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 value of 2.4³ correct to 3 significant figures.
13.82 7.20 13.9 13.8
[1 mark]
2 Circle the equation of a line that is parallel to y = 3x − 2
y = −3x + 2 y = ⅓x − 2 y = 2x − 3 y = 3x + 5
[1 mark]
3 (a) The value of a house increases by 4% each year. In 2023 its value was £215 000.
Work out its value 3 years later.
[3 marks]
3 (b) After a 12% decrease, the value of a car is £11 440. Work out the value of the car before the decrease.
[3 marks]
4 A cuboid measures 6 cm by 4 cm by 3 cm.
4 (a) Work out the length of the longest straight rod that fits inside the cuboid, from one corner to the opposite corner.
[3 marks]
4 (b) Work out the angle between this rod and the base of the cuboid (the 6 cm by 4 cm face).
[2 marks]
5 From a point on level ground 25 m from the foot of a vertical tower, the angle of elevation of the top of the tower is 38°.
Work out the height of the tower.
[2 marks]
6 Priya drives 45 km at an average speed of 60 km/h. She then drives a further 30 km at an average speed of 40 km/h.
Work out her average speed for the whole journey.
[3 marks]
7 The table shows the lengths of 60 fish.
| Length (L cm) | Frequency |
|---|---|
| 0 < L ≤ 10 | 8 |
| 10 < L ≤ 20 | 17 |
| 20 < L ≤ 30 | 21 |
| 30 < L ≤ 40 | 10 |
| 40 < L ≤ 50 | 4 |
7 (a) Work out an estimate for the mean length.
[3 marks]
7 (b) Which class interval contains the median? Give a reason for your answer.
[1 mark]
8 Two cones, P and Q, are mathematically similar. The height of P is 6 cm and the height of Q is 9 cm.
8 (a) The volume of P is 64 cm³. Work out the volume of Q.
[2 marks]
8 (b) The surface area of Q is 99 cm². Work out the surface area of P.
[2 marks]
9 In triangle ABC, AB = 8.5 cm, BC = 11.2 cm and angle BAC = 72°.
9 (a) Work out the size of angle BCA.
[3 marks]
9 (b) Work out the area of triangle ABC.
[3 marks]
10 Solve 2x² + 5x − 4 = 0. Give your answers correct to 2 decimal places.
[3 marks]
11 v² = u² + 2as
Make s the subject of the formula.
[2 marks]
12 Pressure = force ÷ area
12 (a) A force of 450 newtons acts on an area of 0.25 m². Work out the pressure, in newtons per m².
[1 mark]
12 (b) The area is increased to 0.3 m². The pressure stays the same. Work out the new force.
[2 marks]
13 (a) Show that the equation x³ − 6x + 2 = 0 has a solution between x = 0 and x = 1.
[2 marks]
13 (b) Use the iterative formula
xₙ₊₁ = (xₙ³ + 2) ÷ 6
with x₀ = 0.5 to find x₁, x₂ and x₃. Give each value correct to 4 decimal places.
[3 marks]
14 A school has 1200 students: 380 in Year 9, 420 in Year 10 and 400 in Year 11.
A sample of 60 students is stratified by year group. How many Year 10 students should be in the sample?
[2 marks]
15 A radioactive substance has a mass of 80 g. Its mass decreases by 15% every hour.
15 (a) Work out its mass after 5 hours.
[2 marks]
15 (b) After how many whole hours is the mass first less than 20 g?
[2 marks]
16 A circle has centre O and radius 9 cm. The chord AB subtends an angle of 110° at O.
Work out the area of the minor segment cut off by AB. Give your answer correct to 3 significant figures.
[4 marks]
17 The probability that it rains on Monday is 0.3.
If it rains on Monday, the probability that it rains on Tuesday is 0.6. If it does not rain on Monday, the probability that it rains on Tuesday is 0.2.
17 (a) Work out the probability that it rains on exactly one of the two days.
[3 marks]
17 (b) It rains on Tuesday. Work out the probability that it also rained on Monday.
[2 marks]
18 Solve the simultaneous equations
y = x² + 3x − 3 y = 2x + 4
Give your answers correct to 2 decimal places.
[4 marks]
19 a = 6.4, correct to 1 decimal place. b = 2.35, correct to 2 decimal places.
19 (a) Work out the upper bound and the lower bound of a ÷ b.
[3 marks]
19 (b) Use your answers to give the value of a ÷ b to a suitable degree of accuracy. Give a reason for your answer.
[1 mark]
20 The point P(4, 2) lies on the circle x² + y² = 20.
Work out the equation of the tangent to the circle at P. Give your answer in the form y = mx + c.
[3 marks]
21 (a) Solve sin x = 0.4 for 0° ≤ x ≤ 360°. Give your answers to 1 decimal place.
[2 marks]
21 (b) Describe fully the single transformation that maps the graph of y = sin x onto the graph of y = sin x + 2.
[2 marks]
22 A cone has a volume of 300 cm³ and a perpendicular height of 12 cm.
[Volume of a cone = ⅓πr²h]
Work out the radius of the base of the cone. Give your answer correct to 3 significant figures.
[3 marks]
23 y is directly proportional to x³. When x = 3, y = 54.
Work out the value of x when y = 250.
[2 marks]
END OF QUESTIONS — TOTAL: 80 MARKS