Faculty of Science, Technology, Engineering and Mathematics

MU123 Discovering mathematics

MU123

TMA 04 2017J
Covers Units 10, 11 and 12 Cut-off date 24 April 2018

Submission instructions
You will find instructions for completing TMAs in the ‘Assessment’ area of
the MU123 website. Please read these instructions before beginning work on
this TMA.
Reviewing your tutor’s comments on your previous TMA will help you as
you work on this one.

Special instructions
Remember that you need to explain your reasoning and communicate your
ideas clearly, as described in Subsection 5.3 of Unit 1. This includes:
• explaining your mathematics in the context of the question
• the correct use of notation and units
• appropriate rounding.
Your score out of 5 for good mathematical communication (GMC) will be
recorded against Question 8. You do not have to submit any work for
Question 8.

c 2017 The Open University

Copyright WEB 05383 4
16.1
TMA 04 Cut-off date 24 April 2018

Question 1 – 20 marks
This question is based on your work on MU123 up to and including Unit 10.
The trajectory of a rugby ball thrown to a lineout by the hooker in training
can be modelled by the quadratic equation
x2 7x
y=− + + 2,
4 4
where y represents the height in metres of the ball above the ground, and x
represents the horizontal distance in metres of the ball from the position
where it was released by the hooker. Assume that the surface of the playing
field is horizontal and the ball is thrown in straight.
x2 7x
(a) The graph of y = − + + 2 is a parabola.
4 4
(i) Is the parabola u-shaped or n-shaped? How can you tell this from
the equation? [1]
(ii) Use algebra to find the x-intercepts. [4]
(iii) Explain why the y-intercept is 2. [2]
(iv) Find the equation of the axis of symmetry, explaining your method.
Use this information to find the coordinates of the vertex, rounding
your answers where necessary to one decimal place. [4]
(v) Provide a sketch of the graph of the parabola, either by hand or by
using Graphplotter. [3]
You should refer to the graph-sketching strategy box in Section 2.4
of Unit 10 for information on how to sketch and label a graph
correctly.
(b) In this part of the question, you are asked to consider the trajectory of
x2 7x
the ball modelled by the equation y = − + + 2 in conjunction
4 4
with the results that you found in part (a).
(i) Find the height of the ball when it is 3 metres horizontally from the
position where the hooker releases it. [1]
(ii) Use your answer to part (a)(iv) to find the maximum height
reached by the rugby ball. [2]
(iii) What does the y-intercept represent in the context of this model? [1]
(iv) Assuming that the ball is not touched in the air, how far is it
horizontally from where the hooker released it when it first lands
on the ground? Explain your answer. [2]

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Question 2 – 13 marks
This question is based on your work on MU123 up to and including Unit 10.
(a) Use the quadratic formula to solve the equation
5t2 − 11t + 3 = 0.
Give your answers correct to one decimal place. [3]
(b) This part of the question concerns the quadratic equation
3x2 + 6x + 13 = 0.
(i) Find the discriminant of the quadratic expression 3x2 + 6x + 13. [2]
(ii) What does this tell you about the number of solutions of the
equation? Explain your answer briefly. [2]
(iii) What does this tell you about the graph of y = 3x2 + 6x + 13? [1]
(c) (i) Write the quadratic expression x2 − 42x − 7 in completed-square
form. [2]
(ii) Use the completed-square form from part (c)(i) to solve the
equation x2 − 42x − 7 = 0, leaving your answer in exact (surd)
form, simplified as far as possible. [2]
(iii) Use the completed-square form from part (c)(i) to write down the
vertex of the parabola y = x2 − 42x − 7. [1]

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Question 3 – 18 marks
This question is based on your work on MU123 up to and including Unit 11.
A drinks company has two machines that perform the same task: filling
bottles with water. The volumes of water in 20 bottles collected from each
machine were recorded and are given in Table 1.

Table 1 Volume of water

(in millilitres)

Machine 1 Machine 2
490 498
496 502
505 499
510 501
490 499
495 497
500 499
502 504
498 502
497 502
499 501
501 501
500 498
505 501
501 496
501 495
503 500
504 502
499 493
499 501

(a) Enter these data into two lists in Dataplotter

To check that you have entered the values correctly, the mean volume of
water in the bottles for machine 1 is 499.8 millilitres, and the mean
volume of water in the bottles for machine 2 is 499.6 millilitres.
Create boxplots for the two datasets, either using Dataplotter or by
hand. Include either a printout of your boxplots or your complete
hand-drawn boxplots with your answer to this question.
Remember to include all relevant information on your boxplots as set
out in Subsection 1.2 of Unit 11. Using Dataplotter, it is sufficient to
include the key values in the list to the right of the boxplot, rather than
on the boxplot itself. If you draw boxplots by hand, then you should use
squared paper and a common axis for both plots to make it easy to
compare the boxplots. Remember that the mean and standard deviation
are not part of a boxplot. [7]
(b) A boxplot gives you a visual representation of the average value using
the median, and also tells you how the data are spread out based on
the size of the box and the lengths of the whiskers.
(i) How do the average volumes of water in the bottles compare for the
two machines? Use your boxplots from part (a) to explain your
answer. [2]

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 (ii) Are the data more spread out for the volumes of water in the
bottles for machine 1 or for machine 2? Use your boxplots from
part (a) to explain your answer. [2]
(c) Use the boxplot for machine 2 to say whether the data are symmetrical
or skewed. If the data are skewed, then state whether they are skewed
to the left or skewed to the right, explaining your reasoning briefly. [2]
(d) Create a histogram for each of the datasets, using a start value of 490
and an interval of 2. Include either a printout of your histograms or a
sketch drawn by hand with your answer to this question. [3]
If you draw histograms by hand, then you should use squared paper and
the same axis scale for both histograms to make it easy to compare them.
(e) Comment on one aspect of the volumes of water in the bottles that can
be seen more easily on the histograms than on the boxplots. [2]

Question 4 – 19 marks
This question is based on your work on MU123 up to and including Unit 12.
(a) Find the length of the side marked x in the triangle in Figure 1, giving
your answer correct to the nearest cm. [3]

41◦

x 18 cm

Figure 1
(b) Triangle LM N has a right angle at M . The length of side LN is
22.5 cm, and the length of side M N is 18.1 cm, Draw triangle LM N ,
and find ∠M N L, giving your answer correct to the nearest degree. [3]
(c) (i) Find the angle ABC in the triangle in Figure 2, giving your answer
correct to the nearest degree. [5]
B

10.2 cm 6.5 cm

A C
14.1 cm
Figure 2
(ii) Find the area of the triangle ABC in Figure 2, giving your answer
correct to the nearest square cm. [3]
(d) (i) Convert 36◦ to radians, leaving your answer in terms of π. [2]
(ii) Use your answer from part (d)(i) to find the area of a sector of a
circle of radius 9.2 cm and angle 36◦ , giving your answer correct to
two significant figures. [3]

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Question 5 – 10 marks
This question is based on your work on MU123 up to and including Unit 12.
You should use trigonometry, not scale drawings, to find your answers. Give
all answers correct to two significant figures. Remember to use full versions
of earlier answers to avoid rounding errors.
Marcus is standing on his local rugby pitch facing due north. The sideline on
his right-hand side has length 110 metres and runs due north from corner
flag A to corner flag B. The point C where Marcus is standing is 32◦ west of
south of B and 51◦ west of north of A. (You may assume that the rugby
pitch is completely flat and that all distances are in straight lines.)
(a) Draw a diagram showing the points A, B and C (triangle ABC),
marking in the internal angles and the length that you are given. [2]
(b) Find the distance from Marcus to corner flag B, that is, the length of
the side BC. [3]
(c) Add a line to your diagram that shows the shortest distance from
Marcus’s position at C to the sideline AB, marking this point on the
sideline as point D. What angle does CD make with the sideline AB? [2]
(d) Find the distance from Marcus to the sideline, that is, the length
of CD. [3]

Question 6 – 10 marks
This question is based on your work on MU123 up to and including Unit 12.
In this question, you are asked to comment on a student’s incorrect attempt
at answering the question detailed below.
(a) Write out your own solution to the question, explaining your working. [4]
(b) There are two places in the student’s attempt where a mistake has been
made. Identify these mistakes and explain, as if directly to the student,
why, for each mistake, their working is incorrect. [6]

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The Question
A

35.8◦ 50.5◦
D 20 cm C B

Triangle ABD is a right-angled triangle, with ∠ABD = 90◦ . The point C

lies on side BD. CD has length 20 cm. ∠CDA = 35.8◦ and ∠BCA = 50.5◦ .
(i) Find the length of CA, giving your answer to two significant figures.
(ii) Find ∠CAB.
(iii) Use your answer to part (i) to find the length of AB, giving your answer
to two significant figures.

The student’s incorrect attempt

(i) Angle DCB is a straight line angle so ÐDCA = 180° – 50.5° = 129.5°
This means ÐDAC = 180° – 129.5° – 35.8° =14.7°.
Using the sine rule in triangle DCA
CA DC
sin (ÐCDA) = sin (ÐDAC)
CA 20
sin 35.8° = sin 14.7°
20 ´ sin 35.8°
CA = = 46.103 ... cm.
sin 14.7°
The length of CA is 46 cm (to 2 s.f.).

(ii) ÐCAB = 180° – 90° – 35.8° = 54.2°.

(iii) Triangle CAB is a right-angled triangle.

AB is opposite known angle BCA.
Known side CA is the hypotenuse.
So the ratio needed is sine.
sin (50.5°) = 46AB

46 ´ sin (50.5°) = AB
35.494 ... = AB
So the length of AB is 35 cm (to 2 s.f.).

page 7 of 8
Question 7 – 5 marks
Two of the themes that you have met in MU123 are:
• working with data
• algebraic skills.
In this question you are asked to think about your progress so far with one
of these two themes.
(a) Choose one of the two themes mentioned above (either working with
data or algebraic skills). If you plan to continue studying mathematics,
science or engineering, then we recommend that you choose algebraic
skills because this will be an important aspect of later work. Otherwise,
choose either theme.
Write down the theme that you have chosen. Write down one topic in
your chosen theme that you can work with confidently, and one topic in
your chosen theme that you find more challenging. [2]
(If you did not find any of the work in your chosen theme challenging,
then pick two topics that you can work with confidently.)
(b) Describe two steps that you could take to help you to work more
confidently with the topic from part (a) that you find challenging. [2]
(If you did not find any of the work in your chosen theme challenging,
then explain why.)
(c) Give one example from this TMA of your chosen theme where you were
able to check your answer. How did you check it? [1]

Question 8 – 5 marks
A score out of 5 marks for good mathematical communication over the entire
TMA will be recorded under Question 8. [5]

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