Total marks: 80 · Duration: 90 minutes · Tier: Higher
Instructions to candidates
- Use black ink or black ball-point pen. Draw diagrams in pencil.
- Answer all questions.
- You must answer the questions in the spaces provided.
- In all calculations, show clearly how you work out your answer.
- You may not use a calculator.
- 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 marks for questions are shown in brackets.
- The maximum mark for this paper is 80.
- Where a question asks you to circle your answer, choose one of the four options.
Paper
1 Circle the value of 5⁻²
−25 −10 1/25 1/10
[1 mark]
2 Circle the 2nd term of the sequence with nth term n² + 3n
7 10 6 13
[1 mark]
3 Work out 2 1/4 × 1 2/3
Give your answer as a mixed number in its simplest form.
[3 marks]
4 (a) Write 0.00045 in standard form.
[1 mark]
4 (b) Work out (6 × 10⁵) × (4 × 10⁻²). Give your answer in standard form.
[2 marks]
5 Ali and Bea share some money in the ratio 3 : 5. Bea gets £24 more than Ali.
How much money do they share altogether?
[3 marks]
6 (a) Each interior angle of a regular polygon is 150°. How many sides does the polygon have?
[2 marks]
6 (b) Show that a regular polygon cannot have an interior angle of 100°.
[2 marks]
7 A bag contains 40 counters. Each counter is red, blue or green.
The probability of taking a red counter at random is 0.25. The probability of taking a blue counter is twice the probability of taking a green counter.
How many blue counters are in the bag?
[3 marks]
8 (a) A straight line passes through the points (2, 5) and (6, 13). Work out the equation of the line. Give your answer in the form y = mx + c.
[3 marks]
8 (b) Is the line y = −½x + 4 perpendicular to the line in part (a)? Give a reason for your answer.
[1 mark]
9 The box plots for the test marks of two classes can be summarised as follows.
| Lowest | Lower quartile | Median | Upper quartile | Highest | |
|---|---|---|---|---|---|
| Class A | 12 | 24 | 31 | 38 | 47 |
| Class B | 18 | 27 | 30 | 33 | 41 |
9 (a) Work out the interquartile range for Class A.
[1 mark]
9 (b) Compare the distributions of the marks of the two classes.
[2 marks]
10 A rectangle has a length of 12 cm, correct to the nearest centimetre, and a width of 7.4 cm, correct to 1 decimal place.
Work out the upper bound for the perimeter of the rectangle.
[2 marks]
11 Solve the simultaneous equations
3x + 2y = 4 5x − 3y = 13
[4 marks]
12 Expand and simplify (x + 4)(x − 2)(x + 1)
[3 marks]
13 (a) Solve 2^(x + 1) = 32
[1 mark]
13 (b) Work out the value of 8^(2/3) × 4^(−1/2)
[2 marks]
14 (a) Simplify √48 + √27
[2 marks]
14 (b) Work out (3 + √5)(3 − √5)
[1 mark]
14 (c) Rationalise the denominator and simplify 10/√5
[2 marks]
15 Prove that 0.1666… (the digit 6 recurring) can be written as 1/6.
[3 marks]
16 An object starts from rest. It accelerates at a constant rate for 4 seconds until its speed is 12 m/s. It then travels at 12 m/s for 6 seconds. It then decelerates at a constant rate, coming to rest 2 seconds later.
16 (a) Work out the acceleration during the first 4 seconds.
[1 mark]
16 (b) Work out the total distance travelled.
[3 marks]
17 T is directly proportional to the square root of L. When L = 16, T = 6.
17 (a) Work out the value of T when L = 49.
[2 marks]
17 (b) Work out the value of L when T = 9.
[2 marks]
18 (a) Write x² + 8x + 5 in the form (x + a)² + b.
[2 marks]
18 (b) Hence solve x² + 8x + 5 = 0. Give your answers in surd form.
[2 marks]
19 Simplify fully (x² − 9) / (2x² + 5x − 3)
[3 marks]
20 The first three terms of a geometric sequence are √2, 2, 2√2.
Work out the 7th term. Give your answer in the form a√2, where a is an integer.
[2 marks]
21 Show that tan 60° − sin 60° = √3/2
You may use sin 60° = √3/2 and tan 60° = √3.
[2 marks]
22 f(x) = 3x − 1 and g(x) = x²
22 (a) Work out gf(2).
[1 mark]
22 (b) Show that the equation fg(x) = gf(x) has no real solutions.
[4 marks]
23 A, B, C and D are points on a circle with centre O, so that ABCD is a cyclic quadrilateral. Angle ABC = x.
Prove that angle ADC = 180° − x.
You must not use the fact that opposite angles of a cyclic quadrilateral add up to 180°.
[4 marks]
24 OABC is a parallelogram. OA = a and OC = c.
M is the midpoint of BC. N is the point on AC such that AN : NC = 2 : 1.
Prove that O, N and M lie on a straight line.
[4 marks]
25 The line y = x + 1 meets the circle x² + y² = 25 at two points.
Work out the coordinates of the two points.
[3 marks]
END OF QUESTIONS — TOTAL: 80 MARKS