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.A tent's sloping side makes an angle of 45° with the ground. The sloping side is 3√2 m long. Using the exact value of sin 45°, work out the exact height of the tent.
(a)3 m
(b)3√2/2 m
(c)6 m
(d)3√2 m
2.Write these three numbers in order of size, starting with the smallest: 0.7, 3/4, 0.72
(a)3/4, 0.72, 0.7
(b)3/4, 0.7, 0.72
(c)0.7, 0.72, 3/4
(d)0.72, 3/4, 0.7
3.Simplify (y³)⁴, giving your answer as a single power of y.
(a)y¹²
(b)12y
(c)y⁷
(d)y⁸¹
4.The discriminant of a quadratic equation is greater than zero. Which statement about the solutions of that equation is correct?
(a)It has two whole-number solutions
(b)It has one repeated real solution
(c)It has two different real solutions
(d)It has no real solutions at all
5.Simplify √8 + √18, giving your answer in the form k√2.
(a)5√2
(b)√26
(c)7√2
(d)11√2
6.Work out the exact value of (cos 45°)² + (sin 45°)².
(a)2
(b)√2
(c)1
(d)1/2
7.The decimal 0.272727... repeats the block 27 for ever. Write 0.27 recurring as a fraction in its simplest form.
(a)3/10
(b)27/100
(c)3/11
(d)2/9
8.Write down the exact decimal value of the fraction 1/6.
(a)0.1666...
(b)0.16
(c)0.17
(d)0.6
9.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
10.Simplify 2/(x + 1) + 3/(x − 2), giving your answer as a single fraction.
(a)5/(2x − 1)
(b)(5x + 1)/(x² − x − 2)
(c)(5x − 1)/(x² + x − 2)
(d)(5x − 1)/(x² − x − 2)
11.A rectangular lawn is 4 m longer than it is wide. Its area is 96 m². Work out the length of the lawn.
(a)8 m
(b)24 m
(c)16 m
(d)12 m
12.Work out the exact value of cos 0° − sin 30°.
(a)1/2
(b)−1/2
(c)3/2
(d)1
13.A rectangular plywood panel measures 2.4 m by 0.75 m. Work out the area of the panel in square metres, giving your answer as a fraction in its simplest form.
(a)4/5
(b)63/20
(c)9/5
(d)9/50
14.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
15.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°
16.Factorise fully 12x² + 18x.
(a)2x(6x + 9)
(b)6(2x² + 3x)
(c)6x(2x + 3)
(d)3x(4x + 6)
17.A shape is made by joining a rectangle measuring 10 cm by 6 cm to a semicircle of diameter 6 cm along one of the rectangle's shorter sides. Work out the exact area of the shape, in terms of π.
(a)(60 + 3π) cm²
(b)(60 + 4.5π) cm²
(c)(60 + 9π) cm²
(d)(60 + 18π) cm²
18.Work out the value of $1 0^{- 3}$ .
(a)1000
(b)−1000
(c)0.001
(d)−0.001
19.Simplify (2x² + 7x + 3) ÷ (x² − 9), giving your answer as a single fraction.
(a)(2x + 1)/(x + 3)
(b)(2x + 1)/(x − 3)
(c)2x + 1
(d)(2x + 3)/(x − 3)
20.Solve 2x² − 32 = 0.
(a)x = 8
(b)x = 4
(c)x = 16
(d)x = 4 or x = −4

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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