MST125/D
MST1251606F1
Module Examination 2016 Essential mathematics 2 Wednesday 1 June 2016
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.
c 2016 The Open University Copyright
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MST125 June 2016
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 2016
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 3 401 706 × 17 000 018 modulo 17? A 0
B 3
C 6
D 9
E
12
Question 2 Which of the following is a multiplicative inverse of 1123 modulo 59? A 1121
B 1122
C 1125
D 1135
1160
E
Question 3 Given that 13 is a multiplicative inverse of 71 modulo 922, which of the following is a solution of the linear congruence 71x ≡ 92 (mod 922) ? A 104
B 105
C 274
D 275
E
921
Question 4 What is the equation of the directrix of the parabola with equation 3y 2 − 32x = 0 ? A x=−
8 3
B x=−
3 8
C x=
3 8
D x=
8 3
E
x=
32 3
Question 5 A block of mass 5.2 kg rests on a horizontal surface. The coefficient of static friction between the block and the surface is 0.47. What is the maximum magnitude of the static friction force from the surface on the object, in newtons to two significant figures? Take the acceleration due to gravity to be g = 9.8 m s−2 . A 1.1
B 2.4
C 15
D 24
E
110
Question 6 Which option describes the linear transformation represented by the matrix 1 0 ? 0 −1 A A dilation
B A shear
D A rotation
E
4
C A flattening
A reflection
MST125 June 2016
Question 7 Which of the following specifies the glide reflection g formed from the reflection in the y-axis followed by the translation 2 units down? A g(x, y) = (−x, −y − 2)
B g(x, y) = (−x, y − 2)
C g(x, y) = (−x, y + 2)
D g(x, y) = (x, −y + 2)
E
g(x, y) = (x, y − 2)
Question 8 What is the form of the partial fraction expansion of the expression 2x2 − x + 1 ? (x − 1)(x2 + 1) In the options, A, B and C represent constants. A C E
Bx2 + C Ax + 2 x−1 x +1 Bx + C A + 2 x−1 x +1 A B C + + 2 x−1 x+1 x +1
A B + 2 x−1 x +1 A Bx2 D + 2 x−1 x +1
B
Question 9 Which of the following is an integrating factor p(x) for the differential equation dy 6y + =1 dx x A ln(x−6 )
(x > 0) ?
B e6x
C ln(x6 )
D x6
E
e−x
6
Question 10 What is the solution of the initial value problem dy = e4x + sin x, where y = 5 when x = 0 ? dx A y = 14 e4x − cos x
B y = 14 e4x − cos x +
D y = 4e4x − cos x + 2
E
17 4
C y = 14 e4x − cos x +
y = 4e4x − cos x
Question 11 Let P (n) be the statement n is a multiple of 6. Which of the following statements is true? A P (1) AND P (3)
B P (3) AND P (6)
D P (2) OR P (3)
E
MST125 June 2016
C P (6) AND P (14)
P (6) OR P (12)
TURN OVER
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23 4
Question 12 Let P be the following statement. At least one of m and n is odd. Which of the following statements is the negation of P ? A Both m and n are even. B Both m and n are odd. C At most one of m and n is odd. D At most one of m and n is even. E
At least one of m and n is even.
Question 13 Consider the following statement. If n is prime, then n + 2 is prime. Which of the following values of n is a counter-example to this statement? A 3
B 4
C 5
D 6
E
7
Question 14 The graph below is the velocity–time graph of an object moving along a straight line, where v is the velocity (in metres per second) of the object at time t (in seconds). v 2 1
−1
1
2
3
4
5
6
7
t
−2
What is the approximate acceleration (in m s−2 ) of the object at time t = 6? A −2
6
B −0.5
C 0
D 0.5
E
2
MST125 June 2016
Question 15 A projectile is launched with an initial velocity (in m s−1 ) of 2i + 3j, where the Cartesian unit vectors i and j point right and up, respectively. Which of the following gives the velocity v (in m s−1 ) of the projectile at time t (in seconds) after launch? In the options, g represents the magnitude of the acceleration due to gravity (in m s−2 ). A v = −2 i + (gt − 3) j
B v = −2 i + (3 − gt) j
C v=
2 i + (3 − gt) j
E
2 i + (3t − 12 gt2 ) j
v=
2 i + ( 12 gt2 − 3t) j
D v=
Question 16
What are the eigenvalues of the matrix A −3 and −18
B 0 and −21
D 3 and 18
E
3 6 ? 9 18 C 0 and 21
18 and 54
Question 17
Which of the following vectors is an eigenvector of the matrix corresponding to the eigenvalue 3? 4 −1 1 A B C 1 2 −1 Question 18
1 D 1
E
4 −1 , 1 2 1 2
−1 0 The matrix A = can be expressed in the form PDP−1 , where −4 3 1 0 −1 0 1 0 −1 P= , D= and P = . 1 −1 0 3 1 −1 What is A5 ? −1 0 A 0 243 −1 0 D −1024 243
MST125 June 2016
B E
−1 0 243 −243
C
−1 0 1024 −243
−1 0 −244 243
TURN OVER
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Question 19 Consider the set S = {1, 2, 3, 4, 5, 6, 7}. How many subsets of S contain both the elements 3 and 4? Examples of such subsets are {3, 4, 6} and {3, 4, 5, 7}. A 21
B 24
C 25
D 32
E
96
Question 20 What is the probability that when two dice are rolled the total is at least 4? A
1 12
B
5 6
C
31 36
D
11 12
E
61 64
SEE THE NEXT PAGE FOR SECTION B
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MST125 June 2016
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 Consider the curve with parametrisation x = 2 + 4 cos t,
y = −3 + 4 sin t
(0 ≤ t ≤ π/2) .
(a) Calculate the coordinates of the endpoints of the curve.
[2]
(b) Sketch the curve, or describe it geometrically.
[2]
(c) Find the y-coordinate of the point on the curve with x-coordinate 4. Give an exact answer.
[2]
Question 22 A particle, which remains at rest, is acted on by three forces, R, S and T, and no others. The force diagram below shows the angles at which the forces act. The magnitude of the force R is 20 N. T S j
45°
30°
i
R (a) Find expressions for the component forms of the three forces R, S and T, taking the directions of the Cartesian unit vectors i and j to be as shown in the diagram (j is parallel to R), and denoting the magnitudes of S and T by S and T , respectively.
[3]
(b) Hence, or otherwise, find the magnitude S of the force S in newtons to two significant figures.
[3]
Question 23 (a) Find the affine transformation f that maps the points (0, 0), (1, 0) and (0, 1) to the points (−2, 3), (4, 5) and (2, 6).
[2]
(b) Hence find the area of the triangle with vertices (−2, 3), (4, 5) and (2, 6).
[2]
(c) Find the image of the point (1, 1) under f .
[2]
Question 24
Evaluate the integral 4
8
x2 + 2x + 1 dx . x+3
Give an exact answer. MST125 June 2016
[6] TURN OVER
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Question 25 (a) Find the integral sin(2x) cos(5x) dx.
[2]
(b) Hence, or otherwise, solve the following initial value problem, giving your answer in implicit form. dy sin(2x) cos(5x) = dx y2
(y > 0),
where y = 1 when x =
π . 2
[4]
Question 26 Let P (n) be the following statement: 5n > 20n. (a) Show that P (2) is false and P (3) is true.
[1]
(b) Use mathematical induction to show that P (n) is true for all n ≥ 3.
[5]
SEE THE NEXT PAGE FOR SECTION C
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MST125 June 2016
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 function f (x) =
x . (x + 1)2
(a) Find its domain and intercepts.
[2]
(b) Find f (x).
[2]
(c) Find the coordinates of any stationary points of f . Construct a table of signs for f (x), determine the intervals on which f is increasing or decreasing, and determine the nature(s) of the stationary point(s).
[4]
(d) Write down the equations of the asymptotes of f .
[1]
(e) Determine whether f is an even or odd function, or neither.
[1]
(f) Sketch the graph of f .
[2]
MST125 June 2016
TURN OVER
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Question 28 A scene in a film involves a motorcyclist riding off a cliff edge and over a river. Assume that at the instant the motorcyclist leaves the cliff edge, he is 8 m above the river, his speed is V m s−1 and he is travelling in an upwards direction, making an angle of 15◦ with the horizontal. Take the origin to be vertically below the edge of the cliff and on the surface of the river. Take the y-axis to point vertically upwards from the origin and the x-axis to point horizontally from the origin across the river, as shown below. y
|v| = V 15◦ 8m x 45 m
Let i and j be the Cartesian unit vectors in the positive directions of the x- and y-axes respectively. Assume that the motorcyclist and his motorcycle can be modelled as a particle and that the only force acting on it is its weight. Take the magnitude of the acceleration due to gravity to be g = 9.8 m s−2 . (a) Write down an expression for the acceleration of the motorcyclist. Hence show that the position r of the motorcyclist at time t seconds after he has left the cliff edge is given by r = V t cos 15◦ i + (V t sin 15◦ − 12 gt2 + 8) j.
[5]
(b) Assume that the motorcyclist lands on the opposite river bank, after travelling a horizontal distance of 45 m. Find the time, in seconds to two significant figures, that it took the motorcyclist to reach the river bank.
[5]
(c) Determine the speed V , in m s−1 to two significant figures, at which the motorcyclist left the cliff edge.
[2]
12
MST125 June 2016
Question 29 A 10 cm wide strip is to be tiled using the five types of tile shown below. •
Square, 10 cm × 10 cm, in either white or black.
•
Rectangular, 10 cm × 20 cm, in white, black or grey.
Here is an example of a 10 cm × 80 cm strip that has been tiled with these tiles. For each positive integer n, let un be the number of ways of tiling a 10 cm × (10n) cm strip with these tiles. (a) Find the values of u1 and u2 , briefly explaining your answers.
[3]
(b) Explain clearly why the sequence (un ) satisfies the recurrence relation un = 2un−1 + 3un−2.
[3]
(c) Find a closed form for the sequence (un ).
[5]
(d) Find the number of ways of tiling a 10 cm × 50 cm strip.
[1]
[END OF QUESTION PAPER]
MST125 June 2016
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