Program Mathelatic SM025 Question

Program MATHLETICS

MATHEMATICS SM025

PREPARED BY : BOON KOK SIONG & LIM HWEE CHENG
Kolej Matrikulasi Johor KPM Tangkak Johor



MATHEMATICS 2
SM025
PAPER 1

BKS/ HCLIM 2019/20 1

Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
PROGRAM MATHLETICS

SM025/1 SET1

  1. (a) Find e2x ex  ex dx

(b) By using a suitable substitution , evaluate

3

 x 1 x dx

0

2. By using integrating factor method, find the general solution of
ex dy  2ex y  2x .
dx

2 1 dx. by using the trapezoidal rule

Evaluate correct to three decimal places
3. 1 x ln 2x

with five ordinates.

+--
4. Show that the equation 9x2  4y2  36x  8y  4  0 represents an ellipse. Find the

coordinates of the centre, vertices and foci.. Hence sketch the graph of ellipse.

5. Given u  v  3 and the angle between vector u and v is , find u  v . Hence,
~~
~~ ~~

evaluate  4 v  u    u  v  and state the geometrical relationship between these two

~ ~ ~ ~

vectors.

6. Given three vectors a  i  m j (2m 1) k, b  2 i  2 j 2 k and c  3 i  4 j k .

~~ ~ ~~ ~ ~ ~ ~ ~ ~~

a) If a is perpendicular to b find the value of m.
~~

b) Find a vector perpendicular to both b and c .
~~

c) Show that vectors a, b and c form a triangle and find the area of the triangle.
~~ ~

BKS/ HCLIM 2019/20 2

Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
PROGRAM MATHLETICS

SM025/1 SET 2

1. By using integration by part  x sin2 x dx

2. Solve the differential equation dy  y2 x sin 3x given that y  1 when x   .
dx 6

Give your answer in the form y  f 9 .

x

3.
y

Rx

In the figure above, R is the region bounded by y  xex , y  2  x and the x-axis.
Show that the root of the equation xe x  2  x lies between x  0 and x  1. Find
the root of the equation by using Newton-Raphson method with 0.6 as the first
approximation. Give your answer correct to three decimal places.

4. (a) A circle with centre C 3, 2 has equation x2  y2  6x  4 y  12 .

Find the coordinates of the points where the circle crosses y-axis. Find the radius
of the circle.

(b) Show that point P2,5 lies outside the circle. The point Q lies on the circle so

that PQ is a tangent to the circle. Find the length of PQ.

5. Given the vector v  3i  j  2k and w  3i  5 j  5k . Find the magnitude of v  w
and a unit vector in the direction of v  w .

6. The points A2, 1, 2 , B 5, 7, 3 and C3, 3,1 lie on the plane 1. The

equation of second plane 2 , is given as 2x  y  2z  5 .
(a) Find the vector AB and AC .
(b) Determine the Cartesian equation of 1.
(c) Find the acute angle between 1 and 2 , giving your answer in degrees.

BKS/ HCLIM 2019/20 3

Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
PROGRAM MATHLETICS

SM025/1 SET 3

7x2  3x  2
1. Express (x 1)2 (x  2) as partial fractions. Hence, evaluate

4 7 x2  3x  2 dx
(x 1)2 (x  2)
3

2. The gradient of a curve is given by x  y . Find the equation of the curve which

passes through the point  0,1 .

1

3. By using the trapezoidal rule, find the approximate value for x x  1 dx when

0

n  4 , correct to four decimal places.

4. (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) .

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

5. Given the points A1, 2,  2 , B(2, 4, 6) and C 4, 3, 1 . Find the area of

triangle ABC.

6. Given a line l : x  2  t, y  3  4t, z  5  3t and two planes
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
(b) the acute angle between the line l and the plane 1

(c) the acute angle between planes 1 and 2

BKS/ HCLIM 2019/20 4

Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
PROGRAM MATHLETICS

SM025/1 SET 4

1. Find the intersections between the curve y  3  x and line y  1  x  0 .
(a) Sketch the region R bounded by the curves , the line , x-axis and y-axis in the
first quadrant.
(b) Find the area bounded by the region R
(c) Calculate the volume formed when region R is rotated about x-axis through
2 radians

2. The velocity of an airplane departs from rest in the earth’s atmosphere after time,
t, satisfies the differential equation dv  kv  2g where k and g are both positive
dt
constant.

 a. Show that v  2g 1 ekt .
k

b. Find v when t   .

c. Sketch the graph of v against t.

3. Show that the equation ln(2  x)  3x  6  0 has a root between x  1.4 and
x 1.8 . By using Newton Raphson Method and taking x 1.5 ,find the root
correct to 4 decimal places.

4. (a) Find the standard equation of a parabola with its symmetric axis parallel to

the x-axis, vertex at the point 3, 2 and passing through the point 4, 4

(b)Express the equation of parabola x2  4x  24 y  76  0 in standard form.
Hence, determine the coordinates of its vertex and focus.

BKS/ HCLIM 2019/20 5

5. Given u  10i  5 j  5k , v  3i  2 j   k and w  30i 15 j   k . Find
(a)  if u and v are perpendicular.
(b)  if u is parallel to w.

6. The points A(3, 0, 1) , B(1, 2,1) and C(0, 1, 2) lies on the plane 1 .



(i) Find the vectors AB and AC .

(ii) Determine the vector equation and Cartesian equation of 1.
(iii) Find the intersection point between the plane 1 and the line

r  i  2 j  3k  t(2i  j  k)

(iv) Find the angle between the plane 1 and the line

r  i  2 j  3k  t(2i  j  k) .

BKS/ HCLIM 2019/20 6

Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
PROGRAM MATHLETICS

SM025/1 SET 5

1. Find

(a) (ex  ex )2 dx (3 marks)
2e2x (4 marks)
(5 marks)
(b) sin 2x ecos2 x1dx

(c) (ln x)2x1dx

2. Solve the differential equation

cos x dy  (2sin x) y  3x cos x
dx

Given that y     0 (7 marks)
 4 

3. Show that the equation 3  cos x  x3 has a root that lies between 1 and 1.5. Hence

by using Newton Raphson method, find an approximation root for the equation

3  cos x  x3 , correct your answer to three decimal places. (5 marks)

4. (a) Given that x2  20  2(2x  5y)

(i) Show that this equation represents a parabola. (3 marks)

(ii) Find the coordinates of vertex and focus. Hence, sketch the graph of this

parabola. (4 marks)

(b) An ellipse with center at origin passes through the point 2, 0 and

 3 3  . Find the equation of ellipse and state its foci. (5 marks)
1, 2 

BKS/ HCLIM 2019/20 7

5. Given three points R 4,  3, 1 , S 6,  2,  3 and T 7,  3, 0 in plane P.

Find

(a) the vector of RS and RT . (2 marks)

(b) a vector perpendicular to both vectors RS and RT . (4 marks)

(c) the area of parallelogram with edges RS and RT . (3 marks)

6. (a) Using the trapezium rule , find an approximate value 2 2x2 1dx by sub-

x

0

dividing the integration interval [ 0, 2 ] into 5 strips. Giving your answer correct to

3 significant figures. (5 marks)

(b) Find the equation of a circle x2  y2  2gx  2 fy  c  0 which passes through

the points A 0, 1 , B 3,  2 and C 1,  4 . Hence, find an equation of

tangent of the circle at point B.. (8 marks)

7. Given that point A4, 2, 1 and B 1, 1, 3 are two point in three dimension

space.

(a) Find the Cartesian equation of the straight line L which passes through the

points A and B. (5 marks)

(b) The straight line L intersects with the plane  : 3x  4 y  5z  10 at point R.

Determine the coordinate of R. (3 marks)

(c) Determine the acute angle between the straight line L and the plane  .

(4 marks)

BKS/ HCLIM 2019/20 8

MATHEMATICS 2
SM025
PAPER 2

BKS/ HCLIM 2019/20 9

Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
PROGRAM MATHLETICS

SM025/2 SET 1

1. The weights ( in kg) to the nearest integer of ten lecturers in a college are given as

follows :

51, 74, 59, 59, w, 51, 60, 70, w+8, 56

where w is an integer. If the mean weight is 65kg, determine the value of w. Hence,

find the mode and the 80th percentile.

2. Given the digits 2, 3, 4, 5, 6 with no repetition , find the number of
(a) odd number that can be formed.
(b) 5-digit odd number greater than 40000 that can be formed.

3. A and B are two events such that P( A)  0.5 and P(B)  0.6 and P( A  B)  0.85 .

Find (b) P( A  B ') (c) P( A'B) (d) P( A '  B ')
(a) P( A  B)

4. A discrete random variable X has following probability distribution function
m , x  2,1

x2

P(X  x)  x 1, x  1, 2
12

0 , otherwise
(a) Show that m 1
(b) Find P(2  X  2)
(c) Find the mode
(d) Find E( X ) , E(2X  3) and Var(2X  3)

BKS/ HCLIM 2019/20 1
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5. A discrete random variable X has a Poisson distribution with mean 3.2. By using
statistical table,
(a) Find P( X  9)

(b) Determine r if PX  r  0.9554

6. According to a survey, 25% of the government staff overspent their monthly salary.
(a) A random sample of 25 staff was selected. Find the probability that at least two
of staff overspent their monthly salary.
(b) A random sample of 20 staff was selected. Find the probability,
(i) Not more than 5 staff overspent their monthly salary.
(ii) Between 5 and 10 staff overspent their monthly salary.

(c) A random sample of 200 staff is taken. Use the normal approximation to find
the percentage that not more than forty staff overspent their monthly salary.

BKS/ HCLIM 2019/20 1
1

Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
PROGRAM MATHLETICS

SM025/2 SET 2

1. The stem- and-leaf diagram below shows the mass 16 packages.

Stem Leaf
0 256
1 135679
2 247
3 35
4 2
5 9

Key: 2 7 means 27 kg

a) calculate the interquartile range.

b) draw a box and whisker plot to represent the data and hence interpret the
distribution.

2. The operator of two companies has 13 employees in company A and 10
employees in company B. He plans to lay off 5 employees. How many ways can
this are done if
a) all the employees from company A?
b) 3 employees from company A and 2 employees from company B?

c) at least 2 employees from company B?

3. In a small town of Sungai baru, one can hire a bus from one of three transportation
companies, P,Q and R. Of the hirings 30% are from P, 60% are from Q and 10 %
are from R. For buses hired from P , 7 % arrive late, the corresponding 5% and 12%
respectively. Calculate the probability that the next bus hired
(a) will be from P and will not arrive late

BKS/ HCLIM 2019/20 1
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(b) will arrive late
(c) Given that a call is made from tour group for a bus and that is arrives late, find
the probability that is came from R.

4. A continuous random variable has probability density function

f x   kx2 1 x, 0 x 1

0 , otherwise

a) find the value of k .
b) calculate P X  1  .

 3
c) calculate the mean.
d) find the variance.

5. A man observed that over a long period of time, the number of emails he receives
each day is a Poisson distribution with mean 1.5. Calculate the probability that on
one particular day, he will receive
a) no email
b) more than 3 emails.
c) the man is away from home for 3 days. Find the probability that when he
returns, there will be 3 emails in his inbox.

6. Assume that the new born baby’s weight in a hospital is normally distributed with
mean 3.00 kg and standard deviation 0.25 kg.
(a) Find the probability that a baby born in hospital has weight between 2.75kg and
3.20kg.
(b) If 5 babies is picked at random, find the probability that at least one baby has
weight more than 3.25 kg.
(c) If babies whose weight fall in the first quartile is categorized as “small”, what
is the maximum weight for this category?

BKS/ HCLIM 2019/20 1
3

Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
PROGRAM MATHLETICS

SM025/2 SET 3

1. The time taken for 70 students to walk from hostel to class in a certain college are

shown in the following table.

Time ( minute) Number of student

2-4 5

5-7 9

8 - 10 19

11 - 13 21

14 - 16 12

17 - 19 4

(a) Find the mean and mode.
(b) Determine the 40th percentile.
(c) Find the standard deviation.
(d) Calculate the Pearson’s coefficient of skewness. Interpret your answer.

BKS/ HCLIM 2019/20 1
4

2. The table below shows the number of identical black and white marble in three

boxes A, B and C.

Box

Colour A B C

Black 3 6 5

White 7 8 4

(a) All the marbles in box A are arranged in a rows. In how many different
arrangements can this be done ?

(b) A marble is drawn from each of the boxes. If all the marbles drawn are of the
same colour , in how many ways can this be done ?

(c) Four marbles are drawn without replacement from box B. In how many
different ways can the marbles be drawn such that there are equal number of
black and white marbles ?

3. In a college there are 150 students taking course in Chemistry, Physics and Biology.
Among the students, 92 are females. There are 48 students taking Chemistry which
28 are females. Half of the 68 students taking Physics are females.
(a) Construct the contingency table for the given data.
(b) A student is chosen at random. Find the probability that the student
(i) takes biology.
(ii) is a male, given that he takes Biology.
(iii) take Biology or a female.
(c) Two students are chosen at random, find the probability at least one student is
a female and takes Biology.

BKS/ HCLIM 2019/20 1
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4. (a) A discrete random variable X has probability distribution as shown in the table
below.

x 0 1234
P( X  x) 0.3 a 0.1 b 0.2

Determine the values of a and b such that E( X )  2 . Hence find Var(x) .

(b) The probability density function of a continuous random variable X is given by

f (x)  3x2 (2  kx), 0  x  1
0, otherwise

(i) Find the value of the constant k.

(ii) Show that P X  1   3
 2  16

(iii) Calculate the mean and variance of X.

5. It is known that 10 % of the patients with high fever are confirmed to be suffering
from dengue fever.
(a) If 15 patients with high fever are random chosen, find the probability that
(i) less than 6 are confirmed to be suffering from the dengue fever.
(ii) exactly 10 patients with high fever are confirmed to be free of dengue
fever .

(a) If 100 patients with high fever are randomly chosen,
(i) approximate the probability that 9 to 14 patients are confirmed to be

suffering from dengue fever.
(ii) find the value of m such that the probability of more than m patients that

are confirmed to be suffering from dengue fever is 0.025.

BKS/ HCLIM 2019/20 1
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Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
PROGRAM MATHLETICS

SM025/2 SET 4

1. (a) Weight (gram) of a random sample of 25 guava is summarized to

 x  8750 ,  x2  3063700

Find the mean and standard deviation of the weight of guava.
(b) Following table show that the distribution for the number of medical leaves (in

days) taken within a certain period by 65 employees of a company.

Number of Medical Leaves (days) Number of Employees
1-3 4
4-6 6
7-9 8
10-12 12
13-15 18
16-18 11
19-21 6

Calculate

(i) the mean, mode and median. Hence describe the distribution of the data.

(ii) the variance.

(iii) Find m such that 20% of the number of medical leaves exceeds m.

2. In a preliminary round of a cooking competition, there are twelve male and eight

female contestants.Five contestants are chosen to enter the final round. In how

many ways can the finalists be chosen if

(a) They are of the same gender.
(b) The number of males are more than females.
(c) A particular female contestant is disqualified from the final round.

BKS/ HCLIM 2019/20 1
7

3. (a) A and B are two events such that P(A)  3 , P(B)  5 and P(A  B) '  1 .
16 8 4

(i) Show that A and B are not mutually exclusive event.

(ii) Find PA B.

(b) Given two events A and B such that P(A)  1 , P(B ')  1 and PB A  2 .
53 5

Find

(i) P( A  B)

(ii) PA ' B '

4. a) The probability distribution of a discrete random variable Y is showns in the

following table

y 2 1 0 12

P(Y  y) k 3 k2 k2 1
8
16 16 16 16

  Show that k  3. Hence evaluate E 2 Y 2  Y .

(b) A continuous random variable, X has the following probability density function

f (x)  c(2  x) 1 x  2
c 2 x4
0 otherwise

(i) Show that c  0.4 .
(ii) Find

(a) P(1  2X  3)

(b) E(2X  5)

(c) Var(10  2X ) , given E(X 2 )  47 .
6

BKS/ HCLIM 2019/20 1
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5. A van can take either route A or route B for a particular journey. If a route A is
taken, the journey time is normally distributed with mean 46 minutes and a standard
deviation of 10 minutes. However, if route B is taken, the journey time is normally
distributed with mean  minutes and standard deviation of 12 minutes.
(a) For route A, find the probability that the journey takes more than 60 minutes.

(b) For route B, the probability that the journey takes less than 60 minutes is 0.85
.
Find the value of  .

(c) The van sets out at 06:00 and needs to arrive before 07:00 . Determine the
route should it take and justify your answer

(d) One five consecutive days, the van sets out at 06:00 and takes route B. Find
the probability that
(i) It arrives before 07:00 on all five days .

(ii) It arrives before 07:00 on at least three days.

BKS/ HCLIM 2019/20 1
9

Unit Matematik
Kolej Matrikulasi Johor
Kementerian Pendidikan Malaysia
PROGRAM MATHLETICS

SM025/2 SET 5
1. The amount of rainfall for 64 days in a particular year were measured correct to

the nearest mm is given in the following table.

Amount of 0 – 5 5 – 10 10 – 15 15 – 20 20 – 25 25 – 30

rainfall, x

(mm)

Number of 2 5 7 13 Y 16

day

Find the

(a) Value of constant Y . [2 marks]

(b) Mean . [3 marks]

(c) Standard deviation . [4 marks]

(d) Median . [3 marks]

2. Find the number of different arrangments of the eight letters in the word

BINOMIAL which

(a) No restriction imposed. [2 marks]

(b) The first two letters are BI . [2 marks]

(c) Both letters I are next to each other. [2 marks]

3. In a certain college, a total of 200 students take courses in Biology, Physics and

Computer. Among the students, 125 are females. Out of 65 students who take

Physics, 27 are males. One third of the 75 students wo take Biology are males.

(a) Construct a table for the above information. [3 marks]

(b) A student is chosen at random. Find the probability that the student

i) takes Computer. [1 mark]

ii) is a female given that she takes Computer. [2 marks]

BKS/ HCLIM 2019/20 2
0

iii) takes Computer or a female. [2 marks]

4. The number of laptop, X, sold daily by a shop has the probability distribution

function given by P(X  x)  5  x for x  0, 1, 2, 3, 4. [2 marks]
15

(a) Find the probability that at least one laptop sold.

(b) Calculate E( X ) . Hence, find P X  E( X )  5  . [6 marks]
 4

(c) Calculate E(X 2 ) . Hence, find variance of X. [4 marks]

5. Mr. Lee is planning to go fishing this weekend. The number of fish caught per

hour follows a Poisson distribution with mean 0.6 .

(a) Find the probability that he catches at least one fish in the first hour.

[2 marks]

(b) Find the probability that he catches exactly three fish if he fishes for four

hours. [2 marks]

(c) Given that the probability Mr. Lee catches at least one fish is 0.95 if he fishes

for three hours. Find  to the nearest integer where  is the mean number

of fish caught per hour. [3 marks]

6. A continuous random variable X has the probability density function as follows

f (x)  b  3  x, 2x3 
 0, 
 otherwise 

Where b is a constant.

(a) Show that b  3 [3 marks]
2

(b) Find the cumulative distribution function. Hence, find the median, m .

(c) Find E(3X 1) . [6 marks]
[4 marks]

BKS/ HCLIM 2019/20 2
1

7. It was discovered that 5% of the students that played truant were smokers.

(a) If 20 students that played truant are randomly selected, find the probability that

(i) not more than 5 were smokers. [2 marks]

(ii) 15 of them were not smokers. [ 3 marks]

(b) If 200 students that played truant were selected, find

(i) The probability that from 10 to 15 students were smokers.

[3 marks]

(iii) The value of n if the probability of more than m students were

smokers is 0.035. [4 marks]

BKS/ HCLIM 2019/20 2
2