MST125/D 

MST1251509I3

Module Examination 2015

Essential mathematics 2

Thursday 17 September 2015 2.30 pm – 5.30 pm

Time allowed: 3 hours

This paper has THREE sections. You should attempt ALL questions in
Sections A and B, and TWO questions from Section C.
Section A carries 40% of the marks. It has 20 computer marked questions, each
worth 2%.
Section B carries 36% of the marks. It has 6 questions, each worth 6%.
Section C carries 24% of the marks. It has 3 questions, each worth 12%, of
which you should attempt two.
For Section A, mark your answers on the computer marked examination (CME)
form provided, using an HB pencil. Instructions for filling in the CME form are
given overleaf. The assignment number for this examination is MST125 81.
Write your rough work in the answer book(s) provided. It will not be considered
by the examiners.
For Sections B and C, write your answers in pen in the answer book(s)
provided. Start your answer to each question on a new page. Include all your
working. Do not cross out any work until you have written a new attempt.

At the end of the examination

• Make sure that you have completed Part 1 of the CME form, and written
your personal identifier and your examination number on each answer book
used. Failure to do so will mean that your work cannot be identified.
• Make sure that you have entered one answer (A, B, C, D or E) against each
of questions 1–20 in Part 2 of the CME form.
• Attach your signed desk record to the front of your answer book(s) using the
round paper fastener, then attach the CME form and your question paper to
the back of the answer book(s) using the flat paperclip.
It is important that the flat paper clip is used with the CME form
since the computer cannot read CME forms with punched holes.

Copyright 
c 2015 The Open University
2 MST125 September 2015
Instructions for filling in the computer-marked
examination (CME) form
If you do not follow these instructions, then the examiners may not be able to
award you a score for Section A of the examination.
You will find one CME form provided with this paper. The invigilator has a
supply of spare forms.

Writing on the form

• Use an HB pencil.
• To mark a cell, pencil across it, as demonstrated on the form.
• To cancel a mark, pencil in the coloured part of the cell, as demonstrated on
the form.
• If you make any unwanted marks on the form that you cannot cancel clearly,
then ask the invigilator for a new form, and transfer your entries to it.

Completing Part 1
• Write your personal identifier (NOT your examination number) and the
assignment number for this examination, which is MST125 81, in the boxes
provided.
• In the blocks headed ‘personal identifier’ and ‘module and
assignment number’, pencil across the cells corresponding to your personal
identifier and the assignment number given above.

Completing Part 2
• For each question (numbered 1 to 20) in Section A of the examination paper,
mark your answer by pencilling across ONE of cells A, B, C, D or E.
• If you think that a question is unsound in any way, pencil across the
‘unsound’ cell (U), as well as pencilling across an answer cell.
• Do not pencil across any other cell.

MST125 September 2015 TURN OVER 3

SECTION A
Attempt ALL questions in this section. Each question is worth 2%.
Mark your answers in pencil on the CME form.

Question 1
What is the least residue of 900 002 × 450 002 modulo 45?

A 2 B 4 C 44 D 94 E 134

Question 2
Which of the following is a multiplicative inverse of 1021 modulo 43?

A 10 B 1021 C 1022 D 1042 E 1043

Question 3
What is the equation of the ellipse that crosses the x-axis at (±5, 0) and the
y-axis at (0, ±3)?

x2 y 2 x2 y 2 x2 y 2
A + =1 B + =1 C + =1
5 3 25 9 9 25
D 25x2 + 9y 2 = 1 E 5x2 + 3y 2 = 15

Question 4
Which option describes the graph given by the parametrisation
x = 1 + 3t, y = 4 − t (−1 ≤ t ≤ 2) ?

A The line segment joining (−2, 5) and (7, 2)

B The line passing through (−2, 5) and (7, 2)
C The line segment joining (1, 4) and (3, −1)
D The line passing through (1, 4) and (3, −1)
E The line segment joining (1, 4) and (4, 3)

Question 5
A crate rests on horizontal ground. The normal reaction of the ground on the
crate is 25 N vertically upwards. What is the mass of the crate in kilograms to
two significant figures? Take the magnitude of the acceleration due to gravity to
be 9.8 m s−2 .

A 0.26 B 2.5 C 2.6 D 26 E 250

4 MST125 September 2015

Question 6
Which option
 1 describes
 the linear transformation represented by the
1
− √2 √
2
matrix 1 1 ?
√
2
− 2
√

A A scaling B A shear C A flattening

D A rotation E A reflection

Question 7
What is the area of the image
 of the 
unit circle under the linear transformation
3 −2
represented by the matrix ?
−5 4
A π B 2π C 3π D 4π E 5π

Question 8
Let f (x) = Px and g(x) = Qx, where
   
−1 0 1 −2
P= and Q = .
3 2 4 −1

Which of the following matrices represents the composite transformation g ◦ f ?

       
1 0 −1 2 −7 −4 −1 0
A B C D
0 1 11 −8 −7 −2 1 −4
 
5 10
E
2 −2

Question 9
What is the quotient on dividing the polynomial expression x3 − 3 by the
polynomial expression x − 3?

A x2 − 3x − 9 B x2 − 3x + 1 C x2 − 3x + 9
D x2 + 3x + 1 E x2 + 3x + 9

MST125 September 2015 TURN OVER 5

Question 10
The graph of a function f is shown below.

Which of the following could be the rule for f ?

3x 3x 3x + 1
A f (x) = B f (x) = C f (x) =
x2 −1 x2 +1 x2 + 1
D f (x) = 3x(x3 + 1) E f (x) = 3x(x2 + 1)

Question 11
Which of the following is the general solution of the differential equation
dy
= e3x + 4 sin(2x) ?
dx
In the options, c is an arbitrary constant.

A y = 13 e3x − 2 cos(2x) + c B y = 13 e3x + 2 cos(2x) + c

C y = e3x − 2 cos(2x) + c D y = 13 e3x − 2 sin(2x) + c
E y = e3x + 2 sin(2x) + c

Question 12
Which of the following is an integrating factor p(x) for the differential equation
dy y
+ = x − 2 (x > −2) ?
dx x + 2
A p(x) = x − 2 B p(x) = x + 2 C p(x) = x2 − 4
D p(x) = ex+2 E p(x) = ex−2

6 MST125 September 2015

Question 13
Let P (n) be the statement
n > 3.
Which of the following statements is true?

A P (1) OR P (2) B P (1) AND P (2) C P (2) OR P (3)

D P (2) AND P (3) E P (3) OR P (4)

Question 14
What is the contrapositive of the following statement?
If n is odd, then n2 + 1 is even.

A If n2 + 1 is odd, then n is even.

B If n2 + 1 is even, then n is odd.
C If n is odd, then n2 + 1 is even.
D If n is even, then n2 + 1 is odd.
E If n is odd, then n2 + 1 is odd.

Question 15
Consider the following statement.
For each integer n, the integer 6n + 1 is prime.
Which of the following integers is a counter-example to this statement?

A 1 B 2 C 3 D 4 E 5

Question 16
The position of a particle is given in terms of the time t by
r = 3t2 i + (4t − 7) j − sin t k,
where i, j and k are Cartesian unit vectors. What is the acceleration of the
particle in terms of t?

A 4 j − cos t k B 6t i + 4 j − cos t k
C 6 i + sin t k D t3 i + (2t2 − 7t) j + cos t k
 
E 14 t4 i + 23 t3 − 72 t2 j + sin t k

Question 17
An object of mass 20 kg, initially at rest, is subject to a resultant force of
magnitude 25 N in a constant direction. What is its speed in m s−1 after
4 seconds?

A 0.8 B 1.25 C 4 D 5 E 125

MST125 September 2015 TURN OVER 7

Question 18
There are 18 students in a class, taught by four teachers. How many ways are
there to choose a team of representatives consisting of two students and a
teacher?

A 307 B 612 C 648 D 1224 E 1296

Question 19
A die is rolled twice. What is the probability of getting at least one 4?
1 11 1 25 5
A B C D E
6 36 3 36 6

Question 20
What is the general solution of the recurrence relation
un = un−1 + 6un−2 ?
In the options, A and B represent constants.

A un = (A + Bn) × 3n B un = (A + Bn) × (−2)n

C un = A × (−3)n + B × (−2)n D un = A × 3n + B × (−2)n
E u n = A × 3n + B × 2n

SEE THE NEXT PAGE FOR SECTION B

8 MST125 September 2015

SECTION B
Attempt ALL questions in this section. Each question is worth 6%.
Write in pen. Include all your working, as some marks are awarded for this.
Start your answer to each question on a new page of your answer book.

Question 21
For each of the following linear congruences, determine whether it has a solution
and find a solution if one exists.
(a) 11x ≡ 15 (mod 170)
(b) 15x ≡ 11 (mod 35)
[6]

Question 22
The equation x2 − 4y 2 = 8 represents a conic in standard position.
(a) By rearranging the equation in an appropriate way, identify the type of conic. [1]
(b) Find the eccentricity of the conic. [2]
(c) Find the distance between a focus of the conic and the nearest vertex of the
conic. Give an exact answer. [3]

Question 23
A particle, which remains at rest, is acted on by three forces, P, Q and R, and
no others.
The force P acts at an angle of 40◦ to the horizontal, the force Q acts at an angle
of 50◦ to the horizontal and the force R acts horizontally to the right, as shown.
The force P has magnitude 55 N. Let Q = |Q| and R = |R|.
Take the Cartesian unit vectors i and j to be parallel to P and Q, respectively, in
the directions shown.

(a) Find expressions for the component forms of the three forces P, Q and R. [2]
(b) Hence or otherwise find the magnitude Q of the force Q, in newtons to two
significant figures. [4]

MST125 September 2015 TURN OVER 9

Question 24
 √
Find the integral 81 − x2 dx . [6]

Question 25
Solve the initial value problem
dy cos x
= , where y(0) = 0. [6]
dx 2y + e3y
Leave your answer in implicit form.

Question 26
 
4 2
Find the eigenvalue(s) of the matrix . For each eigenvalue find a
3 −1
corresponding eigenvector. [6]

SEE THE NEXT PAGE FOR SECTION C

10 MST125 September 2015

SECTION C
Attempt TWO questions from this section. Each question is worth 12%.
Write in pen. Include all your working, as some marks are awarded for this.
Start your answer to each question on a new page of your answer book.
If you answer three questions, you will be awarded the marks for your best two
answers only.

Question 27
 π/2
(a) Evaluate the integral sin2 x cos3 x dx . [5]
0

8−x
(b) Find the integral dx . [7]
(2x + 1)(x2 + 4)

Question 28
(a) Prove the following statement, where n is a natural number.
If 7n + 2 is odd, then n is odd. [4]
(b) Prove the following statement by using mathematical induction.
3 + 8 + · · · + (n2 − 1) = 16 n(n − 1)(2n + 5), for all integers n ≥ 2. [8]

MST125 September 2015 11

Question 29
A tile slides, under gravity, down a flat rough roof that is inclined at 35◦ to the
horizontal. The coefficient of sliding friction between the tile and the roof is 0.2.
Take the magnitude of the acceleration due to gravity to be g = 9.8 m s−2 . Model
the tile as a particle.

Tile
x

35◦

(a) State the three forces acting on the tile during its motion, and draw a force
diagram representing these forces, labelling them clearly. [3]
(b) Find expressions for the component forms of the three forces, in terms of the
mass m (in kg) of the tile and any unknown magnitude(s) where appropriate.
Take the x- and y-axes to point parallel and perpendicular to the slope,
respectively, in the directions shown in the diagram above. [2]
(c) Hence or otherwise find the magnitude of the acceleration of the tile, in m s−2
to two significant figures. [5]
(d) The tile starts from rest and slides a distance of 3 metres to the edge of the
roof. Find the speed of the tile when it reaches the edge of the roof. Give
your answer in m s−1 to two significant figures. [2]

[END OF QUESTION PAPER]

12 MST125 September 2015