
x
1)
f ( ) 2 ln(x
(a) In a separate diagram, sketch the graph ( )g x and ( )f x . Hence, state the domain and
range for each of the function f and g .
(b) Find f g ( )x . Hence, solve for x such that f g ( )x .
2
1 2x
2
31. If g f x , x and 3 2f x x. Find x
g
2 x
2
f
13 and 3f
32. Given that function x p qx such that f 5 17 . Find the
exact values of p and q.
33. Given two functions lnf x x m and ( )g x e x 2 .
(a) If f 1 x g ( ) , show that the value of m is 2. Hence, find the domain and
x
range for functions ( )g x and g 1 ( ).
x
47 | P a g e
(b) Given ( )h x x 3 , x 3, find h 1 1 . Hence, find g h 1 (1) .
34. Given
f x 1, x 5
x
g x , x x 0
(a) Write down an expression for the composite function g f .x
(b) Find the expression for each of the following inverse functions
(i) f 1 x
1
(ii) g f x
1
x
(c) Verify that g f f 1 g 1 x . Hence, find the domain and range for
1
g f x .
35. (a) Given that lnf x 3x 2
(i) Show that x is one-to-one.
f
(ii) Find the inverse function of x .
f
(iii) State the domain and range of x and f 1 x .
f
13 log x
(b) Given two functions 7f x 4x m and x 7 . If x and x are
f
g
g
n
inverse of each other, express in terms of and .
48 | P a g e
ANSWERS
1. (a) – Refer to Lecturer – (b) D R or ( , ), R , 4
f
f
2. – Refer to Lecturer – ; D ( , ),R [ 1, )
f
f
2
x
3. g ( ) (x 1) 1
ln x ln x
x
x
4. f 1 ( ) , g f 1 ( ) 1
2 2
5. Range = [-4,12] and f is not one to one function.
3x 9
1
x
6. h 2 ( ) , h ( ) ,
x
x
x 3 2
x 1 7
x
3
x
x
7. (a) f 1 ( ) g 1 ( ) (b)
2 4
5
1
8. k 1, f g (0)
2
1
6x 3, x 4
9. h ( )
x
5 2 , x x 1
4
2
x
6
10. (a) a 2 (b) f ( ) 2x 2x
11 1 1
(c) D ( , ), R f 2 , , , 2 or 2 ,
f
1
1
x
11. g ( ) 1
x 2
The words you are searching are inside this book. To get more targeted content, please make full-text search by clicking here.
Discover the best professional documents and content resources in AnyFlip Document Base.
Search