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 x⁵ × x³ ÷ x², giving your answer as a single power of x.
- 2.Solve 5x² − 15x = 0.
- 3.Write 7/12 as a decimal, showing clearly which digit is recurring.
- 4.Simplify (y³)⁴, giving your answer as a single power of y.
- 5.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.
- 6.The discriminant of a quadratic equation is greater than zero. Which statement about the solutions of that equation is correct?
- 7.Work out the value of the discriminant b² − 4ac for the equation 2x² + 3x − 2 = 0.
- 8.Solve 2x² − 32 = 0.
- 9.Triangle DEF has a right angle at E. Angle DFE = 60° and EF = 9 cm. Work out the exact length of DE.
- 10.Expand and simplify (x + 1)(x + 2)(x + 3).
- 11.Rationalise the denominator of 6/√3, giving your answer in its simplest form.
- 12.Factorise x² − 64.
- 13.Work out the exact value of (sin 30°)² + (cos 30°)².
- 14.Put sin 30°, tan 30° and cos 30° in order of size, starting with the smallest.
- 15.Solve x² + 3x − 10 = 0.
- 16.A cube-shaped storage tank has a volume of 15,625 cubic centimetres. Work out the length of one edge of the tank.
- 17.The decimal 0.272727... repeats the block 27 for ever. Write 0.27 recurring as a fraction in its simplest form.
- 18.A rectangle has length (2x + 5) cm and width (x − 2) cm. Work out an expression, in terms of x, for the perimeter of the rectangle. Give your answer in its simplest form.
- 19.Expand and simplify (x − 3)²
- 20.Expand and simplify 2(a + 6) − 2(a − 7)
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