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Faculty of Science, Technology, Engineering and Mathematics
MST124 Essential mathematics 1
MST124
TMA 02 2020B
Covers Units 3, 4, 5 and 6 Cut-off date 13 May 2020
You can submit this TMA either by post to your tutor or electronically
as a PDF file by using the University’s online TMA/EMA service.
Before starting work on it, please read the document Student guidance
for preparing and submitting TMAs, available from the ‘Assessment’
tab of the MST124 website.
If you have a disability that makes it difficult for you to attempt any of these
questions, then please contact your Student Support Team or your tutor for
advice.
The work that you submit should include your working as well as your final
answers.
Your solutions should not involve the use of Maxima, except in those parts of
questions where this is explicitly required or suggested. Your solutions
should not involve the use of any other mathematical software.
Your work should be written in a good mathematical style, as described in
Section 6 of Unit 1, and as demonstrated by the example and activity
solutions in the study units. Five marks (referred to as good mathematical
communication, or GMC, marks) on this TMA are allocated for how well
you do this.
Your score out of 5 for GMC will be recorded against Question 10. You do
not have to submit any work for Question 10.
Copyright
c 2020 The Open University WEB 07737 8
13.1
TMA 02 Cut-off date 13 May 2020
Question 1 – 5 marks
You should be able to answer this question after studying Unit 3.
Use a table of signs to solve the inequality
2x
2 + 3x > 2.
Give your answer in interval notation. [5]
Question 2 – 5 marks
You should be able to answer this question after studying Unit 3.
Radiocarbon dating can be used to estimate the age of material, such as
animal bones or plant remains, that comes from a formerly living organism.
When the organism dies, the amount of carbon-14 in its remains decays
exponentially. Suppose that f(t) is the amount of carbon-14 in an organism
at time t (in years), where t = 0 corresponds to the time of the death of the
organism. Assume that the amount of carbon-14 is modelled by the
exponential decay function
f(t) = Cekt (t ≥ 0),
where C is the initial amount of carbon-14 and k is a constant.
The half-life (or halving period) of a radioactive substance like carbon-14 is
the time that it takes for the amount of the substance to decrease to half of
its original level.
(a) Given that the half-life of carbon-14 is 5730 years, show that the value
of the constant k is −0.000121, correct to three significant figures. [3]
(b) Suppose that some ancient animal bones have been found where the
amount of carbon-14 is known to have decreased to 1/5 of the level that
was present immediately after the death of the animal. How much time
has elapsed since the death of this animal? Give your answer to the
nearest whole number of years. [2]
Hint: Make sure you use an accurate value of k, if possible, not the
rounded value from part (a).
page 2 of 5
Question 3 – 17 marks
You should be able to answer this question after studying Unit 3.
(a) Consider the quadratic function
f(x) = 1
2
x
2 − 4x + 6.
(i) Complete the square in this expression for f to show that it can be
written in the form
f(x) = 1
2
(x − 4)2 − 2. [1]
(ii) Using this completed-square expression, explain how the graph of f
can be obtained from the graph of y = x
2 by using appropriate
translations and scalings.
(You are not asked to submit any sketched graphs in this part, but
you may find it helpful to sketch a graph for yourself.) [3]
(iii) Write down the image set of the function f, in interval notation. [1]
(b) This part of the question concerns the function
g(x) = 1
2
x
2 − 4x + 6 (0 ≤ x ≤ 4).
The function g has the same rule as the function f in part (a), but a
smaller domain.
(i) Sketch the graph of g, using equal scales on the axes. (You should
draw this by hand, rather than using any software.) Mark the
coordinates of the endpoints of the graph. [3]
(ii) What is the image set of g? [1]
(iii) Show that the inverse function g
−1 has the rule
g
−1
(x) = 4 −
√
2x + 4,
justifying each step clearly, and also give its domain and image set. [5]
(iv) Add a sketch of y = g
−1
(x) to the graph that you produced in
part (b)(i). Mark the coordinates of the endpoints of the graph of
g
−1
(x). [3]
Question 4 – 10 marks
You should be able to answer this question after studying Unit 4.
(a) Find all the solutions between −2π and 2π of the equation
sin θ =
1
√
2
,
giving all your answers as exact values in radians. [4]
(b) In triangle ABC (with the usual notation), angle A = 36◦
, side b = 10
and side c = 6.
Use the cosine rule to show that the length of side a is 6.24, rounded to
two decimal places. Without using the cosine rule again, find the
remaining angles B and C, giving your answers to the nearest degree. [6]
page 3 of 5
Question 5 – 10 marks
You should be able to answer this question after studying Unit 4.
(a) Using the exact values for the sine and cosine of both π/4 and π/6, and
the angle difference identity for sine, show that an exact value of
sin(π/12) is given by
√
6 −
√
2
4
. [3]
(b) Use the exact value of cos(π/6) and the half-angle identity for cosine to
show that an exact value of cos(π/12) is given by
p
2 + √
3
2
. [4]
(c) Show that the exact values from parts (a) and (b) satisfy the standard
trigonometric identity sin2
θ + cos2
θ = 1. [3]
Question 6 – 10 marks
You should be able to answer this question after studying Unit 5 and also
Section 7 of the Computer Algebra Guide.
Use Maxima to plot the circle x
2 + y
2 − 2x − 6y − 6 = 0 and the ellipse
25x
2 + 4y
2 − 100x − 32y + 64 = 0 on the same graph. Plotting an ellipse in
Maxima is done in the same way as plotting a circle.
Use Maxima to find the coordinates of the points of intersection between the
circle and the ellipse. State the values of the coordinates rounded to two
decimal places, but do not attempt to use Maxima for rounding.
Include a printout or screenshot of your Maxima worksheet with your
solutions. You are not expected to annotate your Maxima worksheet with
explanations. [10]
Question 7 – 15 marks
You should be able to answer this question after studying Unit 5.
Drones are aircraft without a human pilot aboard and they are increasingly
being used in many civilian applications, including aerial surveys of
pipelines, crops or geology and even for delivery of goods ordered online.
A particular small drone has a speed in still air of 20 m s−1
. It is pointed in
the direction of the bearing 125◦
, but there is a wind blowing at a speed of
7 m s−1
from the south-west.
Take unit vectors i to point east and j to point north.
(a) Express the velocity d of the drone relative to the air and the velocity w
of the wind in component form, giving numerical values in m s−1
to two
decimal places. [7]
(b) Express the resultant velocity v of the drone in component form, giving
numerical values in m s−1
to two decimal places. [3]
(c) Hence find the magnitude and direction of the resultant velocity v of
the drone, giving the magnitude in m s−1
to two decimal places and the
direction as a bearing to the nearest degree. [5]
page 4 of 5
Question 8 – 18 marks
You should be able to answer this question after studying Unit 6.
This question concerns the function
f(x) = 2x
3 − 9x
2 + 12x − 3.
(a) Find the exact x- and y-coordinates of the stationary points of f. [5]
(b) Use the first derivative test to classify the stationary points that you
found in part (a). [5]
(c) Sketch the graph of f, indicating the y-intercept and the points that
you found in part (a). (You should draw this by hand, rather than using
any software, and you can use different scales on the two axes if
appropriate.) [5]
(d) Find the greatest and least values taken by f on the interval [1, 3]. [3]
Question 9 – 5 marks
You should be able to answer this question after studying Unit 6.
An object moves along a straight line. Its displacement s (in metres) from a
reference point at time t (in seconds) is given by
s = 6t
3 − 21t
2 + 12t (t ≥ 0).
(a) Find expressions for the velocity v and the acceleration a of the object
at time t and hence calculate the value of the acceleration at time t = 2. [3]
(b) Find any times at which the velocity of the object is zero. [2]
Question 10 – 5 marks
A score out of 5 marks for good mathematical communication throughout
TMA 02 will be recorded under Question 10. [5]
page 5 of 5
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