KLON PSPM SM025

PROGRAM MUAFAKAT SET 2 SM025

UNIT MATEMATIK
KOLEJ MATRIKULASI JOHOR
KEMENTERIAN PENDIDIKAN MALAYSIA

Section A (25 marks)

1. A group of office workers were questioned about their health and 2 were found to take
5

regular exercise. When questioned about their eating habits, 2 said they always eat
3

breakfast and, of those who always eat breakfast 9 also took regular exercise. Find the
25

probability that a randomly selected member of the group

a) always eats breakfast and takes regular exercise. [2 marks]

b) does not always eat breakfast and does not take regular exercise [3 marks]

c) Determine whether always eating breakfast and taking regular exercise

are statistically independent. Justify you answer. [2 marks]

2. A discrete random variable X has a probability distribution table as follows [2 marks]
Value of X 1 2 3 4 [2 marks]
P(X = x) 3k – 1 k k 1 – 2k [2 marks]
[4 marks]
Where k is a constant.
(a) Determine the value of k .

(b) Find E(X) .
Hence, calculate
(i) P(1  X  4)

(ii) E( 2  X )2

3. The measure of intelligence, IQ, of a group of students is assumed to be Normally
distributed with mean 100 and standard deviation 15.

PROGRAM MUAFAKAT SET 2 SM025

a) Find the probability that a student selected at random has [3 marks]
an IQ less than 91. [5 marks]

b) The probability that a randomly selected student has an IQ
of at least 100 + k is 0.7088.Find, to the nearest integer,
the value of k.

Section B (75 marks)
1(.a) Given f1(x)  2x and f2(x)  ln x.

(ai) Without using curve sketching, show that f1(x)  f2(x) intersect on the interval

of  0.1 , 1  [2 marks]

b) Use Newton-Raphson’s method to estimate the intersection point of y  f1(x) and

y  f2 (x) , with initial value x1  1. Give your answer correct to three decimal

places. [5 marks]

2. Find the equation of a circle that is passing through points (1,2), (-1,2) and (0,-1). Hence,
determine its center.
[5 marks]

3. a) Find the equation in standard form of an ellipse which passes through the point (-1,6)
and having foci at (-5,2) and (3,2). State major vertices and length of major and minor.
[10 marks]

b) From the result obtained in part (a), sketch the graph of the ellipse.

[3 marks]

PROGRAM MUAFAKAT SET 2 SM025

4. Given a line l : x  2  t, y  3  4t, z  5  3t , 1  2x  y  7z  53 and
 2  3x  y  z  1. Find

(a) The point of intersection between the line l and the plane 1 [3 marks]
(b) The acute angle between the line l and the plane 1 [4 marks]
(c) The equation of a plane which perpendicular to plane 1 and 2 [6 marks]

5. The cumulative distribution function of a random variable Y is given by

0 ,y2
F (Y )  ay2  by ,2  y  4

 1 y4

Find

a) a and b. [3 marks]
[3 marks]
b) c such that . PY  c  0.88 [4 marks]
[2 marks]
c) median and 70th percentile [8 marks]

d) the probability density function of Y.

e) the EY ,VarY andVar2Y 3

6. The probability of getting a rotten pear in a Value-mart is 0.10. By using

normal distribution as an approximation , find the probability that out of 500

pears in the market,

a)At least 65 pears are rotten [4 marks]
b)At most 50 pears are rotten [3 marks]

PROGRAM MUAFAKAT SET 2 SM025

7. a) A botanist is studying distribution of roses in a field. His field is divided into a number
of equal sized squares. The mean number of roses per square is assumed to be 3. The roses
are distributed randomly throughout the field. Find the probability that, in a randomly
chosen square there will be

i) Exactly 5 roses [2 marks]
ii) Less than 10 roses [2 marks]

b) The botanist decides to count the number of roses, x , in each of 80 randomly selected
square within the field. The results are summarized as below:

 x  295  x 2  1384

i) Calculate the mean and variance of the number of roses. Give your answer to 1

decimal place. [2 marks]

ii) Explain how the answer from part (i) support the choice of a Poisson distribution

as a model. [1 marks]

iii) Using the mean from part (i), estimate the probability that at most 4 roses will be

found in a randomly selected square. [3 marks]

SM025 KOLEJ MATRIKULASI KELANTAN SESSION 2020/21

ANSWER ALL QUESTIONS TIME: 2 HOURS

SECTION A [25 marks]

1. The probability that a student passes Mathematics is 0.4. If the student passes
Mathematics, the probability that the student will pass Physics is 0.7. If the student
fails Mathematics, the probability the student will pass Physics is 0.63. Let M be
event passes Mathematics and F be event passes Physics.
The following tree diagram describes the above information

(a) Calculate the probability that the student passes [2 marks]
(i) Physics. [3 marks]
[2 marks]
(ii) Mathematics if the student passes Physics.
(b) Determine whether the events of a student passing Mathematics and [3 marks]
[2 marks]
Physics are independent. [2 marks]
[3 marks]
2. The probability density function of a discrete random variable X is
[2 marks]
P( X = x) = kx where x = 1,2,3,4,5,6
2

and k is a constant.

k= 2
(a) Show that 21 .

(b) P(2  x  5) .

(c) F (4) .

(d) E( X ) .

3. Assume that the number of emails received by a student daily has a Poisson
distribution with a mean of 5.

(a) (i) Determine the probability that the student receives between 5
and 13 emails daily.

SM025 KOLEJ MATRIKULASI KELANTAN SESSION 2020/21

(i) If the probability of a student receiving not more than m emails in
a day is 0.616, determine the value of m.
[3 marks]

(b) If 15 days are randomly chosen, by using binomial distribution find the
probability that the student receives between 5 and 13 emails for a period of 9
days.

[3 marks]

SECTION B [75 marks]

1. Show that the equation ln x + x − 4 = 0 has a root between 1 and 3. From the

44 Newton-Raphson formula, show that iterative equation of the root is
2.
X n+1 = Xn (5 - ln xn ) .

1 + xn Hence, if the initial value is x1 = 2 , calculate the

root correct to three decimal places

[6 marks]

Determine the vertices and foci of the ellipse 25x2 + 4 y2 − 250x − 16 y + 541 = 0 .
Sketch the ellipse and label the foci, center and vertices.

[5 marks]

3. Show that the line 2y − 5x + 4 = 0 does not intersect the circle

x2 + y2 + 3x − 2y + 2 = 0. Find centre and radius of the circle. Hence, determine

the shortest distance between the line and the circle.

[12 marks]

4. L : x − 1 = y − 3 = z − 2 [4 marks]
Given the line 2 −1 −3 and [3 marks]
[3 marks]
the planes 1 = 2x − y − 2z = 17 and 2 = −4x − 3y + 5z = 10. Find [4 marks]
(a) The intersection point between L and 1.

(b) The acute angle between 1 and 2 .

(c) The acute angle between L and 2 .

(d) Parametric equations of the line that passes through the point

(2,-1,3) and perpendicular to the plane 2.

SM025 KOLEJ MATRIKULASI KELANTAN SESSION 2020/21

5. The probability distribution function of a discrete random variable X is given as

x x = 1,2,3
17 x = 4,5,6,7
x
f (x)=  34 otherwise


0

(a) Calculate P (2  X  5).

(b) Determine the value of ( )Var X . Hence, calculate the standard [4 marks]
[4 marks]
( )deviation of Y = 5X −1
[2 marks]
6. The time taken by a student (in hours) to study is given by a continuous variable X, [2 marks]
with a cumulative density function of [2 marks]
[2 marks]
0 if x  0 [3 marks]
F ( X ) = 1 − k(10 − x)2 if 0  x  10
[5 marks]
1 if x  10 [5 marks]

where k is a constant.
(a) Determine the value of k.

(b) Find P(3  X  9) .

(c) Determine the probability density function of X for 0  x  10 .

(d) Find the median of X.

(e) Obtain the variance of X.

7. In every delivery of cupcakes to a particular restaurant, 30% will be returned due to
not favoured by cupcakes lovers.

(a) Suppose 20 of the cupcakes are randomly selected from a delivery.
What is the probability that at most 5 will be returned?

(b) Suppose the restaurant will be holding an event which requires an
order of 200 cupcakes from the same supplier.
(i) Approximate the probability that between 56 and 62 of the
cupcakes will be returned.

SM025 KOLEJ MATRIKULASI KELANTAN SESSION 2020/21

(ii) If the probability of observing less than n number of cupcakes
among those delivered which are retuned is 0.992, use the normal
approximation to determine the value of n.
[7 marks]

END OF QUESTION PAPER

Final Answer

SECTION A

1. (a) (i) 0.658 (ii) 0.426

(b) Events is not independent.

2. (a) Show

(b) 3

7
(c) 10

21
(d) 91

21 (ii) m = 5
3. (a) (i) 0.382

(b) 0.0483

SECTION B

1. 2.926

( ) ( )2.
( x − 5)2 ( y − 2)2 = 1, vertices = (5,7) ,(5,−3) , foci = 5,2 + 21 , 5,2 − 21

+
4 25

3. Show

Centre =  − 3 ,1 ,r = 5 , shortest distance between line and the circle = 1.39
 2 2

4. (a) (5,1,−4)

(b) 45

(c)
49.11

(d) x = 2 − 4t , y = −1 − 3t , z = 3 + 5t

5. (a) 7

17

(b) 1041 , 4.24
289

6. (a) k = 1
100

(b) 12

25

SM025 KOLEJ MATRIKULASI KELANTAN SESSION 2020/21

(c)  1 1
 5 50
f ( x ) = − x , 0  x  10

 0 , otherwise

(d) m = 2.93

(e) 50

9
7. (a) 0.4164

(b) (i) 0.2964 (ii) n = 76

PROGRAM KARISMATIK 5.0 SM 025
MATHEMATICS UNIT KMM SET A
SESSION 2020/2021
SM025 TIME: 2 HOURS

INSTRUCTION: Answer ALL questions in Part A and Part B

Part A

1. The events A and B are such that P(A) = 0.3, P(B) = 0.4. Evaluate P(AՍB) in each
of the following cases.
(a) A and B are mutually exclusive.
(b) A and B are independent.
(c) P(A|B) = 0.25

2. The discrete random variable X has the following probability distribution.

x 12 345

P(X=x) 0.3 0.2 0.1 a b

where a, b are positive constants.
(a) Show that a + b = 0.4
(b) Given that E(X) = 2.85, obtain a second equation involving a and b. Hence,

determine the value of a and the value of b.

3. Customers arrive at a shop such that the number of arrivals in a time interval of
duration t minutes follows a Poisson distribution with mean 0.2t.

Determine the probability that the number of arrivals between 10.00 a.m. and 10.30
a.m. is

(a) exactly 5
(b) more than 3

Part B

1. = 4 x sec  x  dx .
Given that I 0  3 

Find estimates for the value of I to 4 significant figures using trapezium rule with 5

strips.

“The only way to learn mathematics is to do mathematics-Paul Halmos”

PROGRAM KARISMATIK 5.0 SM 025
MATHEMATICS UNIT KMM
SESSION 2020/2021
SM025 TIME: 2 HOURS

2. (a) A straight line L, which is perpendicular to the line x + y = 3 passes through the
centre of the circle x2 + y2 – 4x – 2y – 3 = 0. Find

(i) The coordinates of the points of intersection of the line L and the circle.

(ii) The equation of the tangents to the circle which is parallel to the line

x + y = 3.

(b) Find the standard form of the equation of the ellipse
36x2 + 49y2 – 432x – 392y = -316.

3. The random variable Y is normally distributed with mean  and variance  2 . It is
found that P (Y > 150) = 0.0228 and P (Y > 143) = 0.9332. Find the values of 
and  .

4. A plane passes through the points A(-1,-1,-4), B(0, 4, 0) and C(1, 3,-2). The line

4  2
   −1,
r =  1  +     meets the plane at point P.

11  3 

(a) Find the Cartesian equation of the plane,

(b) Determine the coordinates of point P.

(c) Calculate the acute angle between the line and the plane.

5. The successful sale of hats in a local shop can be modelled binomially and the
probability of selling a hat (if the customer enters the shop) is 0.3. Find the
probability that the shopkeeper will have at least two sales in the next ten
customers that visit his shop. Calculate the least number of customers needed in
order that the probability of the shopkeeper getting at least one sale is greater than
99%.

6. The time, X hours, in the evening that Bill spends on his homework has probability
density function f given by

k(2x −1) for 1  x  2
 otherwise
f (x ) = 

 0

where k is a constant.
(a) (i) Find an expression in terms of k and x for F(x), valid for 1 ≤ x ≤ 2, where F

“The only way to learn mathematics is to do mathematics-Paul Halmos”

PROGRAM KARISMATIK 5.0 SM 025
MATHEMATICS UNIT KMM
SESSION 2020/2021
SM025 TIME: 2 HOURS

denotes the cumulative distribution function of X.
(ii) Hence, show that k = 1 .

2
(b) Determine

(i) E(X)
(ii) the median of X,
(iii) the probability that, on a randomly chosen evening, Bill spends longer than

1.5 hours on his homework.

7. In a standard IQ test 5.48% of candidates scored above 120 whilst 91.92% of

candidates scored above 105. IQ scores are given as whole numbers

(a) Assuming the distribution to be normal, find the mean and standard deviation
of the scores.

(b) An employer wishes to only employ candidates whose IQ is in the top 3%.
Find the minimum score that these candidates should achieve in their IQ test.

Answers
“The only way to learn mathematics is to do mathematics-Paul Halmos”

PROGRAM KARISMATIK 5.0 SM 025
MATHEMATICS UNIT KMM
SESSION 2020/2021
SM025 TIME: 2 HOURS

Part A

1. (a) 0.70 (b) 0.58 (c) 0.60

2. (b) 4a + 5b = 1.85, a = 0.15, b = 0.25

3. (a) 0.1606 (b) 0.8488

Part B

1. 17.01
2. (a) (i) (4, 3),(0, -1) (ii) x + y = 7, x + y = -1.

(b) (x − 6)2 + (y − 4)2 =1
72 62

3.  = 146,  = 2 .

4. (a) x – y + z = -4 (b) (-2, 4, 2) (c) 67.79o

5. 0.8507, 13

6. (a) (i) F(x) = kx(x −1) (b) (i) 1.5833 (ii) 1.618 (iii) 0.625

7. (a) Mean = 112, SD = 5 (b) 122

“The only way to learn mathematics is to do mathematics-Paul Halmos”

SM025 UNIT MATEMATIK,
KOLEJ MATRIKULASI PULAU PINANG SESI 2020/2021

SET SUPERB_KMPP

SECTION A [25 marks]

This section consists of 3 questions. Answer all questions.

1. In college X, each student has only one brand of smartphones regardless of brand S,

brand H or brand I. Out of a total of 300 students, 120 of them are male students. 10 out
of 70 students who has brand I smartphone are male students. A total of 80 students have

brand S smartphones and 70 female students have brand H smartphones. A student is

chosen at random from the college X.

(a) Construct a contingency table to represent the information above.
[2 marks]

Hence, find the probability that

(b) a male student who has brand H smartphone is chosen. [1 marks]

(c) a male student or a student who has brand S smartphone is chosen. [2 marks]

(d) a female student is chosen, given that she has brand H smartphone. [2 marks]

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

PX  x  c x 5

3 for x 1,2,3,4 .

(a) Show that c  3 . [2 marks]
14
[5 marks]
(b) Determine the mean and variance for Y  7X  20 [3 marks]

(c) Find P X  E(Y).

3. The probability that a person is cured from pneumonia after being given a new type of
medicine is 0.4.

(a) If a sample of 20 patients is randomly selected,

(i) find the mean and standard deviation of patients that will be cured.

(ii) find the probability that 4 to 12 patients will be cured.

[5 marks]

(b) If a sample of 100 patients is randomly selected, find the probability that less than

66 patients will not be cured. [3 marks]

1

SM025 UNIT MATEMATIK,
KOLEJ MATRIKULASI PULAU PINANG SESI 2020/2021

SECTION B [75 marks]
This section consists of 7 questions. Answer all questions.

1. Given two functions y  ex and y  2  x.
(a) Show that one of the real roots of the equation lies between 0 and 1.

(b) By using the Newton-Raphson method, determine the real root that lies between 0
and 1, correct to three decimal places.
[6 marks]

2. (a) The circle C has radius 5 and touches the y-axis at the point (0, 9) in the second
quadrant.

(i) Write down an equation for the circle C.
(ii) Verify that the point (-10,9) lies on C.
(iii) Find an equation of the tangent to C at the point (-10,9).

(b) The ellipse x2  y2  1 is shifted 4 units to the right and 3 units up to generate the
16 9

new ellipse.

(i) Write the equation of the new ellipse and state its center.
(ii) Find the foci and vertices of the new ellipse.
(iii) Sketch the new ellipse.

[15 marks]

2

SM025 UNIT MATEMATIK,
KOLEJ MATRIKULASI PULAU PINANG SESI 2020/2021

3. Consider two straight lines

 L1 :r 11i  2 j 17k   2i  j  4k and
 L2 :r  5i 11j  ak   bi  2 j  2k

where  and  are parameters, and a and b are constants. Given that L1 and L2 are

perpendicular.
(a) Show that b  3 .

Given further that L1 and L2 intersect, find
(b) the value of a.
(c) the coordinates of the point of intersection.
(d) an equation of the plane containing L1 and L2 in the Cartesian form.

[15 marks]

4. A mobile company sells X smartphones a day with the probability function

P X  x c(25 - x) where c is a constant. It is known that the company will not be

4
selling more than 6 smartphones per day.

(a) Determine the value of constant c . [3 marks]
[2 marks]
(b) Find E(X). [5 marks]

(c) Evaluate P X  E(X ) 3 .

3

SM025 UNIT MATEMATIK,
KOLEJ MATRIKULASI PULAU PINANG SESI 2020/2021

5. The continuous random variable X has a cumulative distribution function

0 , x3
 1 , 3  x  12
F ( x)  9 x  1 ,
c x  12

 1



(a) Find the value of c and k if P X  k  0.65 [5 marks]
[2 marks]
(b) Determine the probability density function of X. [5 marks]
(c) Evaluate E(3X 2  2X 1)

6. A farmer found that the weights of Musang King durian on his farm have a normal
distribution with mean 2.15 kg and standard deviation 0.55 kg.

(a) 250 Musang King durians are chosen at random. Estimate the number of Musang

King durians which have a weight between 1.95 kg and 2,25 kg. [3 marks]

(b) Musang King durian will be rejected to be exported if the weight is less than 1.80 kg.

If a basket has 200 Musang King durians, estimate the number of durians that will

not be exported. [3 marks]

(c) What is the probability that more than 4 Musang King durians will not be exported is

6 durians are chosen at random. [3 marks]

4

SM025 UNIT MATEMATIK,
KOLEJ MATRIKULASI PULAU PINANG SESI 2020/2021

7. In 15 true or false questions, Ali knows exactly the correct answer for 5 questions. For the
rest of the questions, he will toss a fair coin in order to determine the answer.

(a) Find the probability that he managed to answer more than 13 questions correctly.

[4 marks]

(b) Calculate the mean and standard deviation. [4 marks]

SECTION A S ANSWERS
1. (a) 30
50 H I Total
M 80 80 10 120
F 70 60 180
Total 150 70 300

(b) 4
15

(c) 17
30

(d) 7
15

2. (a) - (b) 2, 55 (c) 1
7
3. (a) (i) mean=8, standard deviation = 2.191
(b) 0.8686 (ii) 0.9630

5

SM025 UNIT MATEMATIK,
KOLEJ MATRIKULASI PULAU PINANG SESI 2020/2021

SECTION B

1. (a) - (b) x = 0.443

2. (a) (i) x  52   y 92  25 (ii) - (iii) x  10

(b) (i)  x  42   y  32  1, centre: 4,3

16 9

   (ii) Foci: 4  7,3 & 4  7,3 , Vertices: 0,3 &8,3

(iii)

3. (a) - (b) a 1 (c) 1,7,3 (d) 10x 16y  z 125
4. (a) c  4
(b) E  X   23 (c) 85
91 91
13

5. (a) c  3, k  123 @6.15 (b) f (x)  1 , 3  x  12
20 9
0 , otherwise

(c) 203

6. (a) 0.212, no. of durians = 53 (b) 0.2611, no. of durians = 52
(c) 0.0057

7. (a) 0.0194 (b) mean = 10, standard deviation = 1.8257

6

Model PSPM Set A SM025 Mathematics 2
Session 2020/2021

SECTION A [25 marks]

This section consists of 3 questions. Answer all questions.

1. The letters in the word PAPAYA are to be arranged. A word can be considered formed
without being meaningful. The events Q, R and S are defined as follows.
Q : The word starts and ends with an A.

R : All the P’s in the word are kept together.

S : All the A’s in the word are kept together. 4 marks
Find

(a) P Q and P R.

(b) P Q  R. 2 marks

2. The discrete random variable X has probability

P X  x  052xx,kk , x  2,3,5,
, x  7,8,10,
otherwise.

(a) Find the value of k. 2 marks
2 marks
(b) Construct the table of cumulative distribution of X. 2 marks
4 marks
(c) Find P 4  X  8.
(d) Determine EX and VarX.

3. A lorry owner of car rental company has a fleet of six similar lorries, which he rents out of

the customers on a daily basis. It is known that from past experience, the daily demand for

the lorry has a Poisson distribution with mean 3.56.

(a) Calculate correct to 3 decimal places the probability that in any day

(i) at least two lorries are not rented out. 3 marks

(ii) the demand is not fully met. 2 marks

(b) Find the probability that exactly one lorry is not rented out in exactly two out of 4

days. 3 marks

1 of 4

Model PSPM Set A SM025 Mathematics 2
Session 2020/2021

SECTION B [75 marks]

This section consists of 7 questions. Answer all questions.

1. Given a function y  2x  1 and y  15e2x.

(a) By using change sign, show that the equation 2x  1  15e2x has the solution in the

interval 0,1. 2 marks

(b) By using the Newton-Raphson method and initial value of 0.6, find the root of the

equation 2x  1  15e2x correct to three significant figures. 4 marks

2. Find the equation of a circle that passes through the points A 0,1, B 4,1 and C 1,2.
7 marks

3. A shipment of 7 television sets contains 2 defective sets. A hotel makes a random purchase

of 3 of the sets. If X is the number of defective sets purchased by the hotel, find

(a) the probability of X. 3 marks

(b) P X  1. 2 marks

(c) graph the probability distribution. 2 marks

4. (a) Express the equation of ellipse 4y y  6  36  4x  x2 in standard form. Hence,

determine its centre, vertices and foci. 6 marks

(b) Find the equation of a circle that passing through point 5, 4 and its centre is the

vertex of the parabola y2  8x  6y  25  0. 5 marks

5. The points P 2,2,1 and Q6,7,1 lie on the plane  with Cartesian equation

cx  4y  12z  k, where c and k are constants. 4 marks
 

(a) Find OP and unit vector parallel to vector OP .

(b) Determine the equation of a straight line L, which is perpendicular to  and passing

through P.

7 marks

(c) Given that the shortest distance from point R to plane  is 26 units, where point R lies

on a straight line L. Find the length from point R to Q. 4 marks

2 of 4

Model PSPM Set A SM025 Mathematics 2
Session 2020/2021

6. The following is the graph of the probability density function of continuous random variable

X which only takes values of −2 to 4 inclusive.

(a) Show that a  1 . 2 marks
3
3 marks
(b) Find the probability density function of X. 4 marks
6 marks
(c) Calculate mean of X.

(d) Find P  X  3EX  4 .

7. A van can take either route M or route N for a particular journey. If route M is taken, the

journey time is normally distributed with mean 46 minutes and standard deviation of 10
minutes. However, if route N is taken, the journey is normally distributed with mean 

minutes and standard deviation of 12 minutes.
(a) Find the probability that the journey take more than 60 minutes for route M.

3 marks

(b) The probability that the journey takes less than 60 minutes is 0.85 for route N, find the

value of . 5 marks

(c) The van sets out at 0600 and needs to arrive before 0700. Determine the route should it

take and justify your answer. 4 marks

(d) On five consecutive days, the van set out at 0600 and takes route N. Find the probability

that it arrives before 0700 on all five days. 2 marks

END OF QUESTION PAPER

3 of 4

Model PSPM Set A SM025 Mathematics 2
Session 2020/2021
Answers

Section A 1
11 (b)

1. (a) , 10
53

2. (a) k  10 (b) x 2 3 5 7 8 10 11 154 8943
1 1 (c) (d) ,
F x 10 4 1 16 4
2 25 5 20 25 1250

1

3. (a) 0.714, 0.0701 (b) 0.136, 0.0828

Section B
1. (a) DIY (b) 0.855

2. x2  y2  4x  1  0

3. (a) P X  x  777124 , x  0, (b) 6 (c) DIY
, x  1, 7
, x  2.

   4. (a) x  22  y  32  1, C 2,3, V 4,3, V 0,3, F 2  3,3 , F 2  3,3
4

4. (b) x  22  y  32  10

5. (a) 3, 2 i  2  1 or  2 i  2 j 1 (b) r  2i  2j  k  t 3i  4j  12k (c) 777 units
3 j 3 k 3 3  3 k  

3

6. (a) DIY (b) f x  0x4,162 2 , 2  x  2, 4 1
x , 2  x  4, (c) (d)
otherwise.
3 6

7. (a) 0.0808 (b) 47.52 (c) DIY (d) 0.4437

4 of 4

LET’S SCORE A PROGRAMME KMS SM025

PART A
(25 marks)

1. Two events, M and W, are such that P(M)=0.3, P(W M )  0.5 and P(W M ')  0.25 . Find

(a) P(M W ) (1 marks)

(b) P(M 'W ) (2 marks)

(c) P(W ) (2 marks)

(d) State a reason why the events M and W are dependent (2 marks)

2. A fair die is thrown and the result is observed. If the outcome is one, the die is thrown once

again and the random variable X is the sum of the two throws. If the score is two or more,

then X is the number of the score on the uppermost face of the die. (2 marks)
a) Construct the probability distribution table for X

b) Write the cumulative distribution function of X. (2 marks)

c) Determine the median (1 marks)
d) Find P(2 < x < 5) (2 marks)
e) Find P(x > 5) (2 marks)

3. At a Restaurant R&S, the number of customers has a Poisson distribution with mean 4.5
customers per hour.

(a) Find the probability that

i) Exactly 8 customers dine in during the first hour. (2 marks)

ii) From 10 to 14 customers dine in from 9 am to 10 am. (2 marks)

iii) Less than 5 customers dine in during the first 20 minutes. (2 marks)

(b) If X is the number of customers coming to the restaurant

between 9 pm to 1 am, calculate P(20 < X < 35). (3 marks)

LET’S SCORE A PROGRAMME KMS SM025

PART B
(75 marks)

1

 1. Using the trapezium rule with five ordinates, estimate the value  ln 7  ex dx
1

correct to four decimal places. (6 marks)

2. A conic section is given by the equation x2  2x  6y 13  0. (3 marks)
(a) Express the equation and identify the type of the conic section (5 marks)
in standard form, .
(b) Find the vertex, directrix and focus, hence, sketch.

3. R(-2,4) and S(6,-2) are the endpoints of the diameter of a circle. Obtain

(a) the equation of the circle in standard form and general form. (4 marks)

(b) the equation of the tangents to the circle at points R and S. (5 marks)

4. (a) Given u  i 2 j 3k and v  2 i 4 j 3k , find a vector which is perpendicular

to both vectors, with magnitude 3 . (5 marks)
(b) Given a  2 i j 3k and b  4 i 2 j m k , find m if a and b are

(i) perpendicular (2 marks)

(ii) parallel (2 marks)

(c) Given two straight lines L1 : x 1  y2  z and L2 : x6  y3  z3 ,
3 1 5 1 1 1

Find the angle between L1 and L2. (5 marks)

5. The cumulative distribution function for the continuous random variable X is given by

0 , x0

 x2 , 0 x 1
 , 1 x3

F ( x)   3
x2 1
 x   2
 6

 1 , x3

(a) Calculate the median. (3 marks)

LET’S SCORE A PROGRAMME KMS SM025

(b) Find P  X  1  1  (3 marks)
 2  (3 marks)
(1 marks)
(c) Determine the probability density function of X.

(d) Find the mode.

6. A continuous random variable has a probability density function

2k  x 0 xk
 ,
f (x)   k2

 0, otherwise,where k is a positive constant

(a) Find the mean of X in terms of k. (3 marks)

(b) Show that the standard deviation of X is equal to k (4 marks)
32

(c) Find P(X    2) , where  and  are the mean and standard deviation of X

respectively. (5 marks)

7. (a) The diagram below shows a normal curve.
f(x)

AB x
6 12

Given that the area A is 0.4 and the area B is 0.3, find the mean and

variance of the normal distribution. (6 marks)

(b) The masses of guavas in a farm are normally distributed with mean,  and

standard deviation, . The mass percentages of guava that less than 400g is

15.87% and more than 500g is 6.68%.

(i) Determine the values of  and  . (7 marks)

(ii) Find the probability that a guava has a mass of at most 500g, given that it

is more than 400g. (3 marks)

END OF QUESTIONS PAPER

LET’S SCORE A PROGRAMME KMS SM025

Suggestion answers: (c) 0.325

Part A

1. (a) 0.15 (b) 0.175
(d) P(M W )  P(M )  P(W )

2. (a)

x 234567
P(X=x) 7 7 7 7 7 1

36 36 36 36 36 36

0 x2
 2 x3
 7 3 x4
4 x5
 36 5 x6
14 6 x7
 36
 x7
(b) F ( x)   21

 36
 28

 36

 35

 36
 1

(c) m=4 (d) 14 @ 7 (e) 8 @ 2
36 18 36 9

3. (a) (i) 0.043 (ii) 0.017 (iii) 0.9814
(b) 0.2691

LET’S SCORE A PROGRAMME KMS SM025

Part B

1. n  4, h  1 (1)  1  0.5
42

X f (x)  ln 7  ex 

-1 1.37547

-0.5 1.36209

0 1.33857

0.5 1.29512

1 1.20597

total 2.58144 3.99578

 1 ln 7  ex dx  0.52.58144  2(3.99578)  2.6433(4d.p)

1 2

2.(a) (x 1)2  6( y  2) , parabola (b) V(1,2), F(1,3.5), Dir: y=0.5

(x  2)2  ( y 1)2  25 (b) y  4 x  32 , y  4 x 10
3. (a) 33 3

x2  y2  4x  2 y  20

4. (a) 3   6 3 i 9 3 j 8 3k

a 181 181 181

(b) (i) m  8 (ii) m  6 (c)   72.98o (acute angle)
3 (b) 13

5. (a) m  3  3 24
(d) mode = 1
 2x , 0 x 1
 (c) 8
 3 9

(c) f (x)  1  x , 1 x 3
 3

 0 , otherwise


6. (a) k
3

7. (a)   7.954,  7.722
(b) (i)   440,  40 (ii) 0.9206

BENGKEL COUNTERSTRIKE
KMKJ SEM 2 SESI 2020/2021

NAME:______________________________________________________________

TUTORIAL :_________________________________________________________

DATE:_______________________________________________________________

Answer all the questions.

PART A [25 marks]

1. Given A and B are two events such that P( A)  0.6 and P(B)  0.4 . Probability of

event A or event B but not both is 0.5. [2 MARKS]
(a) Show that P( A  B)  0.25 .

(b) Determine whether A and B are independent events. [2 MARKS]

(c) Find P(B' A' ) [3 MARKS]

2. The probability distribution function for a discrete random variable T with k is a
constant, is given as
P(T  t)  k 3t  5 ,t  1,2,3,4

(a) Show that k  1 and construct a probability distribution table for T. [3 MARKS]
14

(b) Find the mean of T [2 MARKS]

(c) Calculate Var(T) [3 MARKS]

(d) Find the cumulative distribution of the function. [2 MARKS]

3. In a production process of flower vases, it is known that the probability of the flower

vases has defects is 0.145.

(a) In n selected flower vases at random, the probability that none of the flower vases

has defects is at least 0.1. Determine the value of n. [3 MARKS]

(b) Using normal approximation, find the probability that more than ten flower vases

have defects in random sample of size 100. [5 MARKS]

BENGKEL COUNTERSTRIKE
KMKJ SEM 2 SESI 2020/2021

PART B [75 marks]

1. Use Newton-Raphson method with x1  0.7 to find the root of the equation
x  2  2  0, where x  0. Give your answer, correct to three decimal places.
x
[6 MARKS]

2. (a) The centre of a circle lies at the focus of the parabola y2  8y  8x  24  0 . If the

circle passes through the point 3,6, find the equation of the circle.

[6 MARKS]

(b) A conic section has the equation 9x2  5y2  54x  40 y 116  0.

i. Show that the equation of the conic section represents an ellipse. [3 MARKS]

ii. Find the centre, vertices and foci of the ellipse. [4 MARKS]

3. A circle has a centre at the point C(2,4) and radius 5 units. Find the equation of tangent
to the circle at the point P(5,8) .
[4 MARKS]

4. The points A2,3,5, B4,2,3 and C 2,0,7 lie on the plane  . A straight line L
passes through the points P(2,8,9) and Q5,6,4.

(a) Find AB  AC . Hence, obtain the Cartesian equation of the plane  .
[5 MARKS]

(b) Find the vector equation of the straight line L.
[2 MARKS]

(c) Find the acute angle between the straight line L and plane  .
[4 MARKS]

(d) Find the coordinates of the point where the line L meets the plane  .
[3 MARKS]

5. The time taken (in hours) by responses team to attend to complaint is a continuous
random variable X with probability density function

f ( x)   x ,0  x  2
4
0 , otherwise

(a) Find the mode of x and sketch the probability density function graph of f(x)
[4 MARKS]

(b) Find the cumulative distribution function, F(x) and hence find F  1  .
2
[4 MARKS]

(c) Find the expected time to respond to the complaint.
[3 MARKS]

BENGKEL COUNTERSTRIKE
KMKJ SEM 2 SESI 2020/2021

(d)The respond team is efficient if P X    1 is at least 0.990.Determine if the

response team is efficient. [3 MARKS]

6. The continuous random variable X has cumulative distribution function F given by

0 x0
 0 x 1
F (x)   x
x 1
 1


(a) Find the median of X. [2 MARKS]
(b) Find the probability density function of X is f(x). [3 MARKS]
(c) Hence, show that E(X) = 1 .
[3 MARKS]
3

7. It is known that the mean of accidents occurred in a certain road is one in a month.

(a) Define the random variable and identify its distribution. [2 MARKS]

(b) Find the probability that

(i) There is no accident occurring in the next one month. [2 MARKS]

(ii) There are not more than four accidents occurring in the next five months.

[3 MARKS]
(iii) Out of seven months, there are less than two ‘accident-free’ months.

[4 MARKS]

The number of house sold by an estate agent follows a Poisson distribution, with mean
of 2 per week.

(a) Find the probability that in the next 4 weeks the estate agent sells more than 5

houses. [3 MARKS]

The estate agent monitors sales in period of 4 weeks.

(b) Find the probability that in the next twelve of these 4 week periods there are

exactly nine periods in which more than 5 houses are sold. [3 MARKS]

END OF QUESTION PAPER

SM025

SECTION A [25 marks]

1 If S and T are two events and P(T) = 0.4, P(S  T) = 0.15 and P(S’  T’ ) = 0.5, find the
following probabilities

(a) P(S ) 2 marks
(b) P(S’  T) 2 marks
(c) P(S  T’ ) 2 marks
(d) P(S’  T’) 2 marks

2 The discrete random variable X takes integer values from 1 to 4 inclusive with

probability PX  1  p; PX  2  3 ; PX  3  q; and PX  4  r. Given

10

that EX   2and VarX   1. Find

(a) the values of p, q and r. 6 marks
(b) P X    7  4 marks

 2

3 Rose expects to receive 4 letters a week. Find the probability of receiving

(a) one letter this week 2 marks
(b) at least three letters this week 2 marks
(c) 6 letters during the next two weeks. 3 marks

1

SM025

SECTION B [75 marks]

This section consists of 7 questions. Answer all questions.

22

 1 (a) Calculate the exact value of 2x dx using the definite integral exln2 dx , in
11

three decimal places.

2 marks

2

(b) Use the trapezium rule to find 2x dx , taking ordinates with interval 0.25, in
1

three decimal places.

3 marks

(c) Compare your answer in part (a) and part (b) and explain why your answer in
part (b) is an over-estimate.

1 mark 

2 (a) The equation of a circle is x2  y2  3x  4  0. Find 2 marks
(i) the coordinates of its centre 2 marks
2 marks
(ii) its radius

(iii) the coordinates of the point at which it cuts the axis

(b) Show that the line whose equation is 3x  4y  17 touches the circle and find

its coordinates.

3 marks

(c) Show also that this line and the tangent to the circle at the point (3, -2) intersect

at a point on the x-axis and find its coordinate.

3 marks

2

SM025

3 (a) Given the equation x2  4y2  4x 8y  c  0. Find the range of values of c

such that this equation represents an ellipse.

5 marks

(b) The point A, B and C have position vectors i  2 j  3k, i  5 j and 5i  6 j  k
respectively, relative to an origin O.

Show that BA is perpendicular to BC and find the area of the triangle ABC

5 marks

4 The point A has coordinate 7,1,3 and the point B has coordinate 10,2,2.

The line l has vector equation

r  i  j  k   3i  j  k 

where  is a real parameter.

(a) Show that the point A lies on the line l .

3 marks

(b) Find the length of AB .

2 marks

(c) Find the size of the acute angle between the line l and the line segment AB ,
giving your answer to the nearest degree.
Hence, or otherwise, calculate the perpendicular distance from B to the line l ,
giving your answer to 2 significant figures

4 marks

5 A discrete random variable X has the probability distribution function

 q 2 ,x  8, 3
 x


P X  x   q,x 1
5

 0, otherwise



(a) Find the value of q 2 marks
2 marks
(b) Find P  X  3 2 marks

(c) Find E(X)

3

SM025

6 A continuous random variable T (in years) of the lifespan of a device has the

following probability density function.

f  t    2 pt 2 , 0  t  2
 3

 0, otherwise

(a) Determine the value of p. 3 marks
(b) Find E(T) and Var(T). 4 marks
(c) Find the cumulative distribution function of T, F(t) 4 marks
(d) For U  2 T 1determine E(U) and Var(U)
4 marks
5

 7 (a) If X N 100, 2 and P  X  106  0.8849. Find the variance,  2.

5 marks

(b) In a production process of cups, it is known that on average 40 cups have
cracks in every 1000 cups produced.

(i) Find the probability that at most three cups having cracks in a random
sample of 10.

3 marks

(ii) If the probability that no cups in k cups selected at random, having

cracks is 0.12, determine the value of k .

3 marks

(iii) Find the probability that more than two cups have cracks in a random

sample of size 200.

6 marks

END OF QUESTION PAPER
4

Answers 0.25 (b) 0.25 (c) 0.10 SM025
Section A r  1 ,q  1, p  2 (b) 2 (d) 0.85
1. (a)
10 5 5 5
2. (a) (c) 0.1222
0.0733 (b) 0.7619
3. (a)

Section B 2.885 (b) 2.893 (c) Area in part (b) is over estimate
1. (a) since y  2x is concave

2. (a) upwards.
(b)
(i) C  3 ,0  (ii) r  5 (iii) 0,2 , 0,2 , 4,0 , 1,0
 2 
2

3,2 (c)  17 ,0 
 3 

3. (a) c  8 (b) 9 unit2

4. (b) 11 (c) 35o (d) 1.9
(c)  12
5. (a) 2 (b) 1
5 5

6. (a) 9 (b) 3 , 3 (d) 8 , 3
16 2 20 5 125

 0,t  0 (ii) 0.5305
t (ii) 52
t   3

(c) F  ,0  t  2
 8 1,t  2

7. (a) (i) 0.9659

(b) (i) 0.9996

(iii) 0.9761

5

SPECIMEN PAPERSM025

Answerallquestions.

PARTA(25marks)

1.Asurveyon100adultswasdonetofindoutwhethertheyhavee-mailaccountor
nor.The

followingtablesummarizestheresponses.

Yes No

Male 500 100

Female 250 150

(a) Ifanadultisselectedatrandom from these100adults,findtheprobabilitythat
theadult

(i) hasane-mailaccount
(ii) isafemale
(iii) hasnoe-mailaccountgiventhattheadultisafemale.

(4marks)
(b)Aretheevent‘male’and‘female’mutuallyexclusive?Giveexplanation.

(2marks)

2.Theprobabilitydistributionofadiscreterandom variableXisasfollows:
x 0 12

P(X=x) a b 1
3

(a)IfE(X)=191,findthevaluesofaandb (5marks)
(b)FindthecumulativedistributionfunctionofX
(c)FindtheP(0<X≤2) (3marks)
(2marks)

3. Assumethatthenumberofnetworkerrorsexperiencedinadayonalocalarea
network

(LAN)isdistributedasaPoissonrandom variable.Themeannumberofnetwork
errors

experiencedinadayis1.2.Whatistheprobability

(a) zeronetworkerrorswilloccurinanygivenday? (3
(3marks)

(b) twoormorenetworkerrorswilloccurinaweek?
(3marks)

(c)fewerthanthreenetworkerrorswilloccurin2weeks?
marks)

PARTB(75marks)

1. Showthattheequation hasarootintheinterval
UsetheNewton-Raphsonmethodtofindthevalueofthisrootcorrectto3decimal
places.

(6marks)

2. Thecentreofacircleliesatthefocusoftheparabola
Ifthe circlepassesthrough theorigin,findtheequationofthecircle.
(6marks)

3. Astraightline intersectsanellipse at(2,3)and(-2,
-1).

(a) Findthevaluesofaandb.

(b) Hence,findthegeneralequationofaparabolawithavertexatthepoint(a,b),

passingthroughthepoint(2,3).Theaxisoftheparabolaisparalleltothey-
axis.

(10marks)

4. ThecoordinatesofthepointsA,B,CandDare(p,2,1),(0,-1,2),(1,0,q)and(2,-1,-6)
respectively.

(a)Findthevaluesofpandqforwhich

(b)Hence,findtheareaoftriangleABC.

(c)FindtheCartesianequationoftheplane containingthepointsA,BandC.

(d)FindasetofparametricequationsforthelineLpassesthroughthepointDand
whichisperpendiculartotheplane .

(14marks)

5. Thecontinuousrandom variableXhascumulativedistributionfunctiongiven

{ 0 x<0

F(x)= 2x-x2 0≤x≤1
1 x>1
( )(i) FindPX<21
(2marks)

(ii) FindthevalueofksuchthatP(X>k)=34 (3marks)

(iii) FindtheprobabilitydensityfunctionofX.Sketchitsgraph (5
marks)

6. Theprobabilitydistributionfunctionofarandom variableXis

()P(X=x)=k3x forx=0,1,2,3

wherekisaconstant.Determine

a.Thevalueofk.Hence,findP(0<X<3) (4marks)

b.E(X),Var(X)andVar(Y)ifY=2X-3 (8marks)

7. (a) GivenX~N(20,160).Find (3marks)
(i) P(15<X<22) (3marks)
(ii) thevalueofm ifP(X≥m)=0.6736

(b) Inacertaingame,thereisonlythepossibilityofwinningorlosingfor
everytrial.

Iftheprobabilityofwinningis0.3,find
(i) theprobabilityofwinningatleast3timesafter10trials.
(ii) thevalueofn,iftheprobabilityofwinninglessthanntimesis

1.8497 after10 trials.
(6marks)

Iftheprobabilityofwinningis0.4,byusingasuitableapproximation,
findtheprobabilityofwinningbetween40and50timesafter100trials

(5marks)