Total marks: 80 · Duration: 90 minutes · Tier: Higher
Instructions to candidates
- Use black ink or black ball-point pen. Draw diagrams in pencil.
- Answer all questions.
- You must answer the questions in the spaces provided.
- In all calculations, show clearly how you work out your answer.
- You may use a calculator.
- Diagrams are not accurately drawn, unless otherwise indicated. Where a question describes a shape in words, you may find it helpful to sketch it.
Information
- The marks for questions are shown in brackets.
- The maximum mark for this paper is 80.
- Where a question asks you to circle your answer, choose one of the four options.
Paper
1 Circle the expression that is equivalent to (2x³)⁴
8x⁷ 16x⁷ 8x¹² 16x¹²
[1 mark]
2 Circle the reciprocal of 0.4
0.04 2.5 −0.4 4
[1 mark]
3 Circle the transformation that maps the graph of y = f(x) onto the graph of y = f(x) − 3
translation 3 units down translation 3 units right translation 3 units up reflection in the x-axis
[1 mark]
4 (a) Write 360 as a product of its prime factors. Give your answer in index form.
[2 marks]
4 (b) 84 = 2² × 3 × 7
Work out the highest common factor (HCF) and the lowest common multiple (LCM) of 360 and 84.
[2 marks]
5 A shop sells the same shampoo in two sizes.
Small bottle: 400 ml for £3.20 Large bottle: 650 ml for £4.94
Which size is better value for money? You must show your working.
[3 marks]
6 The mass, m kg, of a parcel is 4.7 kg, correct to 1 decimal place.
Write down the error interval for m.
[2 marks]
7 A door code is made from 4 different digits chosen from 0 to 9. The order of the digits matters.
How many different door codes are possible?
[2 marks]
8 The table gives information about the times, t minutes, that some customers waited in a queue. The information is to be shown in a histogram.
| Time (t minutes) | Frequency density |
|---|---|
| 0 < t ≤ 10 | 1.2 |
| 10 < t ≤ 15 | 3.6 |
| 15 < t ≤ 30 | 2.0 |
| 30 < t ≤ 50 | 0.5 |
8 (a) Work out the total number of customers.
[2 marks]
8 (b) Work out an estimate for the number of customers who waited more than 20 minutes.
[2 marks]
9 Mia wants to estimate the number of fish in a lake. She catches 40 fish, marks them and puts them back. A week later she catches 50 fish. 8 of these are marked.
9 (a) Work out an estimate for the number of fish in the lake.
[2 marks]
9 (b) Write down one assumption you have made.
[1 mark]
10 Line L has equation y = 3x + 2
Work out the equation of the line that is perpendicular to L and passes through the point (6, 1). Give your answer in the form y = mx + c
[3 marks]
11 Here are the first five terms of a quadratic sequence.
4 11 22 37 56
Work out an expression for the nth term.
[3 marks]
12 A solid metal ornament is made from a cone on top of a hemisphere. The flat face of the cone and the flat face of the hemisphere fit exactly together. They both have radius 4 cm. The height of the cone is 9 cm.
Volume of a cone = ⅓πr²h Volume of a sphere = (4/3)πr³
12 (a) Work out the volume of the ornament.
[4 marks]
12 (b) The metal has a density of 7.8 g/cm³. Work out the mass of the ornament in kilograms.
[2 marks]
13 In triangle ABC, AB = 7 cm, AC = 9 cm and angle BAC = 64°.
Work out the length of BC. Give your answer to 3 significant figures.
[3 marks]
14 There are 60 students in a year group.
35 students study French. 28 students study Spanish. 12 students study both French and Spanish.
14 (a) How many students study neither French nor Spanish?
[2 marks]
14 (b) A student who studies French is chosen at random. Work out the probability that this student also studies Spanish.
[2 marks]
15 y is inversely proportional to the square of x. When x = 4, y = 5.
15 (a) Write a formula for y in terms of x.
[2 marks]
15 (b) Work out the positive value of x when y = 20.
[1 mark]
16 Ravi bought a car for £18 000. After 3 years it was worth £11 250. The value of the car fell by the same percentage each year.
Work out this percentage.
[3 marks]
17 A ship sails 12 km from port A on a bearing of 060° to a point B. It then sails 8 km on a bearing of 150° to a point C.
17 (a) Show that angle ABC = 90°, and work out the distance AC.
[2 marks]
17 (b) Work out the bearing of C from A.
[3 marks]
18 A sector of a circle has radius 7.5 cm and angle 140°.
Work out the perimeter of the sector.
[3 marks]
19 (a) Solve x² − x − 12 < 0
[3 marks]
19 (b) Write down all the integers that satisfy x² − x − 12 < 0
[1 mark]
20 (a) Write x² − 6x + 11 in the form (x + a)² + b
[2 marks]
20 (b) Write down the coordinates of the turning point of the graph of y = x² − 6x + 11
[1 mark]
21 A bag contains 5 red counters and 3 blue counters. Two counters are taken at random from the bag, without replacement.
Work out the probability that both counters are the same colour.
[3 marks]
22 The table gives the velocity, v m/s, of a cyclist at times t seconds after starting from rest.
| t (s) | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| v (m/s) | 0 | 3.2 | 5.6 | 7.0 | 7.6 |
The graph of v against t is a smooth curve through these points.
22 (a) Use 4 strips of equal width to estimate the distance travelled in the first 8 seconds.
[3 marks]
22 (b) Is your answer to (a) an underestimate or an overestimate? Give a reason for your answer.
[1 mark]
23 Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 8.
[3 marks]
24 Make x the subject of y = (2x + 3)/(x − 4)
[3 marks]
25 f(x) = (2x + 3)/5
Work out f⁻¹(x).
[2 marks]
26 A, B and C are points on a circle. The straight line TAS is the tangent to the circle at A. T and C are on opposite sides of the chord AB.
Angle TAB = 58° Angle ABC = 72°
26 (a) Work out the size of angle ACB. Give a reason for your answer.
[2 marks]
26 (b) Work out the size of angle BAC. Give a reason for your answer.
[2 marks]