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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 = sin–1 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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Question one
a. Parabola
i.
In general, the equation of a parabola is represented by 𝑦 = 𝑎2 + 𝑏𝑥 + 𝑐
1
From the given equation 𝑎 = − 10
ii.

iii.

iv.

The parabola is n-shaped because a < 0 thus has a maximum point.
X-intercepts can be found when y=0
𝑥 2 9𝑥 11
0= −
+
+
10 10 5
Multiply throughout by 10 we get
0 = −𝑥 2 + 9𝑥 + 22
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
2
−9 ± √9 − 4(−1)(22)
𝑥=
2(−1)
−9 ± √169
𝑥=
�...


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