20 exact-answer questions with no calculator: surds, indices, exact trigonometric values, algebraic manipulation and fractions.
🧮 Paper 1 non-calculator warm-up — Higher
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.Simplify x⁽³⁄⁴⁾ ÷ x⁽¹⁄⁴⁾
- 2.A pop-up canopy has two sloping supports that meet at the top. Each support makes an angle of 30° with the ground and is 6 m long. Using the exact value of cos 30°, work out the total width of the canopy's base.
- 3.Simplify x⁶ ÷ x²
- 4.In triangle ABC, angle ABC = 90° and angle BAC = 30°. The hypotenuse AC = 12 cm. Work out the exact length of AB.
- 5.Factorise 3x² + 10x − 8.
- 6.Simplify x⁷ × x⁴, giving your answer as a single power of x.
- 7.Simplify x⁵ × x³ ÷ x², giving your answer as a single power of x.
- 8.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.
- 9.Write 7/12 as a decimal, showing clearly which digit is recurring.
- 10.Decide which of 2³⁰ and 3²⁰ is the larger number, and write down the correct statement.
- 11.Expand and simplify (2x − 1)(x + 5)(x − 2). Write down the coefficient of x in your answer.
- 12.Write 5/6 as a decimal, showing clearly which digit recurs.
- 13.Solve 2x² = 18.
- 14.Factorise fully 56x − 24
- 15.Factorise fully 12x² + 18x.
- 16.Write 0.36 as a fraction in its simplest form.
- 17.Work out √3 × √12, giving your answer as an integer.
- 18.Work out the value of (√5)⁴
- 19.Work out the value of (1/3)⁻²
- 20.A square tile has an area of 144 cm². Work out the side length of the tile.
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.
Similar worksheets worth a look
- ✏️ Paper 1 non-calculator warm-up — Foundation · 20 questions · ~25 min
- 📈 Quadratics: factorise, complete the square, formula · 24 questions · ~45 min
- ⚗️ Ratio and proportion mastery — Higher · 24 questions · ~45 min
- ⭕ Circle theorems practice · 18 questions · ~40 min