Paper 5 (non-calculator), Higher

Find the highest common factor and the lowest common multiple of 72 and 108.

Estimate the value of √(63.8) × 5.12 ÷ 0.0496. Show your rounding.

Work out 2 2/5 ÷ 1 3/5. Give your answer in its simplest form.

Work out (2.5 × 10^6) × (8 × 10^-3). Give your answer in standard form.

Simplify fully (a) 7^5 × 7^2 (b) (7^5)^4 (c) Evaluate 8^(2/3) × 4^(−1/2).

Solve 5(x + 2) = 2x + 19.

Solve x² − 5x − 14 = 0 by factorising.

Prove algebraically that the recurring decimal 0.363636… can be written as 4/11.

Amir and Beth share £450 in the ratio 7 : 2. Amir then gives Beth £25. Work out the new ratio of Amir's money to Beth's money in its simplest form.

y is inversely proportional to x². When x = 2, y = 18. Find y when x = 6.

A line passes through (0, -4) and (5, 6). (a) Find its equation in the form y = mx + c. (b) Find the gradient of a line perpendicular to it.

Work out the exact value of tan 30° × cos 30° + sin 45° × cos 45°.

A, B and C are points on a circle with centre O. Angle AOC = 144°. B is on the major arc. (a) Find angle ABC. (b) D is on the minor arc AC. Find angle ADC. Give a reason for each answer.

In a class of 50 students, 28 study French, 20 study Spanish and 9 study both. (a) Draw/describe a Venn diagram. (b) A student who studies French is chosen at random. Find the probability that they also study Spanish.

A bag contains 8 red and 2 blue counters. Two counters are taken at random without replacement. Work out the probability that they are the same colour.

Make x the subject of a = (5 + x)/(3x − 1).

Write x² + 10x + 7 in the form (x + p)² + q, and state the coordinates of the turning point of y = x² + 10x + 7.

Prove that (n + 5)² − (n + 3)² is always a multiple of 4.

Describe fully the single transformation that maps the triangle with vertices (1, 2), (3, 2), (3, 5) onto the triangle with vertices (2, 1), (2, 3), (5, 3).

Simplify fully (x² − 4x − 21)/(3x² + 9x).

Solve the simultaneous equations 5x + 2y = 24 3x − 4y = 4

Find the nth term of the quadratic sequence 2, 9, 22, 41, 66, …

OPQR is a trapezium with OP = 2a, OR = b and RQ = a. X is the midpoint of PQ. Find OX in terms of a and b, and explain whether OX is parallel to RQ.

Rationalise the denominator and simplify: 8/(√5 − 1).

A regular polygon has 9 sides. Work out the size of each interior angle, and explain why regular polygons with interior angle 100° cannot exist.

Write 540 as a product of its prime factors. Give your answer in index form.

Solve x² − 5x − 6 < 0.

Solve tan x = 1 for 0° ≤ x ≤ 360°.

The probability that a biased coin lands on heads is 0.24. The coin is thrown 250 times. Work out the expected number of heads.

A length is 12.35 m correct to 2 decimal places. Write down the error interval for the length L.

A café offers 5 starters, 8 main courses and 4 desserts. How many different three-course meals are possible?

Expand and simplify (x + 2)(x − 1)(x + 4).

Factorise fully 6x² + 11x − 10.

The graph of y = f(x) has a turning point at (3, -2). Write down the coordinates of the turning point of (a) y = f(x − 4) (b) y = −f(x).

Two parallel lines are crossed by a transversal. One of the angles formed is 117°. Work out the size of the co-interior angle and the alternate angle, giving reasons.