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.Expand and simplify (2 + √3)², giving your answer in the form a + b√3.
(a)7
(b)4 + 5√3
(c)7 + 4√3
(d)5 + 4√3
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.Simplify (5x² + 3x − 2) − (2x² − x + 5)
(a)3x² + 4x + 3
(b)3x² + 2x + 3
(c)7x² + 2x + 3
(d)3x² + 4x − 7
4.Work out the exact value of (sin 30°)² + (cos 30°)².
(a)1/4
(b)3/4
(c)−1/2
(d)1
5.A square petri dish has an area of 121 mm². A scientist wants to know its side length. Work out the side length of the dish.
(a)22 mm
(b)242 mm
(c)60.5 mm
(d)11 mm
6.In a school, 3/5 of the students study French. Of these students, 2/3 also study Spanish. What fraction of all the students study both French and Spanish?
(a)3/5
(b)9/10
(c)2/5
(d)3/4
7.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
8.Simplify (y³)⁴, giving your answer as a single power of y.
(a)y¹²
(b)12y
(c)y⁷
(d)y⁸¹
9.Simplify √8 + √18, giving your answer in the form k√2.
(a)5√2
(b)√26
(c)7√2
(d)11√2
10.Expand and simplify (x + 1)(x + 2)(x + 3).
(a)x³ + 6x² + 11x + 5
(b)x³ + 6x² + 6x + 6
(c)x³ + 5x² + 11x + 6
(d)x³ + 6x² + 11x + 6
11.Factorise fully 5x + 5y − 5
(a)5(xy − 1)
(b)5(x + y + 1)
(c)5(x + y − 1)
(d)5(x + y − 5)
12.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
13.A carton of orange juice holds 1.35 litres. Ruby pours the juice equally into 4 identical glasses. Work out how much juice is in each glass, giving your answer as a fraction of a litre in its simplest form.
(a)27/20
(b)17/50
(c)27/5
(d)27/80
14.Work out the exact value of cos 0° − sin 30°.
(a)1/2
(b)−1/2
(c)3/2
(d)1
15.Factorise fully 12x² + 18x.
(a)2x(6x + 9)
(b)6(2x² + 3x)
(c)6x(2x + 3)
(d)3x(4x + 6)
16.A cube-shaped storage tank has a volume of 15,625 cubic centimetres. Work out the length of one edge of the tank.
(a)5 cm
(b)50 cm
(c)25 cm
(d)125 cm
17.Factorise fully 56x − 24
(a)8(7x − 16)
(b)8(56x − 3)
(c)8(7x + 3)
(d)8(7x − 3)
18.A student writes 0.08 as the fraction 8/10, reading the 8 as if it stood in the tenths column and ignoring the zero. Work out the correct fraction that 0.08 is equal to, giving your answer in its simplest form.
(a)25/2
(b)1/125
(c)4/5
(d)2/25
19.A right-angled triangle has a hypotenuse of 10 cm. One of its other angles is 45°. Work out the exact length of one of the two shorter sides.
(a)5√2 cm
(b)5√3 cm
(c)10√2 cm
(d)5 cm
20.A semicircle has a diameter of 8 cm. Work out the exact area of the semicircle, in terms of π.
(a)4π cm²
(b)8π cm²
(c)16π cm²
(d)32π cm²

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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