Faculty of Science, Technology, Engineering and Mathematics
MU123 Discovering mathematics
MU123
TMA 04
Covers Units 10, 11 and 12
2019B
Cut-off date
13 August 2019
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.
Copyright c 2019 The Open University
19.1
WEB 06686 2
TMA 04
Question 1
Cut-off date 13 August 2019
–
20 marks
This question is based on your work on MU123 up to and including Unit 10.
Marcus competes in the shot put for his local track and field sports club.
When he throws the shot, its trajectory after he releases it can be modelled
by the quadratic equation
y=−
x2 9x 11
+
+ ,
10 10
5
where y represents the height in metres of the shot above the ground, and x
represents the horizontal distance in metres of the shot from the position
where it is released by Marcus. Assume that the surface of the field is
horizontal.
x2 9x 11
(a) The graph of y = − +
+
is a parabola.
10 10
5
(i) Is the parabola u-shaped or n-shaped? How can you tell this from
the equation?
(ii) Use algebra to find the x-intercepts.
(iii) Explain why the y-intercept is
11
5 .
[1]
[4]
[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.
(v) Provide a sketch of the graph of the parabola, either by hand or by
using Graphplotter.
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.
[4]
[3]
(b) In this part of the question, you are asked to consider the trajectory of
x2 9x 11
the shot modelled by the equation y = − +
+
in conjunction
5
10 10
with the results that you found in part (a).
(i)
Find the height of the shot when it is 3 metres horizontally from
the position where Marcus releases it.
[1]
(ii) Use your answer to part (a)(iv) to find the maximum height
reached by the shot.
[2]
(iii) What does the y-intercept represent in the context of this model?
[1]
(iv) How far will the shot be horizontally from the position where
Marcus releases it when it first lands on the field? Explain your
answer.
[2]
page 2 of 9
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 − 14t + 6 = 0.
Give your answers correct to one decimal place.
[3]
(b) This part of the question concerns the quadratic equation
5x2 − 11x + 8 = 0.
(i)
Find the discriminant of the quadratic expression 5x2 − 11x + 8.
[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 = 5x2 − 11x + 8?
[1]
(c) (i)
Write the quadratic expression x2 − 24x − 12 in completed-square
form.
[2]
(ii) Use the completed-square form from part (c)(i) to solve the
equation x2 − 24x − 12 = 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 − 24x − 12.
[1]
page 3 of 9
Question 3
–
18 marks
This question is based on your work on MU123 up to and including Unit 11.
A company wished to know if the training programme that they developed
for a particular task was effective. 20 employees were timed performing the
task before and after the training. The times were recorded and are given in
Table 1.
Table 1 Time spent performing
the task (in minutes)
Before training
After training
27
28
22
26
21
31
29
27
29
29
28
28
28
27
29
28
26
30
26
25
24
23
20
24
21
24
24
23
22
25
23
24
25
22
23
22
23
24
23
22
(a) Enter these data into two lists in Dataplotter.
To check that you have entered the values correctly, the mean number of
minutes that it took to perform the task before training is 27.2 minutes,
and the mean number of minutes it took to perform the task after
training is 23.1 minutes.
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.
page 4 of 9
[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 times compare for performing the task before
training and after training? Use your boxplots from part (a) to
explain your answer.
[2]
(ii) Are the data more spread out for performing the task before
training or after training? Use your boxplots from part (a) to
explain your answer.
[2]
(c) Use the boxplot for before training 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 20
and an interval of 1. 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 time spent performing the task that can
be seen more easily on the histograms than on the boxplots.
Question 4
–
[2]
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]
x
24 cm
38°
Figure 1
(b) Triangle P QR has a right angle at R. The length of side P Q is 24.5 cm,
and the length of side QR is 15 cm. Draw triangle P QR, and find
∠RP Q, giving your answer correct to the nearest degree.
page 5 of 9
[3]
(c) (i)
Find the angle ABC in the triangle in Figure 2, giving your answer
correct to the nearest degree.
[5]
A
.4 c
m
m
17.0 c
32
C
B
24.3 cm
Figure 2
(ii) Find the area of the triangle ABC in Figure 2, giving your answer
correct to the nearest square cm.
(d) (i)
Convert 72◦ to radians, leaving your answer in terms of π.
(ii) Use your answer from part (d)(i) to find the area of a sector of a
circle of radius 8.2 cm and angle 72◦ , giving your answer correct to
two significant figures.
Question 5
–
[3]
[2]
[3]
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.
Tom and his two friends Joe and Colm are playing catch with a beach ball at
the seaside. Colm is standing at a point C, 24 metres due south of Joe, who
is standing at point J. Tom is standing at a point T , 52◦ west of north of
Colm and 27 metres from point C. (You may assume that all distances are
flat and in a straight line.)
(a) Draw a diagram showing the points T , J and C (triangle T JC),
marking the internal angle and the lengths that you are given.
[2]
(b) Find the distance between Tom and Joe, that is, the length of the
side T J.
[3]
(c) When Tom throws the beach ball, the wind catches it so that it lands
on the ground between Joe and Colm (on the line JC) due east of
Tom (T ). Add a line to your diagram that shows the shortest distance
from where Tom is standing (point T ) to where the ball lands at point B
(on the line JC). What angle does the line T B make with the line JC?
[2]
(d) How far is Tom from the ball now? That is, find the length of T B.
[3]
page 6 of 9
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]
The Question
A
40°
D
18
.4
cm
15.2 cm
C
B
Triangle ABC is a right-angled triangle. The length of side BC is 15.2 cm.
∠CAB = 40◦ and ∠BCA = 90◦ . D is a point on side CA, and BD has
length 18.4 cm.
(i) Find ∠CDB and hence ∠ADB, giving your answers to two significant
figures.
(ii) Find the length of DC, giving your answer to two significant figures.
(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) In triangle DBC, the known sides are the hypotenuse and
the side opposite to ÐCDB, so the ratio needed is sine.
15.2
sin (ÐCDB) = 18.4
ÐCDB = sin1
15.2
( 18.4
) = 55.6
¼°
So ÐCDB is 56° (to 2 s.f.)
ACD is a straight angle so is 180°
Therefore ÐADB = 180° 55.6¼° = 124.3¼°
So ÐADB is 120° (to 2 s.f.)
page 7 of 9
(ii) Triangle BCD is a right-angled triangle so
using Pythagoras Theorem
DB2 = CB2 + DC2
DC2 = DB2 CB2
DC = DB CB
DC = 18.4 15.2 = 3.2
The length of DC is 3.2 cm (to 2 s.f.)
(iii) Using the sine rule in triangle ABD
AB
DB
=
sin (ÐADB)
sin (ÐDAB)
AB
18.4
=
sin (120°)
sin (40°)
AB =
18.4 ´ sin (120°)
= 24.7 ¼
sin (40°)
The length of AB is 25 cm (to 2 s.f.)
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?
page 8 of 9
[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.
page 9 of 9
[5]

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