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.
- Calculators may not be used.
- 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.
- Where a question asks for an exact answer, leave it in terms of π or as a surd or fraction.
Paper
1. Work out 2 3/4 ÷ 1 5/6
Give your answer as a mixed number in its simplest form.
(3 marks)
2. A, B, C and D are points on a circle with centre O.
Angle AOB = 136°. C is a point on the major arc AB. D is a point on the minor arc AB, so that ACBD is a cyclic quadrilateral.
(a) Work out the size of angle ACB. Give a reason for your answer.
(2 marks)
(b) Work out the size of angle ADB. Give a reason for your answer.
(2 marks)
3. 20 students recorded how many hours, h, they spent on homework last week.
| Hours (h) | Frequency |
|---|---|
| 0 < h ≤ 2 | 6 |
| 2 < h ≤ 4 | 9 |
| 4 < h ≤ 6 | 3 |
| 6 < h ≤ 10 | 2 |
(a) Work out an estimate for the mean number of hours.
(3 marks)
(b) Explain why your answer to part (a) is only an estimate.
(1 mark)
4. There are 50 students in Year 11 at a small school.
28 students study French. 19 students study Spanish. 7 students study both French and Spanish.
(a) How many students study neither French nor Spanish?
(2 marks)
(b) A student who studies French is chosen at random. Work out the probability that this student also studies Spanish.
(2 marks)
5. (a) Work out the value of 64^(2/3)
(1 mark)
(b) Work out the value of (27/8)^(−1/3)
(2 marks)
(c) Simplify fully (2x³y)⁴
(2 marks)
6. A sector of a circle has radius 6 cm and angle 120° at the centre.
(a) Work out the area of the sector. Give your answer in terms of π.
(2 marks)
(b) Work out the perimeter of the sector. Give your answer in the form a + bπ.
(2 marks)
7. Two cylinders, P and Q, are mathematically similar.
The surface area of P is 50 cm². The surface area of Q is 200 cm². The volume of P is 30 cm³.
Work out the volume of Q.
(3 marks)
8. Prove algebraically that (2n + 3)² − (2n − 1)² is always a multiple of 8 for all integer values of n.
(3 marks)
9. Show that 0.454545… (the digits 45 recurring) can be written as 5/11.
(2 marks)
10. (a) Simplify √75 − √12
(2 marks)
(b) Rationalise the denominator and simplify fully
6 / (3 − √3)
(3 marks)
11. (a) Write x² − 6x + 11 in the form (x + a)² + b, where a and b are integers.
(2 marks)
(b) Write down the coordinates of the turning point of the curve y = x² − 6x + 11.
(1 mark)
(c) Explain how your answer to part (a) shows that x² − 6x + 11 is positive for all values of x.
(1 mark)
12. A right-angled triangle has sides of length x cm, (x + 7) cm and (x + 8) cm. The longest side is (x + 8) cm.
(a) Show that x² − 2x − 15 = 0
(3 marks)
(b) Work out the perimeter of the triangle.
(3 marks)
13. A box contains 5 green pens and 3 yellow pens. Two pens are taken at random from the box, without replacement.
Work out the probability that the two pens are different colours.
(3 marks)
14. Here are the first five terms of a quadratic sequence.
4 9 18 31 48
Find an expression, in terms of n, for the nth term of the sequence.
(3 marks)
15. Solve x² − x − 12 < 0
(3 marks)
16. Make x the subject of the formula
y = (3x + 2) / (x − 4)
(4 marks)
17. The curve y = f(x) has a turning point at (2, −5).
Write down the coordinates of the turning point of each of these curves.
(a) y = f(x + 3)
(1 mark)
(b) y = −f(x)
(1 mark)
(c) y = f(x) + 4
(1 mark)
18. In triangle ABC, angle ABC = 90° and angle BAC = 60°. AC = 10√3 cm.
You may use sin 60° = √3/2 and cos 60° = 1/2.
Work out the exact length of BC.
(3 marks)
19. y is inversely proportional to the square of x. When x = 4, y = 5.
(a) Find a formula for y in terms of x.
(2 marks)
(b) Work out the positive value of x when y = 20.
(2 marks)
20. The functions f and g are such that
f(x) = 2x + 1 g(x) = 1 / (x − 2)
(a) Work out gf(3).
(2 marks)
(b) Find g⁻¹(x).
(2 marks)
(c) State the value of x that must be excluded from the domain of g.
(1 mark)
21. Solve the simultaneous equations
y = 2x − 3 x² + y² = 18
(5 marks)
TOTAL FOR PAPER: 80 MARKS