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.Put sin 30°, tan 30° and cos 30° in order of size, starting with the smallest.
(a)sin 30°, cos 30°, tan 30°
(b)cos 30°, tan 30°, sin 30°
(c)tan 30°, sin 30°, cos 30°
(d)sin 30°, tan 30°, cos 30°
2.Work out the value of $1 6^{3/4}$ .
(a)12
(b)2
(c)64
(d)8
3.A zip-wire is fixed at an angle of 60° to the horizontal ground. The zip-wire is 12 m long. Using the exact value of cos 60°, work out the exact horizontal distance it covers.
(a)24 m
(b)4√3 m
(c)6√3 m
(d)6 m
4.Solve x² − 6x = 0.
(a)x = 0
(b)x = 3
(c)x = 6
(d)x = 0 or x = 6
5.Simplify (y³)⁴, giving your answer as a single power of y.
(a)y¹²
(b)12y
(c)y⁷
(d)y⁸¹
6.A square tile has an area of 72 cm². Work out the exact perimeter of the tile, giving your answer in the form k√2 cm.
(a)144√2
(b)24√2
(c)6√2
(d)12√2
7.A rectangular photo has length (x + 3) cm and width x cm. Its area is 40 cm². Work out the value of x.
(a)x = −8
(b)x = 5
(c)x = 5 or x = −8
(d)x = 4
8.Write 7/12 as a decimal, showing clearly which digit is recurring.
(a)0.588888...
(b)0.58333...
(c)0.585858...
(d)0.58
9.Work out 3/4 − 5/12 exactly, giving your answer in its simplest form.
(a)1/3
(b)−1/6
(c)1/4
(d)7/6
10.A pizza is cut into 12 equal slices. Ben eats 5 slices and Mia eats 3 slices. What fraction of the pizza is left, giving your answer in its simplest form?
(a)1/3
(b)2/3
(c)1/2
(d)7/12
11.The recurring decimal 0.454545... can be written as 0.45 recurring, where both digits repeat forever. Let x = 0.45 recurring. Work out x as a fraction in its simplest form.
(a)1/2
(b)9/20
(c)5/11
(d)4/9
12.Chloe wants to work out 3 1/4 − 1 2/3. She converts both mixed numbers to twelfths, then subtracts the whole numbers and the fraction parts separately, without checking whether she needs to exchange first. Work out the correct value of 3 1/4 − 1 2/3, giving your answer as a mixed number in its simplest form.
(a)1 3/4
(b)2 5/12
(c)8
(d)1 7/12
13.Decide which of 2³⁰ and 3²⁰ is the larger number, and write down the correct statement.
(a)They are equal
(b)3²⁰ is larger
(c)2³⁰ is larger
(d)They cannot be compared without a calculator
14.A student works out the exact area of a circle with radius 4 cm by squaring the radius but forgetting to multiply by π. Work out the correct exact area of the circle, in terms of π.
(a)64π cm²
(b)4π cm²
(c)8π cm²
(d)16π cm²
15.Work out the value of $2^{3} + 3^{2}$ .
(a)17
(b)72
(c)12
(d)−1
16.Expand and simplify 4(2x − 1) − 3(x + 2) − 5x
(a)−10
(b)5x − 10
(c)2
(d)−7
17.Write down the exact decimal value of the fraction 1/6.
(a)0.1666...
(b)0.16
(c)0.17
(d)0.6
18.Expand and simplify √3(2 + √12).
(a)6 + √6
(b)√15 + 2√3
(c)2√3
(d)6 + 2√3
19.Work out the value of $8^{- 2/3}$ .
(a)4
(b)−16/3
(c)−4
(d)1/4
20.3/5 of a 5/6 litre bottle of juice is poured out. Work out the exact volume poured out, in litres.
(a)1/2
(b)43/30
(c)18/25
(d)25/36

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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