MST125/C
MST1251506F1
Module Examination 2015
Essential mathematics 2
Wednesday 3 June 2015
10.00 am – 1.00 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.
c 2015 The Open University
Copyright 2
MST125 June 2015 J
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 June 2015 J
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 14 × 15 modulo 17?
A 5
B 6
C 7
D 8
E
9
Question 2
Which of the following is an integer n such that 144n ≡ 1 (mod 23)?
A 22
B 23
C 24
D 144
E
145
Question 3
What is the equation of the parabola in standard position with focus (2, 0)?
A y 2 = 8x
E
B y = (x − 2)2
C y 2 = 4x
D y 2 = 4(x − 2)
y 2 − 2x = 0
Question 4
What is the equation of the line with parametric equations
x = t − 1,
y = 3t + 2?
A y = 2x + 1
B y = 2x + 3
D y = 3x + 3
E
C y = 3x + 1
y = 3x + 5
Question 5
A paperweight of mass 0.62 kg rests on a horizontal table top. What is the
normal reaction of the table top on the paperweight, in newtons to two significant
figures? Take the magnitude of the acceleration due to gravity to be 9.8 m s−2 .
A 0.62 N, vertically upwards
B 0.62 N, vertically downwards
C 0.62 N, parallel to the table top
D 6.1 N, vertically upwards
E
4
6.1 N, vertically downwards
MST125 June 2015 J
Question 6
A particle, which remains at rest, is acted on by three forces only. Two of the
forces have component forms 2i − 7j and i + 5j, respectively, where i and j are
the Cartesian unit vectors. What is the component form of the third force?
A −3i − 2j
B −3i + 2j
D −6i
E
C 3i − 2j
6i
Question 7
Which of the following transformations is a linear transformation?
A The translation 3 units to the right
B The rotation through π/3 about the point (1, 1)
C The zero transformation
D The reflection in the line y = x + 2
E
f (x, y) = (5y, 1)
Question 8
Which matrix represents the linear transformation that maps the points (1, 0)
and (0, 1) to the points (−3, 5) and (6, 2), respectively?
−3 5
−3 6
2 −6
6 −3
A
B
C
D
6 2
5 2
−5 −3
2
5
6 2
E
−3 5
Question 9
Which of the following is the graph of the function f (x) =
MST125 June 2015 J
x2 + 4x
?
x+5
TURN OVER
5
Question 10
What is the remainder on dividing the polynomial expression 3x2 + x + 1 by the
polynomial expression x − 1?
A −3
B 3
D 3x − 4
C 5
E
3x + 4
Question 11
Which of the following is the general solution of the differential equation
dx
= 4 sin t cos t ?
dt
In the options, c is an arbitrary constant.
1
cos(2t) + c
2
1
x = sin(2t) + c
2
A x = − cos(2t) + c
B x=
D x = − sin(2t) + c
E
C x = 2 cos t + c
Question 12
Which of the following is an integrating factor p(x) for the differential equation
dy
= xy + x2 ?
dx
B p(x) = ex
A p(x) = x
D p(x) = e−x
2 /2
E
2
C p(x) = x2 /2
p(x) = e2x
Question 13
Let P be the following statement:
Every multiple of 8 is even.
Which of the following statements is the negation of P ?
A There exists an even number that is a multiple of 8.
B There exists a multiple of 8 that is even.
C There exists a multiple of 8 that is odd.
D Every even number is a multiple of 8.
E
Every multiple of 8 is odd.
Question 14
A football is kicked along the ground and moves along a straight line. It slows
down at the rate of 2.5 m s−2 and comes to a halt after travelling for 20 m. What
is the speed with which the ball was kicked, in m s−1 ?
A 0
6
B 10
C 20
D 24
E
50
MST125 June 2015 J
Question 15
The position x (in metres) of an object moving along a straight line is given in
terms of the time t (in seconds) by
x = et/2 + 4t.
What is the magnitude of the acceleration of the object after 2 seconds, in m s−2
to two decimal places?
A 0.68
B 1.36
C 4.68
D 5.36
E
6.71
Question 16
A box is pushed with a force P down a straight, rough slope that is inclined at an
angle of 30◦ to the horizontal. The force P acts on the box at an angle of 45◦ to
the horizontal. If W represents the weight of the box, N the normal reaction
from the slope and F the friction force, which of the following diagrams
represents the forces acting on the box?
A
C
B
N
F
N
N
F
F
45°
30°
30°
45°
45°
P
W
D
45°
P
W
W
P
E
N
N
F
30°
45°
45°
W
P
What are the eigenvalues of the matrix
B −1 and −2
F
P
W
Question 17
A −2 and 5
30°
1 3
?
4 2
C 1 and 2
D 2 and 5
E
3 and 4
Question 18
How many ways are there to select three different students from a group of 10, if
the order in which they are considered does not matter?
A 6
B 90
C 120
MST125 June 2015 J
D 360
E
720
TURN OVER
7
Question 19
How many 3-letter sequences of letters from the 26-letter alphabet A, B, . . . , Z
contain at least one A? Examples of such sequences include CBA and AZA.
A 676
B 1951
C 2028
D 2500
E
15 625
Question 20
What is the general solution of the recurrence relation
un = 6un−1 − 8un−2 ?
In the options, A and B represent constants.
A un = A × (−6)n + B × 8n
B un = A × (−2)n + B × (−4)n
C u n = A × 2n + B × 4n
D un = A × 6n + B × (−8)n
E
u n = A × 4n + B
TURN OVER FOR SECTION B
8
MST125 June 2015 J
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
Use Euclid’s algorithm to show that a multiplicative inverse of 32 modulo 77
exists, and find one.
[6]
Question 22
The linear transformation f is formed from the composite of the horizontal shear
with shear factor 3 followed by the (−3, 2)-scaling.
(a) Find the matrix of f .
[3]
(b) Find the images of the points (−4, 2) and (0, 1) under f .
[1]
(c) Given that the triangle with vertices (0, 0), (−4, 2) and (0, 1) has area 2, use
your answer to part (a) to find the area of the image of this triangle under f .
[2]
Question 23
A particle, which remains at rest, is acted on by three forces, F, G and H, and
no other forces.
The force F acts at an angle of 25◦ to the horizontal, and the force G acts at an
angle of 125◦ to F, as shown. The force H acts vertically downwards and has
magnitude 40 N. Let F = |F| and G = |G|.
Take the Cartesian unit vectors i and j to be parallel and perpendicular to F,
respectively, as shown.
(a) Find expressions for the component forms of the three forces F, G and H.
[3]
(b) Hence or otherwise find the magnitude of the force G, in newtons to two
significant figures.
[3]
Question 24
√ 2
x −1
dx .
Find
x
MST125 June 2015 J
[6]
TURN OVER
9
Question 25
Solve the initial value problem
dy
+ 2y = 10x2 (x > 0),
x
dx
y(1) = 3.
[6]
Question 26
Let P (n) be the statement
(n − 1)! > 2n ,
where n ∈ N.
(a) Determine whether P (n) is true or false for each of n = 5 and n = 6.
[2]
(b) Use mathematical induction to show that P (n) is true for all n ≥ 6.
[4]
TURN OVER FOR SECTION C
10
MST125 June 2015 J
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
Consider the equation
4x2 − 25y 2 = 16.
(a) Show that this equation represents a hyperbola in standard position.
[2]
(b) Find the vertices and asymptotes of the hyperbola.
[2]
(c) Sketch the hyperbola, marking the vertices and asymptotes and labelling
them with their coordinates or equations as appropriate.
[3]
(d) Find the eccentricity, foci and directrices of the hyperbola.
[3]
(e) Write down a parametrisation of the part of the hyperbola that lies in the
second quadrant, excluding the point that lies on the x-axis.
[2]
Question 28
9x
dx .
(x + 1)(x − 2)2
π
2
sin3 x cos3 x dx .
(b) Evaluate the integral
(a) Find the integral
[8]
[4]
0
Question 29
9 −20
(a) Express the matrix A =
in the form PDP−1 , where D is a 2 × 2
4 −9
diagonal matrix and P is a 2 × 2 invertible matrix. Find the matrix P−1 as
part of your answer.
10
(b) Find A .
[8]
[2]
(c) Find the general solution of the following system of differential equations.
ẋ = 9x − 20y
ẏ = 4x − 9y.
[2]
[END OF QUESTION PAPER]
MST125 June 2015 J
11