Total marks: 80 · Duration: 105 minutes · Tier: Higher
Instructions to candidates
- Use black ink or black ball-point pen. Draw diagrams in pencil.
- Answer all questions in the spaces provided.
- You must not use a calculator for any question on this paper.
- Where a question asks for an answer in terms of π or as a surd, leave it in that exact form.
- Diagrams are not drawn to scale. Where a question describes a shape in words, you may find it helpful to sketch it.
Information
- The total mark for this paper is 80.
- The number of marks is given in brackets at the end of each question or part-question.
- You are reminded of the need for good English and orderly, clear presentation in your answers.
Paper
1. (a) Write the ratio 2.4 kg : 600 g in its simplest form. (2)
(b) Calculate 7/9 ÷ 14/27, giving your answer as a fraction in its simplest form. (2)
2. The four angles of a quadrilateral are x°, 2x°, (3x − 10)° and (x + 20)°.
Find the size of the largest angle. (3)
3. (a) Express 252 as a product of its prime factors in index form. (2)
(b) Find the smallest positive whole number n such that 252n is a square number. (1)
4. Spinner A has four equal sections numbered 1, 2, 3 and 4. Spinner B has three equal sections numbered 1, 2 and 3. Both spinners are spun and the two numbers are multiplied together.
Find the probability that the product is an even number. (3)
5. The total surface area, A, of a closed cylinder of radius r and height h is given by A = 2πr² + 2πrh.
Make h the subject of the formula. (3)
6. Solve (3x − 1)/4 − (x + 2)/3 = 1. (3)
7. Find all the integer values of x that satisfy −7 < 3x − 1 ≤ 8. (3)
8. (a) The distance from the Earth to the Moon is approximately 3.84 × 10⁵ km. Light travels at approximately 3 × 10⁵ km per second.
How long does light take to travel from the Moon to the Earth? (2)
(b) Calculate (5 × 10⁻³)², giving your answer in standard form. (2)
9. A prism has a cross-section in the shape of a trapezium. The parallel sides of the trapezium are 6 cm and 10 cm, and the perpendicular distance between them is 4 cm. The prism is 15 cm long.
Calculate the volume of the prism. (3)
10. (a) Expand and simplify (3x − 2)² − (x + 4)(x − 4). (3)
(b) Factorise fully 6x²y − 9xy². (2)
11. Solve 3x² − 10x − 8 = 0 by factorising. (3)
12. A cube has edges of length 4 cm.
Find the exact length of a diagonal of the cube that joins one corner to the opposite corner. Give your answer as a surd in its simplest form. (3)
13. (a) Simplify √50 − √18 + √8. (2)
(b) Rationalise the denominator and simplify 12/√3. (2)
14. Prove that the recurring decimal 0.31818… (in which the digits 18 recur) is equal to 7/22. (3)
15. A biased coin has a probability of 0.6 of landing on heads. The coin is thrown twice.
Find the probability that it lands on heads exactly once. (3)
16. AB is a diameter of a circle, and C is another point on the circle. Angle CAB = (2x + 5)° and angle CBA = 3x°.
Find the value of x. You must give a reason for each step of your working. (3)
17. A straight line has equation 2y = 3x − 8.
(a) Write down the gradient and the y-intercept of the line. (2)
(b) Does the point (6, 5) lie on the line? Show how you decide. (1)
18. The time, t days, taken to complete a building job is inversely proportional to the number of workers, w. 6 workers take 10 days.
(a) Find a formula for t in terms of w. (2)
(b) How many workers are needed to complete the job in 4 days? (1)
19. Solve the simultaneous equations
x + y = 7 x² + y² = 29 (5)
20. (a) Simplify fully (x² + 3x − 10)/(x² − 4). (3)
(b) Solve 2/(x + 1) + 3/(x − 2) = 1. Give your answers in the form a ± √b. (4)
21. The graph of y = x² is translated so that its equation becomes y = (x − 3)² + 2.
Write down the coordinates of the minimum point of the new graph, and describe the translation. (2)
22. A histogram is drawn to show the times, t minutes, that 65 customers spent in a shop. The table gives the frequency density of each bar.
| Time (t minutes) | 0 < t ≤ 10 | 10 < t ≤ 15 | 15 < t ≤ 30 |
|---|---|---|---|
| Frequency density | 1.5 | 4 | 2 |
Estimate the number of customers who spent more than 12 minutes in the shop. (3)
23. A sector of a circle has radius 9 cm and angle 80° at the centre.
(a) Find the area of the sector, in terms of π. (2)
(b) Find the length of the arc of the sector, in terms of π. (2)
END OF PAPER — TOTAL: 80 MARKS