Buku Tutorial AM025 (Sesi 2021-2022)

TUTORIAL & PSPM QUESTIONS

MathAMem0a2t5ics 2

SESSION 2021/2022

KOLEJ MATRIKULASI MELAKA

Name : ...........................................................
Practicum :.............................

Contents (AM025)

TOPICS TUTORIAL PAGE PSPM
PSPM (19/20 & 20/21)
1 – PARTIAL DIFFERENTIATION 1–3 (05/06 – 18/19)
2 – INTEGRATION 4–5 PAGE 116
3 – APPLICATIONS OF DEFINITE INTEGRAL 6–7 39 – 42
4 – LINEAR PROGRAMMING 8 – 12 PSPM 2019/2020
5 – MATHEMATICS OF FINANCE 13 – 15 43 – 46
6 – DATA DESCRIPTION 16 – 20 Paper 1
7 – PROBABILITY 21 – 23 47 – 53 (Page 1 – 9)
8 – RANDOM VARIABLES 24 – 29
9 – SPECIAL PROBABILITY DISTRIBUTIONS 30 – 32 54 – 63 Paper 2
10 – CORRELATION & REGRESSION 33 – 36 (Page 1 – 15)
11 – INDEX NUMBER 37 – 38 64 – 72
PSPM 2020/2021
73 – 80
(Page 1 – 17)
81 – 87
FINAL ANSWERS
88 – 96 PSPM 2019/2020

97 – 102 & 2020/2021

103 – 109

110 – 115

Prepared by

PREMA A/P S.KRISHNAN
NOOR HASRYZAL HASNOH
NORAZLIANA MOHMAD KAMAL

SHERALIN ENDION
NABILAH MOHD ABD FATAH

Mathematics AM025
Topic 1: Partial Differentiation – Tutorial

TOPIC 1 : PARTIAL DIFFERENTIATION
f f

1. Find x and y for the following :

(a) f (x, y) = 2x3 − 3y − 4 (b) f (x, y) = 3 (c) f (x, y) = 4x )
x+ y (x2 + y2

(d) f (x, y) = ln(x + y) (e) f (x, y) = ex ln y (f) f (x, y) = (x − y)2

(g) f (x, y) = ln(x2 + y2 ) (h) f (x, y) = 2x2 (i) f (x, y) = x − y
3y3 x+ y

2. Find fx and fy for the following :

1 (b) f (x, y) = x3 + 2y2 (c) f (x, y) = ln(x2 + y2 )
(f) f (x, y) = ey2 ln x
(a) f (x, y) = 3x2 − 3y3

(d) f (x, y) = ex+ y (e) f (x, y) = x
y2

(g) f (x, y) = x2 + y2 (h) f (x, y) = ln(x + x2 + y2 )

3. Find all the second order partial derivatives of the following :

(a) f (x, y) = 3x2 + 5y3 +10x (b) f (x, y) = x2 + 3x + 4y2 + 2y

(c) f ( x, y ) = 10x 3
y2
(d) f (x, y) = (6x − 8y)2

(e) f (x, y) = ey ln x (f) f (x, y) = 2x + y2 − 3y + 3x2

4. Verify that fxy = fyx for the following :

(a) f (x, y) = ln(4x + 3y) (b) f (x, y) = ex + x ln y + y ln x + ln 2

(c) f (x, y) = x3 + 2x + y (d) f (x, y) = y + x
x2 y

(e) f (x, y) = x2 + y2 (f) f (x, y) = y − x3 + 2x

5. Determine the location and nature of all the critical points for the following :

(a) f (x, y) = 4x2 + y2 − 80x + 20y −10 (b) f (x, y) = x3 − 5x2 + 3y2 −12 y +10
32

(c) f (x, y) = x2 + 2 y2 − 20x − 60 y +100 (d) f (x, y) = x3 + y2 − 3x + 6y +10
2

(e) f (x, y) = x3 − y2 − 3x + 4y + 25 (f) f (x, y) = −3x2 − 2y2 + 45x − 30y − 50

(g) f (x, y) = −x2 − 5y2 + 8x −10y −13 (h) f (x, y) = −3x2 − 2y2 + 3x − 4y + 5

(i) f (x, y) = x3 + y3 + 3x2 −18y2 + 81y + 5

6. (a) Maximize f (x, y) =16 − x2 − y2 subject to x + 2y = 6
(b) Minimize f (x, y) = x2 + y2 subject to x + y = 24
(c) Maximize f (x, y) = 20x +10y − x2 − y2 subject to x + 2 y −10 = 0
(d) Maximize f (x, y) = 25 − x2 − y2 subject to 2x + y = 4

1

Mathematics AM025
Topic 1: Partial Differentiation – Tutorial

7. A firm sells two products. The total annual revenue R behaves as a function of the
number of units sold.
R(x, y) = 400x − 4x2 +1960y − 8y2

where x and y , respectively, are the numbers sold of each product. The cost of producing
the two products is

C(x, y) =100 + 2x2 + 4y2 + 2xy

(a) Determine how many of each product should be produced and sold in order to
maximize the annual profit.

(b) What is the total revenue when the profit is maximum?
(c) What is the total cost when the profit is maximum?
(d) What is the maximum profit ?

8. A farmer estimated the annual profit at his farm can be described by the function
P(x, y) = 1600x + 2400y − 2x2 − 4y2 − 4xy

where P equals the annual profit. in ringgit, x equals the number of acres planted with
rice, and y equals the number of acres planted with vegetables. Determine the number of
acres of each crop which should be planted if the objective is to maximize the annual
profit. What is the expected maximum profit ?

9. A store carries two competing brands of bottled water, one from Johor and the other from
Selangor. Assume that the cost for both brands is RM2/bottle. If the Johor water is sold at
RM x/bottle and the Selangor water is sold at RM y/bottle, then the customer will buy
approximately 40 – 50x + 40y bottles of Johor water and 20 + 60x – 70y bottles of
Selangor water each day. How should the owner price the bottled water to generate the
largest possible profit ?

10. A company produces x DVD players and y televisions. The profit function is given as
(x, y) = 60x + 80y − x2 − y2 . If the total production for both products is set at 40 units,

how many of each product should be made to maximize the profit?

ANSWERS

1. (a) f = 6x2 , f = −3 (b) f = − ( 3 )2 , f = − ( 3
x y x +y y
x x + y)2

f = 4 y2 − 4x2 , f = −8xy (d) f = 1 , f = 1
x x2 + y2 2 y x2 + y2 2 x x + y y x + y
( ) ( )(c)

(e) f = ex ln y , f = ex (f) f = 2( x − y) , f = −2( x − y)
x y y
x y

(g) f = 2x , f = 2 y (h) f = 4x , f = − 2x2
x x2 + y2 y x2 + y2 x 3y3 y y4

(i) f = 2 y , f = − 2x
y
x ( x + y)2 ( x + y)2

2. (a) fx = 3 , f y = −9 y2 (b) fx = 3x2 , f y = 4 y
2x

2

Mathematics AM025
Topic 1: Partial Differentiation – Tutorial

(c) fx = 2x , fy = 2y (d) fx = ex+y , fy = ex+y
x2 + y2 x2 + y2

(e) fx = 1 , fy = − 2x (f) fx = e y2 , f y = 2 yey2 ln x
y2 y3 x

x y 1 y
, x2 + y2 x2 + y2 x2 + y2
(g) fx = fy = ( )( )(h) fx = , fy =
x2 + y2
x+ x2 + y2

3. (a) fx = 6x +10 , fxx = 6 , fxy = 0 f y = 15 y2 , f yy = 30 y , f yx = 0

(b) fx = 2x + 3 , fxx = 2 , fxy = 0 f y = 8y + 2 , f yy = 8 , f yx = 0

(c) fx = 10 , f xx = 0 , f xy = − 20 fy = − 20x , f yy = 60x , f yx = − 20
y2 y3 y3 y4 y3

1 27 1 , fxy = − 36

(d) fx = 9(6x − 8y)2 , fxx = 1
(6x −8y)2 (6x −8y)2

1 48 1 , f yx = − 36
(6x −8y)2
f y = −12(6x − 8y)2 , f yy = 1
(6x −8y)2

(e) fx = ey , f xx = − ey , f xy = ey fy = ey ln x , f yy = ey ln x , f yx = ey
x x2 x x

(f) fx = 2 + 6x , fxx = 6 , fxy = 0 f y = 2 y − 3 , f yy = 2 , f yx = 0

4. (a) f xy = f yx = − ( 12 )2 (b) f xy = f yx = 1 + 1 (c) fxy = f yx = 0
+3 y x
2 x y

(d) f xy = f yx =− 2 − 1 (e) fxy = f yx = −xy (f) fxy = f yx = 0
x3 y2
3
( )x2 + y2 2

5. (a) (10, −10) is a minimum point (b) (0, 2) is a saddle point , (5, 2) is a minimum point

(c) (20,15) is a minimum point (d) (−1, −3) is a saddle point , (1, −3) is a minimum point

(e) (−1, 2) is a maximum point , (1, 2) is a saddle point

(f)  15 , − 15  is a maximum point
 2 2 

(g) (4, −1) is a maximum point (h)  1 , −1 is a maximum point
 2

(i) (0,3) is a saddle point , (0,9) is a minimum point , (−2,3) is a maximum point ,

(−2,9) is a saddle point

6. (a)  6 , 12 , 12  , Maximum value = 44 (b) (12,12, −24) , Minimum value = 288
 5 5 5  5

(c) (8,1, −4) , Maximum value = 105 (d)  8 , 4 , 8  , Maximum value = 109
 5 5 5  5

7. (a) x = 20, y = 80 (b) RM112, 000.00 (c) RM29, 700.00 (d) RM82,300.00

8. Rice = 200 acres, Vegetables = 200 acres, Profit = RM400 000.00
9. Johor’s bottled water = RM2.70, Selangor’s bottled water = RM2.50

10. DVD = 15 units, television = 25 units

3

Mathematics AM025
Topic 2: Integration – Tutorial

TOPIC 2 : INTEGRATION

1. Find each indefinite integral

(a) 1 + x7 + 1 − x dx (b)  (2 − 3x)(1+ 5x) dx
x2

(c) x4 −1
x2 dx

2. Find the following indefinite integrals Tips:

(a)  3dx 1. xndx ,when n = −1
x
1dx
(b)  1 dx 2. x = ln x + c
2x
1 1
(c)  1 dx 3. ax + dx = a ln ax + b + c
3x − 2 b

where a and b are constant

3. Find

(a)  3exdx Tips:

(b) 1 1. exdx = ex + c
e2x dx 2. eaxdx = eax + c; a  0

(c)  (3 + ex )(2 + e−x )dx a

( )(d) 2
ex + e−x
dx

4. Integrate the following with respect to x Tips:

(a)  x + 4dx 1. xndx = xn+1 + c , n  −1
n +1
(b)  1 dx
x −1 2. (ax + b)n dx = (ax + b)n+1 + c , n  −1
a(n +1)
(c)  (3x − 2)12dx
3. f '(x)[ f (x)]n dx = [ f (x)]n+1 + c
n +1

5. Find the integral of each of the following

(a) e2x+3dx (b) x2ex3−1dx

(c)  x2 1  ex3 +xdx e3x + x2
 3  e3x + x3 3 dx
+  ( )(d)

6. By using substitution method find

(a)  3 2 dx (b) x2 3 x3 +1dx
− 4x

1 2x
(c) dx
x2 + 2
0

4

Mathematics AM025
Topic 2: Integration – Tutorial

5 10 5

7. Given  f ( x) dx = 5,  f ( x) dx = 10 and  g ( x) dx = 9, find
3
35

5 3

(a)  7 f ( x) dx (b)  g ( x) dx

35

5 10

(c)  12 f ( x) + 3g ( x) dx (d)  f ( x) dx

33

8. Compute the given definite integrals 4  1  2
1  x 
2 (b) + 2 x dx

(a)  ( x2 − x +1) dx

0

(c) 2 1− ex dx
0 ex

ANSWERS

1. (a) x + 1 x8 − 1 − 2 x 3 +c (b) 2x + 7 x2 − 5x3 + c (c) x3 + 1 + c
2 2 3x

8 x3

2. (a) 3ln x + c (b) 1 ln x + c (c) 1 ln 3x − 2 + c
2 3

3. (a) 3ex + c (b) − 1 e−2x + c (c) 7x − 3e−x + 2ex + c (d) 1 e2x + 2x − 1 e−2x + c
2 22

4. (a) 2 ( x + 4) 3 +c (b) 2 x −1 + c (c) 1 (3x − 2)13 + c
2
39
3

5. (a) 1 e2x+3 + c (b) 1 ex3−1 + c (c) ex3 +x + c −1 +c
2 3 3 e3x + x3
6( )(d) 2

6. (a) − 1 ln 3 − 4x + c 1 4 (c) ln 3
2 2
x3 +1 3 + c
( )(b) (d) 15
4

7. (a) 35 (b) −9 (c) 87

8. (a) 8 (b) 38 3 (c) −1 −1
3 4 e2

5

Mathematics AM025
Topic 3: Applications of Definite Integrals – Tutorial

TOPIC 3 : APPLICATIONS OF DEFINITE INTEGRALS

1. (a) Find the area underneath the curve y = x2 + 2 from x = 1 to x = 2 .

(b) Find the bounded between the curve y = x2 − 4 , x-axis and the lines x = −1 and
x=2.

2. Calculate the area of the region bounded by the y – axis and the curve y = ex and
y = 2 + 3e−x .

3. Find the area bounded by the curves y2 = 8x and y = x2 .

4. Find the area bounded by the curve y = x ( x +1)(2 − x) and x – axis.

5. Sketch the curve y = x ( x − 2)( x + 3) . If A1 is the area above x – axis bounded by this

curve and the x – axis while A2 is the area under the x – axis bounded by this curve and
the x – axis, find A1 : A2 .

6. Find the consumer’s surplus at a price level p = RM150 for the price–demand equation

p = D ( x) = 400 − 0.05x .

7. The price–demand equation for a piece of equation is expressed by
y = 30 − 0.5x − 0.02x2 . Find the consumer’s surplus at the output level of 20.

8. Find the producer’s surplus at a price level of p = RM67 for the price–supply equation

p = S ( x) = 10 + 0.1x + 0.0003x2 .

9. Find the consumer’s surplus and the producer’s surplus at the equilibrium price level for
the given price-demand and price supply equations. Include a graph that identifies the
consumer’s surplus and producer’s surplus.

(a) p = D ( x) = 50 − 0.1x ; p = S ( x) = 11+ 0.05x
(b) p = D (q) = −q2 − 4q + 20 ; p = S (q) = q2 + 2q

10. Find the total cost function and the average cost function if the marginal cost function for

the production of x unit piece of equipment are given by C( x) = 3.2 − 0.2x and the fixed

cost is RM2.40.

11. A manager is starting producing a new utensil with the price-demand equation and

marginal cost function given respectively by D ( x) = 15 − x2 and C( x) = 5x + 3. The

output and the price level are fixed in such a way that will maximize the profit. Find the
consumer’s surplus and the maximum profit.

6

Mathematics AM025
Topic 3: Applications of Definite Integrals – Tutorial

ANSWERS

1. (a) 13 units2 (b) 9units2
3

2. 2ln 3 units2

3. 8 unit2
3

4. 37 unit2
12

5. 189 : 64

6. RM625000
7. CS = RM206.67
8. PS = RM9900
9.(a) CS = RM3380 , PS = RM1690 (b) CS = RM13.33 , PS = RM9.33

10. C(x) = 3.2x − 0.1x2 + 2.4 , C ( x) = 3.2 − 0.1x + 2.4

x
11. CS = RM1.58 , maximum profit = RM9.19

7

Mathematics AM025
Topic 4: Linear Programming – Tutorial

TOPIC 4 : LINEAR PROGRAMMING

1. Write inequalities for the statements below :
(a) y is greater than x.
(b) x is not less than y.
(c) x is greater than 3.
(d) the value of y is greater than 100.
(e) the minimum value of x plus y is 10.
(f) the maximum value of y is 8.
(g) 2x exceeds y by at least 5.
(h) the value of x is greater than zero and less than 6.

2. State the inequalities based on the shaded regions below:
(b) y
(a) y

x y=4
x
x=5 (d)
(c) y y y−x=3

x 3 x
x+ y=0 −3

3. Graph: (b) x  0 (c) x  − 1 (d) x  −2 y
(a) y  3 2 (h) 2x + 3y  18

(e) 2 y + 3x  6 (f) y + 3x + 2  0 (g) − 1 x  y − 3
22

4. Construct feasible regions satisfying the systems below:

(a) y  x + 2 (b) 2x − y  1 (c) 2x + 3y  12 (d) 5y  8x +10

2y  7x x+2y  2 y  2x 2x  5

x  0, y  0 x  0, y  0 5y + 2x  5

2y 5

5. Determine the systems which define the shaded regions:

(a) y (b) y

6 28

4 x + 3yy = 84

12 R

R x y = 10
24 8 x

6

Mathematics AM025
Topic 4: Linear Programming – Tutorial

y y= 1x+2 y y=x
(c) 2 (d) 6

9

4 1R

2 34 x x
R 5
−2
−2 (f) y

y
(e)

9

4

2 R 8 x
R −1
−4 3 x
(g) y −2

(h) y

10

1 9
R
x 2R x
−1 1 25 6

−2

6. Solve the linear programming problem below:

(a) Maximize P = 5x + 5y (b) Minimize C = 12x +10y

Subject to 2x + y  10 Subject to y  3x + 5

x+2y 8 x+ y 7

x, y  0 x  0, y  0

(b) Maximize and minimize P = 12x +14y
Subject to −2x + y  6
x + y  15
x − y  −2
x, y  0

7. A small firm builds two types of garden shed. Type A requires 2 hours of machine time
and 5 hours of craftsman time. Type B requires 3 hours of machine time and 5 hours of
craftsman time. Each day there are 30 hours of machine time available and 60 hours of
craftsman time. The profit on each type A shed is RM 60 and on each type B shed is
RM80. Formulate the appropriate linear programming problem and find the number of
Type A and B sheds that can be produced daily in order to maximize the profit.

9

Mathematics AM025
Topic 4: Linear Programming – Tutorial

8. A manufacturer of printed circuits has a stock of 200 resistors, 120 transistors and 150
capacitors and is required to produce two types of circuits. Type A requires 20 resistors,
10 transistors and 10 capacitors. Type B requires 20 resistors, 20 transistors and 30
capacitors. If the profit on type A circuits is RM 5 and that on type B circuits is RM 12,
how many of each circuit should be produced in order to maximize the profit?

9. A farmer has 20 hectares for growing paddy and corn. The farmer has to decide how
much of each to grow. The cost per hectare for padi is RM 30 and for corn is RM 20. The
farmer has budgeted RM 480. Padi requires 1 man-day for hectare and corn requires 2
man-days per hectare. There are 36 man-days available. The profit on padi is RM 100 per
hectare and on corn is RM 120 per hectare. Find the number of hectares of each crop the
farmer should sow to maximize the profits.

10. A chicken farmer can buy a special food mix A at 20 cent per kilogram and a special food
mix B at 40 cent per kilogram. Each kilogram of mix A contains 3000 units of nutrient N
and 1000 units of nutrient M, each kilogram of mix B contains 4000 units of nutrient N
and 4000 units of nutrient M. If the minimum daily requirements for the chickens
collectively are 36 000 units of nutrient N and 20 000 units of nutrient M, how many
kilogram of each food mix should be used each day to minimize daily food costs while
meeting ( or exceeding ) the minimum daily nutrient requirements? What is the minimum
daily cost?

ANSWERS

1. (a) y  x (b) x  y (c) x  3 (d) y  100
(e) x + y  100 (f) y  8 (g) 2x − y  5 (h) 0  x  6
(b) y  4 (c) x + y  0 (d) y − x  3
2. (a) x  5
(b) y (c)
3. (a) y y

y=3 x x
3

x

(d) y (e) y x=−1
2
3
x (f) y

−2 x
x3
2 −2
2y + 3x = 6

x = −2y y + 3x + 2 = 0

10

y (h) Mathematics AM025
(g) 3 Topic 4: Linear Programming – Tutorial

2 y
x
6
3 2x + 3y = 18
x
−1x= y−3 9
22

4. (a) y 2y = 7x (c) y
y = 2x
2 x
4
−2 x
y = x+2
6 2x + 3y = 12

y 2x − y =1 y y=5
(b) 2
(d)
x+2y = 2 x 2
1
12 5y = 8x +10 x=5 x
−1 2 1 2 5y + 2x = 5

−5
4

5. (a) x + y  4 (b) x + 3y  84 (c) y  1 x + 2 (d) y  1 x +1
y  6 − 3x y  −2x +12 2 2
yx
x+ y  4

y0 x  0, y  10 y + 3x  9 6x + 5y  30

x  0, y  0 y 1

(e) 2y  x + 4 (f) x + 2 y  8 (g) −x + y  1 (h) x + y  2

3x + y  9 4y  x−8 2x + y  −2 2x + y  10

x  0, y  0 2x + y  −2 −2x + y  −2 3x + 2y  18

x  0, y  0

6. (a)

(0,4) (4,2) (5,0) Thus, maximum value is 30 at x = 4 , y = 2

P = 5x + 5y 20 30 25

(b)

(0,5) (0,7)  1 , 13  Thus, minimum value is 50 at x = 0 ,
 2 2  y=5

C = 12x +10y 50 70 71

(c) (0,2) (0,6)  13 17  (3,12)
 2 2 
,

P = 12x + 14y 28 84 197 204

Thus, minimum value is 28 at x = 0 , y = 2 and maximum value is 204 at x = 3 , y = 12

7. Objective function, Z = 60x + 80 y

11

Mathematics AM025
Topic 4: Linear Programming – Tutorial

The system of inequalities involved:
2x + 3y  30

5x + 5y  60

x  0, y  0

Vertices Z = 60x + 80 y Thus, 6 type A and 6 type B sheds
should be produced in order to
(0,10) RM 800 maximize the profit that is RM840.

(6,6) RM 840

(12,0) RM 720

8. Objective function, Z = 5x +12y

System of inequalities :
20x + 20y  200

10x + 20y  120

10x + 30y  150

x  0, y  0

Vertex Z = 5x +12y

(0,5) RM 60 Thus, maximum profit RM66 is obtained if

(10,0) RM 50 the company produces 6 type A and 3 type B
(8,2) RM 64 circuits.

(6,3) RM 66

9. Objective function profit, Z = 100x +120y

System of inequalities involved:
x + y  20

x + 2 y  36

30x + 20 y  480

x  0, y  0

Vertex Z = 100x +120y

A(0,18) RM 2 160 Hence, the farmer should grow 4 hectares
of padi and 16 hectares of corn to get the
B(4,16) RM 2 320 maximum profit of RM2320.

C(8,12) RM 2 240

D(16,0) RM 1 600

10. Objective function Cost, Z = 20x + 40y

System of inequalities involved:
3000x + 4000y  36000

1000x + 4000 y  20000

x  0, y  0

Vertex Z = 20x + 40y Hence, 8 kg of Food Mix A and 3 kg of Food
(0,9) Mix B should be used to minimize daily food
(8,3) RM 3.60 cost. The minimum daily food cost is RM 2.80.
(20,0) RM 2.80
RM 4.00

12

Mathematics AM025
Topic 5: Mathematics of Finance – Tutorial

TOPIC 5 : MATHEMATICS OF FINANCE

1. Find the simple interest for each of the following information
(a) RM 500 at a simple interest rate 4% per year for 2 years
(b) RM 6000 at a simple interest rate 6.5% per year for 1 year
(c) RM 600 at a simple interest rate 15% per year for 3 months
(d) RM 3000 at a simple interest rate 4% per year for 6 months

2. Mr. Muhamad borrows RM 5000 from Bank X to buy a computer. How much he
should pay if he settle the debt within four months at a simple interest of 11% per
year.

3. Mr. Umar Ubaidah invests RM 5500 in Bank Y that gives simple interest rate at 8%
per year. After 6 months, he withdraws RM 5550 from his account for Hari Raya
Aidilfitri expenses. Find the balance of his account.

4. Miss. Rushidah receives a loan of RM40000 from PTPTN to continue her study in
University Putra Malaysia. She should settle her debt at the end of 10 years with
simple interest rate at 4.5% per year. Find how much she should pay?

5. Find the proceed, the discount and equivalent simple interest rate to simple discount

rate for each of the following

Loan Discount rate Period

(a) RM25000 8% 9 months

(b) RM10000 8.25% 4 months

(c) RM20000 5% 60 days

(d) RM30000 6% 6 months

6. Find the interest rate per year that is compounded every 3 months if the investment of
RM 6000 will be RM 6750.35 within one year.

7. Miss Norshahida invests RM 4800 in a financial institution that pays a compound

interest at a rate of 9% per year compounded every 3 month. Calculate her investment

after 51 years.
2

8. Bank XX offers their customers a simple interest rate of 6% per year for the first 3
years. After that the bank offers their customers a compound interest at a rate of 7.5%
per year compounded every 3 months. If Rashid saves RM 5000 for 6 1 years, find
2
how much he has saved at the end of the period.

9. Mr. Azizul invests in Bank M and after 5 years his investment becomes RM26862.72.
The bank pays interest at an annual rate of 3% compounded every 6 months. Find the
total money invested by him.

10. Mr. Khairul gets an interest RM 1729.34 after investing RM 5000 for 5 years in Bank
A. Determine the compound interest rate if the interest is compounded every 4 month.

13

Mathematics AM025
Topic 5: Mathematics of Finance – Tutorial

11. Find the effective rate equivalent to an annual interest rate of 8% compounded

(a) every 6 months, (c) every 4 months,

(b) quarterly, (d) monthly

12. Miss Faziatul makes a monthly investment of RM 320 with an annual interest rate of
6% compounded monthly. How much had she already invested at the end of 3 years?

13. Miss Zaipah borrows RM 35000 from Bank B that charges an annual interest rate of
8% compounded monthly and she has to settle the debt within 5 years. Find her
monthly payment.

14. At the end of every month, Mr. Zaki saves RM 200 in an account that pays an annual
rate of 10% compounded monthly. After 3 years, he adds RM 60 to his savings per
month. Show that the total amount after 6 years is RM 22129.17.

15. A one-year old car (PROTON WAJA) is sold for RM 59000. Mr. Kong decides to
buy that car by paying 25% down payment and the balance is settled by taking a loan
from Bank C which offers a simple interest rate at 6.25%. Monthly payment will be
settled within 5 years.
(a) Find the total interest that he has to pay.
(b) Find the total amount of monthly payments.
(c) Find the monthly payment if an annual interest rate of 6.25% is charged and
the interest compounded every month.

16. Mr. Fahmi buys a new car for RM 85000 by paying RM 30000 as down payment and

the balance will be settle by taking a loan from one of the following:
❖ BIJAK Financial Institution that offers an annual interest rate of 10%

compounded monthly for 10 years
❖ BAGUS Institution that offers a simple interest rate of 6 % per year within 10

years.

(a) Find the monthly payment for each institution.

(b) Hence, from (a), which institution should Fahmi choose?

17. An investment of RM 1000 will accumulate to RM 1331 in 4 years at an annual
simple interest rate k% for the first 2 years and simple interest rate 2k% for the last
2 years. Find the value of k% .

18. At an effective interest rate of j%, RM 10 will accumulate RM 200 at the end of
(x+2y) years and RM 10 will accumulate RM 500 at the end (2x+y) years. How much
will RM10 accumulate at the end of (x - y) years?

19. Mr. Khairil plans to accumulate RM 100787.00 in N years. He deposits RM 500.00
from first year to the 10th year and make no deposits from the 11th years to the N

years. If the interest rate is 9% compounded annually, calculate the value of N.

20. In order to save RM 100000 for her retirement, Rosnani wants to put a fixed amount
every year into a fund that pays interest at an annual rate of 8% compounded
annually. If she plans to save in this way for 15 years, how much should she save
every year?

14

Mathematics AM025
Topic 5: Mathematics of Finance – Tutorial

21. Mr. Joe buys a house for RM 200000. He makes a 10% down payment and settle the
balance by taking a loan for 20 years with annual interest rate of 4% compounded
monthly. Find his monthly payment. Find the interest.

22. Mr. Zack needs RM 5000 three years from now. How much should he invest in every
six-month for three years, with an interest rate 8% compounded semiannually? Find
the interest.

ANSWERS

1. (a) RM 40.00 (b) RM 390.00 (c) RM 22.50 (d) RM 60.00
2. RM 5183.33
3. RM 170.00
4. RM 58000.00
5. (a) RM 23500.00 ; RM 1500.00 ; 8.51%

(b) RM 9725.00 ; RM 275.00 ; 8.48%
(c) RM 19833.33 ; RM 166.67 ; 5.04%
(d) RM 29100.00 ; RM 900.00 ; 6.19%
6. 11.96%
7. RM 7831.31
8. RM 7652.42
9. RM 23146.73
10. 6 %
11. (a) 8.16% (b) 8.24% (c) 8.22% (d)8.30%
12. RM 12587.55
13. RM 709.67
14. RM 22129.17 (shown)
15. (a) RM 13828.13 (b) RM 967.97 (c)RM 860.63
16. BIJAK RM 726.83 ; BAGUS RM 733.33 ; choose BIJAK
17. 5.16%
18. RM 310.00
19. 40 years
20. RM 3682.95
21. RM 1090.76, RM 81782.40
22. RM 753.81, RM 477.14

Tips for Topic 5
1. We need to make sure that we are capable to distinguish all the formulae in

mathematics of finance especially when applying the compounded interest.
2. While we are trying to solve the problem which related to the future and present

value, we need to understand the problem and try to relate which formula is relevant
with the problem.

15

Mathematics AM025
Topic 6 : Data Description – Tutorial

TUTORIAL 6 : DATA DESCRIPTION

1.

Class Lower Upper Lower Upper Class

limit limit boundary boundary mark
1–4
5–8
9 – 12

13 - 16

Complete the above table. Determine the class width used.

2. Determine which of the following are population or sample.
(a) The age of each member in a family.
(b) The income of 500 residents in a town of 1000 residents.
(c) The weight of 100 packages send overseas.
(d) The marks obtained by students in a Mathematics class.

3. Determine whether the following are quantitative or qualitative data.
(a) The number of persons in a family.
(b) The color of 100 cars.
(c) The standard of living of 100 families in a town.
(d) The distance covered by marathon runners.
(e) The telephone bills to be paid.

4. Which of the following are grouped data and ungrouped data?
(a) the following data represent the speed of vehicles along a road
40 50 45 42 60 55 54 53 56

(b) The following data represent the number of children in 30 families.
Number of children 0 1 2 3 4 5
Number of families 3 5 7 6 5 4

(c) The following is the number of minutes taken to go from home to school

for a group of students.

Number of minutes taken 11-15 16-20 21-25 26-30

Number of students 7 12 15 6

5.

Class Frequency Cumulative frequency

10

29.5 – 32.5 30
32.5 – 35.5 13
47

49 149

28

5

18 200

Complete the above table by filling in the empty spaces.

16

Mathematics AM025
Topic 6 : Data Description – Tutorial

6. If the mean for the data set 9, 11, 14, 15, 20, x, 21, y is 18.
Find the value for x + y.

7. A set of numbers 13, 2 ,14, p,11, 3, q, 12, 6 has mode of 6 and median of 7

(a) Find the possible values of p and q
(b) Find the mean for the above set of numbers

8. (a) The table given below shows a number in each of 20 families

Number of children 0 1 2 3 4 5

(x)

Number of family 2 7 5 4 1 1

(f)

(i) Find the mean and median of the number of children
(ii) Find the variance.
(b)

Class intervals Frequency ( f )

1-5 5

5-9 3

9-13 6

13-17 9

17-21 2

21-25 4

For each of the above data, find the

(i) mean

(ii) median

9. The frequency distribution table below shows the distance cycled by Shaq in a
week for a duration of 50 weeks.

Distance Frequency ( f )
1-5 4
6-10 10
11-15 5
16-20 12
21-25 13
26-30 6

Calculate the mean and standard deviation.

17

Mathematics AM025
Topic 6 : Data Description – Tutorial

10. In a survey the amount of time devoted by 80 students to leisure activities in a
week is shown in the following frequency table

Time (hour) Frequency ( f )
1-5 6
6-10 19
11-15 24
16-20 14
21-25 15
26-30 2

Find the mean, median and mode. Hence, state the skewness of the distribution.

11. Given the information for two data as below:

Data Mean Standard deviation
Data I 230 102
Data II 230 56

Determine the coefficient of variation for the above data and interpret the
values.

12. Describe the symmetriness and skewness of the following data.
(a) A data with a mean, median and mode of 6, 5.8 and 2 respectively.
(b) A data with a mean, median and mode of 7, 8.8 and 14 respectively.

13. The waiting time in minutes, for 100 passengers at a bus station is given in the
frequency table below.

Class Frequency ( f )
1-10 5
11-20 39
21-30 23
31-40 18
41-50 6
51-60 7
61-70 2

Given that  fx = 2650 and  fx2 = 89725 . Calculate

(a) the mode and the median
(b) the coefficient of variation.

14. (a) The following of data represents the number of goals scored by a
football team in 9 matches.

130251431

Find the median and the range.

18

Mathematics AM025
Topic 6 : Data Description – Tutorial

(b) is 371 . 12 2 =450 ,
The sample variance of a set of data 11 If i =1 x i determine

the sample mean.

15. The table below shows the cumulative distribution for the duration of calls (in
minutes) of 60 callers to a counselor on a particular week.

Duration of calls (in minutes) Number of callers
<5 0
<15 3
<25 10
<35 25
<45 37
<55 49
<65 60

Calculate the mode, median and range for the above data.

16. Types of equipment X Y

Mean 1525 days 19 days

Standard deviation 250 days 3 days

The table above shows the mean life spans and their standard deviation of 2
types of equipments, X and Y. Find the coefficient of variation of each type of
equipment.

ANSWERS

2. (a) population (b) sample
(c) sample (d) population
qualitative
3. (a) quantitative (b) quantitative
(c) qualitative (d)
(e) quantitative Ungrouped

4. (a) Ungrouped (b) 8.22
(c) Grouped

6. 24

7. (a) p = 6, q = 7 or q = 6, p = 7 (b)

19

Mathematics AM025
Topic 6 : Data Description – Tutorial

8. (a) (i) 1.9 , 2 (ii) 1.67
(b) (i) 12.66 (ii) 13.22

9. 16.8 , 7.66

10. 14.19 , 13.63 , 12.17, positively skewed

11. CV Data I = 44.35% , CV Data II = 24.34%

12. (a) positively skewed (b) negatively skewed

13. (a) 17.3 , 23.11 (b) 52.94%

14. (a) 2 , 5 (b) 2.57

15. 32.27 , 39.17 , 65

16. 16.39% , 15.79%

20

Mathematics AM025
Topic 7: Probability – Tutorial

TOPIC 7 : PROBABILITY

1. A unbiased die and a fair coin are tossed together. List down all the possible outcomes in
the sample space, S. Find the probability of obtaining:
(a) a number 3 and a tail,
(b) a head and a number greater than 2,
(c) a tail and an odd number

2. One marble is drawn at random from a bag containing four white marbles, five red
marbles and six green marbles. Find the probability that the marble is
(a) red marble
(b) purple marble
(c) not a white marble
(d) white or green marble

3. In a class, there are 35 of them. Everybody must participate in either drawing or singing
competition that will be held on the same day. The monitor has decided that 24 of them
will participate in drawing competition and 20 of them will participate in singing
competition. How many students will represent in both competitions?

4. A bag contains 25 packets of nuts, 9 packets of groundnut, 15 packets of hazelnut and a
packet of almond. What is the probability of Chris choose randomly a packet groundnut
or almond?
Tips: It is mutually exclusive events because a packet of nut cannot be groundnut and
almond on the same time.

5. In a class of 24 male students, 9 are in both team A and team B. There are 16 students
involved in team A and 10 involved in team B. Find the probability that a student chosen
at random
(a) is only in team A
(b) is in either team A or team B
(c) is in neither team A nor team B

6. A fair blue dice and a fair red dice are rolled and the scores on the dice are added. Find
the probability that
(a) the sum of the scores is 7 or 10
(b) the scores on both dice are equal
(c) the scores are unequal

7. An automobile manufacturer is concerned about a possible recall of their best selling
four-door car. From past records, recalls were related to defects detected with
probabilities of 0.14, 0.30, 0.20, 0.36 respectively to the brake system, transmission, fuel
system, and some other areas.
(a) What is the probability of defect in the brake or fueling system if the probability of
defects in both systems simultaneously is 0.18?
(b) What is the probability that there are no defects in either brakes or the fuel system?
(c) What is the probability of defects in both transmission and other areas if the
probability of defects in transmission or other areas is 0.35?

21

Mathematics AM025
Topic 7: Probability – Tutorial

8. A unbiased cubical dice is labeled so that it has two 2, one 5 and three 6. The dice is
tossed once and the events A and B are defined as follows:
A: the number showing is 2
B: the number showing is even

(a) Find P ( A), P ( A  B) and P ( A' B)

(b) Write down two pairs of non-mutually exclusive events.

9. Given that P(E) = 0.3, P(E  F) = 0.28 and P(E ' F) = 0.32 . Find
(a) P(E  F ')
(b) P(E | F )

10. Given that A and B are two events such that P(A) = 8 , P(A B) = 1 and P(A | B) = 4 .
15 3 7

Find

(a) P ( B), P ( B | A) and P ( B | A')

(b) State the reason whether A and B are
(i) independent
(ii) mutually exclusive

11. There are 60 students in the sixth form of a certain school. Mathematics is studied by 27

of them, Physics by 20 and 22 students study neither Mathematics nor Physics.

(a) Find the probability that a randomly selected student studies both Mathematics and

Physics.

(b) Find the probability that a randomly selected Mathematics student does not study

Physics.

A student is selected at random.
(c) Determine whether the event ‘studying Mathematics’ is statistically independent of

the event ‘not studying Physics’.

12. The table below shows the information of a sample of a 100 people in a study of genetics
in Malacca Matriculation College.

Color Blind (C) Male (M) Female (F) Total
Not Color Blind(C’) 4 1 5
40 55 95
Total 44 56 100

Find the probability that a person
(a) is color blind, given that the person is a male.
(b) is not color blind, given that the person is a female.

22

Mathematics AM025
Topic 7: Probability – Tutorial

13. In a lecture class of 100 students, 12 are left-handed. If two students are chosen at
random, what is the probability that:

(a) both are left-handed
(b) neither are left-handed
Extend your probability tree diagram to show the possible outcomes if three students are
chosen. What is the probability that the second student and the third student are chosen is
left-handed and right-handed respectively?

14. Given that A and B are two events such that P(A) = 0.1 and P(B) = 0.8 . Find
(a) P(A  B) if A and B are independent events
(b) P(A | B) if A and B are not independent events and P(A  B) = 0.7

15. A bag contains 3 blue marbles and 2 yellow marbles. 3 marbles are randomly drawn from
the bag one after the other without replacement. Find the probability of drawing
(a) the second marble is blue
(b) the third marble is blue
(c) the second marble is yellow given that the first marble is blue

ANSWERS

1. (a) 1 (b) 1 (c) 1 2.(a) 1 (b) 0 (c) 11 (d) 2
12 3 4 3 15 3

3. 9 4. 2 5. (a) 7 (b) 17 (c) 7
5 24 24 24

6. (a) 1 (b) 1 (c) 5 7. (a) 0.16 (b) 0.84 (c) 0.31
4 6 6

8. (a) 1 , 1 , 1 (b) A and B, A’ and B 9. (a) 0.02 (b) 7
332 15

10. (a) 7 , 5 , 15 (b) (i) A and B are not independent
12 8 28

(ii) A and B are not mutually exclusive

11. (a) 3 (b) 2 (c) the two events are independent 12. (a) 1 (b) 55
20 3 11 56

13. (a) 1 (b) 58 , 8 14. (a) 0.82 (b) 0.25 15. (a) 3 (b) 3 (c) 1
75 75 75 5 52

23

Mathematics AM025
Topic 8: Random Variables – Tutorial

TOPIC 8 : RANDOM VARIABLES

1. State which of the following are discrete or continuous random variables.
(a) The number of copies of the New Straits Times sold each month.
(b) The amount of ink used in printing.
(c) The actual number of ounces of laundry detergent in a one-gallon bottle.
(d) The number of patients what are admitted per week in a particular hospital.

2. The probability distribution function of a discrete random variable X is given by

0.1 , x = 1,3,5, 7

P(X = x) = 0.2 , x = 2, 4, 6

 0 otherwise.

Prove that X is a discrete random variable with the probability distribution function of P.
3. The discrete random variable X has probability function

 kx , x = 2,3
 ,
P( X = x ) =  x2 +1 x = 4,5
2kx otherwise
0x, 2 −1


20
(a) Show that the value of k is 33 .

(b) Find the probability that X is less than 3 or greater than 4.

4. The discrete random variable X has probability distribution function

P( X = x) =  x, x = 1,3,5,7
 m , x = 2,4,6
 x2

m

(a) Find the value of m

(b) Determine P ( X − 4  1)

(c) Find P ( X  3.5)

5. T =  0, 1, 2, 3, 4 is a discrete random variable with the probability distribution

function as below : a , t = 0, 1
 , t=2
P( T =t ) =  a+b , t = 3, 4

 b

Given that P ( T  2 ) = 5 , find the values of a and b .

9

24

Mathematics AM025
Topic 8: Random Variables – Tutorial

6. The discrete random variable X has cumulative distribution function F(x) defined by

F ( x) = 2 + x , x = 1, 2,3, 4 and 5

7

(a) Find P ( X  3) .

(b) Show that P ( X = 4) = 1 .

7
(c) Find the probability distribution for X.

7. The cumulative distribution function of a discrete random variable X is

given as

0 , x0
1 , 0 x1

 10 , 1 x 2
 3
2 x3
F (x) =  10
 2, 3 x4
 x4
5
9
110 ,
,

(a) Find the probability distribution of X.

(b) Calculate P (2  X  3) and P (1  X  4) .

8. A discrete random variable X has the following probability distribution :

x1 2 3 4
1
P(X = x) 1 1 11
6
4 8 24

(a) Calculate P ( X  3) .

(b) Find the cumulative distribution function , F ( x) .

9. The discrete random variable X has the probability distribution shown in the table

below:

x2 4 6

P ( X = x) 0.3 0.6 0.1

Calculate: E ( X ), E (3X + 2), Var ( X − 8), Var (5X + 3) .

25

Mathematics AM025
Topic 8: Random Variables – Tutorial

10. A discrete random variable X can take any one of five possible values and has the

following probability distribution. Find

x02 5 8 n

P ( X = x) 0.1 0.25 0.2 2p p

(a) the value of p

(b) the value of n , given that E ( X ) = 5.7

( )(c) E X 2 and the variance of X.

11. Given that X is a discrete random variable and k is a constant. If E (3X + k ) = 26 and
E (2k − X ) = 3, find E ( X ) .

12. A continuous random variable X has probability density function f(x) where

f (x) = 3 x2 , for 1  x  3
26
0 , otherwise

Find: (b) P ( X  2.5) (c) P (1.6  X  2)

(a) P ( X  2.5)

13. The continuous random variable X has probability density function f (x) where

 1, 0 x 1
 4

f (x) =  3 (3 − x), 1  x  3

 8

0, otherwise



(a) Find P (0.5  X  1.5) . (b) Find P ( X  2) .

14. The continuous random variable W has probability density function g , given by

g ( w) =  aw + b , 0w4
 bw + a , 4w5


Find the values of a and b if it is given that P ( 2  W  4 ) = 23 . Hence, find

51

(a) P ( W  3 ) ,

(b) value of u if P ( W  u ) = 39 .

68

15. A continuous random variable X has cumulative distribution function

 0 , x  0,
 , 0  x  4,
F ( x) =  1 x2 + 3 x
40 20 , x  4.

 1

Find the probability density function.

26

Mathematics AM025
Topic 8: Random Variables – Tutorial

16. The continuous random variable X has probability density function given by

f (x) =  3 (1+ x2 ) , −1 x 1
8 ,
 0
, otherwise

(a) Find E ( X ) .

(b) Show that 2 = 0.4 .

(c) Find P (−  X  )

17. The continuous random variable X has probability density function given by

f ( x ) = 0.5 ( 2 − x) , 0 x2
 , otherwise
 0

(a) Find the cumulative distribution function, F ( x) of X.

(b) Determine P( 1 < X < 2 ).

18. The continuous random variable X has a probability density function given by

3 x 2 , 0 x2
8
f (x) =

 0, otherwise

(a) Find the cumulative distribution function of X and sketch its graph.
(b) Find P(X  1)

19. The continuous random variable X has probability density function

1 + x , 1 x 3
 6
f (x) =

0, otherwise.

(a) Calculate the mean of X.

(b) Find the cumulative distribution function, F ( x) .

20. A continuous random variable X has probability density function f (x) given by:

f (x) = 2(1− x) , 0  x  1,
 , otherwise.
 0

Find :

(a) E ( X ) (b) Var ( X ) (c) P ( X  E(X )).

21. The probability density function of X is given by :

f (x) = 2x , 0  x  1,
 , otherwise.
 0

Find E ( X ) and Var ( X ) . Hence, find

(a) E (2X + 5) (b) E ( X + 4) (c) Var ( X + 3) (d) Var (3X − 2)

27

Mathematics AM025
Topic 8: Random Variables – Tutorial

22. Given X and Y are two independent random variables such that E ( X ) = 20 ,

E ( X 2 ) = 425 , E (Y ) = 25 and E (Y 2 ) = 641 . Find

(a) E (2X + 3) (b) E (2X + 3Y ) (c) Var (3Y + 6) (d) Var (3X − 2Y )

ANSWERS (b) Continuous (c) Continuous (d) Discrete
5. a = 1 , b = 2
1. (a) Discrete 4. (a) 72 1 8
49 (b) 3 (c) 9 99

3. (b) 99 (c) X 123 45
5 11
311 77
6. (a) 7

P(X=x) 777

X 0 1 23 4 (b) 1 , 7
2 10
7. (a) P(X=x) 1 2 15 1

10 10 10 10 10

0 ,x 1
 ,1  x  2
 1
,2  x  3
4
 3 ,3  x  4
5 (b) F (x) =  ,x 4 9. 3.6, 12.8, 1.44, 36
8. (a) 6  8

5

 6

1

10. (a) 0.15 (b) 12 (c) 46.8, 14.31 11. 7

12. (a) 0.5625 (b) 0.4375 (c) 0.150 29 3
13. (a) 64 (b) 16

14. a = 1 ,b = 1 29 (b) u = 3 15. f (x) =  1 (x + 3) , 0 x4
18 17 (a) 68  20
 0 , otherwise

0 , x0
 x2 ,0  x  2
16. (a) 0 (c) 0.54 17. (a) F ( x ) =  x − 1 (b) 1
4
 4 , x2
 1
y

 0, x  0 1
 0
18. (a) F (x) =  x3 , 0 x2 1
 8 (b) 8
x
 1, x  2

12 3
28

Mathematics AM025
Topic 8: Random Variables – Tutorial

0 , x1

19. (a) 21 (b) F (x)=  1 ( x + 3) ( x −1) , 1 x3 (c) 2.162
9  12

 1 , x3

1 1 4
20. (a) 3 (b) 18 (c) 9

21. E ( X ) = 2 ; Var ( X ) = 1 (a) 19 (b) 14 (c) 1 (d) 1
3 3 18 2
3 18

22. (a) 43 (b) 115 (c) 144 (d) 289

29

Mathematics AM025
Topic 9: Special Probability Distribution – Tutorial

TOPIC 9 : SPECIAL PROBABILITY DISTRIBUTION

1. If W ~ B(20, 0.25) , determine the following probability by using binomial formula.

(a) P(W  3) (e) P(4  W  9)

(b) P(W = 5) (f) P(5  W  10)

(c) P(W  4) (g) P(5  W  11)

(d) P(W  6) (h) P(7  W  10)

2. If X ~ B(15, 0.75) find (c) P(X  7)
(a) P(X  6) (d) P(X  5)
(b) P(X = 8)

3. Four unbiased dice are thrown. Find the probability that there are:
(a) Exactly two sixes
(b) At least two sixes

4. Six coins are tossed. Find the probability that there are:
(a) Not more than four heads
(b) At least two heads

5. Find the mean and variance of the binomial distribution B(8, 0.25)

6. It is claimed that 95% of first class mail with the same city is delivered within two days of
the time of mailing. Six letters are randomly sent to different locations.
(a) What is the probability that all six will arrive within two days
(b) What is the probability that exactly five will arrive within two days
(c) Find the mean number of the letters that will arrive within two days
(d) Find the variance number and standard deviation of the letters that will arrive within
two days.

7. At the local swimming club, the expected number of people that can swim a mile is 4.5
and the variance is 3.15. Find the probability that at least three people can swim a mile.

8. In a local youth group , the expected number of people who can play a musical instrument
is 4 and the variance is 3.2 . Find the probability that:
(a) Five people can play a musical instrument
(b) Less then six people can play a musical instrument.

9. If X ~ P0 (5.5) , find :

(a) P(X  7) (b) P(X  0) (c) P(X  11) (d) P(X  3)

10. Find the values of a, b, c and d given that X ~ P0 (2.5) :

(a) P(X  a) = 0.8912 (c) P(X  c) = 0.9997

(b) P(X  b) = 0.5438 (d) P(X  d) = 0.0420

30

Mathematics AM025
Topic 9: Special Probability Distribution – Tutorial

11. The number of letters arriving at a particular address each day can be modeled by a
Poisson Distribution with a mean of seven letters per day, find the probability that
tomorrow:
(a) no letters will arrive
(b) four letters will arrive
(c) at least two letters will arrive
(d) fewer than five letters will arrive

12. A television repair company uses a particular spare part at a rate of 4 per week. Assuming
that requests for this spare part occur at random, find the probability that
(a) exactly 6 are used in a particular week
(b) at least 10 are used in a two week period
(c) exactly 6 are used in each of 3 consecutive weeks.

13. The mass, M grams, of a batch of commerative coins is such that M ~ N(50,9) . Each
coin is weighed individually before packaging and will be rejected if its mass is less than
47g. What percentage of coins would you expect to be rejected?

14. Find the following probabilities
(a) If X ~ N (5,1) , find P(X  6.5)
(b) If X ~ N(65,81) , find P(53  X  72)
(c) If X ~ N (25, 25) , find P(19  X  26.5)

15. Using, Z ~ N(0,1) find k such that :
(a) P( Z  k) = 0.85

(b) P( Z  k) = 0.1

16. The scores , X , in a Mathematics Competition are such that X ~ N(80, 20) . Find :
(a) The probability that Ali, a student chosen at random, scores between 75 and 90 on the
test.
(b) The probability that Erica, a student chosen at random, scores less than 72.
(c) The probability that Intan, a student chosen at random, scores more than 85, given
that she scored less than 90.
(d) The score needed to obtain a high Distinction Certificate, if those are awarded to the
top 8 % of the students.

17. If X ~ N (100, 2 ) and P(X  106) = 0.8849 . Find the variance  2 .

18. The numbers of days taken for a sick leave by a factory’s workers is normally distributed
with a mean of 12 days and variance  2 in a year. If the probability of less than 10 days
off is 0.1587, find the variance  2 . Finally, find a if. P( X −12  a) = 0.950

19. If X ~ B(200, 0.7) , use the normal approximation to find

(a) P(X  130) (d) P(X  152)

(b) P(136  X  148) (e) P(141  X  146)

(c) P(X  142)

31

Mathematics AM025
Topic 9: Special Probability Distribution – Tutorial

20. 10% of the chocolates produced in a factory are mis-shapes. In a sample of 1000
chocolates find the probability that the number of mis-shapes is
(a) less than 80
(b) between 90 and 115 inclusive
(c) 120 or more.

21. It is estimated that 1/5 of the population of Malaysia watched last year’s World Cup Final
on television. If random samples of 100 people are interviewed, calculate the mean and
variance of the number of people from these samples who watched the World Cup Final
on television.
Use normal distribution tables to estimate, to 2 significant figures, the approximate
probability of finding, in a random sample of 100 people, more than 30 people who
watched the World Cup Final.

22. A lorry load of potatoes has, on average, one rotten potato in 6. A greengrocer tests a
random sample of 100 potatoes and decides to turn away the lorry if she finds more than
18 rotten in the sample. Find the probability that she accepts the consignment.

ANSWERS

1. (a) 0.2252 (b) 0.2023 (c) 0.7748 (d) 0.6173

(e) 0.5713 (f) 0.5812 (g) 0.3789 (h) 0.2004

2. (a) 0.0042 (b) 0.0393 (c) 0.0042 (d) 0.999

3. (a) 0.1157 (b) 0.1319

4. (a) 0.8906 (b) 0.8906

5. mean = 2 variance = 1.5

6. (a) 0.7351 (b) 0.2321 (c) 5.7 (d) 0.285, 0.534

7. 0.8732

8. (a) 0.1746 (b) 0.8042

9. (a) 0.8095 (b) 1 (c) 0.9890 (d) 0.2017

10. (a) a = 4 (b) b = 3 (c) c = 9 (d) d = 5

11. (a) 0.0009 (b) 0.0912 (c) 0.9927 (d) 0.1730

12. (a) 0.1042 (b) 0.2834 (c) 0.0011

13. 15.87%

14. (a) 0.9332 (b) 0.6905 (c) 0.5028

15. (a) k = 1.44 (b) k = 1.64

16. (a) 0.856 (b) 0.0367 (c) 0.1203 (d) 86.31

17.  2 = 25 18.  2 = 4 , a = 3.92

19. (a) 0.9474 (b) 0.6598 (c) 0.5910 (d) 0.0268 (e) 0.2113

20. (a) 0.0154 (b) 0.8149 (c) 0.0197

21. mean = 20, variance = 16, 0.0043

22. 0.6879

32

Mathematics AM025
Topic 10: Correlation and Regression – Tutorial

TOPIC 10 : CORRELATION AND REGRESSION

1. Based on the diagram, discuss the value of linear Pearson’s Coefficient of correlation for
each of the diagram.

(a) y (b) y

xx x
x x
xx
x
x

xx

2. The diameter of the longest lichens growing on gravestone were measured as below :

Age of 9 18 20 31 44 52 53 61 63 63 64 64 114 141
gravestone 2 3 4 20 22 41 35 22 28 32 35 41 51 52
(x years)
Diameter
of lichen

(y mm)

(a) Calculate the values of x and y .
(b) Calculate the linear Pearson’s coefficient of correlation.
(c) Calculate the coefficient of determination and interpret.

3. Calculate linear Pearson’s correlation coefficient for the following :

x6 8 10 12 14 16 18 20 22

y 4.1 8.6 3.2 10.4 12 8.7 9.4 13.1 16

4. The systolic blood pressure of 10 men of various ages are given in the following table.

Age, x (year) Systolic Blood pressure, y (mm mercury)

37 110

35 117

41 125

43 130

42 138

50 146

49 148

54 150

60 154

65 160

(a) Find the equation of the regression line of systolic blood pressure on age.

(b) Use your regression line to predict the systolic blood pressure for a man who is :

(i) 20 years old (ii) 45 years old

(c) Comment on the accuracy of your predictions in (i) and (ii).

33

Mathematics AM025
Topic 10: Correlation and Regression – Tutorial

5. The table below gives the height of a bean shoot in cm (y) and the number of days since it
was planted (x).

Number of days, x Height, y (cm)
40 9.6
45 10.5
50 11.2
55 12.3
60 13.4
65 14.3
70 15.2

(a) Calculate the line of regression of y on x.
(b) Estimate the height of the shoot exactly 8 weeks (56 days) after planting.
(c) Why would it not be sensible to use the regression equation to estimate the height of

the shoot 3 month after planting?

6. The given data relate to the price and engine capacity of new cars in January 2000.

Car model Price (RM), y Capacity (cc), x
A 3900 1000
B 4200 1270
C 5160 1750
D 6980 2230
E 6930 1990
F 2190 600
G 2190 650
H 4160 1500
J 3050 1450
K 6150 1650

Calculate the line of regression of y on x.

7. Obtain the regression equation for y on x in the form y = a + bx where n = 8 if :
x = 38.45 , x2 = 211.513 , y = 21.1 , y2 = 63.19 , xy = 111.775

Use your regression line to estimate the value of y when x = 3 .

8. Given the following data

X : Mathematics marks 59 61 63 65 67 69

Y : Statistics marks 64 66 67 67 68 69

Find the linear regression equation y = a + bx from the above data.

34

Mathematics AM025
Topic 10: Correlation and Regression – Tutorial

9. The Management of a production company intends to analyze the relation between the
cost for the research and development and its profit. The information regarding to the data
last six years are shown in the following table.

Years The cost for research & development Proft (million of ringgit)
(million of ringgit)
1994 5 31
1995 11 40
1996 4 30
1997 5 34
1998 3 25
1999 2 20

[Remark: Show the detail solution]

(a) Find the linear regression equation y = a + bx from the data.

(b) Use the equation from (b) to estimate the profit if the cost for research and
development is RM 8 million.

(c) How much is the changing percentage in the company profit could be related to the
linear regression equation?

10. The income and expenses annually (in thousand of ringgit) for seven family given in
the table below.
Income (x) Expenses (y)
15 5
21 7
25 9
28 8
35 9
39 11
49 15

(a) Find the linear regression to explain the data by using the least square method.
(b) Estimate the expenses if the annually income is RM 30,000.
(c) Calculate the linear Pearson’s, r, and explain your answer.

11. A car manufacturer company wished to carry out research on the depreciation of one of
its model. Eight cars are taken for samples. The information of ages and prices are shown
in the table below.
Age 8 3 6 9 2 5 6 3
Price 16 74 38 19 102 36 33 69

(a) Fine the linear regression equation with age as independent variables and price as
dependent variable.

(b) Estimate the price of this car model which is 7 years.

35

Mathematics AM025
Topic 10: Correlation and Regression – Tutorial

ANSWERS

1. (a) Positive correlation between variable x and y
(b) Negative correlation between variable x and y

2. (a) x = 56.9 , y = 27.7

(b) r = 0.8804
Strong positive correlation between the age of gravestone and diameter of lichen.

(c) Coefficient of determination, r2 = 0.7551. This means that 77.51% of the variation in
the diameter of lichens is explained by the variation in the age of gravestone.

3. r = 0.805

4. (b) y = 62.59 +1.58x (c) (i) y = 94.19 (ii) y = 133.69

(d) (i) Extrapolation – not accurate, linear model may not continue.

(ii) Interpolation – likely to be reasonably accurate.

5. (a) y = 1.91+ 0.19x (b) y = 12.55 cm

(c) At 3 months, extrapolation would be used and therefore results may be inaccurate as

linear model may not continue.

6. y = 235.82 + 3.02x

7. y = 0.773 + 0.388x , y = 1.937

8. y = 38.488 + 0.4429x

9. (a) y = 20 + 2x
(b) RM 36 million
(c) For every 1 million unit increase in research and development cost (x), the profit (y)
will increase 2 million which means the percentage of change in profit is 200%.

10. (a) y = 1.1414 + 0.2642x
(b) RM9,067.40
(c) r = 0.9587
The expenses (y) have strong positive relation with the income (x).

13. (a) y = 107.9657 −11.3506x
(b) RM 28,511.50

36

Mathematics AM025
Topic 11: Index Number – Tutorial

TOPIC 11 : INDEX NUMBER

1. Price per item (RM) Quantity consumed per
week (kg)
Item
1992 1998 1992 1998
A 1.10 1.20
B 0.99 1.29 3.8 3.7
C 0.80 1.20
15.0 18.0

1.0 1.2

The above table shows the prices per item and quantities consumed per week for three
items used in a factory. Using 1992 as the base year, find the following indices:-

(a) the simple relative price index for each item.
(b) the simple relative quantity index for each item.
(c) the simple aggregate price index
(d) the simple relative average price index
(e) the simple aggregate quantity index
(f) the simple relative average quantity index
(g) the Laspeyres price per index
(h) the Laspeyres quantity index
(i) the Paasche price index
(j) the Paasche quantity index

2. Selling price for terrace house at Shah Alam from year 1980 to 1983 are as follows :

Year 1980 1981 1982 1983
Price 50,000 55,000 60,000 70,000

By taking year 1980 as the base year, calculate the simple relative price index for the
terrace house from year 1980 to 1983

3. Table below shows the total of computers sold by a company from year 1990 to 1993.

Year 1990 1991 1992 1993
Selling 200 450 880 1400

Calculate the relative quantity index for computers sold by that company in year 1993 by
taking year 1990 as the base years.

(a) Calculate the relative quantity index for computers sold by that company in year 1991
by taking year 1993 as the base years.

37

Mathematics AM025
Topic 11: Index Number – Tutorial

4. Price list for 3 types of food A, B and C are as follows :

Food Types 1990 (RM/kg) 1993 (RM/kg)
A 5.00 7.00
B 3.00 1.50
C 7.50 7.50

Calculate the simple aggregate price index for those foods for year 1993 (take year 1990
as the base year). Give a brief explanation about your answer.

5. Calculate the simple aggregate quantity index in year 1995 for 3 types of printer X, Y,
and Z by taking year 1990 as reference.

Printer Types Quantity, 1990 Quantity, 1995
X 200 800
Y 450 860
Z 750 1200

6. With reference to question 4 in this tutorial, calculate the average relative price index for
those foods in year 1993 by taking year 1990 as the base year. Give a brief explanation
about your answer.

7. Calculate the average relative quantity index in year 1995 for 3 types of printer X, Y, and
Z by taking year 1990 as reference. Give a brief explanation about your answer.

Printer Types Quantity, 1990 Quantity, 1995
X 200 800
Y 450 860
Z 750 1200

8. The table below shows the mean price per unit and the number of printers sold by MAJU

company for the year 2003 and 2004.

Model Mean Price ( RM / unit ) Sales ( ‘000)

2003 2004 2003 2004

A 395 413 20 35

B 615 597 45 65

C 348 389 40 70

Calculate : (b) The Paasche quantity index
(a) The Laspeyres price index

ANSWERS

1. (a) 109.09, 130.30, 150 (b) 97.37, 120, 120 (c) 127.68 (d) 129.8 (e) 115.66
(j) 115.89
(f) 112.46 (g) 126.63 (h) 115.23 (i) 127.35
5. I1995 = 204.29
2. 1980 = 100 ; 1981 = 110 ; 1982 = 120 ; 1983 = 140 (b) 158.8

3. (a) 700 (b) 32.14 4. I1993 = 103.23

6. I = 96.67 7. I = 250.33 8.(a) 102.4

38

MATHEMATICS 2 (AM025) – PSPM Questions

T 1 P DOPIC : ARTIAL IFFERENTIATION

PSPM QUESTIONS

1. Find x and y that minimize the function f ( x, y) = 16x2 + 8y2 subject to the constraint

x + y = 12 by using the method of Lagrange multipliers. [PSPM 05/06]

2. The yearly profit function for an agricultural company is given as

P ( x, y) = 800x +1200 y − x2 − 2 y2 − 2xy

where P represents profit in RM, x represents the number of acres for pineapple and y
represents the number of acres for durian.
(a) Determine the number of acres for each product to maximize the profit.
(b) Hence, calculate the maximum profit.

[PSPM 05/06]

3. If f ( x, y) = xy4 − 2x2 y3 + 4x2 − 3y, show that fxy ( x, y) = f yx ( x, y) . [PSPM 06/07]

4. The profit function for a firm is P ( x, y) = 80x − 2x2 − xy − 3y2 +100 y, where x and y

are number of units sold for commodity M and commodity N respectively. The
maximum output capacity for both commodities is 12 units. Use the method of
Lagrange multiplier to find x and y so that the profit is maximized. Hence, find the
maximum profit.

[PSPM 06/07]

5. If f ( x, y) = e2x+3y , show that fxx ( x, y) + f yy ( x, y) = 13 f ( x, y) . [PSPM 07/08]

6. Determine the maximum or minimum or saddle point of the function

f ( x, y) = 6x2 − 2x3 + 3y2 + 6xy by using the second derivative test. [PSPM 07/08]

7. Find x and y that minimise the function f (x, y) = 2x2 + y2 + xy subject to the

constraint x + y = 4 . Use the method of Lagrange multipliers to find the minimum

value of the function. [PSPM 08/09]

8. A yearly profit earned by selling x units of item A and y units of item B is given as

P ( x, y) = 200 − 2x2 − 2 y2 + 400x + 700 y . Find the yearly sales of item A and item B

that give maximum profit. Hence, find the annual revenue that gives maximum profit

if the annual cost function of production of both items is given by

C ( x, y) = x2 + y2 + 200x +100 y +100 . [PSPM 08/09]

 9. Show that the function z = ln ( x − a)2 + ( y − b)2 , where a and b are constants,

satisfies the equation 2z + 2z =0. [PSPM 09/10]
x2 y2

39

MATHEMATICS 2 (AM025) – PSPM Questions

10. Farid draws on x units of shirt A and y units of shirt B. The demand functions for shirt

A and shirt B are pA ( x) = 24.8 − 0.1x and pB ( x) = 11.6 − 0.2 y respectively. Both

functions give the selling price for each shirt in RM. The cost function for

manufacturing x units of shirt A and y units of shirt B is C ( x, y) = 20x + 5y + 0.1xy

(a) State the total revenue and the total profit functions by selling x units of shirt
A and y units of shirt B.

(b) Determine the maximum profit and hence, obtain the quantity and price of
shirt A and shirt B.
[PSPM 09/10]

11. If f ( x, y) = ex+2y , show that 2 f + 2 f = 5 f ( x, y) . [PSPM 10/11]
x2 y2

12. Use the method of Lagrange multiplier to find the maximum value for the function

f ( x, y) = 2x3 + 8y3 subject to the constraint x + y = 6 . [PSPM 10/11]

13. Determine the maximum or minimum or saddle point of the function

f ( x, y) = xy − x2 − y2 − 3x − 3y + 8 by using the second derivative test. [PSPM 11/12]

14. Find x and y that minimise the function f ( x, y) = 16x3 + 4y3 subject to the

constraint x + y − 3 = 0 . Use the method of Lagrange multiplier to find the minimum

value of the function. [PSPM 11/12]

15. The cost function which is given as C ( x, y) = x2 + y2 + xy − 20x − 25y +1500 for

inspection of an assembly operation depends on the number of inspections x and y .

How many inspections of x and y should be made in order to minimise the cost?

Hence, determine the minimum cost. [PSPM 12/13]

16. Use the method of Lagrange multiplier to find the maximum value for the function

F ( x, y) = x + 5y − 2xy − x2 − 2 y2 subject to the constraint 2x + y = 4. [PSPM 12/13]

( )17. 2V 2V
Given V ( x, y) = ln x2 + y2 . Calculate x2 + y2 . [PSPM 13/14]

18. Determine the stationary points of f ( x, y) = x3 − 3x2 + 2xy − y2 and determine the

types of each point. [PSPM 13/14]

8x2 y2
x2 + y2
( ) ( )19.
If F(x, y) = ln x2 + y2 , show that x2Fxx + y2Fyy − 2 = −2 . [PSPM 14/15]

20. Find the maximum value for the function F ( x, y) = 8x3 + 2 y3 subject to the constraint

x + y = 3 by using the Lagrange multiplier method. [PSPM 14/15]

40

MATHEMATICS 2 (AM025) – PSPM Questions

21 Given F ( x, y) = x3e2 x−3 y and U ( x, y) = 1 Fxy + 1 Fy . Find U (3, 2) . [PSPM 15/16]
x2 3

22. Find x and y that minimize the function f ( x, y) = 3x2 + y2 + 2xy + 7 subject to the

constraint 3x + 2 y = 90 by using the Lagrange multiplier method. Hence, find the

minimum value of the function. [PSPM 15/16]

23. Find x and y that minimize the function C ( x, y) = 6x2 +12 y2 subject to the constraint

x + y − 90 = 0 by using the method of Lagrange multiplier. Hence, find the minimum

value of the function. [PSPM 16/17]

24. Determine the four critical points of the function f ( x, y) = 2x2 + y3 − x2 y − 3y .

Hence, determine the maximum, minimum or saddle points of the function using the

second derivative test. [PSPM 16/17]

25. Find the maximum values of the function f ( x, y) = xy3 subject to the constraint

x + y = 8 by using the Lagrange multiplier method. [PSPM 17/18]

26. Given f ( x, y) = 2x + 4 y − x2 − y2 + 5 . Find the stationary point. Hence, determine the

nature of the stationary point. [PSPM 18/19]

ANSWERS TOPIC 1 (PSPM)

1. x = 4, y = 8
2. (a) 200 acres for pineapple and 200 acres for durian to maximize the profit.

(b) RM200 000
4. x = 5 , y = 7, Maximum profit =RM868
6. (0,0) is a minimum point, (1-1) is a saddle point.
7. x = 1 , y = 3, Minimum value =14.
8. 100 of item A and 175 of item B should be selling in order to maximize the profit.

The annual revenue =RM159 675

10. (a) R ( x, y) = 24.8x − 0.1x2 +11.6y − 0.2y ,
 ( x, y) = 4.8x − 0.1x2 + 6.6 y − 0.2 y2 − 0.1xy

(b) Maximum profit =RM82.80
The quantity required for shirt A is 18 units and for shirt B is 12 units in order
to achieve its maximum profit.
The price for shirt A is RM23 and for shirt B is RM9.20.

12. Maximum value =1728
13. Maximum point at (-3,-3)
14. x = −3, y = 6

x = 1, y = 2
Minimum value = 48

41

MATHEMATICS 2 (AM025) – PSPM Questions

15. In order to minimize the cost, 5 units of inspection x and 10 units of inspection y

should be made. Minimum cost, C(5,10) = RM 1325

16. Maximum value is − 3
4

17. 0

18. ( 0, 0) and  4 , 4  ,
 3 3 

(0,0) , maximum point.  4 , 4  , saddle point.
 3 3 

19. Shown

20. Maximum value is 216 when x = −3 and y = 6

21. −54

22. 1807

23. x = 60, y = 30 , minimum value 32400

24. (0,1) is a minimum point, (0, −1) is a saddle point, (3, 2) is a saddle point,

(−3, 2) is a saddle point

25. The maximum value is 432 when x = 2 and y = 6 .

26. (1, 2) is a maximum point

42

MATHEMATICS 2 (AM025) – PSPM Questions

T 2 IOPIC : NTEGRATION

PSPM QUESTIONS

2 [PSPM 05/06]

1. Evaluate x3 x4 + 5 dx .

1

4

2. A function is f defined by f ( x) = x +1 − 2 . Evaluate  f ( x) dx . [PSPM 06/07]
−4

3. 1 .
Evaluate 1 + e− x dx [PSPM 08/09]

 xex2 , x  1 3

4. Given a function g defined by g ( x ) =  ( ln x )2 . Evaluate  g ( x) dx .

 , x 1 −1
x

[PSPM 09/10]

1 [PSPM 11/12]

 5. Find 3e3x+2dx. Hence, evaluate 6e3x+2dx.
0

6. Integrate the following:

3e−2x+1dx (b)  1   2 1  3 x +1
 (a)  t − t2   t + t  dt
(c) 2 3x2 − 3 dx
[PSPM 11/12]

4
7. (a) Express (x +1)(x2 − 9) in the form of partial fractions.

10 4

(b) Use the result in part (a) to evaluate 4 (x +1)(x2 − 9) dx correct to 4 decimal
places.
[PSPM 12/13]

8. Solve ( )4 − 4x e3+2x−x2 dx [PSPM 13/14]

3 3
( )( )9.
Evaluate x2 − 3 x3 −9x −6 dx [PSPM 13/14]
[PSPM 13/14]
−1

55 8

10. Given that  f ( x) dx = 8,  f ( x) dx = 5 and  f ( x) dx = 15 .

14 1

8

Show that  f ( x) dx = 12

4

43

MATHEMATICS 2 (AM025) – PSPM Questions

11. Evaluate 1  3e2 x − 3  dx . [PSPM 14/15]
 e2x 


0

12. (a) Given 2 f (u) du = −5 , 2 h ( u ) du =4 and 5 f (u) du =8.

  

11 2

(i) Evaluate 2  f (u) − 3h(u)  du .
3 
  4 

1

(ii) Find the value of p if 5  f (u) − 3pu du = 39 .



1

Solve x2
3 − 2x3
( )(b)  5 dx
2

[PSPM 14/15]

( )1 [PSPM 15/16]

13. Solve  3x2 x3 + 2 dx .

0

14. (a) Given f '(x) = 6− 3 − x3 and f (1) = 7 . Determine f (x).
x2

2 − e−3x −1
dx .

3
( ) ( )(b)
Show that 2 − e−3x −1 = e3x . Hence, solve
2e3x −1


[PSPM 15/16]

1 3
( )15.
Evaluate x 2x2 +1 dx by using appropriate substitution. [PSPM 16/17]

0

73 3

16. (a) Given that  f ( x) dx = 6 ,  f ( x) dx = 3 and  g ( x) dx = −2 . Evaluate the
11 1

following:

7

(i)  8 f ( x) dx

3

3

(ii)   f ( x) − 3g ( x) − 6x2  dx
1

(b) Evaluate  2  2 − e2x+1   dx [PSPM 16/17]
0 e3x 

17. Solve  (1 7x )4 by using an appropriate substitution. [PSPM 17/18]
− 4x

44

MATHEMATICS 2 (AM025) – PSPM Questions

99 6

18. (a) Given  f ( x) dx = 15 ,  f ( x) dx = 12 and  g ( x) dx = 18 .
26 2

Show that 6 ( x) + g ( x) = 18
 dx
 3 f 2 

2

11 11 9

(b) Suppose  f ( x) dx = k ,  f ( x) dx = 3k and  g ( x) dx = 8 .
97
7

9

If   f ( x) + g ( x) dx = 10 , find the value of k .

7

2

(c) Evaluate the integral xex2 dx by using substitution method.

0

[PSPM 17/18]

( )( )Solve 3
19. (a) e−3x − ex e−3x + 3ex
(b) dx

20. (a) 77 10
(b)
Given that  g ( x) dx = 15 ,  g ( x) dx = 11 and  g ( x) dx = 25 .
15 1

10

(i) Show that  g ( x) dx = 21.

5

5

(ii) Find  2g ( x) dx .

7

[PSPM 18/19]

Solve  (2 − 3x)4 dx

18x2
3 − 2x3
( )Integrate 4 dx

[PSPM 18/19]

ANSWERS TOPIC 2 (PSPM)

1. 13.59 2. 1 3. ln ex +1 + c

(ln 3)3 5. 282.05

4. @ 0.442
3

6. (a) −3e−2x+1 + c (b) 2 t 7 2 t − 3 +c (c) 1 ln 2
2 2 3
2+
73

7. (a) − 1 + 1 + 1
2(x +1) 3(x + 3) 6(x − 3)

(b) 0.1364

45

MATHEMATICS 2 (AM025) – PSPM Questions

8. 2e3+2x−x2 + c 9. 320 10. Shown 11. 8.2866

3

12. (a) (i) −18 (ii) p = −1 1 +c
3 − 2x3
( )(b) 48 4

13. 1.578

14. (a) f (x) = 6x + 3 − x4 − 7 (b) Shown , 1 ln 2e3x −1 + c
18
x44

15. 5

16. (a) (i) 24 (ii) −43 (b) −1.685

17. −7  1 + 1  + c
16 − 
 3(1− 4x)3 2(1− 4x)2 

18. (a) shown (b) k = 1 (c) 1 (e4 −1)
2

19. (a) ( )− 1 e−3x + 3ex 4 + c
12

(b) (i) Shown

(ii) −22

20. (a) − 1 (2 − 3x)5 + c

15

1 +c
3 − 2x3
( )(b) 3

46

MATHEMATICS 2 (AM025) – PSPM Questions

T 3 A D IOPIC : PPLICATIONS OF EFINITE NTEGRAL

PSPM QUESTIONS

1. The demand function D(q) and the supply function S(q) are given respectively as

p = D (q) = 20 − q2 and p = S (q) = q2 + 6q where q represents the number of

products and p represents the price in RM.
(a) Determine the market equilibrium point.
(b) Sketch the demand function and the supply function on the same graph and

shade the consumer’s and producer’s surplus region.
(c) Hence, find the consumer’s surplus and the producer’s surplus.

[PSPM 05/06]

2. The company produces x units of product. The marginal cost and marginal revenue

functions are given as C '( x) = 2x − 20 and R '( x) = 100 − 4x respectively.

(a) Find the demand function and the cost function if the fixed cost of the
company is RM 200.

(b) Find the number of products produced to obtain the maximum profit and
determine the maximum profit.
[PSPM 06/07]

3. Given that the demand and supply functions of a commodity are

p1 (x) = 12 −  x  and p2 (x) =  x  + 5 respectively, where x is the quantity
 50   20 

produced.

(a) Find the equilibrium point for the commodity.
(b) Shade the consumer’s and producer’s surplus regions on the same graph.
(c) Hence, find the consumer’s surplus and the producer’s surplus.

[PSPM 06/07]

4. The demand function D ( x) and the supply function S ( x) are given as

y = D ( x) = 16 − x2 and y = S ( x) = x2 + 4x

where x represents the numbers of item and y represents the price in RM.

(a) Determine the market equilibrium point ( x0 y0 ) .

(b) On the same graph, sketch the demand and the supply functions and shade the
consumer’s and the producer’s surplus regions.

(c) Hence, determine the consumer’s and the producer’s surplus.
[PSPM 07/08]

5. Sketch the region bounded by the curves y = xex2 , y = x2 , x  0 and the line x = 2 .

Find its area. [PSPM 08/09]

47

MATHEMATICS 2 (AM025) – PSPM Questions

6. The demand function D ( x) and the supply function S ( x) are given as

y = D ( x) = 113 − 4x2 and y = S ( x) = (2x +1)2 , where x represents the numbers of

item and y represents the price in RM.
(a) Find the consumer’s surplus and producer’s surplus.
(b) Sketch the demand and supply curves and shade the consumer’s and

producer’s surplus regions on the same graph.
[PSPM 08/09]

7. The price (RM) and the marginal cost function of a firm are p = 2000 − 4x and

C( x) = 32x respectively, where x is the output unit. The fixed cost is RM1000.

Determine
(a) the revenue function and the profit function
(b) the number of units that will maximise profit and the price at this level of

production.
(c) the profit and the average cost when the revenue is maximized.

[PSPM 08/09]

8. A region R is bounded by the curve y = x ( x − 2) and line y = x .

(a) Sketch the graphs and shade the region R.
(b) Find the area R.

[PSPM 09/10]

9. The demand and supply functions for an item are

y = D ( x) = 50 − x2 and y = S ( x) = 3x +10 ,

respectively, where x represents the number of item and y represents the price in RM.
(a) Find the market equilibrium point.
(b) On the same graph, sketch the demand and supply curves and shade the

consumer’s and producer’s surplus regions.
(c) Determine the consumer’s surplus and producer’s surplus.

[PSPM 09/10]

10. The marginal revenue function and the marginal cost function of a product are

R '( x) = 498 −10x and C '( x) = 6x − 350,

respectively, where x is the quantity of production and the fixed cost is RM 7000.
(a) Find the revenue function, cost function and profit function
(b) Calculate the level of production that will maximize the profit and hence,

determine the maximum profit.
(c) Find the total revenue and price when the profit is maximized.

[PSPM 09/10]

11. Given x = 2 y2 and x + y = 3 .

(a) Sketch the curve and straight line on the same axes and find the points of
intersection.

(b) Calculate the area of the region bounded by the curve and the line.
[PSPM 10/11]

48