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
- Questions get harder as the paper goes on. Do not spend too long on one question; come back to it if you have time.
- Check your answers if you have time at the end.
- Give answers to 3 significant figures unless the question asks for a different degree of accuracy.
Paper
1. Work out
(3.6 × 10⁵) × (8.5 × 10⁻²) ÷ (1.2 × 10⁻³)
Give your answer in standard form.
(2 marks)
2. Green paint is made by mixing blue paint and yellow paint in the ratio
blue : yellow = 3 : 5
Tom has 1.2 litres of blue paint and 1.8 litres of yellow paint.
Work out the greatest amount of green paint Tom can make.
(3 marks)
3. Priya buys a new car for £18 500.
The value of the car falls by 15% in the first year.
After the first year, the value falls by 9% each year.
Work out the value of the car 4 years after Priya bought it. Give your answer to the nearest pound.
(3 marks)
4. A ladder is 6.5 m long. It leans against a vertical wall on horizontal ground. The foot of the ladder is 1.8 m from the bottom of the wall.
(a) Work out how far up the wall the ladder reaches. Give your answer to 3 significant figures.
(2 marks)
(b) For the ladder to be safe, the angle between the ladder and the ground must be between 70° and 80°.
Is the ladder safe? You must show how you get your answer.
(2 marks)
5. Solve the simultaneous equations
5x + 2y = 11 3x − 4y = 30
(3 marks)
6. An alloy is made by melting together 240 cm³ of copper and 60 cm³ of zinc.
The density of copper is 8.96 g/cm³. The density of zinc is 7.13 g/cm³.
Assume that the volume of the alloy is the sum of the two volumes.
Work out the density of the alloy. Give your answer correct to 3 significant figures.
(3 marks)
7. The waiting times, t minutes, of 80 customers at a bank were recorded. A histogram was drawn to show the data. The table gives the class intervals and the frequency density of each bar of the histogram.
| Waiting time (t minutes) | Frequency density |
|---|---|
| 0 < t ≤ 5 | 2.4 |
| 5 < t ≤ 10 | 4.0 |
| 10 < t ≤ 20 | 2.6 |
| 20 < t ≤ 30 | 1.5 |
| 30 < t ≤ 50 | 0.35 |
(a) Work out the frequency for each class interval.
(2 marks)
(b) Estimate the number of customers who waited more than 15 minutes.
(2 marks)
(c) Work out an estimate for the mean waiting time.
(3 marks)
8. Solve 3x² − 7x − 2 = 0
Give your solutions correct to 2 decimal places.
(3 marks)
9. A coach travels 125 km, correct to the nearest 5 km. The journey takes 1.6 hours, correct to 1 decimal place.
(a) Work out the upper bound for the average speed of the coach. Give your answer correct to 3 significant figures. You must show your working.
(3 marks)
(b) Sam says, "The average speed of the coach was definitely more than 75 km/h."
Is Sam correct? Give a reason for your answer.
(1 mark)
10. In triangle ABC:
AB = 9.2 cm AC = 7.5 cm angle BAC = 48°
(a) Work out the length of BC. Give your answer correct to 3 significant figures.
(3 marks)
(b) Work out the area of triangle ABC. Give your answer correct to 3 significant figures.
(2 marks)
11. A solid is made from a cylinder and a hemisphere. The cylinder has radius 4 cm and height 10 cm. The hemisphere has radius 4 cm and sits exactly on top of the cylinder.
(a) Work out the volume of the solid. Give your answer correct to 3 significant figures.
[Volume of a sphere = (4/3)πr³]
(3 marks)
(b) The solid is melted down and all of the metal is made into spheres, each of radius 1.5 cm.
Work out the greatest number of complete spheres that can be made.
(2 marks)
12. A cyclist starts from rest. The table shows her velocity, v metres per second, at different times, t seconds.
| t (seconds) | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| v (m/s) | 0 | 5.2 | 8.6 | 10.4 | 11.0 |
The velocity increases throughout the 8 seconds, but it increases more slowly as time goes on.
(a) Use 4 strips of equal width to work out an estimate for the distance the cyclist travels in the first 8 seconds.
(3 marks)
(b) Is your answer to part (a) an underestimate or an overestimate of the true distance? Give a reason for your answer.
(1 mark)
13. The number of bacteria, N, in a sample after t hours is given by
N = a × bᵗ
where a and b are constants.
When t = 0, N = 500. When t = 3, N = 1372.
(a) Work out the value of a and the value of b.
(3 marks)
(b) Work out the first whole number of hours after which the number of bacteria is more than 5000.
(2 marks)
14. (a) Show that the equation x³ + 2x − 5 = 0 has a solution between x = 1 and x = 2.
(2 marks)
(b) Using
xₙ₊₁ = ∛(5 − 2xₙ)
with x₀ = 1.3, find the values of x₁, x₂ and x₃. Give each value correct to 4 decimal places.
(3 marks)
15. OAB is a triangle.
OA = a OB = b
P is the point on AB such that AP : PB = 1 : 2. M is the midpoint of OB. R is the point such that OR = 2a, so R lies on OA extended beyond A.
(a) Write down the vector AB in terms of a and b.
(1 mark)
(b) Find the vector OP in terms of a and b. Give your answer in its simplest form.
(2 marks)
(c) Prove that M, P and R lie on a straight line.
(3 marks)
16. The functions f and g are such that
f(x) = 3x − 5 g(x) = x² + 2
(a) Work out fg(−2).
(2 marks)
(b) Find f⁻¹(x).
(2 marks)
(c) Solve fg(x) = gf(x). Give your solutions correct to 3 significant figures.
(4 marks)
17. A circle has equation x² + y² = 45. The point P(3, 6) lies on the circle. O is the origin.
(a) Find an equation of the tangent to the circle at P. Give your answer in the form ax + by = c, where a, b and c are integers.
(3 marks)
(b) The tangent meets the x-axis at the point Q. Work out the area of triangle OPQ.
(2 marks)
18. A bag contains 6 red counters and 4 blue counters. Asha takes two counters at random from the bag, without replacement.
(a) Work out the probability that at least one of the counters is blue.
(3 marks)
(b) Given that at least one of the counters is blue, work out the probability that both counters are blue.
(2 marks)
TOTAL FOR PAPER: 80 MARKS