Total marks: 80 · Duration: 90 minutes · Tier: Higher
Instructions to candidates
- Use black ink or ball-point pen.
- Answer all questions.
- You must show all your working. Answers given without working may not gain full marks.
- You may use a calculator.
- If your calculator does not have a π button, take the value of π to be 3.142 unless the question says otherwise.
- Diagrams are NOT accurately drawn. Where a question describes a shape in words, you may find it helpful to sketch it before you start.
- The total mark for this paper is 80. The marks for each question are shown in brackets.
Information
- Section A has shorter structured questions. Section B has two longer multi-step problems.
- Give answers to 3 significant figures unless the question asks for a different degree of accuracy.
Paper
Section A — Structured Questions (48 marks)
1. A train travels from London to Edinburgh, a distance of 534 km. The train leaves London at 07 40 and arrives in Edinburgh at 12 25.
(a) Work out the time taken for the journey. Give your answer in hours and minutes.
(2 marks)
(b) Work out the average speed of the train in km/h. Give your answer to 3 significant figures.
(2 marks)
(c) The train company claims that its trains travel at an average speed of at least 120 km/h on this route. Using your answer to part (b), comment on this claim.
(1 mark)
2. ABC is a triangle with angle ABC = 90°. AB = 8.4 cm and BC = 5.7 cm. D is the point on AC such that BD is perpendicular to AC.
(a) Work out the length of AC. Give your answer correct to 3 significant figures.
(2 marks)
(b) Work out the size of angle BAC. Give your answer correct to 1 decimal place.
(2 marks)
(c) Work out the area of triangle ABC.
(2 marks)
(d) Work out the length of BD. Give your answer correct to 3 significant figures.
(3 marks)
3. Jamal invests £4500 in a savings account. The account pays 3.2% compound interest per year.
(a) Work out the amount in the account at the end of 6 years. Give your answer to the nearest penny.
(3 marks)
(b) Work out the number of whole years it takes for the amount in the account to be more than £6000.
(2 marks)
4. Mei wants to estimate the number of fish in a pond. She catches 40 fish, marks each one, and puts them all back. A week later she catches 50 fish. 8 of these fish are marked.
(a) Work out an estimate for the number of fish in the pond.
(2 marks)
5. The table gives information about the masses, in kilograms, of 80 athletes.
| Mass (m kg) | Cumulative frequency |
|---|---|
| m ≤ 40 | 0 |
| m ≤ 50 | 8 |
| m ≤ 55 | 18 |
| m ≤ 60 | 32 |
| m ≤ 65 | 50 |
| m ≤ 70 | 64 |
| m ≤ 75 | 74 |
| m ≤ 80 | 78 |
| m ≤ 85 | 80 |
You may draw a cumulative frequency graph on graph paper, or work from the table by assuming the masses are spread evenly within each interval.
(a) Find an estimate for the median mass.
(1 mark)
(b) Find an estimate for the interquartile range.
(2 marks)
(c) Find an estimate for the number of athletes with a mass greater than 72 kg.
(2 marks)
(d) One of the 80 athletes is chosen at random. Work out an estimate for the probability that this athlete has a mass between 58 kg and 68 kg.
(2 marks)
6. The waiting times, t minutes, of 120 customers at a post office were recorded. A histogram was drawn for the data. The table gives the class intervals and the frequency density of each bar.
| Waiting time (t minutes) | Frequency density |
|---|---|
| 0 < t ≤ 10 | 2.4 |
| 10 < t ≤ 20 | 3.6 |
| 20 < t ≤ 25 | 5.2 |
| 25 < t ≤ 30 | 3.6 |
| 30 < t ≤ 50 | 0.8 |
(a) Work out the frequency for each class interval.
(3 marks)
(b) Work out an estimate for the number of customers who waited more than 22 minutes.
(2 marks)
7. In triangle PQR, PQ = 11.4 cm, angle QPR = 71° and angle PRQ = 52°.
Work out the length of QR. Give your answer correct to 3 significant figures.
(3 marks)
8. The functions f and g are such that
f(x) = 4x + 1 g(x) = x² − 3
(a) Work out fg(3).
(2 marks)
(b) Find gf(x). Give your answer in the form ax² + bx + c.
(3 marks)
(c) Solve f(x) = g(2).
(2 marks)
9. Solve the simultaneous equations
3x + 2y = 13 x² + 2y = 17
(5 marks)
Section B — Extended Response (32 marks)
10. A clothing company makes and sells hoodies. The total cost, £C, of making n hoodies is given by
C = 8n + 4500
The company sells each hoodie for £35.
(a) Write down an expression, in terms of n, for the total income from selling n hoodies.
(1 mark)
(b) The company makes a profit when the total income is greater than the total cost. Find the least number of hoodies the company must sell to make a profit.
(3 marks)
Market research shows that the number of hoodies, n, the company can sell depends on the selling price, £p, according to the formula
n = 2400 − 40p
(c) Show that the profit, £P, made by selling hoodies at £p each is given by
P = −40p² + 2720p − 23700
(3 marks)
(d) By completing the square, or otherwise, find the selling price that gives the greatest profit. You must show your working.
(4 marks)
(e) Work out the greatest profit the company can make.
(2 marks)
The finance director says: "The price that gives the greatest profit is not necessarily the best choice. A lower price would sell many more hoodies for only a small loss of profit."
(f) Use your answers and the profit formula to comment on the finance director's statement. In your answer you should compare the profit and the number of hoodies sold at your price from (d) and at a price of £30.
(3 marks)
Total for Question 10: 16 marks
11. A container is in the shape of a frustum of a cone. The frustum is made by removing a small cone from the top of a large cone, and then turning the remaining shape upside down so that it stands on its smaller circular face.
The top circular face of the container has radius 12 cm. The bottom circular face has radius 8 cm. The vertical height of the container is 15 cm.
[Volume of a cone = (1/3)πr²h, where r is the radius of the base and h is the vertical height]
(a) Show that the volume of the container is 1520π cm³.
(4 marks)
The empty container is placed under a tap that leaks water at a constant rate of 12 cm³ per minute.
(b) Work out how long it takes to fill the container completely. Give your answer in hours and minutes, to the nearest minute.
(3 marks)
The container is now full of water. A solid metal sphere of radius 6 cm is lowered into the container until it is completely under the water.
[Volume of a sphere = (4/3)πr³]
(c) Work out the volume of water that overflows. Give your answer in terms of π.
(3 marks)
The sphere is removed. At a depth of d cm above the bottom of the container, the radius of the water surface is r cm. The radius increases steadily from 8 cm at the bottom to 12 cm at the top.
(d) (i) Find a formula for r in terms of d.
(2 marks)
(ii) Ellie says, "If the container is filled to half of its height, it holds half of its total volume."
Show that Ellie is wrong. You must show your working.
(4 marks)
Total for Question 11: 16 marks
TOTAL FOR PAPER: 80 MARKS