Maths assignment

TMA 04 Cut-off date 11 August 2020

Figure 1

(i) Find the equation of the axis of symmetry of the parabola given by

y = 2x2 + 2x - 24, explaining your method.

(ii) Use your answer to part (b)(i) to find the coordinates of the vertex [2]

of the parabola given by y = 2x2 + 2x - 24. [2]

Question 1 – 18 marks

This question is based on your work on MU123 up to and including Unit 10. (a) This part of the question concerns the quadratic equation

-2x2 + x - 3 = 0.

(i) Find the discriminant of the quadratic expression -2x2 + x - 3.

(ii) What does this tell you about the number of solutions of the [2]

equation? Explain your answer briefly. [2]

(iii) What does this tell you about the graph of y = -2x2 + x - 3?

(b) Figure 1 shows part of the graph of the quadratic function y = 2x2 + 2x - 24. [1]

(c) This part of the question concerns the quadratic function

y = x2 + 8x + 10.

(i) Write the quadratic expression x2 + 8x + 10 in completed-square

form.

(ii) Use the completed-square form from part (c)(i) to solve the equation x2 + 8x + 10 = 0, leaving your answer in exact (surd) [2]

form.

(iii) Use the completed-square form from part (c)(i) to write down the [3]

vertex of the parabola y = x2 + 8x + 10.

(iv) Provide a sketch of the graph of the parabola y = x2 + 8x + 10, [1]

either by hand or by using a suitable graphing software package.

If you intend to go on to study more mathematics, you are advised to sketch by hand. Whichever method you choose, you should refer to the graph-sketching strategy box in Subsection 2.4 of Unit 10 for information on how to sketch and label a graph correctly.

Question 2 – 12 marks

This question is based on your work on MU123 up to and including Unit 10.

Lydia is a computer programmer for a small startup company which manufactures artisan fudge. She is creating an economic model to predict how much fudge should be produced to maximise profit in the company’s first year. If too little fudge is made, the company will not meet consumer demand and will miss the opportunity to sell more. If too much fudge is made, there will be an excess of unsold fudge, which will incur a cost to the company. Lydia’s computer program models the situation as a quadratic equation of the form

y = ax2 + bx + c,

where y represents the profit (in thousands of pounds) the company is predicted to make, and x represents the amount of fudge (in tonnes) produced in the year. The values of a, b and c are determined by the computer code, depending on various economic measures and results of market research which Lydia inputs manually.

(a) Lydia wants to check that her code is working correctly. She inputs some test values and runs the code to simulate the model. The quadratic equation output by her code is y = 12.197x2 + 5.825x + 9.215.

By considering the coefficient of x2, explain why Lydia’s code must [3]

contain an error.

(b) Lydia corrects the error, and runs the code again. This time the quadratic equation output by the code is y = -15.217x2 + 18.379x - 0.829.

In this part of the question, you are asked to consider the parabolic graph modelled by the equation y = -15.217x2 + 18.379x - 0.829. [1]

(i) Explain why the y-intercept is -0.829.

(ii) Using the context of the question, give one possible reason why the [1]

y-intercept is negative. [1]

(a) Sam considers displaying the data as a dotplot. Do you think this would be a good choice for representing this data? Give a reason for your

answer. [2]

(iii) Use the quadratic formula to find the x-intercepts. Give your

answers correct to three decimal places.

(iv) The chief executive officer (CEO) of the company wants to set a target of producing 1.2 tonnes of fudge in the first year. Should [4]

Lydia support or advise against this target? Explain your answer.

(v) The CEO sets a target of making £4000 profit in the first year. The company decides to produce 0.4 tonnes of fudge. According to Lydia’s model, will the company exceed or fall short of its target? [2]

Explain your answer.

Question 3 – 20 marks

This question is based on your work on MU123 up to and including Unit 11.

Sam works in the public transport department for a city council. A new tram service has just been launched which links the city centre to the airport. The tram service duplicates an existing bus service, which also runs from the city centre to the airport. Sam would like to compare journey times of 15 journeys from the city centre to the airport for each mode of transport. The raw data is shown in Table 1.

Table 1 Journey times (in minutes) to the airport [3]

Bus Tram

27 26

27 31

28 32

29 32

30 32

30 33

30 33

31 34

32 35

33 35

36 35

37 36

39 36

40 37

59 40

(b) Sam decides to create a boxplot for each mode of transport as displayed in Figure 2.

Figure 2

With reference to Subsection 1.2 of Unit 11, state three ways in which

Sam could improve the clarity of their presentation of the boxplots.

(c) Sam can’t remember which boxplot belongs to which mode of transport.

State one way in which they could check by looking at the raw data for [3]

each mode of transport.

(d) Sam realises that the top boxplot represents the data for the tram and the bottom boxplot represents the data for the bus. They try to interpret what the data is telling them by making the statements below. For each one, state whether you agree or disagree with Sam’s interpretation, and explain why they are correct or incorrect. [1]

(i) ‘On average, the tram is quicker than the bus.’

(ii) ‘The tram is more reliable than the bus, since there is less [2]

variability in tram journey times.’

(iii) ‘More than half of bus journeys take longer than the median bus [2]

journey time.’

(e) Use the boxplot for the bus to say whether the data are symmetrical or skewed. If the data are skewed, then state whether they are skewed to [2]

the left or skewed to the right, explaining your reasoning briefly. [2]

(i) Which histogram belongs to the tram data – Figure 3 or Figure 4?

Give a reason for your answer.

(ii) Comment on one aspect of the performance of the two modes of transport that can be seen more easily on the histograms than on [2]

the boxplots.

(iii) Comment on one aspect of the performance of the two modes of transport that can be seen more easily on the boxplots than on the [2]

histograms. [2]

(f) Sam creates histograms for each of the two modes of transport, shown in Figure 3 and Figure 4 below.

Figure 3

Figure 4

Question 4 19 marks

(a) Figure 5 shows a sketch of a triangle, where all side lengths are measured in centimetres. Find the length of the side marked x, giving

your answer correct to the nearest centimetre. [3]

Figure 5

(b) Triangle DEF has a right angle at E. The length of side DE is 8.1km, and the length of side EF is 1.7km. Find ?EDF, giving your answer correct to the nearest degree.

You may find it useful to sketch triangle DEF in your answer. [3]

(c) (i) Figure 6 shows the triangle ABC, where all side lengths are measured in centimetres. Find ?ABC, giving your answer correct

to the nearest degree. [5]

Figure 6

(ii) Find the area of triangle ABC in Figure 6, giving your answer

correct to the nearest square centimetre. [3]

(d) (i) Convert 105? to radians, leaving your answer in terms of p. [2]

(ii) Use your answer from part (d)(i) to find the area of a sector of a circle of radius 4.6 centimetres and angle 105?, giving your answer

correct to two significant figures. [3] Question 5 11 marks

You should use trigonometry, not scale drawings, to find your answers.

The Isle of Raasay can be accessed by ferry from the small port of Sconser on the Isle of Skye. The ferry terminal on Raasay lies 4.5 kilometres away from Sconser (as the crow flies) in an approximately north-northeast direction. However, the ferry can’t sail in a straight line to Raasay and needs to sail around a headland which stands in the way. The ferry journey can be approximated by the following simplified mathematical model.

The ferry sails out from Sconser in a straight line for 1.5 kilometres in an approximately east-northeast direction. From this position, the ferry captain has sailed sufficiently far away to clear the headland, and can see both terminals. She measures that the angle between the lines of sight to each terminal is 84?.

(a) The ferry captain would like to sketch a diagram of the situation, using the point R for the position of the ferry terminal on Raasay, the point S for the position of the ferry terminal at Sconser, and the point F for the position of the ferry. Sketch a diagram showing the points R, S and F, marking in the angle and the lengths that you are given. Join the three

points with line segments to make the triangle RSF.

(b) The ferry captain would like to calculate the distance between the ferry and the terminal on Raasay. She realises that in triangle RSF she has two side lengths and an angle. She erroneously concludes that she can solve her problem with a single direct application of the Cosine Rule. Explain, as if directly to the captain, why she can’t use the Cosine Rule [2]

directly to solve her problem.

(c) (i) Use the Sine Rule to find the angle at R. Give your answer correct [2]

to the nearest degree.

(ii) Use your answer to part (c)(i) to find the angle at S. Give your [3]

answer correct to the nearest degree.

(iii) Use the Cosine Rule and your answer to part (c)(ii) to find the distance between the ferry and the ferry terminal on Raasay. Give [1]

your answer correct to two significant figures. [3]

Question 6 10 marks

In this question, you are asked to comment on a student’s attempt at answering the question detailed below. Read the question and the student’s incorrect attempt first, then answer the questions below.

The student’s incorrect attempt

We first calculate the angle at A, using the fact that all three angles in triangle ACM add up to . So we have180° A = 180 ? (90 + 33) = 57.

So the angle A is 57°.

Using the Sine Rule in triangle ABM, we have

a = sin M m

sin A

so a = sin M m ´ sin A = sin 3066° ´ sin 57° = 27.54...

So the length of BM is 27.5 m (to 1 d.p.).

The length of BM is the same as the length of AM, and triangle ACM is right-angled, so we have

cos 33° = adjhyp = 27.5 CM so CM = = 32.78...cos 3327.5 °

Therefore, to 1 decimal place, Mads should stand at least 32.8 m away from the centre of the building.

(a) Using what you know about right-angled triangles, and without performing any calculations, explain how you know that the student’s

answer for the length CM must be wrong.

(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, [1]

why, for each mistake, their working is incorrect.

(c) Do you think the student’s approach to solving the problem is the most [4]

efficient method? Give a reason for your answer. [1]

(d) Write out your own solution to the problem, explaining your working. [4]

TMA 04 Cut-off date 11 August 2020

Figure 1

(i) Find the equation of the axis of symmetry of the parabola given by

y = 2x2 + 2x - 24, explaining your method.

(ii) Use your answer to part (b)(i) to find the coordinates of the vertex [2]

of the parabola given by y = 2x2 + 2x - 24. [2]

Question 1 – 18 marks

This question is based on your work on MU123 up to and including Unit 10. (a) This part of the question concerns the quadratic equation

-2x2 + x - 3 = 0.

(i) Find the discriminant of the quadratic expression -2x2 + x - 3.

(ii) What does this tell you about the number of solutions of the [2]

equation? Explain your answer briefly. [2]

(iii) What does this tell you about the graph of y = -2x2 + x - 3?

(b) Figure 1 shows part of the graph of the quadratic function y = 2x2 + 2x - 24. [1]

(c) This part of the question concerns the quadratic function

y = x2 + 8x + 10.

(i) Write the quadratic expression x2 + 8x + 10 in completed-square

form.

(ii) Use the completed-square form from part (c)(i) to solve the equation x2 + 8x + 10 = 0, leaving your answer in exact (surd) [2]

form.

(iii) Use the completed-square form from part (c)(i) to write down the [3]

vertex of the parabola y = x2 + 8x + 10.

(iv) Provide a sketch of the graph of the parabola y = x2 + 8x + 10, [1]

either by hand or by using a suitable graphing software package.

If you intend to go on to study more mathematics, you are advised to sketch by hand. Whichever method you choose, you should refer to the graph-sketching strategy box in Subsection 2.4 of Unit 10 for information on how to sketch and label a graph correctly.

Question 2 – 12 marks

This question is based on your work on MU123 up to and including Unit 10.

Lydia is a computer programmer for a small startup company which manufactures artisan fudge. She is creating an economic model to predict how much fudge should be produced to maximise profit in the company’s first year. If too little fudge is made, the company will not meet consumer demand and will miss the opportunity to sell more. If too much fudge is made, there will be an excess of unsold fudge, which will incur a cost to the company. Lydia’s computer program models the situation as a quadratic equation of the form

y = ax2 + bx + c,

where y represents the profit (in thousands of pounds) the company is predicted to make, and x represents the amount of fudge (in tonnes) produced in the year. The values of a, b and c are determined by the computer code, depending on various economic measures and results of market research which Lydia inputs manually.

(a) Lydia wants to check that her code is working correctly. She inputs some test values and runs the code to simulate the model. The quadratic equation output by her code is y = 12.197x2 + 5.825x + 9.215.

By considering the coefficient of x2, explain why Lydia’s code must [3]

contain an error.

(b) Lydia corrects the error, and runs the code again. This time the quadratic equation output by the code is y = -15.217x2 + 18.379x - 0.829.

In this part of the question, you are asked to consider the parabolic graph modelled by the equation y = -15.217x2 + 18.379x - 0.829. [1]

(i) Explain why the y-intercept is -0.829.

(ii) Using the context of the question, give one possible reason why the [1]

y-intercept is negative. [1]

(a) Sam considers displaying the data as a dotplot. Do you think this would be a good choice for representing this data? Give a reason for your

answer. [2]

(iii) Use the quadratic formula to find the x-intercepts. Give your

answers correct to three decimal places.

(iv) The chief executive officer (CEO) of the company wants to set a target of producing 1.2 tonnes of fudge in the first year. Should [4]

Lydia support or advise against this target? Explain your answer.

(v) The CEO sets a target of making £4000 profit in the first year. The company decides to produce 0.4 tonnes of fudge. According to Lydia’s model, will the company exceed or fall short of its target? [2]

Explain your answer.

Question 3 – 20 marks

This question is based on your work on MU123 up to and including Unit 11.

Sam works in the public transport department for a city council. A new tram service has just been launched which links the city centre to the airport. The tram service duplicates an existing bus service, which also runs from the city centre to the airport. Sam would like to compare journey times of 15 journeys from the city centre to the airport for each mode of transport. The raw data is shown in Table 1.

Table 1 Journey times (in minutes) to the airport [3]

Bus Tram

27 26

27 31

28 32

29 32

30 32

30 33

30 33

31 34

32 35

33 35

36 35

37 36

39 36

40 37

59 40

(b) Sam decides to create a boxplot for each mode of transport as displayed in Figure 2.

Figure 2

With reference to Subsection 1.2 of Unit 11, state three ways in which

Sam could improve the clarity of their presentation of the boxplots.

(c) Sam can’t remember which boxplot belongs to which mode of transport.

State one way in which they could check by looking at the raw data for [3]

each mode of transport.

(d) Sam realises that the top boxplot represents the data for the tram and the bottom boxplot represents the data for the bus. They try to interpret what the data is telling them by making the statements below. For each one, state whether you agree or disagree with Sam’s interpretation, and explain why they are correct or incorrect. [1]

(i) ‘On average, the tram is quicker than the bus.’

(ii) ‘The tram is more reliable than the bus, since there is less [2]

variability in tram journey times.’

(iii) ‘More than half of bus journeys take longer than the median bus [2]

journey time.’

(e) Use the boxplot for the bus to say whether the data are symmetrical or skewed. If the data are skewed, then state whether they are skewed to [2]

the left or skewed to the right, explaining your reasoning briefly. [2]

(i) Which histogram belongs to the tram data – Figure 3 or Figure 4?

Give a reason for your answer.

(ii) Comment on one aspect of the performance of the two modes of transport that can be seen more easily on the histograms than on [2]

the boxplots.

(iii) Comment on one aspect of the performance of the two modes of transport that can be seen more easily on the boxplots than on the [2]

histograms. [2]

(f) Sam creates histograms for each of the two modes of transport, shown in Figure 3 and Figure 4 below.

Figure 3

Figure 4

Question 4 19 marks

(a) Figure 5 shows a sketch of a triangle, where all side lengths are measured in centimetres. Find the length of the side marked x, giving

your answer correct to the nearest centimetre. [3]

Figure 5

(b) Triangle DEF has a right angle at E. The length of side DE is 8.1km, and the length of side EF is 1.7km. Find ?EDF, giving your answer correct to the nearest degree.

You may find it useful to sketch triangle DEF in your answer. [3]

(c) (i) Figure 6 shows the triangle ABC, where all side lengths are measured in centimetres. Find ?ABC, giving your answer correct

to the nearest degree. [5]

Figure 6

(ii) Find the area of triangle ABC in Figure 6, giving your answer

correct to the nearest square centimetre. [3]

(d) (i) Convert 105? to radians, leaving your answer in terms of p. [2]

(ii) Use your answer from part (d)(i) to find the area of a sector of a circle of radius 4.6 centimetres and angle 105?, giving your answer

correct to two significant figures. [3] Question 5 11 marks

You should use trigonometry, not scale drawings, to find your answers.

The Isle of Raasay can be accessed by ferry from the small port of Sconser on the Isle of Skye. The ferry terminal on Raasay lies 4.5 kilometres away from Sconser (as the crow flies) in an approximately north-northeast direction. However, the ferry can’t sail in a straight line to Raasay and needs to sail around a headland which stands in the way. The ferry journey can be approximated by the following simplified mathematical model.

The ferry sails out from Sconser in a straight line for 1.5 kilometres in an approximately east-northeast direction. From this position, the ferry captain has sailed sufficiently far away to clear the headland, and can see both terminals. She measures that the angle between the lines of sight to each terminal is 84?.

(a) The ferry captain would like to sketch a diagram of the situation, using the point R for the position of the ferry terminal on Raasay, the point S for the position of the ferry terminal at Sconser, and the point F for the position of the ferry. Sketch a diagram showing the points R, S and F, marking in the angle and the lengths that you are given. Join the three

points with line segments to make the triangle RSF.

(b) The ferry captain would like to calculate the distance between the ferry and the terminal on Raasay. She realises that in triangle RSF she has two side lengths and an angle. She erroneously concludes that she can solve her problem with a single direct application of the Cosine Rule. Explain, as if directly to the captain, why she can’t use the Cosine Rule [2]

directly to solve her problem.

(c) (i) Use the Sine Rule to find the angle at R. Give your answer correct [2]

to the nearest degree.

(ii) Use your answer to part (c)(i) to find the angle at S. Give your [3]

answer correct to the nearest degree.

(iii) Use the Cosine Rule and your answer to part (c)(ii) to find the distance between the ferry and the ferry terminal on Raasay. Give [1]

your answer correct to two significant figures. [3]

Question 6 10 marks

In this question, you are asked to comment on a student’s attempt at answering the question detailed below. Read the question and the student’s incorrect attempt first, then answer the questions below.

The student’s incorrect attempt

We first calculate the angle at A, using the fact that all three angles in triangle ACM add up to . So we have180° A = 180 ? (90 + 33) = 57.

So the angle A is 57°.

Using the Sine Rule in triangle ABM, we have

a = sin M m

sin A

so a = sin M m ´ sin A = sin 3066° ´ sin 57° = 27.54...

So the length of BM is 27.5 m (to 1 d.p.).

The length of BM is the same as the length of AM, and triangle ACM is right-angled, so we have

cos 33° = adjhyp = 27.5 CM so CM = = 32.78...cos 3327.5 °

Therefore, to 1 decimal place, Mads should stand at least 32.8 m away from the centre of the building.

(a) Using what you know about right-angled triangles, and without performing any calculations, explain how you know that the student’s

answer for the length CM must be wrong.

(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, [1]

why, for each mistake, their working is incorrect.

(c) Do you think the student’s approach to solving the problem is the most [4]

efficient method? Give a reason for your answer. [1]

(d) Write out your own solution to the problem, explaining your working. [4]