Total marks: 100 · Duration: 2 hours · Tier: Higher
Instructions to candidates
- Use black ink or ball-point pen.
- Answer all questions.
- Answer the questions in the spaces provided. You must show all your working.
- Calculators may be used.
- If your calculator does not have a π button, take the value of π to be 3.142 unless the question instructs otherwise.
- 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 total mark for this paper is 100.
- The marks for each question are shown in brackets.
- This mock follows the layout of Specification A (4MA1) Paper 1H. Both 4MA1 Higher papers allow a calculator.
Paper
1 A car is bought for £16 500. In the first year its value falls by 18%. In each of the next two years its value falls by 11%.
Work out the value of the car after 3 years. Give your answer to the nearest penny.
(3)
2 A map has a scale of 1 : 50 000. A lake has an area of 6 cm² on the map.
Work out the real area of the lake in km².
(3)
3 ℰ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
A = {multiples of 3} B = {factors of 30}
(a) List the members of A ∩ B. (1)
(b) Find n(A ∪ B). (1)
(c) List the members of A′ ∩ B. (1)
4 A boat is out at sea. From the top of a vertical cliff 85 m high, the angle of depression of the boat is 23°. The bottom of the cliff is at sea level.
Work out the horizontal distance from the bottom of the cliff to the boat. Give your answer correct to 3 significant figures.
(3)
5 A car travels 125 m, correct to the nearest 5 m, in 9.2 seconds, correct to 1 decimal place.
Work out the lower bound for the average speed of the car. Give your answer correct to 3 significant figures. You must show your working.
(3)
6 The region R is defined by the inequalities
y ≥ 2, y ≤ x, x + y ≤ 8
(a) Find the coordinates of the three vertices of R. (3)
(b) The point (x, y) lies in R. Find the greatest possible value of 2x + 3y. (2)
7 Solve 3x² − 7x − 5 = 0. Give your solutions correct to 3 significant figures.
(3)
8 The table shows the numbers of goals scored by a team in 20 matches.
| Goals | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Frequency | 3 | 6 | 5 | 4 | 2 |
(a) Work out the mean number of goals per match. (3)
(b) After 5 more matches, the mean number of goals per match for all 25 matches is 2. Work out the total number of goals scored in the 5 extra matches. (2)
9 Triangles P and Q are mathematically similar. The area of P is 32 cm² and the area of Q is 72 cm². One side of P is 8 cm long.
Work out the length of the corresponding side of Q.
(3)
10 A sector of a circle has radius 7.5 cm and angle 135°.
Work out the perimeter of the sector. Give your answer correct to 3 significant figures.
(3)
11 T = 2π√(l/g)
Make l the subject of the formula.
(3)
12 f(x) = 3/(2x − 1)
(a) State the value of x that must be excluded from the domain of f. (1)
(b) Find f⁻¹(x). (3)
(c) Find ff(1). (2)
13 A ship sails 8.5 km from port P on a bearing of 040° to a point Q. It then sails 6 km on a bearing of 125° to a point R.
(a) Show that angle PQR = 95°. (2)
(b) Work out the distance PR. Give your answer correct to 3 significant figures. (3)
(c) Work out the bearing of R from P. Give your answer correct to the nearest degree. (2)
14 A box contains 4 red beads and 6 white beads. Three beads are taken at random from the box, without replacement.
Work out the probability that exactly 2 of the beads are red.
(3)
15 A particle moves along a straight line. Its displacement, s metres, from a fixed point O at time t seconds is given by
s = t³ − 9t² + 24t
(a) Find an expression for the velocity, v m/s, at time t. (2)
(b) Find the two times at which the particle is instantaneously at rest. (2)
(c) Find the acceleration of the particle when t = 5. (2)
16 d is directly proportional to the square of t. When t = 3, d = 45.
Work out the positive value of t when d = 125.
(3)
17 A pyramid has a square base of side 10 cm. Its vertex is vertically above the centre of the base, at a height of 12 cm.
(a) Work out the length of one of the sloping edges of the pyramid. (2)
(b) Work out the angle between a sloping edge and the base. (2)
18 The table gives information about the times, t minutes, that 80 people waited at a clinic.
| Time (t minutes) | 0 < t ≤ 10 | 10 < t ≤ 15 | 15 < t ≤ 20 | 20 < t ≤ 30 | 30 < t ≤ 50 |
|---|---|---|---|---|---|
| Frequency | 12 | 18 | 25 | 20 | 5 |
(a) A histogram is to be drawn. Work out the frequency density for each class. (2)
(b) Estimate the median waiting time. (2)
19 Show that (n + 3)² − (n − 1)² = 8(n + 1) for all values of n.
(3)
20 Solve the simultaneous equations
y = 2x − 1 x² + xy = 12
Give your solutions correct to 3 significant figures.
(5)
21 (a) Find the magnitude of the vector (5, −12). (1)
(b) The vectors (k, 6) and (2, 3) are parallel. Find the value of k. (2)
22 (a) The price of a coat including VAT at 20% is £270. Work out the price before VAT. (2)
(b) A shop buys a bag for £36 and sells it for £45. Work out the percentage profit. (2)
23 180 = 2² × 3² × 5 and 168 = 2³ × 3 × 7
(a) Find the highest common factor (HCF) of 180 and 168. (2)
(b) Find the lowest common multiple (LCM) of 180 and 168. (2)
24 A sphere has a volume of 500 cm³.
[Volume of a sphere = (4/3)πr³]
Work out the radius of the sphere. Give your answer correct to 3 significant figures.
(3)
25 A is the point (−2, 5) and B is the point (4, 1).
Find an equation of the perpendicular bisector of AB.
(4)
26 (a) A country has a population of 6.7 × 10⁷ and an area of 2.43 × 10⁵ km². Work out the population density, in people per km². Give your answer to the nearest whole number. (2)
(b) The mass of an electron is 9.11 × 10⁻³¹ kg. How many electrons would have a total mass of 1 gram? Give your answer in standard form correct to 3 significant figures. (2)
END OF PAPER — TOTAL: 100 MARKS