MODUL MATHMAZING

FOR ME

Name : ……………………………………….
Class : ……………………………………….

“The least you can do in math is TRY
and WRITE, if you hesitate then you
will get an EGG ”

Prepared by: Mdm. Nurul Bilqis Bt. Baharuddin
AM015 2021/2022

Page | 1

CONTENTS

CHAPTER TOPICS PAGES
3
1 NUMBER SYSTEMS AND EQUATION
4–5
2 INEQUALITIES AND ABSOLUTE VALUES 6

3 SEQUENCES 7–8

4 MATRICES AND SYSTEM OF LINEAR 9 – 10
11 – 12
EQUATIONS 13 - 14
14 – 16
5 FUNCTIONS AND GRAPHS 17 – 18

6 POLYNOMIALS

7 LIMITS

8 DIFFERENTIATION

9 APPLICATIONS OF DIFFERENTIATION

SO LET’S GET STARTED!!

Page | 2

CHAPTER 1 Answer: −4,3)

NUMBER SYSTEMS AND EQUATION Answer: 

1. Simplify the following using the real number line.

a) (−,3) −4,6)
b) (−,1−2,)

2. Given A = x : −2  x  5, x  , B = (2,6) , C = x : −3  x  4, x  

Find Answer: 3, 4
a) B  C Answer: −3x : −2  x  5 or −32,5

b) A  C

3. Find the value of x , given ln (5x + 2) − ln 2x = 3 . Give your answer correct to 4 decimal

places. Answer: 0.0569

4. Solve log 3(2x −1) + log 2 4 = 5 . Answer: 14

5. Solve log 1 = −3. Answer: 3
x 27

6. Find the value of x , given log5 x + logx 5 = 2 log5 x . Can we accept both of the answer?

Why? Answer: 5 , 1
5

7. Simplify 25  4n −16  4n−1 . 21
4n+3 − 8  4n+1 Answer:

32

( )8. Solve 5 5x = 6 − 5−x. Answer: -1 , 0

9. 121(x+3)2 = 1111x+24 3

2−2 Answer: or -2
10. Simplify 4 + 2 4 − 2
2
11. Solve 3x − 4 − x − 4 = x + 4 −2 2

12. Solve Answer:
7
6(√ − 3 + 1) = 6√ −3+30
√ −3−1 Answer: 4
Answer: 12

Page | 3

CHAPTER 2

1. Solve the following inequalities and write the answer in interval form

(a) 5  2x + 3  6 3

Answer: [ 1, ]

2

24  2 (x − 5)  36 Answer: [ 41, 59)
(b) 3

2. Find the set values of

 x − 1 2  x + 1
 2 4.
ANS: 0,2

3. Solve

2x −1  7 ANS: { : < 11}
(a) 3

( ) ( )3 2x −1  4 x − 3 ANS: { : ≤ − 29}

(b)

4. Find the solution set of the following inequality

3x(x −1)  10(x +1) ANS: {x : x  − 2  x  5}

3

5. Solve x − 9  x2 + x  20 ANS: { : −5 < ≤ −3} ∪ { : 3 ≤ < 4}

6. Solve − 2 ≤ 2−5 ≤ − 1 ANS: 8 ≥ ≥ 11

34 5 20

7. Solve y2 ≥ 64-12y ANS: y : y  −16  y  4

8. Solve the equation

x =3 ANS: -3 or 3

(a)

x + 4 = 5x +8 ANS: {x:-1}

(b)

Page | 4

9. Find the possible values for x if x2 − 2x + 2 = 5 ANS: 3,-1

10. Solve the equation ANS: { : − 3 , −5}

2x −1 = 3x + 4 , 5

a) ANS: { : − 25 , −3}

x − 5 = 2x +10 3
b) 2

11. Solve

2X + 3  6, ANS:{ : − 9 ≤ ≤ 3}

(a) 22

2x +1  3x + 2 : ANS: { : ≥ − 3}

(b) 5

12. Find the interval of x that satisfy the inequality x − 2  2x + 3 ANS: (-∞, -1/3)

x2 − 9x +19  1 ANS: s.s = x : 3  x  4  5  x  6

13. Solve

14. Determine an inequality in the form x− p r where p and r are constant such

that its solution set is x : 3  x  7 ANS: x − 5  2

Page | 5

CHAPTER 3

SEQUENCES

1. The 7th term of an arithmetic sequence is five times the second term. The
two terms differ by 20. Find the first term and the common difference.
ANS: 1, 4

2. Find the sum of the first twelve terms of a geometric sequence that has a

1

first term of and the 8th term of 243. ANS: 265720/9

9
3. The sum of the first n terms of a series is 3n2 . Find the first 3 terms of

the series. ANS: 3,9,15

4. Given the arithmetic series 3 + 7 +11+15 + ... + 79 . Find the sum of all

the terms of the above series. ANS: 820

5. An arithmetic sequence has 11 terms. The first and last terms are 5 and

75 respectively.

a) Find the 7th term. ANS: 47

b) Find S6 ANS: 135

6. 3, p, q,375 are four consecutive terms of a geometric sequence. Find

a) the values of p and q ANS: p= 15, q=75

b) S4 . ANS: 468
7. Given a , a + 1 , a − 2 are three consecutive terms of a geometric

sequence, find a and the common ratio. ANS: -1/4, -3

7

8. The sum of the first three terms of a geometric sequence is and the

4

7

sum of the next three terms is . Find the common ratio of the

32

sequence. ANS: 1/2

9. Encik Jamal saves RM800 in his saving account every year. If the bank

pays an interest rate of 5% per annum, find the total saving (including the

interest) after 12 years. ANS: 13370.39

10. Billy Gates takes an interest free loan to buy a uPhone. He repays the

loan in monthly installments. For the first month, he repays RM110. For

the second month, he repays RM160. He repays RM50 more every month

until the loan is paid off. If his final monthly instalment is RM860, find

(a) the number of months it takes to settle the loan, ANS: 16 months

(b) the amount of the loan ANS: RM 7760

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CHAPTER 4

MATRICES AND SYSTEM OF LINEAR EQUATIONS

2 0 3  13 8 23 
1. Given that R = 1 1 2 and S = 2x + y 6 
15  . Find x and y if S = RRT .

4 1 5  23 15 5x + 3y

ANS: x= -18, y=44

 5 4 −2
 5 −2 such that A2 + pA + qI = 0 where I is an identity
2. Given that matrix A =  4

−2 −2 2 

matrix 3x3 and 0 is a zero matrix 3x3. Determine the values of p and q.

ANS: p=-11, q=10

3. Find the set of values of y if the determinant of matrix, A = is always
negative. ANS:

4. Find the value of for a matrix B = if the cofactor is 2.
ANS: -2

 0 1 2
5. Given Q = −1 1 0 and QP = Q-1, where P and Q are square matrices. Find P -1.

 2 0 1

312
ANS: [−1 1 −2]

225

6. Given A = 3 1 if A2 – 5A + 7I = 0 where I is the identity matrix and 0 is the null
−1 2,

matrix. Show that A−1 = 5 I − 1 A.Hence, find A-1. ANS: [21//77 −31//77],
77

Page | 7

5 0 3  2 0 −3
Given A = 2 1 1 and B = −1 1 
7. 1  . Find AB and hence find A-1.

3 0 2 −3 0 5 

5 0 3 x 4
If 2 1  y  6 ,
1  = find the value of x, y and z. ANS: x=2, y=4, z=-2

3 0 2  z  2

 1 2 3
8. Find the inverse matrix of A = −1 1 4 by using adjoint method

 2 0 1

1 −2 5

13 13 13

ANS: −1 = 9 − 5 − 7
13 13 13

[− 2 4 3
13 13 ]
13

9. When we subtract 2 times the second number from the first number, then we get -5.
When we multiply the first number by three and add to 4 times the second number,
the sum is 15. Let x be the first number and y be the second number. Construct a
system of linear equation for the above conditions and transform the system into a
matrix equation. Hence, solve the system by the inverse matrix method. [Hint: Find
the inverse by using adjoint method] ANS: x= 1, y=3

10.  3 −1 2  ANS:  D −1 =  − 2 −1 5 
Find the inverse of the matrix D =  − 2 2 1. − 3 −1 7

 1 0 1  2 1 − 4

11. An animal breeder sells x cows and y goats to a farmer at a price of RM30 and RM20

respectively for each of the animals. If the breeder was paid RM660 for a total of 27

cows and goats, find the number of cows and goats that were sold by the breeder. (use

G-J elimination method) ANS: x=12, y=15

12. Solve the following system of linear equation given by using Gauss – Jordan
Elimination method.

a−b+c =9
4a + 2b + c = 6
9a + 3b + c = 13

ANS: a=2, b=-3, c=4

Page | 8

CHAPTER 5

FUNCTIONS AND GRAPHS

1) Given f (x) = 3 + 2x , find g(x) if (g  f )(x) = 1+ 2x , x  −2.
2+ x

ANS: g(x) = 2(x − 2)
x +1

2) Given ( f  g)(x) = x 2 +1.

a) Find g(x) if f (x) = 1− 2x. ans: -x2/2
ans: x=±1/2
b) Hence, find x if g(x) = − 1 .
8

3) Find the inverse of the following functions and state the domain

and range for f (x) and f -1(x):

a) f (x) = 2x-3 + 6

b) f (x) = log2 (x -1) + 5

3

4) Show that f (x) = 2e- x + 3 and g(x) = - ln æ x - 3ö are inverses
çè 2 ÷ø

of each other by using the composite function properties.

 x3 , x  −2

5) Sketch the graph of f (x) =  x ,−2  x  2 and hence find its
x −1 , x  2



domain and range. Ans: D f = (−, ) R f = (−,−8) [0, )

6) Sketch the graph of f (x) = −(x − 3)(x + 5) , − 4  x  1.

Hence find its domain and range. ANS: D f = (−4,1] R f = (7,16]

Page | 9

7) Determine the inverse of the following function if exist

f (x) = x2 + 4x + 3 ; x  −2

8) Determine the inverse of the following function if exist

f (x) = x + 2

9) Without sketching the graph, determine the domain and range of
f −1(x) , given f (x) = x2 + 3 ;x  1 ans: domain= [4,∞) , range= [1,∞)

10. The function f is defined as f : x → 5x + 2 + 3
a) Find f −1

b) Sketch the graph of f and f −1 on the same axis. Hence,

state the domain and range for f and f −1 . Df = R f −1 = − 2 ,  
Rf = D f −1 5 
Ans:
= 3,

11. The function f is defined as f : x → 2ln x −1

a) Find f −1 . +1

Ans: ⅇ 2

b) Sketch the graph of f and f −1 on the same axis. Hence, state the

domain and range for f and f −1 . D f = R f −1 = (0,)
Ans: Rf = Df −1 = (−,)

12. The function f is defined as f : x → 2ex −1

a) Find f −1 . Ans: ( +1)

2

b) Sketch the graph of f and f −1 on the same axis. Hence, state the

domain and range for f and f −1 . D f = R f −1 = (−,)
Ans: Rf = Df −1 = (−1,)

Page | 10

CHAPTER 6

POLYNOMIALS

3x3 + 4x + 11
1) Divide x2 - 3x + 2 by using long division.

12x3 - 11x2 + 9x + 18

2) Divide 4x+ 3 by using long division.

3. The remainder when x3 + ax2 + bx + 2 is divided by ( x + 2) is twice the

( )remainder when the same polynomial is divided by x − 1 . Show that
( )a − 2b = 6. If also the remainder is 2 when the polynomial is divided by x + 1 ,

find the values of a and b. ans: a= -4, b= -5

4. The polynomial P(x) = ax3 − 9x2 + bx + 6 ( )is exactly divisible by x − 3 , and
on division by ( x −1) the remainder is 6.

a) Find the values of a and b. ans: a= 2, b=7

b) Factorize P(x) completely ans: (x-3)(2x+1)(x-2)

5. a) Show that -3 is a zero of P( X ) = x3 + 2x2 − 5x − 6.

b) Find all the zeros of the expression. Ans: -3,-1,2

6 Determine whether the number given in the bracket is a zero of the
polynomial.

(a) P(x) = 4x3 − 8x2 + x + 13;  3 
 2 

P(x) = 2x4 + 3x3 + 25x + 7;(−2).

(b)

Page | 11

7. If a polynomial P(x) = ax3 + 2x2 + 3x + b has factors (x-1) and (x+1)

a) Find the values of a and b. ans: a=-3, b=-2

b) Factorize P(x) completely ans: (x-1)(x+1)(-3x+2)

8. Given Q(x) = 3x3 + mx2 + nx + 5 . Q(1)=0 and Q(-1)=1.

Find the values of m and . ans: m=-9/2, n=-7/2

−x
9. Express (2x + 3)(x − 4) as partial fractions

Ans:  −x =− 3 − 4
(2x + 3)(x − 4) 11(2x + 3) 11(x − 4)

x2 +1
10. Express (x −1)(x + 2)(x − 2) as partial fractions

Ans:  x2 +1 =− 2 + 5 + 5

(x −1)(x + 2)(x − 2) 3(x −1) 12(x + 2) 4(x − 2)

2x + 5
11.Express x 2 (x +1) as partial fractions

Ans:  2x + 5 = − 3 + 5 + 3
x2 (x +1) x x2 (x +1)

x(4x − 9)
12.Express (x −1)3 as partial fractions

Ans:  x(4x − 9) = 4 − 1 − 5
(x −1)3 (x −1) (x −1)2 (x −1)3

13. Given that ( x + 2) is a factor of P(x) = 2x3 + 5x2 − x − 6 , factorize P(x)

x2 −3
completely and hence express P(x) in partial fractions.

( )14. Given that x − 3 is a factor of P(x) = x3 − 3x2 + 2x − 6 , factorize

x2
P(x) completely and hence express P(x) in partial fractions.

Page | 12

CHAPTER 7

LIMITS

1. Find the following limits

(a) lim − 5 = (b) lim x2 + 3 = (c) lim (x − 2)5 =
x→0 x→2 x→1

Ans: a) -5, b) 7, c=-1

x3 , x  −1

2. Given the function as, f ( x) =  0 , x = −1

 x , x  −1


Determine (b) lim f (x) =
(a) lim f (x) = x→−1+

x→−1−

Ans: a) -1, b)1

3. Evaluate the following limits, if exist.

lima) x3 + 4x2 + 2x + 8 limb) x3 − 3x2 −10x + 24
x ⎯⎯→ −4 x+4 x ⎯⎯→ 2 x2 − 4

ans: a) 18, b) -5/2

4. Evaluate the following limits, if exist. limb) x +1
x ⎯⎯→ −1 3 − x − 2
lima) x −1 −1

x ⎯⎯→ 2 x − 2

ans: a) ½, b) -4

5. The diagram shows the graph of a function f. Use the graph to find each of the
following limits, if it exists.

y=f(x)

3
2

12

a) lim f (x) (b) lim f (x) (c) lim f (x)
x→1 x→2 x→−

ans: a) 2, b) does not exist, c) -∞

6. Given f (x) = x−2 x2, determine the limit of f (x) as x → 2 , if it exist.
,
x−2

Ans: does not exist

,
Page | 13

7. Find the following infinite limit if exists

a) lim x2 + 4 b) lim 3 c) lim 5
x→0− x x→−5+ 5 + x x→3+ 3 − x
x
Answer: a) -∞, b) ∞, c) does not exist
8. y

x = -3 x=3

The diagram show the graph of y = f(x). Find the following limits

a) lim f (x) b) lim f (x) c) lim f (x) d) lim f (x)

x→3+ x→3− x→−3+ x→−3−

Answer: a) -∞, b) ∞, c) -∞ , d) ∞

lim9) a)Evaluate 4x2 − 3x + 2 . limb) Evaluate 4x2 − 3x + 2
x→− .
x→+ x+9 x+9

Ans: a) 2, b) -2

10)Find the limit

a) lim 2 x4 − x2 + 8x b) lim 2x4 − x2 + 8x
−5x4 + 7
x→+ x→− −5x4 + 7

ans: a) -2/5, b) -2/5

lim 5x5 - 3x3 - 4x ans: ∞
11. Evaluate x4 - 3x2 ans: √3

x®+¥

lim 3x3 + 5
12. Evaluate x3 + 7

x®+¥

Page | 14

x3 , x  −1

13. Sketch the graph of f ( x) =  0 , x = −1 and hence, determine

 x , x  −1


a) lim f (x) b) lim f (x)

x→−1− x→−1+

ans: a) -1, b) 1

CHAPTER 8

DIFFERENTIATION

1. Find dy for the function y = 8x − 3 by using the first principle method. Ans: 4
dx
√8 −3

2. Find dy for the function y = 2 + x by using the first principle method. Ans: −4 + 1
dx x2 3

3. Differentiate f (x) = (2x2 -1)2 (2 - 3x)3 4. Differentiate f (x) = (1- 2x)9(3 - x)3

Ans: (2 2 − 1)(2 − 3 )2(−42 2 + 16 + 9) Ans: −3(3 − )2(1 − 2 )8(−8 + 19)

5.Differentiate ( x − 3) 2 y = ln x
x2
y = (x + 2)2 6. Differentiate

Ans: 10( −3) Ans: 1−2
( +2)3 3

7. Find d2y y= e2x
dx2 of x +1

Ans: ⅇ2 (2 2+2 +1)
( +1)3

8. Find the second derivative of f (x) = x2 − 2x . Ans: − 1
3
( 2−2 )2

9. Differentiate the following function with respect to x.

a) y = e x−1 b) f ( x) = 1− x c) y = 1
ex − e−x
ex

ans: a) ⅇ√ −1 , ) −2 , ) −(ⅇ −ⅇ− )
ⅇ (ⅇ −ⅇ− )2
2(√ −1)

a) y = e x−1 b) f ( x) = 1− x c) y = 1
ex − e−x
ex

Page | 15

10. Differentiate the following function

a) y = x2 ln 2x b) y = ln  2x +1  c) y = x2
 1− x  ln x

ans: ) (1 + 2 ), ) 3 , ) (2 ln −1)
(2 +1)(1− ) (ln )2

11. Find dy x2 ln y = e2y + 3xy. ans: (3 −2 )
2−2 ⅇ2 −3
, if

dx

12. If 3x2y + y2 = x + 9, show that  3x2 + 2y  dy = 1.
1− 6xy dx

13. By taking logs both sides and using implicit differentiation, find the derivative of

y = 1− x2 . Ans: − (3 2+2 +1)
1+ 2x (1+2 )3∕2

( )14. Differentiate y = 3 − e3 + 3x − ln (3x) + e3x − e3 ln e3x dy = 3 − 1 + 3e3x − 9e3
ans: dx x
15. Find dy of the following functions
dx

( )y = ln x + x −1 b) y= e−x + ex
e−x − ex
a)

Ans: a) 1 , ) 4ⅇ 2
2√ 2− (1−ⅇ 2 )2

Page | 16

CHAPTER 9

APPLICATIONS OF DIFFERENTIATION

1. Find the equation of the tangent to the curve with the equation y = 1 x3 − 3 x at the
8

point where x = 4 . Ans: y = 1 x −19
2

2. Determine the point where the gradient of the tangent to the curve f (x) = 1− 3x2

is equal to 5. Then find the equation of the tangent to the curve. Ans:

 − 5 , − 13  y = 5x + 37
6 12  12
,

3. Find the equation of the tangent and normal to the curve 2x2 + y2 = 33 at the point

(2, 5). ans: y = − 4 x + 33 @ 5y = −4x + 33 , y = 5 x + 5 @ 4 y = 5x +10
55 42

4. Find the value of x coordinate of the point where the normal to f (x) = x2 − 3x +1

at x = −1intersects the curve again. Ans: 21/5

5. Given y = 3x4 − 4x3 , find if exist the maximum and minimum point using first
derivative test. Ans: (0,0) is not minimum or maximum, (1,-1) is a minimum point.

6. Consider the curve with equation y = x4 − 8x2 − 5 . Find the stationary points on the
curve and determine their nature by using first derivative test.
Ans: (0,-5) is a maximum point, (2, -21) is minimum point, (-2, -21) is minimum poin

7. Find the coordinates of the stationary points on f (x) = −x3 − x2 and determine the
nature of these points by using second derivative test.
Ans: (0,0) max, (-2/3, -4/27) min

( )8. Find the coordinates of the stationary points on y = (1− x) 6x − x2 − 2 and

determine the nature of these points.
Ans: (2/3,14/27) max, (4,-18) min

9. A computer repair shop states the cost and revenue functions such that the cost

function: C ( x) = 4x2 − 40x + 300 and the revenue function, R ( x) = 100x − 3x2 . Let x be

the number of computers, find
(a) average cost
(b) marginal cost,
(c) marginal revenue and
(d) demand function
(e) profit function
(f) marginal profit
(g) profit function, if the government charge RM5 taxed
for each computer.

Page | 17

10. The profit function and the average cost function for a product are given as

 ( x) = −0.4x2 + 60x − 3000 and C ( x) = 0.5x − 20 + 3000 ,

x

where x is the number of units of the product. Find
(a) the cost function
(b) the revenue function
(c) the marginal revenue and the demand function

11. The demand function , in ringgit, for x units of a pot is given by
p(x) = 1000 – 5x

(a) Find the quantity to achieve the maximum revenue. Ans: 100

(b) Find the maximum revenue. Ans: RM 50 000

12. The average cost function and demand function , in ringgit, of a company that

produce. Computer chips are C(x) = 4000 + 8 and p(x) = 60 – 0.002x, respectively.
x

Find the maximum profit. Ans: RM 334000

13. A company produces and sells pots each year with cost function,
C(x) = 1000 + 5x and demand function, p(x) = 60 – 0.004x where x is the number
of pots and C(x) and p(x) are in ringgit. Determine
a) the profit function.
b) the maximum profit and the number of pots that need to be produced and

sold each year to achieve maximum profit, if each pot sold is taxed RM3.
ANS: b) 6500 units, RM 168000

( )14. Each month, the average cost function, C(x) , and the revenue function, R x

, in Ringgit Malaysia, of a company that manufactures computer

are C ( x) = 6000 +1000 and R(x) = 4000x − 20x2. Find

x

a) The cost function, the demand function and the profit function.

b) The maximum profit and the selling price to ensure maximum profit, if each

computer sold is taxed RM40.

ANSWER CHECK WITH ME

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