MATHEMATICS 2 (AM025) – PSPM Questions
12. The total profit and the marginal revenue functions of a company assembling personal
computers are given as P ( x) = 600x −10x2 − 500 and R( x) = 500 − 2x respectively.
Find the demand function and the fixed cost. [PSPM 10/11]
13. The demand function D (q) and the supply function S (q) are given as
p = D (q) = 30 − q2 and p = S (q) = 2q2 + q , where q represents the number of
products and p represents the price in RM. Determine the market equilibrium point.
Hence, find the consumer’s surplus and producer’s surplus. [PSPM 10/11]
14. An electronic company produces calculators for the local market. The company finds
that the marginal cost function is C '( x) = 50 and the demand function is
p ( x) = 100 − 0.02x , where x represents the number of calculators and y represents
the price per calculator in RM and the fixed cost is RM20000.
(a) Find the cost, revenue and profit functions.
(b) Calculate the level of production that will maximise the profit. Hence, find the
maximum profit.
(c) Determine the price per unit when the profit is maximum.
[PSPM 11/12]
15. The demand and supply functions for an item are given as y = D ( x) = 30 − x and
y = S ( x) = 6x + x2 respectively, where x represents the number of items and y
represents the price in RM.
(a) Determine the market equilibrium point ( x0, y0 ) .
(b) On the same graph, sketch the demand and supply functions and shade the
consumer’s surplus and producer’s surplus region.
(c) Determine the consumer’s and the producer’s surplus.
[PSPM 11/12]
16. Given y = x3 − x2 − 6x and y = 6x .
(a) Find the points of intersection of the above curve and the straight line.
(b) Sketch the curve and the straight line on the same axes.
(c) Calculate the area of the region bounded by the curve and the straight line.
[PSPM 11/12]
17. The demand function D (q) and the supply function S (q) are given as
p = D (q) = 200 − 0.1q and p = S (q) = 20 + 0.2q where q represents the quantity of
product and p represents the price in RM. Determine the market equilibrium point.
Hence, find the consumer’s surplus and producer’s surplus. [PSPM 12/13]
18. Let R be the region enclosed by the curve y = x2 − 4 and a straight line y=3x. The two
graphs intersect at points P and Q.
(a) Find P and Q.
(b) Sketch the curve and straight line on the same axes, shade the region R.
(c) Calculate the area of the region R.
[PSPM 12/13]
49
MATHEMATICS 2 (AM025) – PSPM Questions
19. The demand and supply curves for a consumers’ product are given by the equations
D (q) = 70 − q2 and S (q) = 10 + 7q respectively in RM; q is quantity, measured in
thousands of unit produced. Find the price and quantity at equilibrium point. Hence,
calculate the consumer’s surplus and producer’s surplus. [PSPM 13/14]
20. Given two functions 18x = 231− y and x = y − 2x with x and y represents quantity
4
and price respectively.
(a) State the demand function and the supply function.
(b) Determine the market equilibrium point.
(c) Evaluate the consumer’s surplus and producer’s surplus.
[PSPM 14/15]
21. The demand function and the supply function are given as p = D (q) = 60 − q2 and
p = S (q) = q2 + 2q + 20 respectively, where q represents the number of products and
p represents the price in RM.
(a) Determine the market equilibrium point.
(b) Sketch the graph of demand and supply functions on the same plane, and
shade the consumer’s surplus and producer’s surplus region.
(c) Find the consumer’s surplus and producer’s surplus.
[PSPM 15/16]
22. Given the demand function p = 60 − 2x2 and the supply function p = x2 + 9x + 30 ,
where x and p represents the quantity and price respectively.
(a) Determine the market equilibrium point.
(b) On the same graph, sketch the demand and the supply functions and shade the
consumer’s and producer’s surplus region.
(c) Hence, determine the consumer’s and the producer’s surplus.
[PSPM 16/17]
23. The demand and the supply functions of a consumer’s product are given by the
equations y = D ( x) = −0.1x2 +1.5x +160 and y = S ( x) = 0.2x2 − 0.5x + 80
respectively, where x represents the quantity of a product and y represents the price
in RM.
(a) Determine the market equilibrium point ( x0, y0 )
(b) Sketch the demand and the supply functions and shade the consumer’s surplus
and the producer’s surplus region.
(c) Determine the consumer’s surplus and the producer’s surplus.
[PSPM 17/18]
50
MATHEMATICS 2 (AM025) – PSPM Questions
24. Given p −130 = −q2 − 4q and p −10 = q2 +10q are demand and supply functions in
RM.
(a) Find the market equilibrium point.
(b) Sketch the demand and supply functions on the same axes. Label the
consumer’s and producer’s surplus.
(c) Determine the consumer’s surplus and the producer’s surplus.
[PSPM 18/19]
25. The diagram shows part of the graph of y = 2 − x and y = x2 . Find the area of the
shaded region.
y y = x2
2
y =2−x
x
02
[PSPM 18/19]
ANSWERS TOPIC 3 (PSPM)
1. (a) (2,16)
(c) CS = RM5.33 , PS = RM17.33
2. (a) p ( x) = 100 − 2x, C ( x) = x2 − 20x + 200
(b) 20 units, RM1000
3. (a) (100,10)
(c) CS = RM100 , PS = RM250
4. (a) (2,12)
(c) CS = RM5.33 , PS = RM13.33
5. 24.13 unit2
6. (a) CS = RM114.33 , PS = RM138.83
7. (a) R ( x) = 2000x − 4x2 , ( x) = −20x2 + 2000x −1000
(b) x = 50 , RM1800
(c) RM− 751000 , RM 4004
8. (b) 4.5 unit2
9. (a) (5, 25)
(c) CS = RM83.33 , PS = RM37.50
10. (a) R ( x) = 498x − 5x2 , C ( x) = 3x2 − 350x + 7000 , ( x) = −8x2 + 848x − 7000
(b) x = 53 , RM15472
(c) RM12349 , RM233
51
MATHEMATICS 2 (AM025) – PSPM Questions
11. (a) ( 2,1) and 9 ,− 3
2 2
(b) 125 @5.21 unit2
24
12. D ( x) = 500 − x , fixed cost = RM 500
13. (3, 21) , CS = RM18 , PS = RM 40.50
14. (a) C ( x) = 50x + 20000 , R(x) =100x − 0.02x2 , ( x) = 50x − 0.02x2 − 20000
(b) x = 1250 , RM11250
(c) RM75
15. (a) ( x0, y0 ) = (3, 27)
(c) CS = RM4.50 , PS = RM45
16. (a) (0,0),(-3,-18),(4,24)
(c) 937/12
17. The market equilibrium point is (600,140)
CS = RM18000 , PS = RM36000
18 (a) (4,12),(-1,-3)
(c) 20.83
19. The market equilibrium point is (5, 45) , CS=RM 83.33 , PS=RM 87.50
20. (a) Demand function, D ( x) = 231−18x ,
Supply function, S ( x) = 4x2 + 2x
(b) 11 ,132
2
(c)
21. (a) CS = RM 272.25 , PS = RM 473.92
(4, 44)
(b) S (q) = q2 + 2q + 20
60
CS E.P = (4, 44)
D (q) = 60 − q2
44
PS
20
q
4
(c) CS = RM 42.67 , PS = RM 58.67
22. (a) (2,52)
(b) graph
(c) RM 10.67 , RM 23.33
23. (a) (20,150)
(c) CS = RM233.33 , PS = RM966.67
52
MATHEMATICS 2 (AM025) – PSPM Questions
24. (a) (5,85)
(b) p
135 p = q2 +10q +10
CS ( 5, 85 )
85
p = −q2 − 4q +130
PS
10 q
2
(c) CS = RM 133.33 , PS = RM208.33
25. 7 unit2 or 1.17 unit2
6
53
MATHEMATICS 2 (AM025) – PSPM Questions
T 4 L POPIC : INEAR ROGRAMMING
PSPM QUESTIONS
1. Determine the system of inequalities satisfying the feasible region in the following
figure:
y
20
(12, 8)
R 8 20 x [PSPM 05/06]
5
2. A publishing company produces two types of books X and Y. The company has a
production time of 3 hours 30 minutes for evaluation process and 3 hours for printing
process. The production time and cost requirements for each book are given in the
following table .
Production Time (minute) Production
Cost
Book Evaluation Printing
(RM/unit)
X Process Process 10
Y 15
46
53
(a) Determine the objective function and formulate the system of inequalities
satisfying the given constraints.
(b) Sketch the feasible region that satisfies the system of inequalities in (a).
(c) By using the corner-point method, determine the number of each book to
minimize the cost. Hence, state the minimum cost.
[PSPM 05/06]
3. On the same graph, shade the feasible region satisfying the following system of
inequalities:
4x + 3y 24
x+ y 4
2x y
x4
[PSPM 06/07]
54
MATHEMATICS 2 (AM025) – PSPM Questions
4. Miss Maria sells her home-made biscuits. As a start, she makes 2 types of biscuits
which are Biscuit A and Biscuit B. She is capable of making at most 15 kilograms(kg)
of biscuits per day. Miss Maria makes Biscuit A at least 25% of biscuit B. At the
same time she has an agreement to supply at least 5 kg of biscuit B per day. The cost
of producing 1 kg of Biscuit A and 1 kg of Biscuit B are RM 3.00 and RM 4.50
respectively. The profit obtained for producing 1 kg of Biscuit A and 1 kg of Biscuit
B are RM 10.00 and RM 13.00 respectively. Assume that x is the weight (kilogram
per day) of Biscuit A and y is the weight (kilogram per day) of Biscuit B.
(a) State the objective function to minimize the cost and formulate the system of
inequalities satisfying the given constraints.
(b) Sketch the feasible region satisfying the system of inequalities in part (a).
(c) Determine the weight (kilogram per day) to minimize the cost and state the
minimum cost by evaluating the objective function at each vertex of the
feasible region. Hence, find the profit obtained.
[PSPM 06/07]
5. Find the system of inequalities satisfying the feasible region R given in the graph
below.
y
14
10
(2, 10)
R
4R
(6, 2)
x
0 7 12
[PSPM 07/08]
6. A chemical company produces two brands of fertilizer, Regular Brand and Super
Brand. Regular Brand contains the mixture of material A, B and C in the ratio of 1:3:6
(by weight) and Super Brand contains the mixture of material A, B and C in the ratio
of 3:4:3. Each month the company receives supply of 6, 9 and 12 tonnes of material
A, B and C respectively. The profit obtained for producing each tonne of Regular
Brand fertilizer and each tonne of Super Brand fertilizer are RM150 and RM240
respectively. The material requirement and the profit for each brand are given in the
following table.
Brand Material (tonne) Profit
A BC (RM/tonne)
Regular Brand 0.1 0.3 0.6
Super Brand 0.3 0.4 0.3 150
6 9 12 240
Supply
55
MATHEMATICS 2 (AM025) – PSPM Questions
Assume that x is the number of tonne for Regular Brand and y is the number of tonne
for Super Brand fertilizer.
(a) Determine the objective function to maximize the profit and formulate the
system of inequalities satisfying the given constraints.
(b) Sketch the feasible region satisfying the system of inequalities in part (a).
(c) By evaluating the objective function at each vertex of the feasible region,
determine the number of tonne for each brand of fertilizer to maximize the
profit. Hence, state the maximum profit obtained.
[PSPM 07/08]
7. On the same graph, sketch the graph and shade the feasible region satisfying the
following system of linear inequalities :
2x + 5y 20
−2x + y 10
x + y 13
x 1
y2
[PSPM 08/09]
8. A company produces 500 products of M and N. The minimum capital allocated is
RM500. The cost of producing a unit of product M and N are RM 2 and RM 1
respectively. At least 200 units of product N must be produced, while not more than
250 units of product M. The selling price per unit of product M and N are RM 5.50
and RM 3.80 respectively. Assume x and y represent the number of units of product
M and N produced respectively.
(a) Formulate the given information in the form of linear programming model
(b) Shade the feasible region satisfying the system of linear inequalities in part (a)
(c) Determine the number of units of product M and N that will maximize sales
and state the maximum sales.
[PSPM 08/09]
9. Find the system if inequalities the feasible region R in the following graph.
y
8
7 , 10
3 3
3
(2, 2)
R x
[PSPM 09/10]
1
46
56
MATHEMATICS 2 (AM025) – PSPM Questions
10. A company manufactures two products namely product A and product B. The
production of these two products involves the following three processes: fabricating,
assembling and painting. The weekly capacity for these processes are 72 hours, 30
hours and 40 hours respectively. Every unit of product A requires 2 hours of
fabricating, 1 hour of assembling and 1 hour of painting. Every unit of product B
requires 3 hours of fabricating, 1 hour of assembling and 2 hours of painting. The
profits obtained from the sale of each unit of product A and product B are RM 15 and
RM 20 respectively.
The work-hours requirement and the profit for each unit of product A and B are
summarized in the following table.
Product Work-hours per unit Profit
(RM/unit)
A Fabricating Assembling Painting
B 2 15
3 11 20
12
Assume that the company can sell all the products it produces and let x be the
number of units of product A produced and y be the number of units of product B
produced.
(a) Determine the objective function to maximize the profit and formulate the
given information in the form of linear programming model.
(b) Shade the feasible region satisfying the system of inequalities in part (a).
(c) By evaluating the objective function at each vertex of the feasible region,
determine the number of units of each product that will maximize the weekly
profit. Hence, state the maximum profit obtained.
[PSPM 09/10]
11. Determine the maximum value for the objective function,
z = 0.04x + 0.06y ,
subject to the following constraints
x + y 8000
y 3000
x 1500
x y
x, y 0
[PSPM 10/11]
12. A factory produces product X and product Y. Product X needs 30 minutes of labour
time, while product Y needs 15 minutes. The materials needed for each product X and
each product Y are 2.5 kilograms and 2 kilograms respectively. The testing process
for product X is 3 minutes, while for product Y is 4 minutes. In any one week, only
60 hours of labour time are allocated and 500 kilograms of materials are available.
Owing to cost and availability, the testing equipment must be used at least 8 hours.
Due to existing orders, at least 40 units of product X and 80 units of product Y must
be produced. The profits from each unit of X and Y produced are RM 6.00 and RM
4.50 respectively.
57
MATHEMATICS 2 (AM025) – PSPM Questions
If x is the number of units of product X and y is the number of units of product Y
produced,
(a) Summarise the above information in a suitable table.
(b) Determine the objective function to maximize the profit and formulate the
given information in a form of linear programming model.
(c) On the same graph, shade the feasible region satisfying the system of
inequalities in part (b).
(d) Find the weekly production for each product that will,
(i) maximize the profit and state the expected maximum profit.
(ii) minimize the profit and state the minimum profit.
[PSPM 10/11]
13. Find the system of inequalities satisfying the feasible region R as in the graph.
y
( 4, 3)
3
2
R
1
0 24 x
12
[PSPM 11/12]
14. An airline company has two types of airplane, KT1 and KT2. The company has a
contract with a travel agency to provide seats for at least 2,000 first-class, 1,500
business-class and 2,400 economy-class passengers. The operating costs for KT1 is
RM15,000 per km and can accommodate 40 first-class, 40 business-class and 120
economy-class passengers. Whereas the operating costs for KT2 is RM12,000 per km
and can accommodate 80 first-class, 30 business-class and 40 economy-class
passengers. Assuming x and y represent the number of airplanes KT1 and KT2 to be
utilized respectively.
(a) Determine the objective function that minimizes the operating cost and
formulate the given information in the form of linear programming model.
(b) Shade the feasible region satisfying the system of inequalities in part (a).
(c) By evaluating the objective function at each vertex of the feasible region,
determine the number of each type of airplane that should be utilized to
minimize the operational cost.
[PSPM 11/12]
58
MATHEMATICS 2 (AM025) – PSPM Questions
15. Determine the system of inequalities satisfying the feasible region R in the following
graph.
y
11
10
8
R
3
24 8 9 11 x
[PSPM 12/13]
16. The Warna Company owns a small paint factory that produces both interior and
exterior house paint for wholesale distribution. Two raw materials P and Q are used to
manufacture the paints. The maximum supply of P is six thousand litres a day and Q
is eight thousand litres a day. The daily requirement of the raw materials per thousand
litres of interior and exterior paint are summarised in the following table.
Paint Type Raw Material (thousand litre)
PQ
Exterior 12
Interior 21
A market survey has established that the daily demand for interior paint cannot exceed
that of exterior paint by more than one thousand litres. The survey also shows that the
maximum demand for interior paint is limited to two thousand litres a day. The
wholesale price per thousand litres is RM 30,000 for exterior paint and RM 10,000 for
interior paint. Let x be the number of thousand litres of exterior paint produced daily
and y be the number of thousand litres of interior paint produced daily.
(a) Determine the objective function to maximise the income and formulate the
given information in the form of linear programming model.
(b) Plot the graph and shade the feasible region satisfying the system of
inequalities in part (a) on a graph paper.
(c) By evaluating the objective function at each vertex of the feasible region,
determine how much interior and exterior paint should the company produce
to maximise the income.
[PSPM 12/13]
59
MATHEMATICS 2 (AM025) – PSPM Questions
17. A manufacturer used two machines, M1 and M2 to produce a particular item. The
items then are graded into Grade A and Grade B. The manufacturer decides to produce
600 items of Grade A and 630 items of Grade B for every 30 days. It cost RM1200
per day to run M1 and RM1800 per day to run M2. Each day, M1 produce 12 items of
Grade A and 18 items of Grade B; M2 produce 15 items of Grade A and 9 items of
Grade B.
(a) Determine the objective function that minimizes the cost and formulate the
given information in the form of linear programming model.
(b) Shade the feasible region satisfying the system of inequalities in part (a).
(c) By evaluating the objective function at each vertex of the feasible region,
determine the number of days for each machine to run that will minimize the
cost.
[PSPM 13/14]
18. A farmer has 80 hectares of land for planting corns and cabbages. He must grow at
least 10 hectares of land for corns and 20 hectares of land for cabbages to meet the
market demands. His work force and equipment will only allow him to plant a
maximum of 50 hectares of land for corns. The profit for planting corns is RM1,800
per hectare and for cabbages is RM500 per hectare. Let x and y be the number of
hectares of land for planting corns and cabbages respectively.
(a) Determine the objective function that maximizes the profit and formulate the
given information in the form of linear programming model.
(b) Plot the graph and shade the feasible region satisfying the system of
inequalities in (a) on the graph paper.
(c) By evaluating the objective function at each vertex of the feasible region,
determine the number of hectares of corns and cabbages should be planted to
maximize the profit and state the profit.
[PSPM 14/15]
19. ABC company manufactures biscuit with two flavours, vanilla and chocolate. The
company has 852 kg of dough to make the two types of biscuits. The weight of each
piece of vanilla flavoured biscuit is 20 gm and the weight of each piece of chocolate
flavoured biscuit is 30gm. The manufacturing cost and the selling price of a piece of
vanilla flavoured biscuit are RM0.44 and RM0.70 respectively. The manufacturing
cost and the selling price of a piece of chocolate flavoured biscuit are RM0.80 and
RM1.10 respectively. At least 10,000 pieces of vanilla flavoured biscuits and 12,000
pieces of chocolate flavoured biscuits must be produced. The number of chocolate
flavoured biscuits produced must be more than the number of vanilla flavoured
biscuits. Assume x represents the number of vanilla flavoured biscuits and y
represents the number of chocolate flavoured biscuits.
(a) Determine the profit for each piece of vanilla flavoured biscuit and the profit
for each piece of chocolate flavoured biscuit. Hence, determine the objective
function to maximize the total profit and formulate the given information in
the form of the linear programming model.
(b) Plot the graph and shade the feasible region satisfying the system of
inequalities in part (a) on a graph paper.
(c) By evaluating the objective function at each vertex of the feasible region,
determine the number of biscuits for each flavor that should be produced to
maximize the total profit.
[PSPM 15/16]
60
MATHEMATICS 2 (AM025) – PSPM Questions
20. Each person needs at least 16 units of protein, 24 units of carbohydrate and 18 units of
fat in a week. Food type A consists of 4 units of protein, 12 units of carbohydrate and
2 units of fat per kg. Food type B consists of 2 units of protein, 2 units of
carbohydrate and 6 units of fat per kg. The supply of food type B in a week is not
more than 12 kg. Price per kg of food type A and type B are RM170 and RM80
respectively. Let x and y represent the weight in kg of food type A and type B
respectively.
(a) Determine the objective function and formulate the given information in the
form of linear programming model.
(b) Plot the graph and shade the feasible region satisfying the system of
inequalities in part (a) on the graph paper.
(c) By evaluating the objective function at each vertex of the feasible region,
determine the weight in kg of food type A and type B that should be bought to
minimize the cost. Hence, state the minimum cost.
[PSPM 16/17]
21. Encik Ali sells two types of fried nodules. A pack of plain fried noodles uses 120 g of
prawns and 300 g of chicken, while a pack of special fried noodles uses 240 g of
prawns and 200 g of chicken. He has 8.4 kg of prawns and 12 kg of chicken to make x
packs of plain fried noodles and y packs of special fried noodles. The number of plain
fried noodles packs cannot be more than two times the number of special fried
noodles packs. The profit for one pack of plain fried noodles is RM2 and the profit for
one pack of special fried noodles is RM3.
(a) Determine the objective function to maximize the profit and formulate the
given information in the form of linear programming model.
(b) Plot the graph and shade the feasible region satisfying the system of
inequalities in part (a) on the graph paper.
(c) By evaluating the objective function at each vertex of the feasible region,
determine the number of packs of plain fried noodles and special fried noodles
to maximize the profit and state the profit.
[PSPM 17/18]
22. Write all the inequalities for the feasible region R :
y
6
4 8 10 x
R2
03
[PSPM 18/19]
61
MATHEMATICS 2 (AM025) – PSPM Questions
ANSWERS TOPIC 4 (PSPM)
1. x + y 20, y 2x −16, x 5, y 8, y 0
2. (a) Z = 10x +15y, 4x + 5y 210, 6x + 3y 180, x 0, y 0
(c) The minimum cost is RM 300 when 30 units of book X and 0 units of book Y
were produced.
4. (a) Z = 3x + 4.5y, x + y 15, x 1 y, y 5, x 0, y 0
4
(c) The minimum cost is RM 26.25 when 1.25kg of Biscuit A and 5kg of Biscuit
B were made. The profit obtained is RM77.50
5. x 2, y −2x +14, 6y −5x + 60, y 1 x, 7 y −4x + 28
3
6. (a) Z = 150x + 240 y, 0.1x + 0.3y 6, 0.3x + 0.4 y 9, 0.6x + 0.3y 12, x, y 0
(c) The maximum profit is RM 5220 when 6 tonnes of Regular Brand and 18
tonnes of Super Brand were produced.
8. (a) Z = 5.5x + 3.8y, x + y 500, 2x + y 500, y 200, x 250, x, y 0
(c) The maximum sales is RM2325 when 250 units of Product M and 250 units of
product N were produced.
9. y 2, x, y 0, y x +1, 2y −x + 6, y −2x + 8
10. (a) Z = 15x + 20y, 2x + 3y 72, x + y 30, x + 2y 40
(c) Hence, the firm should produce 20 units of product A and 10 units of product
B in order to achieve the maximum profit, which is RM 500.
11. The maximum value is RM380 when x = 5000, y = 3000
12. (b)
Z = 6x + 4.5y, 30x +15y 3600, 2.5x + 2 y 500,3x + 4 y 480, x 40, y 80
(d) (i) Maximum profit is RM960 when the factory produces 40 units of
product X and 160 units of product Y.
(ii) Minimum profit is RM645 when the factory produces 40 units of
product X and 90 units of product Y.
13. y 3 , x 12 , x 0 , y 0 , y x +1 , 4 y + x 16
14. (a) Z = 15000x +12000y
40x + 80y 2000 , 40x + 30 y 1500 , 120x + 40 y 2400 , x, y 0
(c) In order to minimise the operating cost RM 570,000, the company should use
30 airplanes of type KT 1 and 10 airplanes of type KT 2.
15. y = −2( x − 4) y −2x + 8
y = −5( x − 2) y −5x +10
y = −1( x −11) y −1 x+3
3
y 8 , x 8 , x 0, y 0
y −x +11
62
MATHEMATICS 2 (AM025) – PSPM Questions
16. (a) Z = 30000x +10000y
(b) x + 2y 6, 2x + y 8, y x +1, y 2, x 0, y 0
y
8 y = x +1
2x + y = 8
3
2 y=2
R x+2y =6
x
−1 4 6
(c) The company should produce 4000 litres of exterior paint and 0 litres of
interior paint.
17. (a) Z = 1200x +1800y , 12x +15y 600 , 18x + 9y 630 , x 30 , y 30
(b) 70
18x + 9y = 630
x = 30
40
30 y = 30
R 12x +15y = 600
3035 50
(c) To minimize the cost, the manufacturer should run M1 for 30 days and M2 for
0 days.
18. (c) In order to maximize the profit, the farmer should plant 50 hectares of corns
and 30 hectares of cabbages. The maximum profit is RM 105,000.
19. (c) 17040 of vanilla flavoured biscuits and 17040 chocolate flavoured biscuits
should be produced to maximize the total profit which is RM 9542.40.
20. (c) 1 kg of food type A and 6 kg of food type B should be bought to minimize the
cost. Therefore, the minimum cost is RM 650.
21. (c) The maximum profit is RM117.50 when 25 packs of plain fried noodles and
22.5 packs of special fried noodles were produced.
22. 3y −2x + 6 , 5y −3x + 30 , x 0 , y 0 , x 8 , y 4
63
MATHEMATICS 2 (AM025) – PSPM Questions
T 5 M FOPIC : ATHEMATICS OF INANCE
PSPM QUESTIONS
1. If the interest rate is 5% compounded quarterly, find the amount of annuity of RM 2000
payable at the end of each quarter for a period of 6 years. Calculate the total interest.
[PSPM 05/06]
2. (a) Find the effective rate which is equivalent to 8% compounded semiannually.
(b) Find the nominal rate compounded monthly which is equivalent to 10% effective.
[PSPM 05/06]
3. (a) A sum of RM 10000 was invested at 9% compounded annually. How long will it
take for the investment to double its value?
(b) Ahmad deposits RM 10000 now in a bank at an interest rate of 12% compounded
quarterly. He will invest another RM 10000 after four years. What will be the
future value of his account eight years from now?
[PSPM 05/06]
4. Mr. Raju needs RM 5740 to pay for the down payment of a new car. He makes a loan
from a bank that charges a bank discount rate of 12% for a period of 18
months.
(a) Calculate the simple interest rate equivalent to the bank discount rate.
(b) Obtain the amount of loan made by Mr. Raju.
[PSPM 06/07]
5. Mr. Alias was given three options by a bank for a loan of RM 200,000. The three
options are as follows:
Option A Simple interest rate of 9% for a period of 5 years
Option B Interest at 7.5% compounded monthly to be paid at the end of
the fifth year
Option C Equal monthly payment of RM 4822 for a period of 5 years at
interest rate of 15.6% compounded monthly
(a) Find the effective rate for option C
(b) Find the amount Mr. Alias as to pay for each option.
(c) Find the interest Mr. Alias as to pay for each option.
(d) Which is the best option for Mr. Alias? Give the reason for your answer.
[PSPM 06/07]
6. (a) Find the total saving of RM 3000 at 8% compounded quarterly at the end of 5
years.
(b) A Finance Company A offers its depositors an interest rate of 8% compounded
monthly while a Finance Company B offers its depositors an interest rate of
8.15% compounded quarterly. Which company make the better offer by
comparing the effective rates?
[PSPM 07/08]
64
MATHEMATICS 2 (AM025) – PSPM Questions
7. Six years ago, Alia deposited RM k in an account that paid an interest rate of 3.75%
compounded monthly. Today, an amount of RM 37556.50 has been accumulated in the
account. She plans to add more savings in the account by making monthly deposits of
RM 300 for the next 10 years. During this period, the given interest rate is 4.05%
compounded monthly.
(a) Find the value of k.
(b) How much the overall interest will be obtained at the end of the 10 years period?
(c) After the 10 years period Alia withdraws all the saving and buys an annuity
scheme that promises a return with the rate of 6% compounded quarterly. How
much equal withdrawal (RM) at the end of every three months can she make for
the period of 15 years?
(d) Calculate the effective rate equivalent to the nominal rate in part (c).
[PSPM 07/08]
8. Investment scheme A pays 8.35% compounded monthly and investment scheme B pays
8.40% compounded every four months. Which investment scheme gives a better
return?
[PSPM 08/09]
9. RM 1000 is invested in a fund for 3 years at 6% simple interest rate for the first year,
6% effective interest rate for the second year and 6% nominal interest rate compounded
quarterly for the third year. Find
(a) the amount of interest earned in the second year.
(b) the accumulated value of the investment at the end of the third year.
[PSPM 08/09]
10. A housewife makes deposits into a fund at the end of each year for 20 years. She
deposits RM 1000 annually for the first 10 years and RM (1000 + M ) for the next 10
years. The fund earns interest at an effective rate of 4%. The amount of the fund at the
end of 20 years is RM 50000.
(a) Find the value M.
(b) If the housewife decides to keep the RM 50000 in the same fund, how many years
will she have to wait to earn interest of RM 24000?
[PSPM 08/09]
11. A father has a 12 year-old son. He wants to have RM S by the time his son reaches 18
years old and intends to invest RM P now in an account that pays 4% interest rate
compounded monthly. If the interest accumulated is RM 4261, find P and S.
[PSPM 09/10]
12. A customer borrowed RM 5000 for 6 months from a bank which charged a bank
discount rate of 6%.
(a) Find the bank discount and the customer’s proceed.
(b) Determine the simple interest rate that is equivalent to the given discount rate.
[PSPM 09/10]
65
MATHEMATICS 2 (AM025) – PSPM Questions
13. Two years ago, Encik Mahmud deposited RM 3000 ia a savings account at 10% interest
rate compounded every six months. At the same time, he deposited RM 150 at the end
of every month in another account at 6% interest rate compounded monthly. Today, he
intends to withdraw all his money from both accounts to pay part of the 10% down
payment for a new car. The cash price of the car is RM 85000.
(a) Calculate the total withdraws from both accounts.
(b) Find the additional amount to settle the down payment.
(c) Encik Mahmud plans to take up a loan from a finance company to settle
the 90% balance of the price of the car. The company charges 5% interest
compounded monthly and the loan is payable in 8 years. Find the monthly
payment.
[PSPM 09/10]
14. Three years ago Ah Lee deposited RM T in an account which gave a simple interest of
6.25%. Today he withdraws RM 100000 from the account and leaves a balance of
RM19343.75.
(a) Find the value of T and hence, state the interest he obtained for the three years
deposit.
(b) Ah Lee invests the RM 100000 in an investment scheme which promises a return
of 5.15% compounded monthly for a period of five years.
(i) Find the return for the investment.
(ii) Determine the effective rate equivalent to the above nominal rate.
[PSPM 10/11]
15. Five years ago, Shamsuddin deposited RM 4444 every six months in account A which
gave an interest rate of 4.025% compounded semi-annually. For the same period of
time he deposited RM 50000 in account B which gave an interest rate of j %
compounded quarterly and obtained an interest of RM 11312.24.
(a) Find the value of j.
(b) Find the amount in account A today. Hence, calculate the interest obtained.
(c) If he closed both accounts and invested the entire amount in an annuity scheme
for 10 years which gave an interest rate of 4.55% compounded monthly, find the
equal monthly withdrawal the that he would get. Hence, calculate the interest
obtained from this annuity scheme.
[PSPM 10/11]
16. Mary has to pay RM1502 every month for 5 years to settle her car loan at interest rate
of 3.75% compounded monthly. What is the original value of the loan and the total
interst that she has to pay? [PSPM 11/12]
17. Muthu borrowed a sum of money RMX from a bank that charged a bank discount rate
of 15%. If the amount of bank discount was RM 210, find X and the total proceeds he
obtained for the loan of 200 days. [PSPM 11/12]
18. Daniel deposited RMY into an account that paid an interest rate of 6% compounded
monthly. Five years later, he was sent overseas for two years. Prior to his departure, he
had already made arrangements with the bank to pay his son RM 500 every month for
two years. All his savings would be depleted when the final RM 500 was made.
Find
(a) the amount in his account when he left the country.
66
MATHEMATICS 2 (AM025) – PSPM Questions
(b) the value of Y.
(c) the total interest earned by his savings account for the two years while he was
overseas.
(d) the effective rate that is equivalent to the above nominal rate.
[PSPM 11/12]
19. (a) Ameera borrowed RM10400 for 5 years from a bank that charged bank discount
rate of d % . If the proceed she received was RM7280, find the simple interest
rate that is equivalent to the bank discount rate.
(b) Ammar deposited RM50000 into an account that paid interest rate 5%
compounded quarterly. He intended to keep the account untouched for 10 years.
However, after four years he had to withdraw 50% from the account. Find the
amount in the account 6 years after the withdrawal.
[PSPM 12/13]
20. Nadira bought a new house at a price of RM250,000. She paid RM25,000 as a down
payment. The balance was borrowed from a bank that charged an interest rate of 4.2%
compounded monthly for 25 years.
(a) Find her monthly payment.
(b) Find the amount of interest charge.
(c) If Nadira missed the first 3 monthly payments, how much should she pay on her
fourth payment to settle all the outstanding arrears?
(d) Immediately after making the 30th payment, Nadira decided to settle the
outstanding balance in a single payment. Calculate the amount.
(e) Calculate the effective rate which is equivalent to the above nominal rate.
[PSPM 12/13]
21. A 60-day loan RM6000 at bank discount rate, d % obtains a bank discount RM140.
What is the simple interest rate equivalent to the discount rate d % ?
[PSPM 13/14]
22. An investor borrowed RM50000 at simple interest rate of 3.28% for a period of five
years. He invested 50% of the loan into Fund A that promises an interest rate of
3.75% compounded quarterly and 50% in Fund B that promises an interest rate of
3.25% compounded monthly. The total amount of both funds at the end of the fifth
year will be used to settle the loan.
(a) Obtain the amount that he has to pay to settle the loan at the end of the five years.
(b) Obtain the amount of each fund at the end of the fifth year. Hence, find the
balance, after he has settled the loan.
(c) Obtain the effective interest rate equivalent to the interest rate given in Fund B.
[PSPM 13/14]
23. Samy want to create a fund of RM 50000 at the end of 10 years. He has two options.
Option A Option B
He deposits RM X now at an interest He deposits RM Y at the end of each year
rate of 8% compounded every six for ten years at interest rate of 12%
months for the first five years and at compounded annually.
interest rate of 10% compounded
quarterly for the next five years.
67
MATHEMATICS 2 (AM025) – PSPM Questions
The fund created is used to pay 20 equal semi-annually payments of RM Z which the
first payment starts six month later.
(a) Find X. Hence, find the interest rate compounded annually to obtain the fund
of RM50000.
(b) Find Y.
(c) Find Z given interest rate of 10% compounded semi annually.
[PSPM 13/14]
24. Azman deposited RM6,000 in a bank that offered simple interest at 7.5% per annum.
After four years, he withdrew a sum of RMA from the account. If the balance in the
account two years after the withdrawal was RM4,600, determine the value of A.
[PSPM 14/15]
25. Ali needs RM10,000 now. How much should he borrow from a bank which charges
3.5% discount rate for 180 days? Hence, find the amount of bank discount and
determine the simple interest rate that is equivalent to the discount rate.
[PSPM 14/15]
26. (a) On 2nd of June 2010, Lim Seng deposited RM3, 500 into an account that pays
interest at 8% compounded quarterly. On 2nd of June 2013, he added another
RMQ into the same account. Find the value of Q if the accumulated amount in
the account on 2nd of June 2020 is RM15,000.
(b) Find the effective rate which is equivalent to the above nominal rate.
[PSPM 14/15]
27. Three years ago, Aleesya deposited RM3,000 in a savings account earning 10%
interest compounded every six months. At the same time, she deposited RM150 at the
end of every month in another account earning 5% interest compounded monthly.
Today, she withdraws all her money from both accounts to settle the 10% down
payment for a new car with the price of RM155,000.
(a) She found out that the total amount of money withdrawn was not sufficient to
settle down payment of the car. How much additional money is needed?
(b) In order to settle the remaining 90% of the price of the car, she took a loan for
nine years from a financial institution that charged an interest rate of 3.85%
compounded monthly. Calculate her monthly payments.
[PSPM 14/15]
28. Mahani invested RM X in an investment scheme that promises a return of 9%
compounded quarterly. The value of her investment after 5 years was RM40,573.24.
Find X. [PSPM 15/16]
29. Five years ago, Faisal deposited RM7,000 in an account that pays simple interest rate,
r. Today, he deposited another RM1,000 into the same account. If the amount in the
account two years from today is RM12,080, find r. [PSPM 15/16]
68
MATHEMATICS 2 (AM025) – PSPM Questions
30. Shereen is considering the following three banks to apply for a car loan :
Bank P Bank Q Bank R
Interest rate 4.75% Interest rate 4.65% Interest rate 4.7%
compounded every six compounded every compounded every four
months. month. months.
Which bank gives the lowest interest by comparing the equivalent effective rate of
each bank? [PSPM 15/16]
31. An entrepreneur plans to buy a shop lot for RM450,000 five years from today. He
deposits RMQ every month into an account that offers 6% interest compounded
monthly which enables him to pay 10% down payment of the shop lot. The balance is
financed by a bank that charges 3% interest compounded monthly with payback
period of 30 years.
(a) Determine Q.
(a) Calculate the monthly payment for the loan.
(b) Calculate the interest earned from his savings, and the interest charged on his
loan.
[PSPM 15/16]
32. Five years ago, a sum of money RMQ was deposited in a savings account which gave
6% simple interest. The accumulated amount today is RM6,500. How much is the
amount of Q and how many years from today will the savings become RM8,600?
[PSPM 16/17]
33. Encik Abu invests RM5,000 in a trust fund for 5 years 8 months. This investment
offered an interest rate 6% compounded every three months for the first 3 years and
7% compounded monthly for the rest of the period.
(a) Calculate the amount received at the end of the investment period.
(b) Calculate the interest earned at the end of the investment period.
[PSPM 16/17]
34. On Ahmad’s 6th birthday, his father had deposited RM8,500 in a savings account that
pays 6% interest compounded monthly. Ahmad’s father plans to give him the
accumulated amount as a gift on his 21th birthday.
(a) How much will Ahmad receive on his 21th birthday?
(b) How old is Ahmad when the accumulated amount is RM29,872.15?
(c) Find the effective rate which is equivalent to the above nominal rate.
[PSPM 16/17]
35. Encik Salem purchased a house by paying RM75,000 as down payment and the
balance was borrowed from a bank that charged interest of 6% compounded monthly.
The monthly payment for that loan for 25 years is RM1,500.
(a) Find the cash price of the house.
(b) Calculate the amount of interest charged.
(c) If Encik Salem has not paid his first 4 monthly payments, how much should he
pay on his fifth payment to settle all the outstanding arrears?
69
MATHEMATICS 2 (AM025) – PSPM Questions
(d) Immediately after making the 180th repayment, Encik Salem decided to settle
the balance of the loan. Find the balance.
[PSPM 16/17]
36. Encik Ahmad invested an amount of RM13,900 divided into two investments which
are investment A with RM X and investment B with RM Y at a simple interest rate of
14% and 11% respectively. If the total amount of simple interest earned from both
investments in 2 years was RM3,508, determine the values of RM X and RM Y.
[PSPM 17/18]
37. David borrowed RM X from a bank that charged a bank discount of 6.5% for 180
days. If the amount of bank discount was RM210.50, find RM X and the proceed he
obtained. Hence, find the simple interest rate equivalent to bank discount rate.
[PSPM 17/18]
38. Encik Azhar deposited RM500 every quarter for five years in an account that paid
3.85% interest compounded quarterly. He withdrew RM5000 at the end of the fifth
year and the balance was left in that account.
(a) Determine the amount accumulated at the end of five years.
(b) How much is the balance in the account at the end of eight years?
(c) Find the effective rate which is equivalent to the above nominal rate.
[PSPM 17/18]
39. Aisyah purchased a semi-detached house at a price of RM280,000. She paid RM X as
a down payment and borrowed the balance from a finance company that charged 6%
interest compounded monthly.
(a) If Aisyah paid RM1,533.44 at the end of every month for 25 years, calculate
the amount of down payment RM X that she made.
(b) Calculate the instalment price and the interest incurred.
[PSPM 17/18]
40. (a) Ah Chong borrowed RM8,000 for 4 months from ABC bank which charged a
bank discount rate of 9%. Find the bank discount, proceed and hence
determine the simple interest rate that equivalent to the given bank discount
rate.
(b) Ahmad deposited RM25,000 into an account that paid an interest rate of 3%
compounded quarterly. After five years, he had withdrawn RM9,029.60 from
the account and kept the balance for two years. Find the amount in the account
two years after the withdrawal.
[PSPM 18/19]
70
MATHEMATICS 2 (AM025) – PSPM Questions
ANSWERS TOPIC 5 (PSPM)
1. RM 7576.17
2. (a) 8.16% (b) j = 9.57%
3. (a) t = 8.043 (b) RM 41797.89
4. (a) 14.63% (b) RM 7000.00
5. (a) 16.77%
(b) Option A : RM 290000.00 (c) Option A : RM 90000.00
Option B : RM 290658.88 Option B : RM 90658.88
Option C : RM 289320.00 Option C : RM 89320.00
(d) Choose Option C since the interest is the lowest
6. (a) RM 4457.84
(b) Finance Company A : 8.3% , Finance Company B : 8.4%
Company B gives a better return.
7. (a) RM 30000 (b) Interest = RM 34561.77
(c) RM 2553.61 (d) 6.14%
8. Scheme A gives a better return.
9. (a) RM 63.60 (b) RM1192.55
10. (a) RM 1684.30 (b) 10 years
11. P = RM 15738.24, S = RM 19999.24
12. (a) RM 150 , RM 4850
(b) 6.2%
13. (a) RM 7461.31
(b) RM 1038.69
(c) R = RM968.48
14. (a) RM 100 500 , RM 18843.75
(b) (i) RM 29297.92
(ii) 5.27%
15. (a) j = 4.1% (b) RM 48688.38 , RM 4248.38
(c) RM 1142.68; RM 27120.98
16. RM 82058.93; RM 8061.07
17. X = RM 2520.00; RM 2310.00
18. (a) RM 11281.43 (b) Y = RM 8363.74
(c) RM 718.57 (d) 6.17%
19. (a) d = 6% , r = 8.57% (b) RM41090.49
20. (a) RM1212.62 (b) RM138786
(c) RM4876.00 (d) RM211577.46
(e) 4.28%
21. 14.33%
22. (a) RM 58200
(b) Fund A = RM 30129.43 , Fund B = RM 29404.75
RM 1334.18
(c) 3.3%
23. (a) RM 20613.86 , 9.27%
(b) RM 2849.21
(c) RM 4012.13
24. RM3800
25. RM10,178.12, RM 178.12, 3.56%
26. (a) RM 4176.77 (b) 8.24%
71
MATHEMATICS 2 (AM025) – PSPM Questions
27. (a) RM 5666.71 (b) RM 1530.40
28. RM 26, 000
29. 8%
30. Bank Q gives the lowest interest
31. (a) RM644.98
(b) RM1707.50
(c) RM6,301.20 , RM209, 700.00
32. Q = RM 5000, 7 years
33. (a) RM 7201.04
(b) RM 2201.04
34. (a) RM 20859.80
(b) Ahmad will get the accumulated amount when he is 27 years old
(c) 6.17%
35. (a) RM 307810.30
(b) RM 217189.70
(c) RM 7575.38
(d) RM135110.18
36. X = RM 7500 , Y = RM 6400
37. X = RM 6476.92 , Proceed = RM 6266.42 , r = 6.72%
38. (a) RM10969.41
(b) RM 6696.57
(c) 3.91%
39. (a) RM 41999.59
(b) RM 502, 031.59 , RM 222,031.59
40. RM240 , RM7760 , 9.28% (b) RM21, 231.98
72
MATHEMATICS 2 (AM025) – PSPM Questions
TOPIC 6 : DATA DESCRIPTION
PSPM QUESTIONS
1. The mean of a set numbers 5, p, 9, 6, 2p, 9, 7, 5, 6, 10 is 6.9, where p is a constant.
Obtain the value of p. [PSPM 05/06]
2. The following table shows the production of cars for April/May 2004 for a sample of
40 days.
Production (unit) Number of days
50 – 54 8
55 – 59 12
60 – 64 8
65 – 69 5
70 – 74 4
75 – 79 3
Calculate the Standard deviation (c) Mode and interpret its meaning.
(a) Mean (b) [PSPM 05/06]
3. A sample consists of numbers 5, 9, 8, x, 11, 2, 8, 10, 4 and (x – 1) as a mean 7.0 with
constant x. Find the value of x. Hence, find the standard deviation. [PSPM 06/07]
4. The following frequency table shows the consultation time (rounded to the nearest
minute) needed by a doctor for a patient in a day.
Consultation Time (minutes) Number of patients
5–9 5
10 – 14 8
15 – 19 9
20 – 24 3
25 – 29 5
(a) Calculate the mean, mode and median
(b) Using the answer in (a), determine the skewness of the data distribution.
[PSPM 06/07]
5. Marks obtained by thirty students in a test are shown in the table below.
Marks Number of students
15 4
16 6
17 5
18 8
19 7
(a) Calculate the mean, median and mode.
(b) State the shape of the distribution of the marks.
[PSPM 07/08]
73
MATHEMATICS 2 (AM025) – PSPM Questions
6. The table below shows the frequency distribution of monthly income for workers in a
certain factory.
Monthly Income (RM) Number Of Workers
400 and less than 500 9
500 and less than 600 15
600 and less than 700 20
700 and less than 800 25
800 and less than 900 18
900 and less than 1000 13
Calculate the standard deviation of the workers’ monthly income. Hence, calculate the
coefficient of variation given that the mean of the workers’ monthly income
RM717.00. [PSPM 07/08]
7. The following table indicates the number of books bought by students in a semester.
Number of books 1 2 3 4 5 6 7
Number of students 2 10 1 4 3 3 2
(a) Find the mean, median and mode.
(b) Determine the percentage of students who bought books more than the mean
value.
[PSPM 08/09]
8. The following table shows the cumulative frequency distribution for the ages (years)
of 80 customers entering a mini market on a particular day.
Ages of customers Cumulative frequency
15-19 8
20-24 20
25-29 38
30-34 58
35-39 70
40-44 76
45-49 80
(a) Calculate the mean and median.
(b) State the skewness of data distribution and give a reason for your answer.
[PSPM 08/09]
9. A survey on 12 household on the amount(RM) spent on food per day in a certain
residential area are given as follows.
18 25 18 38 60 71 22 28 35 35 35 35
Calculate the mean, median and mode. Hence, state the shape of the distribution.
[PSPM 09/10]
74
MATHEMATICS 2 (AM025) – PSPM Questions
10. The following frequency table shows the total sales (RM’00) for 95 dealers of a direct
selling company for the month of March 2009.
Total Sales (RM’00) Number Of Dealers
100 and less than 150 5
150 and less than 200 12
200 and less than 250 18
250 and less than 300 28
300 and less than 350 15
350 and less than 400 10
400 and less than 450 7
(a) Calculate the mean and standard deviation for the dealers’ total sales in March
2009.
(b) If the coefficient of variation for dealers’ total sales of April 2009 is 34.55%,
determine which of the two months whereby the total sales are more
consistent.
[PSPM 09/10]
11. The ages (to the nearest year) of a sample of women giving births at a clinic is given
as follows.
27 28 28 29 30 30 31 32 36 37
Find
(a) the median and mode age of the women.
(b) the percentage of women whose age is older than the mean age.
[PSPM 10/11]
12. Table below shows the amount (RM) spent by a random sample of customers at a
grocery store.
Amount (RM) Number of Customer
40 and less than 60 5
60 and less than 80 7
80 and less than 100 7
100 and less than 120 18
120 and less than 140 23
140 and less than 160 14
160 and less than 180 10
180 and less than 200 16
Find (a) the mean (b) the mode (c) the standard deviation
the amount spent by the customer. [PSPM 10/11]
13. The blood types O, A, B and AB for 50 randomly selects students are given as below.
O AB A O A O A A A A
O A AB A O A O O O O
OB O AOBOAAB
AO B OAAAOAO
O O A A A B B O O AB
(a) Construct a frequency distribution table and find the mode for the given data.
(b) Can we find the mean and median value for this set of data? Justify your
answer. [PSPM 11/12]
75
MATHEMATICS 2 (AM025) – PSPM Questions
14. The following frequency distribution table shows the speed in kilometers per hour
(km/h) for 100 cars travelling at KESAS highway.
Speed (km/h) Frequency
80 - 89 14
90 - 99 41
100 - 109 21
110 - 119 16
120 - 129 8
Calculate the mean, mode, median and standard deviation for the speed of the cars.
[PSPM 11/12]
15. Marks obtained by 10 students for a quiz are as follows.
15 12 10 13 8 17 20 8 5 11
Calculate the mean, median and mode of the students marks. Hence, state the shape of
the data distribution. [PSPM 12/13]
16. The table below shows the annual income (RM’000) of 200 workers at ABX
company.
Income (RM’000) Number Of workers
60-64 20
65-69 10
70-74 40
75-79 50
80-84 40
85-89 40
(a) Calculate the mean, median and standard deviation of the workers’ annual
income.
(b) Given the coefficient of variation for ABY company is 8.75%. Determine the
annual income of workers from which company is more consistent.
[PSPM 12/13]
17. The following data represent the number of pencils bought by nine students from a
book store in campus:
5 m −1 1 4 13 11 m 10 4
where m is a constant. If the mean number of pencils bought is 7, find the value of m.
Hence, obtain the median and standard deviation. [PSPM 13/14]
18. The table below shows the cumulative frequency distribution for the number of
children who successfully completed a crossword puzzle within the stated time, x
interval.
Time (in minutes) 2 4 6 8 10 12
Cumulative frequency 4 8 11 13 18 20
Calculate
(a) the mean, median and mode.
(b) standard deviation.
[PSPM 13/14]
76
MATHEMATICS 2 (AM025) – PSPM Questions
19. The following data is arranged in an ascending order
5,8,(3m + 2),15,(7m − 4), 22, 26,32
If the median is 16, determine the value of m. Hence, the mean and standard
deviation. [PSPM 14/15]
20. (a) Complete the following frequency distribution table which shows the length of
babies(cm) at birth at Hospital A.
Length (cm) Frequency Cumulative Class Mid -Point
frequency
35-40 1
40-45 2
8
13
12
4
(b) Calculate the mean, median and standard deviation for the length of babies
(cm) at birth at Hospital A. Hence, state with reason the shape of the data
distribution.
(c) Calculate the coefficient of variation for the given data. If the coefficient of
variation for babies’ length at Hospital B is 15.2%, which hospital’s data is
more disperse?
[PSPM 14/15]
21. The number of study hours per week spend in the library of 10 students, are given as
follows:
3 1.5 1.5 5 3 q 7 1.5 5 9
(a) Given mean is 4.2, find the value of q.
(b) Find the median and mode. Hence, determine the skewness of the distribution.
[PSPM 15/16]
22. The following frequency table shows the sales value (RM’000) for 95 distributors of a
direct selling company for March 2015.
Sales (RM’000) Number of Distributors
100 and less than 150 5
150 and less than 200 12
200 and less than 250 18
250 and less than 300 28
300 and less than 350 15
350 and less than 400 10
400 and less than 450 7
Total 95
(a) Calculate the mean and standard deviation for the distributors’ sales value for
March 2015.
(b) If 50% of the distributors’ sales value is at least RM d, determine the value of
d.
(c) Determine the distributors’ sales value that is more consistent between March
2015 and April 2015 if the coefficient of variation for distributors’ sales value
for April 2015 is 34.55%.
[PSPM 15/16]
77
MATHEMATICS 2 (AM025) – PSPM Questions
23. A survey on the total daily expenditure (RM) on foods has been conducted on 20
households in a certain residential area. The following results are obtained as follows.
25 38 40 18 22 45 35 28 24 50
42 65 60 35 40 18 25 30 35 55
For the given data,
(a) state the variable of the survey and determine whether it is qualitative or
quantitative.
(b) find the mean, median and mode. Hence, describe the skewness of the data.
[PSPM 16/17]
24. Suppose a market researcher surveyed a sample of 170 listeners who liked pop music
and obtained the following distribution.
Age (in years) Number of Listeners
15 – 19 9
20 – 24 16
25 – 29 27
30 – 34 44
35 – 39 42
40 – 44 23
45 – 49 7
50 – 54 2
Total 170
(a) Calculate the mean, median, mode and standard deviation for the above
distribution. Hence, state the shape of the data distribution.
(b) Another researcher carried out the same survey and obtained the coefficient of
variation for the age of people listening to pop music is 30%. Which survey
data is more dispersed?
[PSPM 16/17]
25. A sample of 15 seedlings has the following heights (in mm) as follows:
54 21 39 23 49
30 28 21 39 37
40 27 12 34 39
Calculate the mean, median and mode of the seedling’s height. Hence, verify and state
the shape of the data distribution. [PSPM 17/18]
26. A company gives an aptitude test to 100 applicants that applied for the position of a
senior technician. The scores are as follows:
Score 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100 Total
Frequency 6 24 35 20 15 100
For the above frequency distribution, calculate the
(a) mean, median and mode.
(b) standard deviation.
(c) coefficient of variation.
[PSPM 17/18]
78
MATHEMATICS 2 (AM025) – PSPM Questions
27. The given data represent the number of kilometres per litre for 20 four-wheel-drive
vehicles in an expedition.
4.8 6.8 4.8 5.6 6.4 7.2 6.4 7.2 4.8 6.4
6.8 6.0 6.0 6.4 4.8 6.0 6.4 6.4 4.8 5.6
(a) Find the mean, median and mode. Hence, explain the skewness of the data
distribution.
(b) Calculate the standard deviation. Hence, compute the coefficient of variation
of the data.
(c) The coefficient of variation for a second set of data is 15.78%, determine
which set of data is more consistent.
[PSPM 18/19]
28. The following table shows the frequency distribution for the age of customers (year)
entering a mini market on a particular day. Calculate the mean, mode, median and
standard deviation. Hence, determine the coefficient of variation of the data
distribution.
Age of Customers Frequency
15 – 19 8
20 – 24 12
25 – 29 18
30 – 34 20
35 – 39 12
40 – 44 6
45 – 49 4
[PSPM 18/19]
ANSWERS TOPIC 6 (PSPM)
1. p = 4
2. (a) x = 61.25 (b) s = 7.642
(c) Mode = 57 , The most frequent output on April 2004 is 57 units.
3. x = 7.0 standard deviation = 2.789
4. (a) Mean = 16.167 ; mode = 15.214 ; median = 15.611
(b) Positively skewed
5. (a) Mean =17.26, Median =17.5 , Mode =18 (b) Negatively skewed
6. (a) r = RM605.00 (b) Standard deviation =RM 149.11, COV = 20.8%
7. Mean, 3.52 @ 88 , Median x = 3 ,Mode, xˆ = 2 , 48% bought books more than mean.
25
8. (a) 30.125 , Median in the 4th Class, Median= 30
(b) Right skewed since Mean>Median.
9. Mean = RM 35, Median = RM 35.00 , Mode = 35 , symmetric
− (b) 28.43%
10. (a) x = 274.47 , sd = 78.039
11. (a) Median = 30 ,Mode = 28 and 30 (b) Mean = 30.8 ,
The percentage = 40%
12. (a) RM131.80 (b) RM127.14 (c) RM 39.50
79
MATHEMATICS 2 (AM025) – PSPM Questions
13. (a) Mode is blood “O”
(b) No, we cannot find the mean and median for this set of data because the blood
types is a qualitative variables
14. 100.8 , 95.24 , 98.28 , 11.517
15. x = 119 = 11.9 , Median location= 5.5th x = 11.5, x = 8 positively skewed.
10
16. (a) Annual mean salary of workers is RM77000
Median annual salary is RM77500
The standard deviation of the annual salary is RM 7601.90
(b) Coefficient of variation for ABX= 9.87 .The company of ABY is more
consistent compare with ABX company.
17. m = 8 , x = 7 , s = 3.87
18. (a) Mean, x = 5.6 or 28 , Median, x = 5.33 , Mode, xˆ = 9
5
(b) s = 3.5
19. m = 3 , x = 17 s = 9.2273
20. (b) 53.125cm,53.462cm, 5.7943
X x → almost symmetric or slightly skewness
(c) COV (Hospital A) = 10.91%
Hospital B is more dispersed.
21. (a) q = 5.5 (b) Median = 4 , Mode = 1.5 , Positively skewed
22. (a) 274.47 (RM '000) , 78.04 (RM '000)
(b) d = 272.32 (RM '000) (c) cvMarch = 28.43%
23. (a) Variable : Total daily expenditure on foods (quantitative variable)
(b) RM 36.50, RM 35, RM 35 , the data is positively skewed
24. (a) x = 32.91 , x = 33.25 , xˆ = 33.75 , s = 7.65 , skewed to the left.
(b) 23.25% , the another researcher’s survey data is more dispersed.
25. x = 32.87mm , x = 34mm , xˆ = 39mm , skewed to the left.
26. (a) x = 52.8, x = 51.43, xˆ = 48.46 (b) 22.567 (c) 42.74%
27. (a) x = 5.98 , Median = 6.2 , Mode = 6.4 , Negatively skewed
(b) s = 0.8154 , Coefficient of Variation = 13.64%
(c) The first data is more consistent compare to second data.
28. x = 30.13 , Mode = 30.5 , Median, 30
Std. dev, s = 7.93 , Coefficient of Variation = 26.32%
80
MATHEMATICS 2 (AM025) – PSPM Questions
T 7 POPIC : ROBABILITY
PSPM QUESTIONS
1. Given events A and B with P (A) = 3 , P (A|B) = 3 and P (B|A) = 2 . Find
45 5
P (A B) and P (B|A') . [PSPM 05/06]
2. A box contain 7 black marbles and 17 white marbles.
(a) If two marbles are taken at random without replacement, find the probability
that at least one marble is black.
(b) If three marbles are taken at random without replacement, find the probability
that two marbles are black.
[PSPM 06/07]
3. Given events A and B with P (A) = 0.4, P (B|A) = 0.7 and P (A' B) = 0.3 . Find
(a) P (A B)
(b) P (B)
(c) P (A B)
(d) State with a reason whether events A and B are independent.
[PSPM 06/07]
4. A student could travel to campus using three routes, A, B or C. The probability that
the student chooses route A or B are 1 or 2 respectively. The probability that the
45
student arriving at the campus on time if he chooses route A, B or C are, 3 , 1 or 4
42 5
respectively.
(a) Sketch the tree diagram for the above data.
(b) Calculate the probability that the student arrives at the campus on time.
(c) What is the probability that the student will arrive on time at the campus if he
chooses route B?
(d) Given that the student arrives late at the campus, what is probability that he
chooses route C?
[PSPM 07/08]
5. A survey on preferences of drink has been conducted on one hundred households in a
certain town. The survey result shows that 60 and 40 households are of higher and of
lower income groups respectively. Out of 65 households who prefer brand A, 35 of
them are of higher income group. If one household is selected at random, find the
probability that
(a) the household comes from higher income group or prefers brand A.
(b) the household prefers brand A given that it comes from a higher income group.
Hence, determine whether the household from the higher income group is
independent of brand A preferences.
[PSPM 07/08]
81
MATHEMATICS 2 (AM025) – PSPM Questions
6. The probability for a person in a certain village being infected by SARS virus is 0.7. A
total of 20% of the people who have been infected by SARS virus are free from any
symptoms. While a total of 10% of those who are not infected do show the symptoms
of SARS virus.
(a) What is the probability that a villager is free from any symptoms?
(b) A villager has been tested and is free from any symptoms. What is the
probability that he has not been infected by the SARS virus?
[PSPM 08/09]
7. Given event A is a defect of type A and event B is a defect of type B that occurs on an
electrical device produced by a factory. The probability of A to occur is 1 and the
8
probability of both A and B to occur is 1 . If A and B are independent events, find
20
the probability that an electrical device produced.
(a) has defect of type B.
(b) has defect of type A or type B.
(c) has defect of type B and not type A.
(d) do not have both defects of type A and B.
(e) has defect not type B, if it is known that it has defect not of type A.
[PSPM 08/09]
8. Students at a college have to register their subjects online. For the second session of
2009, 30% of the students use cyber café computers, 40% use their personal computers
and the rest use the college laboratory computers to register. It is found that of those
using cyber cafes to register, 55% fail to register and of those who use personal
computer, 10% fail to register. Result shows that 95% of the students who use the
college computer laboratory register successfully.
(a) Draw a probability tree diagram to represent the above information.
(b) Obtain the probability that a student fails to register and uses cyber café
computer or personal computer.
(c) Given that a student fails to register, what is the probability that the student
uses the personal computer to register?
(d) Given that a student register successfully, what is the probability that the
student uses the cyber café to register?
[PSPM 09/10]
9. A factory produces certain item using three machines A, B and C with probability
1 , 1 and 1 respectively. The percentages of defective items produced by machine A
4 3 12
is 8%, machine B is 15% and machine C is 12%. An item is selected at random.
(a) Find the probability that it is a defective item.
(b) It is found to be defective. What is the probability that it is produced by
machine A?
[PSPM 10/11]
82
MATHEMATICS 2 (AM025) – PSPM Questions
10. A business reporter forecasts the performance of a company listed on a stock exchange
by looking at the daily share prices. He classifies the performance in a day as ‘no
change’, ‘increase’ or ‘decrease’. The following is the record of the forecast and the
actual performance for 270 trading days.
Forecast Actual Performance
Performance
No change No change Increase Decrease Total
82 114
Increase 21 20 12 83
Decrease 3 73
106 45 17 270
Total
27 43
92 72
A day is randomly selected from the 270 days it was traded. Obtain the probability that
(a) the forecast is correct.
(b) the chosen day is forecasted as an ‘increase’ but the actual performance is ‘no
change’.
(c) the forecast is correct, given that the forecast is ‘no change’.
(d) it was forecasted as an ‘increase’, given that the forecast is correct.
[PSPM 10/11]
11. A factory has 100 workers where 60% of them are males. Sixteen of the male workers
are non-Malay, while 36 of the female workers are Malays.
(a) Construct a contingency table to illustrate the above information.
(b) A worker is selected at random,
(i) what is the probability that the worker is Malay?
(ii) what is the probability that the worker is female or Malay?
(iii) what is the probability that the worker is male, if it is known that the
worker is non-Malay?
[PSPM 11/12]
12. Two bags marked M and N contain identical pen drives. Bag M contains 10 green pen
drives and 8 red pens drives. Beg N contains 7 green pen drives and 6 red pen drives.
A pen drive is randomly selected from bag M and put into bag N. Then, another pen
drive is randomly selected from bag N.
(a) Draw a tree diagram to represent the above information.
(b) If a red pen drive is selected from bag N, find the probability a green pen drive
is selected from bag M.
[PSPM 12/13]
13. A and B are two events such that P(A) = 0.1, P(B) = 0.5, P( A B) = 0.1 and
P(A B) = 0.55 . Find
(a) (i) P(A B)
(ii) P(A' B' )
(b) Determine whether
(i) A and B are independent.
(ii) A and B are mutually exclusive.
[PSPM 12/13]
83
MATHEMATICS 2 (AM025) – PSPM Questions
14. Events E and F are independent such that P(E) = 2 and P(E F) = 1 . Calculate
33
(a) P(F) .
(b) P(E F)'.
[PSPM 13/14]
15. A new bus terminal has been operating for two months. A record of 500 buses from
Johor Bahru and Pulau Pinang gives the following information on their arrival status
(early, on time or late).
50 buses arrived early
275 buses arrived on time
150 buses from Johor Bahru arrived on time
285 buses from Johor Bahru
60 buses from Pulau Pinang arrived late
Suppose a record of the bus using the terminal is chosen at random. Calculate the
probability that the bus is
(a) from Penang if it is known that has arrived early.
(b) from Johor Bahru or it has arrived early.
(c) arrives late from Johor Bahru or the bus is on time from Pulau Pinang.
(d) arrives early given that the bus is from Penang or is late given that the bus is
from Johor Bahru.
[PSPM 13/14]
16. A and B are two events such that P( A) = 3 , P(B) = 5 and P(A B)' = 1 .
16 8 4
(a) Show that A and B are not mutually exclusive event.
(b) Find P( A B).
[PSPM 14/15]
17. A social worker wants to distribute 14 red and 16 green T-shirts to an orphanage. The
T-shirts comes in small, medium and large sizes. Among the green T-shirts, six are of
small size and six are of large size. Whilst, among the red T-shirts, five are of medium
size. There are 12 T-shirts of small size.
(a) Construct a contingency table based on the given information.
(b) A T-shirt is selected at random, what is the probability that it is
(i) not green given that it is of large size?
(ii) neither red nor medium size.
[PSPM 14/15]
18. The students at a certain university have to register subjects online. For 2015 intake,
30% of the students used cyber café computers, 40% used personal computers and the
rest used university computers. 55% of students who used the cyber café computers
and 10% of students who used the personal computer failed to register online.
However, 95% of the students who used the university computers were able to
register successfully.
84
MATHEMATICS 2 (AM025) – PSPM Questions
(a) Draw a tree diagram to represent the above information.
(b) Find the probability that the student used personal computer and failed to
register.
(c) Obtain the probability that a student failed to register was using either cyber
café computer or personal computer. Hence, find the probability that the
student failed to register.
(d) Find the probability that a student used the cyber café computer or able to
register successfully.
[PSPM 15/16]
19. Suppose P ( A B) = 0.2 , P( B) = 0.8 and P( A B ') = 0.4 . Calculate P( A) . Hence,
determine whether A and B are independent. [PSPM 16/17]
20. A group of 200, 100 and 200 students from the matriculation, diploma and STPM
programme respectively, were interviewed for the placement based on the degree
chosen by them. Out of 200 males students, 80 are from the matriculation programme
and 90 are from the diploma programme.
(a) From the above information, construct a tree diagram.
(b) Using the tree diagram, if one student is selected randomly, find the
probability that
(i) the student is male and from the STPM programme.
(ii) the student is from the STPM programme.
[PSPM 16/17]
21. A group of 100 people were classified into two categories; wore glasses and left-
handed. From this group, 36 wore glasses, 38 were left-handed, 17 wore glasses and
were left-handed. A person was selected at random from this group. Find the
probability that this person
(a) wore glasses but was not left-handed.
(b) was classified in only one of the category.
[PSPM 17/18]
22. Given that P ( A B) = 0.25 , P ( A) = 0.5 and P ( A C ') = 0.9 . Events A and C are
mutually exclusive. Based on the information, find
(a) P (C ) . Hence, calculate P ( A B)
(b) P ( A' B) . Hence, calculate P ( B | A')
[PSPM 17/18]
23. The events A and B are independent with P ( A) = x , P ( B) = ( x + 0.3) and
P ( A B) = 0.04 . Determine the value of x. Hence, find P ( A B) ' and P ( A' B) .
[PSPM 18/19]
85
MATHEMATICS 2 (AM025) – PSPM Questions
ANSWERS TOPIC 7 (PSPM)
1. 19 , 4
20 5
2. (a) 35 (b) 357
69 2024
3. (a) 0.28 (b) 0.58 (c) 0.7
(d) Events A and B are not independent
4. (b) 267 (c) 1 (d) 4
400 2 19
5. (a) 9 (b) 7 (c) not independent
10 12
6. (a) 0.41 (b) 27
41
7. (a) 0.4 (b) 19 (c) 0.35 (d) 0.525 (e) 0.6
8. (b) 0.205 40
(d) 0.1731
(c) 2
11
9. (a) 2 (b) 1
25 4
10. (a) 17 (b) 7 (c) 41 (d) 9
27 90 57 34
11. (b) (i) 0.8 (ii) 0.84 (iii) 4
5
12. (b) 0.5712 (ii) P(A' B' ) = 0.45
13. (a) (i) P(A B) = 0.05
(b) (i) A and B are independent events.
(ii) A and B are not mutually exclusive.
14. (a) 0.5 (b) 1 (c) 0.48 (d) 0.543
15. (a) 0.6 6
(b) 0.63
16. (a) Shown (b) 1
10
17. (a) Contingency Table
(b) (i) 1 (ii) 2 or 0.4
35
86
MATHEMATICS 2 (AM025) – PSPM Questions
18. (a) 0.55 F
C F'
0.3 0.45 F
0.1 F'
0.4
L
0.9
0.3 0.05 F
U
(b) 0.04 0.95 F '
(c) 0.205 , 0.22
(d) 0.945
19. P ( A) = 0.24 A and B are not independent
20. (a) tree diagram
(b) (i) 0.06 (ii) 0.4
(b) 2
21. (a) 19
100 5
22. (a) P (C ) = 0.1, P ( A B) = 0.75
(b) P ( A' B) = 0.25, P ( B | A') = 0.5
23. x = 0.1 , P ( A B) ' = 0.54 , P ( A' B) = 0.36
87
MATHEMATICS 2 (AM025) – PSPM Questions
TOPIC 8 : RANDOM VARIABLES
PSPM QUESTIONS
1. A discrete random variable X has probability distribution as shown in the following
table.
x 01248
P(X = x) 1 k 1 1 1
16 4 8 16
Find
(a) the value of k if E ( X ) = 2
(b) Var ( X )
(c) Var (3X + 5)
[PSPM 05/06]
2. The continuous random variable X has the probability density function,
t , −1 x 0
f ( x) = t (1+ 6x) , 0 x 1
0 , otherwise
where t is a constant.
(a) Show that t = 1
5
(b) Evaluate P − 1 X 1
2 2
(c) Find E ( X )
( )(d) Calculate the standard deviation of (2X − 3) if E X 2 = 13
30
[PSPM 05/06]
( )3. A discrete random variable X has E ( X ) = 8 and Var ( X ) = 4 . Calculate E X 2
and E (2X −1)2 .
[PSPM 06/07]
4. A continuous random variable X has the probability density function,
a , 0 x9
f ( x) = x
0 , otherwise
(a) Determine the value of a
(b) Evaluate P (1 X 4)
(c) Calculate E ( X )
88
MATHEMATICS 2 (AM025) – PSPM Questions
(d) Find Var ( X ) . Hence, calculate Var (2x − 3)
[PSPM 06/07]
5. A discrete random variable X has the following probability distribution function,
X −1 0 1 2
P(X = x) q13 r
48
( )Given E ( X ) = 3 and E X 2 = 3 , find
42
(a) the values of q and r (b) E ( X − 3)2
[PSPM 07/08]
6. A continuous random variable X has the probability density function,
kx2 , 0x2
, 2x4
f ( x ) = 1 , otherwise
4
0
(a) Show that k = 3
16
(b) Calculate
(i) E ( X )
(ii) P 4 X 3
5
( )(iii) Var (3x − 5) given that E X 2 = 118
15
[PSPM 07/08]
7. The probability density function of a random variable Y is given as
f (y) = ky 2 , 0 y 2,
2
0 , otherwise.
(a) Show that k = 3
4
(b) Hence, calculate E(Y −1)2 .
[PSPM 08/09]
89
MATHEMATICS 2 (AM025) – PSPM Questions
8. The probability distribution of a discrete random variable X is tabulated as follows:
x −1 0 1 23 4
1k 1
P(X = x) 12 12 k −1 1 1 6
12 12 12
(a) Determine the value of k .
(b) Find P (0 x 3)
(c) Calculate E ( X ) and E(X − 2)2
(d) Hence, calculate Var(3x − 2)
[PSPM 08/09]
9. The probability density function for discrete random variable X is as follows.
P( X = x) = kx2 , x = −1,1, 2
k , x = 3, 4
( x +1)
Find the value of k and hence determine the value of E 4( X −1) .
[PSPM 09/10]
10. The probability density function of a continuous random variable X is given as
f ( x) = k ( x −1)2 , 0 x 1
, otherwise
0
Determine the value of k . Hence,
(a) find the cumulative distribution function F ( X ) for X.
(b) calculate P X 1
2
(c) evaluate E ( X +1)
[PSPM 09/10]
11. The number of childbirths X a career woman may have is given by the probability
mass function, P(X = x) = (6 − x)(1+ x) , x = 0,1, 2,3, 4
50
(a) Construct the probability distribution table for X.
(b) Find the probability a randomly chosen career woman has 3 childbirths.
(c) Calculate the mean number of childbirths. Hence, determine the probability
that a randomly chosen career woman has more than the mean number of
childbirths.
[PSPM 10/11]
90
MATHEMATICS 2 (AM025) – PSPM Questions
12. The time spent (in hours) by students at a library is a continuous random variable X
with probability density function given by
f ( x) = kx2 ( 6 − x ) ; 0 x6 where k is a constant.
0 ; otherwise
(a) Show that k = 1 .
108
(b) Find the probability that
(i) a randomly chosen student spends at least 3 hours at the library.
(ii) a randomly chosen student spends at most 30 minutes at the library.
(iii) two randomly chosen students both spend at least 3 hours at the library.
(c) Determine the expected time a student spends at the library.
[PSPM 10/11]
13. The probability distribution of a discrete random variable, X is tabulated as follows:
X 0 1234
P(X = x) 1 1 3 1 1
16 4 8 4 16
Calculate
(a) P(1 X 3)
(b) E ( X ) . Hence, determine Var (3 − 2X ) .
[PSPM 11/12]
14. The probability density function for a continuous random variable, X is given as
f ( x) = 1 (kx − x2 ), 0 x6
36
0,
otherwise.
(a) Determine the value of k.
(b) Obtain the cumulative distribution function, F (x).
Hence, find P(X 3) and P(1 X 4).
[PSPM 11/12]
15. The table below shows the probability for a discrete random variable X.
X =x 1 2 3 4 5
0.1
P(X = x) r 0.05 0.2 s
Given E ( x) = 3.2 , find
(a) the values of r and s.
(b) E (3X −1)
(c) Var (1− 3X )
[PSPM 12/13]
91
MATHEMATICS 2 (AM025) – PSPM Questions
16. X is a continuous random variable with probability density function
x +1 , −1 x m
m x3
4
f (x) = 3 − x ,
4
0 , otherwise
(a) Show m = 1
(b) Find E ( X )
(c) F ( x) is the cumulative distribution function for f ( x) . Given F (1) = 1 ,
2
approximate the value of k such that F(k) = 1 . Given the answer correct to 4
4
decimal places.
[PSPM 12/13]
17. The probability density function of a continuous random variable X is
f (x) = 2 (x + 1), 0 x3
15
0,
otherwise
(a) Find E ( 3 X )2 .
(b) Calculate P(X 1).
[PSPM 13/14]
18. The probability distribution of a discrete random variable X is shown in the following
table.
X=x 2 3 4 56
P (X = x) 0.15 0.25 2k 0.1 + 3k 0.4 - 3k
where k is a constant.
(a) Show that k = 0.05
(b) Find the mean of X. Hence, calculate P(X E(X ) −1).
(c) Calculate Var(4X − 3).
[PSPM 13/14]
19. The probability distribution of a discrete random variable Y is shown in the following
table.
y -2 -1 0 12
k+2 1
P(Y = y) k 3 k−2 16 16
8 16 16 [PSPM 14/15]
( )Show that k = 3 . Hence, calculate E 2 Y 2 − Y .
92
MATHEMATICS 2 (AM025) – PSPM Questions
20. The probability density function of a continuous random variable X is given as
cx 2 , 0 x3
f ( x) = 0, otherwise.
(a) Show that c = 1 .
9
(b) Calculate
(i) P(1 x 2).
(ii) Var (3X − 2)
[PSPM 14/15]
21. A random variable X has the following probability distribution:
X =x -2 -1 1 3
P(X = x) 2k 0.5 k 0.2
( )If k = 0.1, find E ( X ) and E X 2 . Hence, calculate Var ( X ) .
[PSPM 15/16]
22. A continuous random variable, X has the following probability density function
c (2 − x), 1 x<2
f (x) = c, 2x4
0, otherwise
where c is a constant.
(a) Show that c = 0.4.
(b) Find
(i) P (1 2X 3)
(ii) E (2X − 5)
( )(iii) Var (10 − 2X ) , given E X 2 = 47 .
6
[PSPM 15/16]
23. A random variable X has probability distribution as follows. 5
0.05
X 01234
P ( X = x) 0.1 0.3 0.4 0.1 0.05
Find the cumulative distribution function. Hence, compute P (1 X 5) .
[PSPM 16/17]
93
MATHEMATICS 2 (AM025) – PSPM Questions
24. (a) The probability distribution function for a discrete random variable X is as
follows.
P( X = x) = kx2 , x = −1,1, 2
k ( x +1), x = 3, 4
Find the value of k and hence, determine the value of E 4( X −1) .
(b) The continuous random variable T has a probability density function given as
1− t, 0 t 1
f (t ) = 1− t, 1t 2
0, otherwise
and illustrated by the graph below.
f (t)
1
1 2 t
Determine k if F (k ) = 0.52 .
[PSPM 16/17]
25. The probability distribution of the discrete random variable Y is given in the following
table.
Y=y 0 1 2 3 4
P(Y = y) 1 1 1 k 1
12 6 4 6
(a) Find the value of k.
(b) Given that E (Y ) = 7 , find Var (3Y +1) .
3
[PSPM 17/18]
26. The total amount of petrol pumped at a petrol station per week, Y is a continuous
random variable (measured in 10000 litres) with a probability density function given
as
f ( x) = ky (1− y )2 , 0 y 1
elsewhere.
0,
(a) Show that the value of k is 12.
(b) Find the probability that the station will pump at most 6000 litres in a
particular week.
(c) Find the probability that the station will pump at least 8000 litres in a
particular week.
94
MATHEMATICS 2 (AM025) – PSPM Questions
(d) Determine E (Y ) .
(e) Find the cumulative distribution function, F ( y) .
[PSPM 17/18]
27. (a) The following table shows the probability distribution for a discrete random
variable, X.
X =x 1 2 3 4 5
P(X = x) 1 1 m 1 1
12 3 3 12
(i) Show that m = 1 .
6
(ii) Determine E ( X ) and Var ( X ) . hence, find Var (4 − 3X ) .
(b) The probability density function of a continuous random variable X is given as
f (x) = x2 , 0 x3
9
0 , otherwise
Calculate P (0 x 1) and E ( X ) .
[PSPM 18/19]
ANSWERS TOPIC 8 (PSPM)
1. (a) k = 1 (b) 7 (c) 63
2 2 2
2. (b) 7 (c) 2 (d) 1.0456
20 5
E( )3.X2 = 68 , E (2X −1)2 = 241
4. (a) a = 1 (b) 1 (c) 3 (d) 36 , 144
6 3 55
5. (a) q = 1 , r = 1 (b) 6
84
6. (b) (i) 9 (ii) 359 (iii) 25.2375
4 500
7. (a) k = 3 (b) 0.4
4
8. (a) k = 4 (b) 0.417 (c) 37 (d) 363
12 16
9. k = 1 , 20
15 3
0, x0
0 x 1 (b) 1 (c) 5
10. k = 3 (a) F ( X ) = x3 − 3x2 + 3x, 8 4
x 1
1,
95
MATHEMATICS 2 (AM025) – PSPM Questions
11. (a)
x 01234
P(X = x) 31 6 61
25 5 25 25 5
(b) 6 = 0.24 (c) E(X ) = 11 = 2.2 , 11 = 0.44
25 5 25
12. (b) (i) 11 or 0.6875 (ii) 5 or 0.0022 (iii) 121 = 0.4727 (c) 3.6
16 2304 256
0, x 0
x2
13. (a) 5 (b) 2, 4 14. (a) k = 6 (b) F (x) = 12 − x3 , 0 x 6, 1 , 2
8 108 2 3
1, x 6
15. (a) r = 0.2, s = 0.45 (b) 8.6 (c) 14.94
16. (b) 1 (c) k = 0.4142
17. (a) 35.1 (b) 0.8
18. (b) 4.2, 0.6 (c) 32.96
19. 5.5
20. (a) Shown (b) (i) 0.259 (ii) 3.0375
( )21. E ( X ) = −0.2 , E X 2 = 3.2 , Var ( X ) = 3.16
22. (a) Shown (b) (i) 3 or 0.15 (ii) 1 (iii) 2.89
20 3
0, x 0
0.1, 0 x 1
0.4, 1 x 2
23. F ( x) = 0.8, 2 x 3
0.9, 3 x 4
0.95, 4 x 5
1, x5
P (1 X 5) = 0.55
24. (a) k= 1, E 4( X −1) = 20 (b) k = 1.2
15 3
25. (a) k=1 (b) 25
3 2
26. (a) shown (b) 0.8208 (c) 0.0272 (d) 4000litres
0, y 0
( )(e) ( ) 2
F y = y 6−8y +3y2 , 0 y 1
1, y 1
27. (a) (i) Shown (ii) E ( X ) = 3.Var ( X ) = 4 , Var (4 − 3X ) = 12
3
(b) 1 , 9 or 2.25
27 4
96
MATHEMATICS 2 (AM025) – PSPM Questions
T 9 S P DOPIC : PECIAL ROBABILITY ISTRIBUTIONS
PSPM QUESTIONS
1. A random variable X is normally distributed with mean µ and variance 2 . Given that
P( X 68.37) = 0.02 and P(X 50.85) = 0.03 , find the µ and .
[PSPM 05/06]
2. (a) Given X ~ B(n, 0.35) with variance 2.275.
(i) Find the value of n.
(ii) Hence, calculate P(X = 5) and P(X 3) .
(b) The monthly expenditure of 800 accounting students in a matriculation college
is normally distributed with mean RM170 and standard deviation RM121.
(i) How many students spend between RM150 and RM200 a month?
(ii) Find the value of r, given that 70% of the students spend more than
RM r a month.
[PSPM 05/06]
3. Given X ~ P0 () , with λ integer and P(X = 0) = 0.0498 . Find the value of λ. Hence,
find P( X 3) .
[PSPM 06/07]
4. (a) A random variable X has a normal distribution with the mean µ and standard
deviation 2. If P(X 12) = 0.0228 , find the value of µ.
(b) A type of antibiotics has a probability of 0.95 to cure a contagious disease.
(i) If 5 patients are given the antibiotics, find the probability that at least
one patient will be cured.
(ii) If 500 patients are given the antibiotics, find the value of n such that
the probability that at least n patients will be cured is 0.9.
[PSPM 06/07]
5. The number of claims approved by an insurance company follows a Poisson
distribution with the mean of 7 claims per week. Find the probability that the
insurance company approves
(a) between 3 to 8 claims in a week.
(b) less than 2 claims on a given day.
[PSPM 07/08]
6. Six out of ten families in a district own a Malaysia made car.
(a) If twenty families are selected at random from the district, calculate the
(i) expected number of families who own a Malaysian made car.
(ii) Probability that from 3 to 7 families do not own a Malaysian made car.
(b) If one hundred families are selected at random from the district, find the
probability that less than 66 families own a Malaysian made car.
[PSPM 07/08]
97
MATHEMATICS 2 (AM025) – PSPM Questions
7. In a shipment by company, probability of obtaining a wooden furniture that does not
meet order specifications is 0.2.
(a) If 20 units of wooden furniture are chosen at random from particular shipment,
(i) State the mean and variance of the distribution for the number of unit
of wooden furniture that do not meet order specifications.
(ii) Calculate the probability that not more than 3 units of wooden
furniture do not meet order specifications.
(b) A sample of 200 units of wooden furniture is chosen from a particular
shipment. Approximate the probability that at most 36 units of wooden
furniture do not meet order specifications.
[PSPM 08/09]
8. (a) According to a poll, 5% of the government staffs invest in a trust fund.
A random sample of 25 government staffs is selected. What is the probability
that at least 2 of them invest?
(b) The total monthly investment for 500 government staff is normally distributed
with the mean RM165 and standard deviation RM200.
(i) Find the number of government staffs whose investment does not
exceed RM100.
(ii) If 83 government staffs invest at least RM Y, find the value of Y.
[PSPM 09/10]
9. The amount spent by college students for lunch in a day at a campus café is normally
distributed with mean and standard deviation RM 2.00. It is found that 2.5% of the
students spend more than RM 12.
(a) Show = RM 8.08
(b) Find the probability that a randomly chosen college student spends between
RM7.50 and RM10.00 for lunch.
(c) Ten college students are chosen at random. Find the probability that
(i) Two of them spend more than RM 12.00 for lunch.
(ii) Not more than two of them spend more than RM 12.00 for lunch.
[PSPM 10/11]
10. A factory is using 12 old machines. The probability that a machine is still functioning
at the end of a day is 0.7. Each machine functions independently. Calculate the mean
and variance for the number of machines that are still functioning at the end of a day.
[PSPM 11/12]
11. One thousand students sat for a Statistics test. The marks obtained are normally
distributed with mean 45 and standard deviation 9.
(a) If the passing mark is 55, estimate the number of students who pass the test.
(b) If 70% of the students pass the test, obtain the minimum passing mark. State
your answer to the nearest integer.
[PSPM 11/12]
12. A study on the reading habit is done in a community. It is found that 15% of its
population read at least one book in a month. Let X represents the number of people
in the community who read at least one book in a month.
(a) If 20 people is chosen at random, find the probability not more than five
people read at least one book in a month.
98
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Buku Tutorial AM025 (Sesi 2021-2022)
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