MODUL PERFECT SCORE

PREPARED BY: MDM NURUL BILQIS BINTI
BAHARUDDIN

AM015 2021/2022

CONTENTS

CHAPTER TOPICS PAGES
3–4
1 NUMBER SYSTEMS AND EQUATION 5–6

2 INEQUALITIES AND ABSOLUTE VALUES

3 SEQUENCES 6–7

4 MATRICES AND SYSTEM OF LINEAR 8 – 10

EQUATIONS

5 FUNCTIONS AND GRAPHS 11 – 12

6 POLYNOMIALS 13 – 14

7 LIMITS 15 - 16

8 DIFFERENTIATION 17 – 18

9 APPLICATIONS OF DIFFERENTIATION 19 – 20

SO LET’S GET STARTED!!

2

CHAPTER 1

NUMBER SYSTEM AND EQUATIONS

1. Given A = x : −4  x  4, x  Z , B = x : x  −1 x  1, x  R and
C = x : −3  x  3, x  R .
Find (a) A  B (b) A  B (c) A C
ANSWER: a) {-3,-2,-1,2,3}, b) (−, −1 0 1, ) c) −2, −1, 0,1, 2,3

2. Given A = x : −2  x  5, x  R , B = (2, 6) and = { : −3 ≤ ≤ 4, ∈ }
Find (a) A  B (b) B  C (c) A  C
ANSWER: a) (2,5) b) 3, 4 c) −2,5 −3

3. If = (−8, 1], = [−6, 3] and = (−9, −1). Find

(a) ∩ ∩ (b) ∪ ∪

ANSWER:a) [-6,-1] b) (-9,3)

4. 5+ 6 (b) 3 + 3 + 3
Simplify (a) 3 − 2 2+ 3

ANSWER: a) 3, b) 7√3 + 8√2

5. If x = 3 + 2 and y = 3 − 2 . Show that 1 + 1 = 1.
x +1 y +1

ANSWER: SHOWN

6. Solve x + 9 = x − 7 + 2

ANSWER: 16

7. Solve the equation 2x −1 − x + 3 = 1

ANSWER: 13

( )2

8. If x and y are rational and x − y = 14 − 6 5 , find the possible values of x and

y.

ANSWER: x=5, y=9 ;x=9, y=5

9. Rewrite ln  1  − ln  1  as a single logarithm.
 x   x3 

ANSWER: 2 ( )

10. Solve log8 ( x + 3) + log8 (9 − x) =12
3

ANSWER: 1,5

3

11. Find the value of x if 2 logb x = 2 logb (1− a) + 2 logb (1+ a) − logb 1 − a 2
 a 

ANSWER: a

12. Solve log4 x = logx 5

ANSWER: 4.454, 0.225

13. Solvefor for 2 + 2 ln(3x + 7) = ln(x2 + 3)2
ln x ln x

ANSWER: −7+√73

4

14. Show that

( )log a−1 + b−1 = log (a + b) − log a − log b

(a)

1 log3 64 − 2 log3 4 + log3 18 = 2
2
(b)

x−2 − y−2
15. Simplify x−1 + y−1

ANSWER: −


6x + 6−x = 10
3
16. Solve

ANSWER: − 6 3 , 6 3

e2x − e−2x 1
e2x + 2e−2x
, find the exact value of

4
x17. If =

ANSWER: 1 2 , 0.1733
4

18. Show that the equation 32x+1 − 32x−1 = 24 can be written in the form 8 (32x ) = 24 .

3

xHence, obtain the value of which satisfies the equation.

ANSWER: 1

19. Solve the equation 52x+2 − 5x+1 = 5(5x ) −1 .

ANSWER: -1

20. If 5a = 3b = 45c , prove that ab = c(2a + b).

ANSWER: PROVED

4

CHAPTER 2

INEQUALITIES & ABSOLUTE VALUES

1. Solve (a) x  2 − 3x  12 + x (b) 2(2x − 3)  3x(x − 5)

ANSWER: { : − 5 < < 1⁄2},{ : 1 < < 6}
2 3

2. Solve x + 4  x2 − 2  1− 2x

ANSWER: { : −3 < ≤ −2}

3. Solve x 2 −1  2x

ANSWER: { : −0.4142 ≤ ≤ 2 ⋅ 4142}

4. Solve (a) x −1 = 2x − 3 (b) 3 x + 5 = 2x +1

2

ANSWER: 4/3, { : = − 3 , 8}

7

5. Solve (a) x2 − 2x = 1 (b) x − 4 = 4 x + 2

ANSWER: a) {1,1±√2}, b) -4/5,-4

6. Solve (a) −3x + 6 −15  −30 (b) 2x − 7  4x +1

2

ANSWER: a) ∅ , b) { : ≥ 21} , [21 , ∞)

7. Solve (a) 3x − 5  2x − 3 (b) 5x − 6  2x +1

ANSWER: a){ : ≤ 8 ∪ ≥ 2} , (−∞, 8] ∪ [2, ∞), b)(−∞, 5) ∪ (7 ∕ ∞)
5
5 7 3

8. Solve x − 3  1 (5 − x)

2

ANSWER: { : 1 < < 11}

3

9. Solve 5 + 2x  −1 or 8 − 3x  2 . Give your answer in interval notation.

ANSWER: (−∞, −3) ∪ [2, ∞)

10. Given f (x) = x2 − 3x − 4 . Find the set of values of x which satisfy the inequality

f (x)  11.

4

ANSWER: { : < − 3 > 9}

22

11. If x is an integer, what is the solution to x − 2 +1 5 ?

ANSWER: {-2,-1,0,1,2,3,4,5,6}
12. If │x + 4│ ≤ 3, find the values of p and q for which p ≤ 2x – 4 ≤ q .

ANSWER: p=-18, q=-6
13. Determine the set of values of which satisfies both inequalities

1 ( + 3) ≥ 1 (3 + 1)and1 < 8 − < 2( + 1)

24

ANSWER: (2,5]

5

14. If 2 2 + 1 > 3 , show that |4 − 3| > 1.
15. Find the values of and given that | − | < and the solution set is (3, 7)

ANSWER: p=5, r=2

CHAPTER 3

SEQUENCES

1. 3 p +10, 20, p2 + 2 are the first three terms of an arithmetic series.

Show that 4 is one of the possible values of p . Hence, calculate the tenth

terms and the sum of the first 10 terms of the resulting arithmetic series.

ANSWER: p=4, 4, 130

2. The sum of the first five terms of an arithmetic series is 115 and the second
term is nine times the eight term. Find
(a) the first term and the common difference,
(b) the sum of the first 23 terms of the arithmetic series.
ANSWER: a) 31,-4 b) -299

3. The sum of the first n terms of an arithmetic series is given by Sn = 4n 2 + n .
Find the first term and the common difference. Hence, find the 11th term of the
arithmetic series.
ANSWER: 5, 8, 85

4. Find the value of n if the sum of the first n terms of arithmetic sequences

63,54,45,....is −10773 .

ANSWER: 57

5. The fifteenth term is 59 and the sum of the first fifty terms is four times the
sum of the first twenty terms. Find the first term and the common difference.
ANSWER: 31, 2

6. The sum of the first 8 terms of an arithmetic sequence is 60 and the sum of the
next 6 terms is 108. Find the 25th term of this arithmetic sequence.
ANSWER: 153/4

7. If the pth and the qth term of an arithmetic sequence are 1 and 1
qp

respectively, prove that the sum of the first pq terms must be 1 ( pq +1) .

2
ANSWER: PROVEN

8. Nurul saves her money to buy a new car. The first month she saved RM600.
Every month thereafter, her saving increases by RM150. Find,

6

a) How much money she will saved in the 12th month
b) The total savings within 12 months
ANSWER: a) 2250, b) 17100

9. Safa save RM1200 now and every year she saved RM100 more than she did
in the preceding year. How many years for the amount of money will be worth
RM16500?
ANSWER: 10 years

10. The sum of the first three terms of a geometric series is 16 and the sum of the

next three terms is 128. Determine the first term and the sum to 10th terms of

geometric series.

ANSWER: 16/7, 2338 2
7

11. The fourth and the seventh terms of a geometric series are 1 and 1

2 16

respectively. Determine the 12th term and the sum of the first ten terms of the

geometric series.

ANSWER: 512, 1023/128

12. The first term of a geometric progression is 2 + 2 and the second term is

2 − 2 . Find, in the simplest form, the third and fourth terms.

ANSWER: 10 − 7√2, 58 − 41√2

13. The sum of the first 3 terms of a geometric progression is 336 and the sum of
the 5th, 6th and 7th terms is 21. Given that the common ratio is positive, find the
first term and the common ratio.
ANSWER: 192, 1/2

14. For the year 2016, the number of students taking the MBBS course in St.
Mary College is 500. It is known that the number of students taking this
course increases by 10% every year. Find the minimum number of lecturers
required to teach this course in 10 years’ time given that a lecturer can only
handle a maximum of 100 students?
ANSWER: 12

15. A rubber ball is dropped from a height of 486 meters, and each time, it
rebounds one-third of the height from which it last fell. How far has the ball
traveled by the time it strikes the ground for seventh time?
ANSWER: 971.34 meter

7

CHAPTER 4

MATRICES & SYSTEM OF LINEAR EQUATIONS

2 2 3
1. Given A = 1 5 4

3 1 4

Find

(a) the determinant of A (b) the minor of A (c) the adjoint of A

 16 −8 −14   16 − 5 − 7
 −1  A=  8 −1 −5
ANSWER: a) 6, b) =  5 −4  , c) Adjoint

 −7 5 8   − 14 4 8 

2. If A  1 0 −1  such that A = −3 ,
= 3 x−2 4 

 1 1 x − 3

(a) determine the values of x
(b) find the value of

(i) AT (ii) A2
ANSWER: a) 0, 4, b) i. -3, ii.9

1 2 3
3. Given A= 4 5 6 . Find

1 0 2
a) M12 and c22
b) Evaluate the determinant of A by using cofactor expansion of the first row.
c) Find Adj A
d) Find A−1 using adjoint method.

10 −4 −3T 10 −2 −5
ANSWER: a) 2, -1, b) -9, c) AdjA = −2  = −4 
−1 6  −1 2  d)

−5 2 −3 −3 6 −3

−10 4 1 
10 −2 −5  9 9 3 
1 −4   1 −2 
A−1 = − 9 −3 −1 2  =  2 9 3
6 9
−3
 5 −2 1 
 9 9 3 

1 3 
4. Find the inverse of 9 −2 by using adjoint method.

ANSWER: [29//2299 −31//2299]

8

5. Given the matrices P =  3 6 3  and Q =  −1 −1 4  Find PQT . Hence, find
3 −3 0  −1 −2 5 .

9 3 3  1 1 − 3

P −1 .

          1 −1 −1 1
3
1 0 0 = (−1 −2 1)
ANSWER: 3 (0 1 0) 4 5
−3
001 −1 −1
1

33 3

           = −1 −2 1
3 3 3

4 5 −1)
3
(3

6. Given that A = −3 −2 and A2 + mA + nI =0 where m and n are real numbers, where
 
 2 −2 

I is a 2 2 identity matrix and 0 is a 2 2 zero matrix. Find the values of m and n . Hence,

find A−1 and A3 .
ANSWER: m=5, n=10, [−−11//55 −13//510] , [350 −2300]

7. If A=  2 − 2  , find A-1 .
3 − 4

ANSWER:  2 − 1
 3 − 1

2

8. If A =  2 −1 1  , find A-1 .
1 −1 − 1

 2 − 2 −1

1 3 − 2
ANSWER: 1 4 − 3

0 − 2 1 

 7 16 −10 1 −2 −6
  1 
9. It is given P =  5 11 −8  and Q = −1 2  . Find QP and and write down Q −1

−1 −1 1  2 −3 −1

k  4 
  −3
. Use this matrix to find the values of k,l and m given that Q  l  = .

m −2

ANSWER: 3I, find on your own Q−1 , k=0, l=1, m=-1

9

10. Solve the following system of linear equations by using the inverse matrix

3x + 2y + 5z = 31
4x + 3y + 7z = 44
2x + y + 5z = 21

ANSWER: x=3.5, y=6.5, z=1.5

1 2 0  2 2 − 2
11. Given A =  3 2 1 and B =  1 −1 1  .
 2 4 1  − 8 0 4 

(a) Find AB and hence determine A−1 .

(b) A hawker sells three types of cakes, cheese, chocolate and strawberry cakes.
Amy pays RM5 for 1 cheese cake and 2 chocolate cakes. Nurul pays RM10 for 3
cheese cakes, 2 chocolate cakes and 1 strawberry cake. Siti pays RM13 for 2
cheese cakes, 4 chocolate cakes and 1 strawberry cake. Form a system of linear
equations based on the above information. Write the system of linear equations
in matrix form and hence find the price of each type of cake.

1/2 1/2 −1/2
ANSWER: a) 4I, [1/4 −1/4 1/4 ] b) cheese cake= RM 1, choco= RM 2,

−2 0 1
strawberry= RM3

12. A triangular nursery plot PQR has a perimeter of 50 m. The length of side PR is 1 m
less than twice the length of side PQ. The sum of the lengths of PQ and PR is 10 m
less than three times the length of side QR. By representing PQ = x m , QR = y m

and PR = z m , write the above information in the matrix equation. Use the Gauss-
Jordan elimination method to determine the values of x, y and z.

ANSWER: x= 12m, y=15m, z=23m

13. In a triangle, the largest angle is 20o more than four times the smallest angle and
the largest angle is twice the sum of the other two angles. Denoting the largest angle
as x, the smallest angle as y and the third angle as z. Rewrite the above information
in the matrix equation. Use the Gauss-Jordan elimination method to determine the
values of x, y and z.
ANSWER: x= 120, y=25, z=35

10

CHAPTER 5

FUNCTIONS & GRAPHS

1. Given that f (x) = 3x ; x  2.
x−2

a) Find composite function (f  f )(x).

b) If ( f g)(x) = 2x +1, find the function g(x).
ANSWER: a) 9 , ≠ −4 b) 2 +1

+4 −1

2. a) Given that (g f )(x) = 1 and f (x) = 2 − x , find g ( x) .

x −1

b) Functions f and g are defined by f (x) = x −1 and g(x) = 3x + 6 . Find

(g  f )(x) and state its domain and range.

ANSWER: a) 1 b) x-1, [1,∞), [6,8)
1−

3. Given the function f(x) =e2x + 1 and g(x) = ln(x – 3). Find fg and gf .

ANSWER: = ⅇ(( − 3)2) gf= (ⅇ2 +1 − 3)

4. If f (x) = 3x −1, x  −5 , find f −1(x) .
x+5

ANSWER: 5 +1 , ≠ 3
3−

5. Write an expression to describe the graph below and state its range.

6. Sketch the graph of f (x) = 3x2 − 6x + 5, x  1. Then, state it domain and range.

7. Sketch the graph of the following function and hence, state its domain and range.

a) f(x) = ex – 1 b) f(x) = ln(2x + 1)

8. If f is given by f (x) = 4 − 3x

11

a) Find f −1(x)

b) Sketch the graph of f for −1  x  4 . Hence, sketch f −1(x) on the same axes.

c) State its domain and range

ANSWER:

Df = R −1 = [−1, 4)
f

Rf = D −1 = [−8, 7)
f

9. Given that f (x) = (x − 2)2 −1 for x  2 . Find f −1 and state its domain and range.
Show the relationship between the graphs of f and f −1 on a diagram.

ANSWER:Range of = [−1, ∞) Domain of = [2, ∞)

10. If f (x) = 3 − e−2x , find f −1 (x) . Show both the graphs of f and f −1 on the same
coordinate axes.

ANSWER: GIVE YOURS

11. Given that f (x) = ln(x − 3) and g(x) = 4x + 9 .
(a) Find f −1 (x) . State its domain and range.
(b) Show the relationship between f and f −1 on a diagram.
(c) Find (g  f −1 )(x) and sketch its graph.

ANSWER: a) D f −1 = (− , ) R f −1 = (3, )

12. Two function f and g are defined as follows.

f (x) = ln(x − 2) and g(x) = 3x + 5
(a) Find ( f  g −1 )(x) .
(b) Sketch the graph and determine the domain and range of ( f  g −1 )(x) .

( ) ( )ANSWER: D f g−1 = 11,  R f g−1 = − , 

12

CHAPTER 6

POLYNOMIALS

1. Given that polynomial P ( x) = x3 + x2 − 8x −12 is divisible by ( x − 3) . Find another

factor of P(x).
ANSWER: ( − 3)( + 2)2

2. By using the long division, find the remainder when 6x 3 − 5x 2 + 9x − 11 is divided by
(2x − 1)(x + 2).
ANSWER: 36x-25

3. Polynomial P(x) = 2x 3 + ax 2 − x + b has x + 1 as a factor and leaves a remainder
12 when divided by x − 3 . Find the values of a and b and hence find all the factors of

P(x).
ANSWER: a= -5, b=6 , (x+1), (x-2), (2x-3)

4. When the polynomial P(x) is divided by x −1, the remainder is 5 and when it is divided
by x − 2 , the remainder is 7. Find the remainder when P(x) is divided by ( x −1)( x − 2
).
ANSWER: 2x+3

5. When the polynomial P(x) is divided by ( x +1) , the remainder is 5 and when it is
divided by ( x − 4) , the remainder is 15. Find the remainder when P(x) is divided by
( x +1)( x − 4) .

ANSWER: 2x+7

6. Find the values of the constants p, q and r for which
(x −1)(x − 2)(x + p) = qx3 − 7x2 +14x + r. Hence, solve the equation
qx3 − 7x2 +14x + r = 0.
ANSWER: p=-4, q=1, r=-8, ; x=1,2,4

7. Given x +1 is a factor of polynomial f (x) = ( p + 1)x3 − 2 px2 − 5x + p . When f (x) is

divided by x + 2 , the remainder is p + q .

(a) Find the values of p and q . Hence factorise f (x) completely.

 (b) Given that f (x) = (3x −1) Q(x) − x2 −1 , find Q(x) and the real roots of Q(x) = 0

ANSWER: a) p=2, q=-30 (3x-1)(x-2)(x+1) b) Q(x)=2 2 − − 1 x=-1/2, 1

8. Express 7x − 4 in partial fractions.

x(x −1)(x + 2)

ANSWER: 7x − 4 = 2 + 1 − 3
x(x −1)(x + 2) x x −1 x + 2

13

9. Express 2x +1 in partial fractions

(x + 1)2 (2x − 5)

ANSWER: 2x +1 = −12 + 1 + 24
(x + 1)2 (2x − 5) 49(x + 1) 7(x + 1)2 49(2x − 5)

x2

10. Express (x + 3)(x −1)2 in partial fractions.

ANSWER: 2 = 9 + 7 + 1
( −1)2( +3) 16( +3) 16( −1) 4( −1)2

2x −13 in partial fractions.
( )11. Express
(x + 1) x2 + 4

ANSWER: 2 −13 = − 3 + 3 −1
( +1)( 2+4) ( +1) ( 2+4)

12. Given (x + 3) is one factor of P(x) = 9 −12x −11x 2 − 2x 3 . Factorise completely P(x),

and express 13x + 18 as a sum of partial fractions.

P(x)

ANSWER: 13 +18 = 1 − 3 + 2
( +3)2(−2 +1) ( +3) ( +3)2 (−2 +1)

14

CHAPTER 7

LIMITS

x2 − x + 3
lim
1. Evaluate the following limits (a) x→1 x +1 (b) lim 1
x→2 3x3 −1

ANSWER: a) 3/2, b) √23
23

( )2. x2 − 4
Evaluate the following limits.(a) lim 3x3 − 5x +1 (b) lim
x→2 x→2 x + 2 − 2

ANSWER: a) 15, b) 16

lim x −1 lim (2x − 5)( x −1)
3. Evaluate the following limits.(a) x→1 x3 −1 (b) x→1 2x2 + 3x − 5

ANSWER: a) 1/6, b) -3/14

lim 1  1 + 1 −x2 + 7x −10
x  x−2 2 
4. Evaluate the following limits. (a) x→0 (b) lim
x→2
x2 − 6x +8

ANSWER: a) -1/4, b) -3/2

lim x−3

5. Evaluate the following limits(a) x→3 x2 − 9 x2 + 6x +8
(b) lim
x→−2 x + 2

ANSWER: a) does not exist, b) -2

e2x −1 x  0 . Find

6. Given that g ( x) =  ex −1

 x + 2 x  0

a) lim g(x) b) lim g(x) c) lim g(x)
x→0− x→0+ x→0

ANSWER: a) 2, b) 2, c) 2

15

lim 2x + 1 x2 −1
lim
(a) x→+ x 2 − 3 b) x→+ 3x + 5
7. Determine the following limits:

ANSWER: a) 0, b) 1/3

x2 −1 lim x2 + x +1 − x

8. Evaluate (a) lim . (b) x→− 3x +1

x→+ 4x2 + 5

ANSWER: a) ½, b) -2/3

9. Determine the following limits: lim x4 + 2 (b) lim 6x5 + 2x4 + 3x2
x2 +1
(a) x→−
x→−

ANSWER: a) 1, b) -∞

10. Find the limits (if it is exists) 2x −3 −3
(b) lim (c) lim
6
(a) lim x→3 x − 3 x→3 (x − 3)2

x→2 2 − x

ANSWER: a) ∞, b) does not exist, c) -∞

11. Find the limits (if it is exists)

1+ x (b) x2 +1 x+3 3x
(a) lim lim (c) lim (d) lim
x→4+ x − 2 x→3− x2 − 9
x→−0.5− 2x +1 x→1− 1 − x2

ANSWER: a) -∞, b) ∞, c) -∞, d) ∞

16

CHAPTER 8

DIFFERENTIATION

1. By using the first principle, differentiate the following;

a) y = x2 +2x b) y = x ANS: a) 2x+2, b) 1

2√

2. Differentiate the following using the first principle:

a) f (x) = 1 b) f (x) = 2 + x ANS: a) (2 − +24)2, b) −4 +
2x + 4 x2 3

1

3. Differentiate the following functions

a) y = ex ex ( )b) y = x2 + 3x − 5 ln(x2 + 3x − 5)
+ e−x

ANS: a) 2 ⋅b) (2 + 3)[1 + ( 2 + 3 − 5)]
(ⅇ +ⅇ − )2

4. If y = x2ex+1 . Find d2y in its simplest form. ANS: ⅇ +1( 2 + 4 + 2)
dx2

5. Differentiate y = ex ln 3x ANS:ⅇ [3−12 + (5−4 ) 3 ]
1− 4x (1−4 )2

e2x (2 + 3ln x) ANS: ⅇ2 [3 +1 5 +(2+3 )(2 2−2 +10)]
( 2+5)2⋅
6. Differentiate y = x2 + 5

7. Find dy for y =  3x − 2 2 x −1. ANS: √ − 1 (3 − 2) [15 32− 122( −2+16) −8]
dx  x 


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

3
( 2−2 )2

9. Given that x2 − 2xy + 2 y2 = 4 . Solve for x and y if dy = 0. ANS: x,y=±2
dx

dy dy at (0 , 0)
dx
10.Find dx of a curve xexy = e2x − e3y , hence determine the value

ANS: 1/3

dy ANS:
11. Find dx if ln(xy ) = ey ( ⅇ −1)

12. Given ye2x = 2x + 1. Prove that dy = 2e−2x − 2y . ANS: PROVEN

dx

17

CHAPTER 9

APPLICATIONS OF DIFFERENTIATION

1. A curve has the equation ln y = 3y2 − x . Find dy at the point (4,-1). Hence, obtain the
dx

equation of the tangent at that point.

ANS: dy = − 1 , y = −x − 1
dx 5 55

2. A curve has the equation ln y2 = 4 y3 − x−2 . Find dy at the point (-1,2). Hence, obtain the
dx

equation of the tangent at that point.

ANS: dy = 2 , y = 2x + 96
dx 47 47 47

3. Find the equation of a normal line at a curve 4xy2 − 3x = 5y when y = 1.
y −1 = 35(x − 5)

ANS:
 y = 35x −174

4. Find the value of k such that the line y = 2x + k is a normal to the curve y = 2x2 − 3

ANS: k= -87/32

5. Find all the minimum and maximum points of the curve, y = x4 − 2x2 . By using first
derivative test, determine their nature.
ANS: (-1,-1) and (1,-1) are min points and (0,0) is a max point

6. Find all the minimum and maximum points of the curve, y = x3 + x2 − 2x . By using first
derivative test, determine their nature.
ANS: (-1.215,2.113) is a max points and (0.549,-0.631) is a min point

7. For each of the following function, find the first and second derivative at the point (0,0).
Hence, determine the nature of the two points. From the result, what can we conclude?
a) f ( x ) = x3
b) f ( x ) = x4
ANS: a) ( 0,0 )is not an extremum point b) ( 0,0 )is a local mi nimum point

8. Given that f ( x ) = e−x + ex . Find the stationary and determine its nature by using second

derivative test.

ANS: stationary point is (0,2) which is a relative minimum point

9. A company produces televisions every month with the production average cost C(x) and

revenue cost R(x) given by C(x) = 3000 + 500 and R(x) = 200x −10x2 . Find the cost function,

x

demand function and profit function
ANS: C(x)= 3000+500x, P(x)=2000-10x, Profit=1500x-10x2-3000

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10. Given the cost function, C(x) = 50 000 + 60x + 0.002x2 and the revenue function,

R(x) = 300x − 0.003x2 . Find

(a) the profit function,
(b) the average profit function,

(c) (1000) and interpret,

(d) the marginal profit function.

ANS: a) -50000+240x-0.005x2 b) − 50000 + 240 − 0.005 , c) RM185, d) 240-0.01x


11. A company produces a car air conditioner every month with the production cost C(x) and

the demand function P(x) given by C(x) = 6000 +1000x and P(x) = 4000 − 20x . Find

(a) The revenue function and the number of air conditioner should be produced
every month to maximize the revenue

(b) The number of air conditioner to be produced monthly to maximize the profit
and state the value of the profit.

(c) The selling price to maximize the profit.
(d) Maximum profit if each unit of air conditioner is taxed at RM40.

ANS: a) 4000x-20x2, 100 units, b) 75 units, RM 106500, c) RM2500, d) RM 103,520

12. The average cost function and the demand function for a company are given as
C(x) = 3x −150 + 2500 and p = 300 − 2x respectively, where x is the number of items
x
produced.

(a) Find the cost function, revenue function and profit function.
(b) Determine the number of items that should be produced to maximize the

profit.
(c) Find the price per unit when profit is maximized.
(d) Calculate the increase in cost when the number of items produced is

increased from 32 to 35 units.

ANS: a) 3 2 − 150 + 2500, 300 − 2 2, 450 − 5 2 − 2500 b) 45, c) RM210, d)
RM153

13. Innovative Sdn. Bhd, makes toys. The demand function and average cost function in
ringgit are p(x) = 80 − 0.004x and C(x) = 8 + 4000 respectively , where x is the
x
number of toys made. Find
(a) the cost function
(b) the maximum revenue and the number of toys that need to be sold to achieve
this,
(c) the maximum profit
(d) the maximum profit if each toy sold is taxed RM2.

ANS: a) 8x+4000, b) 10000, RM 400 000, c) RM 320 000, d) RM 302 250

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14. A television manufacturer produces x sets per week so that the total cost of

production is given by the relation C ( x) = x3 −195x2 + 660x +15000 . Find how many

television sets must be manufactured per week to minimize the total cost.
ANS: 110 sets
15. From a research, the price of mineral water K is RM1.50 and the daily demand is 1200

bottles. When the price of the mineral water K is reduced to RM1.20 per bottle, the daily
demand increase to 1500 bottles.

a) Find the demand function by assuming it is a linear function.
b) By using second derivative test, calculate the maximum revenue and the price

per bottle at this level.
c) If the cost of each bottle of mineral water K is 30sen and the daily fixed cost is

RM650, find the cost function.
d) By using second derivative test, calculate the maximum profit if each mineral

water K sold is taxed at 5 cents.
ANS: a) -0.001x+2.7, b) RM 1822.50, c) 0.3x+650, d) RM 730.63

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