Question 1 (Unit 1 ) – 25 marks In parts (a), (b)(i), (c) and (d)(i) you must not use your computer to solve the problems, though you may of course use it to check your answers. The solutions that you...

Question 1 (Unit 1 ) – 25 marks
In parts (a), (b)(i), (c) and (d)(i) you must not use your computer to solve
the problems, though you may of course use it to check your answers. The
solutions that you submit to your tutor must show your working, and
contain explanations of how you obtained your answers. You may,
however, quote any formulae from the MST209 Handbook that you need,
provided that you reference it clearly.
In parts (b)(ii), (b)(iii), (d)(ii) and (d)(iii) you are expected to use your
computer. We have suggested that you use certain Mathcad worksheets
associated with Unit 1 ; you are not obliged to do so, but you will find that
these worksheets are convenient because, when using them, you can
consult the ‘Mathcad tips’ pop-ups if you can’t remember what to do.
page 2 of 22
When you use Mathcad, it is important that you make your tutor aware of
the results that you wish to be considered: you should not leave your tutor
to interpret Mathcad printouts.
(a) Real variables x and y are related by the equation
ln(3y + 2)=5 ln(x - 1) - ln(2 - x) - x.
(i) Determine the range of values of x and y for which the
expressions on each side of this equation are defined. [2]
(ii) Find y explicitly as a function of x, that is, express the equation
in the form y = f(x), simplifying your answer as far as possible. [3]
(b) (i) Show that the expressions 2v
3 sin(3t + p/3) and
v
3 sin(3t)+3 cos(3t) are equivalent. [2]
(ii) Use Mathcad to graph the function f(t) = v
3 sin(3t)+3 cos(3t)
PC on the interval -p = t = p. On the same graph, plot the function
g(t) = 0.8t
2
. How many solutions does the equation f(t) = g(t)
have on the interval -p = t = p? Briefly justify your answer. [3]
(iii) Use Mathcad to find, correct to four decimal places, all of the
PC solutions of the equation f(t) = g(t) on the interval -3 = t = -1.
(You can use the Mathcad worksheet 20901-02 Solving
numerically.xmcd, where you will need to follow the instructions
concerning solve blocks in the Mathcad tips pop-up.) Submit to
your tutor just one page of Mathcad printout, on which you
should clearly identify the input and your solutions. [3]
(c) If xy2 + 3x
2y
2 - x
3y = 5, use implicit differentiation to determine
dy/dx, expressing your answer in the form
dy
dx = f(x, y),
that is, an expression that involves terms in both x and y. [4]
(d) (i) Without using Mathcad, determine the indefinite integral of the
function
f(x) = 1
(x + 3)(x - 2) (-3 (ii) Use Mathcad to obtain the indefinite integral in part (d)(i), and
PC comment on the result that you obtain. (If you wish, you can use
the Mathcad worksheet 20901-05 Integration.xmcd, which has
a Mathcad tips pop-up.) [2]
(iii) Evaluate the integral
% 2p/3
p/2
sec(2x) dx.
(There is no need to evaluate your answer numerically.)
Compare your answer with that obtained using Mathcad. [3]
PC
page 3 of 22
TMA MST209 01 Part 2 Cut-off date 28 November 2012
Questions 2 to 6 below, on Units 2, 3 and 4, form the second part of
TMA 01. Your overall grade on TMA 01 will be based on the sum of your
marks on these questions and on the question in Part 1.
In order to encourage you to present your solutions to the TMA questions
in a good mathematical style, there are 5 presentation marks on this TMA
given for how you:
• use correct mathematical notation;
• define any symbols that you introduce in formulating and solving a
problem;
• give references for standard formulae and derivations;
• include comments and explanations within your mathematics;
• explicitly state results and conclusions, giving answers to an
appropriate degree of accuracy and interpreting answers in the
context of the question;
• draw diagrams and graphs;
• annotate your Mathcad worksheets.
These features are seen as being essential to complementing your
mathematical skills. Your tutor will have made comments on how to
achieve the threshold requirement for these objectives in the first part of
this TMA. The presentation marks will be put in the box for Question 7
on the TMA form (PT3).
Please send your answers to Questions 2 to 6 to your tutor. Your tutor
should have kept the PT3 for this assignment, so there is no need to send
another. (If your tutor has returned your original PT3 by mistake with
your answer to Question 1, send it back with your answers to Questions 2
to 6.) Your copy of the form will be returned to you with your answers to
these questions.
Question 2 (Unit 2 ) – 14 marks
In each of parts (a) and (b) you must solve the problem by hand, and the
solution that you submit to your tutor should contain all your working.
(a) Consider the differential equation
2y +
dy
dx tan(2x)=cos(2x) (p/4 Which of the methods of finding analytic solutions of differential
equations described in Unit 2 could you use to solve this equation?
Give reasons for your answer.
Find the general solution of the differential equation, expressing y
explicitly as a function of x. Hence find the particular solution of the
differential equation that satisfies the initial condition y(p/4) = 1. [8]
(b) Consider the differential equation
e
t
dy
dt = y
3
(0

Which of the methods of finding analytic solutions of differential
equations described in Unit 2 could you use to solve this equation?
Give reasons for your answer.
page 4 of 22
Find the general solution of the differential equation, expressing y
explicitly as a function of t. Hence find the particular solution of the
differential equation that satisfies the initial condition y(0) = 1. [6]
Question 3 (Unit 2 ) – 9 marks
This question is concerned with the use of Euler’s method to find a
numerical solution to the initial-value problem
dy
dx = 2x
2 - 3y
2
, y(0) = 0.
In part (a) you may use a computer or calculator only to perform
numerical calculations. In part (b), on the other hand, you are expected to
use one of the MST209 Mathcad worksheets. You may find it helpful to
use the same worksheet in part (c).
(a) Use Euler’s method with a step size of 0.1 to find an approximation to
the value of y(0.3), where y(x) is the solution to the given initial-value
problem. Carry out your calculations using at least five decimal
places. Show all your working, and quote your final answer to four
decimal places. [3]
(b) Use the Mathcad worksheet 20902-02 Euler’s method.xmcd
PC associated with Unit 2, Activity 2.3, to calculate approximations to
six-decimal-place accuracy to the value of y(1), where y(x) is the
solution to the given initial-value problem, with step sizes h = 0.01,
0.001 and 0.0001. (You may have to edit the worksheet, by entering
the appropriate right-hand side for the differential equation, the
appropriate initial values and the number (three) of step sizes.)
Submit to your tutor the Mathcad printouts that show your edited
inputs and the output. [3]
(c) The value 0.526 709 of the solution y(1), which is correct to six
decimal places, has been obtained using a different numerical method.
Using the three approximate values for y(1) that you have obtained
PC using Euler’s method in part (b), confirm that ‘absolute error is
approximately proportional to step size’ (page 80 of Unit 2 ) when
Euler’s method is used for this initial-value problem with step sizes
h = 0.01, 0.001 and 0.0001. Find the constant of proportionality
correct to one decimal place.
(Hint: You may find it helpful to construct a table of the following
form.
Step size Approximation Correct value Absolute error
Absolute error
Step size
0.01 0.526 709
0.001 0.526 709
0.0001 0.526 709
The approximate values and absolute errors may be obtained from the
Mathcad worksheet.) [2]
(d) Use your answer to part (c) to predict the size of the absolute error in
calculating an approximation to y(1) using a step size of 0.000 001. [1]
page 5 of 22
Question 4 (Unit 3 ) – 23 marks
In parts (a)–(c) you must solve the problem by hand, and you must show
your working in your solution. In part (d) you are expected to use a
computer.
(a) Determine the general solutions of the following linear second-order
homogeneous differential equations.
(i)
d
2y
dx2
+ 8
dy
dx + 7y = 0
(ii) d
2y
dx2
+ 8
dy
dx + 16y = 0
(iii) d
2y
dx2
+ 8
dy
dx + 25y = 0 [6]
(b) Find a particular integral of the inhomogeneous differential equation
d
2y
dx2
+ 8
dy
dx + 25y = 9e
-4x - 125x.
Hence write down the general solution of this equation. [6]
(c) Find the particular solution to the initial-value problem
d
2y
dx2
+ 8
dy
dx + 25y = 9e
-4x - 125x, y(0) = 8
5
, y!
(0)=1. [5]
(d) Use Mathcad to plot the particular solution to the initial-value
PC problem in part (c) for x between 0 and 2. The particular solution to
the initial-value problem in part (c) can be split into terms arising
from the complementary function and those from the particular
integral. Use Mathcad to plot both of these functions. [4]
(e) Using part (d), or otherwise, identify the approximate solution to the
initial-value problem in part (c) for large values of x. Give a very brief
justification. [2]
Question 5 (Unit 4 ) – 21 marks
Note that throughout this question, vectors are shown in bold type
(e.g. v) or with an over-arrow (e.g.
-?OA). When writing your solutions, if
use of an over-arrow is inappropriate, then you should use underlining to
show a vector quantity (see Subsection 1.2 on page 155 of Unit 4 ). If you
type your assignments, then vectors must be in bold type. If you fail to
distinguish vectors in this way, you will certainly lose some of the
presentation marks available for this assignment.
In this question you should quote all numerical answers correct to two
decimal places.
page 6 of 22
The dimensions of a badminton court are as shown below in plan view.
O
A
X
6.096 m
?
?
5.1816 m
?
?
? 6.7056 m ?
1.9812 m ? ?
?
j
?i
k?
A player at the point O smashes the shuttlecock from a point Y at a
height of 3.00 m vertically above O. Assume that the shuttlecock then
travels in a straight line directly over the net at the midpoint A, which is
at a height of 1.55 m, before bouncing at the point X. The unit vector k is
directed vertically upwards.
(a) Taking the origin at O and axes as shown in the figure, write down
the position vectors of the points Y and A. [2]
(b) Determine the position vector of any point on the line Y A, and hence
find the position vector of the point X. Deduce that the shuttlecock
cannot land in the shaded area of the court. [7]
(c) Find the distance travelled by the shuttlecock between the point of
the smash and hitting the floor. [2]
(d) Find the dot product of the vectors
--?XO and
--?XY , and hence determine
the angle below the horizontal at which the shuttlecock travels before
landing. Give your answer in degrees correct to two decimal places. [4]
(e) Find the cross product of the vectors
--?Y X and
--?Y O, and use this result
to determine the area of the triangle OY X and a unit vector
perpendicular to --?Y X and
--?Y O. [6]
Question 6 (Unit 4 ) – 3 marks
Consider the vector v shown in the following diagram.
?
?
i
j
?
?
?
v?
Find the i- and j-components of v in terms of the magnitude of v and ?,
simplifying your answers as far as possible.