AS Level Maths 9709 Paper 1: Pure Mathematics Revision with Worked Examples

Pure Mathematics revision can feel comfortable when the chapter title tells you exactly what to do. A mixed paper removes that clue. To prepare, practise recognising the structure of a problem: whether a quadratic needs rearranging, whether an angle is in radians, or whether the question asks for a gradient rather than a coordinate. The worked examples below are original practice exercises.

Check your paper and qualification route

The Cambridge Mathematics 9709 syllabus for 2026–2027 gives Paper 1 a duration of 1 hour 50 minutes and 75 marks. Paper 1 is part of each AS and A Level route. Confirm the accompanying papers with your centre; the AS combination of Papers 1 and 2 cannot be carried forward towards the full A Level.

Start with the algebra that connects topics

Use a short diagnostic before writing a large revision timetable. Try expanding brackets, solving a quadratic, rearranging a fraction, working with indices and interpreting a function. Record the first line where your working stops being valid. An error with a negative sign may affect five later topics, so correcting that habit can be more useful than repeatedly rereading those chapters.

Worked example: completing the square

Write y = 2x² − 8x + 5 in completed-square form. First factor 2 from the terms containing x: y = 2(x² − 4x) + 5. Since x² − 4x = (x − 2)² − 4, this becomes y = 2(x − 2)² − 8 + 5 = 2(x − 2)² − 3.

The minimum point is (2, −3), because a squared real expression cannot be negative and its coefficient is positive. Expand the answer to check the original expression. For the roots, set y = 0: (x − 2)² = 3/2, so x = 2 ± √(3/2). Keep both signs. The completed-square form now gives both the turning point and a route to solving the equation.

Worked example: radians and sector area

A sector has radius 6 cm and angle 1.2 radians. Its arc length is rθ = 6 × 1.2 = 7.2 cm. Its area is ½r²θ = ½ × 36 × 1.2 = 21.6 cm². Check the units: arc length is a length, while sector area is an area. An answer with cm² for the arc signals a mismatch even if the arithmetic is correct.

If the angle were given in degrees, convert it before using these formulas. For example, 60° equals π/3 radians. Do not enter 60 directly into rθ and expect the same result. Write the angle unit next to the given value so the conversion decision happens before the calculation.

Worked example: tangent to a curve

For y = x³ − 3x, find the tangent at x = 2. Differentiate: dy/dx = 3x² − 3. At x = 2 the gradient is 9. Find the point using the original curve: y = 8 − 6 = 2. The tangent through (2, 2) is y − 2 = 9(x − 2), giving y = 9x − 16.

Worked example: a definite integral

Find the integral of 3x² from x = 1 to x = 2. An antiderivative is x³. Evaluate the upper value and subtract the lower: 2³ − 1³ = 7. Since 3x² is positive throughout this interval, the integral also gives the area under this graph between those limits. If a curve crosses the axis, inspect the sign before treating a definite integral as total geometric area.

Build a weekly cycle around your errors

1. Begin with a short mixed set and record the method you choose for each question.
2. Use the next session to repair one recurring algebra or interpretation error.
3. Attempt unfamiliar questions on that skill without looking at the worked examples.
4. Finish with a timed mixed section and compare the causes of lost marks with the previous week.

Keep sessions on your other AS component in the timetable; Paper 1 preparation is only part of the qualification. Our guide to the jump to AS and A2 helps with workload planning. For feedback on method choice and working, explore Mathematics tutoring and our A Level course. Bring an attempted solution so the lesson can focus on the exact step you need to understand.