20 exact-answer questions with no calculator: surds, indices, exact trigonometric values, algebraic manipulation and fractions.

🧮 Paper 1 non-calculator warm-up — Higher

Maths · GCSE Higher · 20 questions · ~25 min

MathsUKwww.geekhero.co.uk

On Higher, Paper 1 asks for answers in exact form — a surd left as a surd, sin 60° left as √3/2, a fraction left as a fraction — and a rounded decimal scores nothing. This sheet drills exactly that. Twenty questions across simplifying surds and rationalising denominators, the index laws including fractional and negative indices, exact values of sin, cos and tan at the standard angles, expanding and factorising, and arithmetic with fractions and mixed numbers. Give yourself twenty-five minutes and no calculator, and write every answer in its exact form even when a decimal feels more natural. If you find yourself reaching for the calculator out of habit, that is the habit this sheet exists to break.

1.Work out the value of (1/3)⁻²
(a)6
(b)9
(c)1/9
(d)−9
2.A ball is thrown in the air. Its height, h metres, above the ground after t seconds is given in this table: when t = 0, h = 0; when t = 1, h = 15; when t = 2, h = 20; when t = 3, h = 15; when t = 4, h = 0. Use the table to find the two times, in seconds, at which the ball is at ground level.
(a)t = 0 or t = 4
(b)t = 2
(c)t = 0
(d)t = 1 or t = 3
3.Factorise fully 12x² + 18x.
(a)2x(6x + 9)
(b)6(2x² + 3x)
(c)6x(2x + 3)
(d)3x(4x + 6)
4.Solve x² − 100 = 0.
(a)x = 10 or x = −10
(b)x = 10
(c)x = 100 or x = −100
(d)x = 50
5.Factorise 3x² + 10x − 8.
(a)(3x − 4)(x + 2)
(b)(3x + 2)(x − 4)
(c)(x − 2)(3x + 4)
(d)(3x − 2)(x + 4)
6.Solve 5x² − 15x = 0.
(a)x = 0 or x = 15
(b)x = 3
(c)x = 0 or x = 3
(d)x = 5 or x = 3
7.Solve x² − 8x + 3 = 0 by completing the square. Give your answers in surd form.
(a)x = 4 ± √13
(b)x = −4 ± √13
(c)x = 8 ± √13
(d)x = 4 ± √19
8.Work out the value of $1 6^{3/4}$ .
(a)12
(b)2
(c)64
(d)8
9.Simplify (5² × 2³)³ ÷ (5³ × 2²)²
(a)2
(b)2⁵
(c)2²
(d)2¹³
10.A square tile has an area of 144 cm². Work out the side length of the tile.
(a)12 cm or −12 cm
(b)72 cm
(c)12 cm
(d)36 cm
11.Work out the value of $1 0^{- 3}$ .
(a)1000
(b)−1000
(c)0.001
(d)−0.001
12.Simplify (x² − 9) ÷ (x² + 5x + 6).
(a)(x + 3)/(x + 2)
(b)(x − 3)/(x + 2)
(c)x − 3
(d)−9/(5x + 6)
13.Expand and simplify √3(2 + √12).
(a)6 + √6
(b)√15 + 2√3
(c)2√3
(d)6 + 2√3
14.Rationalise the denominator of 10/(4 − √6), giving your answer in its simplest form.
(a)4 + √6
(b)4
(c)4 − √6
(d)4 + 10√6
15.Factorise fully 56x − 24
(a)8(7x − 16)
(b)8(56x − 3)
(c)8(7x + 3)
(d)8(7x − 3)
16.Work out the exact value of tan 30° + tan 30°.
(a)2√3/3
(b)√3/3
(c)√3
(d)2√3
17.Subtract, giving your answer as a single fraction in its simplest form: 4/(x − 1) − 2/(x + 3)
(a)(2x + 11)/((x − 1)(x + 3))
(b)(2x + 14)/((x − 1)(x + 3))
(c)2/((x − 1)(x + 3))
(d)(2x − 10)/((x − 1)(x + 3))
18.Robert is converting 0.999... (with the 9s recurring forever) into a fraction. He lets x = 0.999... . Multiplying by 10 gives 10x = 9.999... . Subtracting x from 10x gives 9x = 9, so x = 1. Which statement correctly explains this result?
(a)Only approximately 1, since it never truly reaches 1.
(b)Only works because the recurring digit is 9.
(c)Exactly 1: the subtraction has no rounding at any step.
(d)Wrong: 10x minus x gives 9x = 8.999..., not 9.
19.Factorise x² − 64.
(a)(x − 4)(x + 16)
(b)(x − 8)(x − 8)
(c)(x + 8)(x + 8)
(d)(x − 8)(x + 8)
20.Simplify √8 + √18, giving your answer in the form k√2.
(a)5√2
(b)√26
(c)7√2
(d)11√2

What is on this worksheet?

The sheet holds 20 questions drawn from the MathsUK bank — the content areas covered: Number, Algebra, Geometry and measures (statements N7, N8, N10, A4, A18, G21). It is pitched at GCSE Higher and takes about 25 minutes to work through in full. It is built for independent practice, with full answers at the end for self-marking.

How to use the sheet well

Print it or open it on screen — both work. Printing is A4; the screen view fits phones and tablets.
Do all 20 questions before checking — about 25 minutes is the guide, but there is no time pressure.
Check the answers — press “Show answers” or print the answer page separately.
Redo the questions you got wrong — twice as effective as doing 20 fresh ones.
“New questions” — builds a fresh sheet on the same statements, so you can practise again without repeats.

Why this sheet helps

MathsUK worksheets use questions graded by difficulty and a fair spread of correct-answer positions (the answer is not always (a)) — so the student really has to think about each question rather than guess a pattern. Every question is tagged to a DfE content statement and checked before it enters the bank. The answers come with a step-by-step explanation, not just a value — so a wrong answer becomes a lesson.

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