Total marks: 100 · Duration: 90 minutes · Tier: Higher
Instructions to candidates
- Use black ink. You may use an HB pencil for graphs and diagrams.
- Answer all the questions.
- You must not use a calculator in this paper.
- Read each question carefully before you start your answer.
- Show your working. Marks may be given for a correct method even if the answer is wrong.
- Diagrams are not drawn accurately 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 OCR J560 Paper 5, the Higher tier non-calculator paper.
Paper
1 (a) Calculate 4.8 × 0.25 [1]
(b) Calculate 2.73 ÷ 0.3 [2]
(c) Calculate 3/8 + 5/12. Give your answer as a fraction in its simplest form. [2]
(d) Write 0.04857 correct to 2 significant figures. [1]
2 A laptop costs £640. In a sale its price is reduced by 15%. After the sale, the new price is increased by 10%.
(a) Work out the price of the laptop after the 10% increase. [3]
(b) Sam says, "Overall, the price has gone down by 5%."
Explain why Sam is wrong. [2]
3 Each exterior angle of a regular polygon is 24°.
Work out the sum of the interior angles of the polygon. [4]
4 60 students were asked whether they like maths (M) and whether they like science (S).
32 like maths. 25 like science. 10 like both.
(a) Draw a Venn diagram to show this information. Show the number of students in each region. [2]
(b) A student is chosen at random. Find the probability that this student likes neither subject. [1]
(c) A student who likes science is chosen at random. Find the probability that this student also likes maths. [2]
5 At a café, 3 coffees and 2 cakes cost £11.10. 2 coffees and 3 cakes cost £10.65.
Work out the cost of one coffee and the cost of one cake. [5]
6 Bus A leaves the station every 12 minutes. Bus B leaves the station every 18 minutes. Both buses leave the station at 9 am.
When is the next time that both buses leave the station together? [3]
7 (a) Calculate (2 × 10⁸) ÷ (5 × 10³). Give your answer in standard form. [2]
(b) Find the value of 27^(−2/3). [2]
(c) Simplify fully (4x⁶y²)^(1/2). [2]
8 (a) Here are the first five terms of a quadratic sequence.
3 9 19 33 51
Find an expression for the nth term. [3]
(b) The first two terms of a Fibonacci-type sequence are a and b. Each term after that is the sum of the two terms before it.
The 3rd term is 11 and the 5th term is 29. Find the values of a and b. [4]
9 A rectangle has length (x + 3) cm and width (2x − 1) cm. Its area is 60 cm².
Find the perimeter of the rectangle. [5]
10 (a) Expand and simplify (2 + √3)². [2]
(b) Simplify (1 + √2)/(√2 − 1). Give your answer in the form a + b√2, where a and b are integers. [3]
11 The length of each side of a square is 6 cm, correct to the nearest centimetre.
Find the error interval for the area, A cm², of the square. [3]
12 (a) Write x² − 4x + 1 in the form (x − p)² − q. [2]
(b) Write down the coordinates of the turning point of the graph of y = x² − 4x + 1. [1]
(c) Solve x² − 4x + 1 = 0. Give your answers in the form a ± √b. [2]
13 Line L passes through the points (−1, 4) and (3, −4).
(a) Find the equation of line L. [3]
(b) Line M is perpendicular to line L and passes through the point (4, 1). Find the equation of line M. [2]
(c) Find the coordinates of the point where L and M intersect. [1]
14 The cumulative frequency table shows the finishing times, t minutes, of 80 runners in a race.
| Time (t minutes) | t ≤ 20 | t ≤ 25 | t ≤ 30 | t ≤ 35 | t ≤ 40 | t ≤ 45 |
|---|---|---|---|---|---|---|
| Cumulative frequency | 6 | 20 | 44 | 64 | 74 | 80 |
Assume that the times are spread evenly within each interval, or draw a cumulative frequency graph.
(a) Estimate the median time. [2]
(b) Estimate the interquartile range. [2]
15 Two bottles are mathematically similar. The small bottle is 10 cm tall and the large bottle is 15 cm tall. The small bottle holds 400 ml.
How much does the large bottle hold? [3]
16 A, B and C are points on a circle. TA is the tangent to the circle at A, and T is on the opposite side of AB from C.
Angle TAB = 58° and angle ABC = 71°.
(a) Find angle ACB. Give a reason for your answer. [2]
(b) Find angle BAC. Give a reason for your answer. [2]
17 Prove that the sum of the squares of any two consecutive odd numbers is always 2 more than a multiple of 8. [4]
18 f(x) = (x + 1)/2 and g(x) = 4x − 3
(a) Show that fg(x) = gf(x) for all values of x. [3]
(b) Find f⁻¹(x). [2]
19 A bag contains n red counters and 3 blue counters. Two counters are taken at random from the bag, without replacement.
The probability that both counters are blue is 1/7.
(a) Show that n² + 5n − 36 = 0. [3]
(b) Find the value of n. [2]
20 (a) Triangle PQR has a right angle at Q, angle QPR = 45° and PR = 8 cm. Find the exact length of PQ. [2]
You may use sin 45° = cos 45° = 1/√2.
(b) Triangle XYZ has XY = 6 cm, XZ = 10 cm and angle YXZ = 30°. Find the area of triangle XYZ. [2]
You may use sin 30° = 1/2.
21 The graph of y = f(x) has a maximum point at (3, 5).
Write down the coordinates of the maximum or minimum point of each of these graphs.
(a) y = f(x) − 2 [1]
(b) y = f(2x) [1]
(c) y = −f(x) [1]
22 Make x the subject of a(x − b) = cx + d. [3]
END OF QUESTIONS — TOTAL: 100 MARKS