Higher Maths practice questions, papers and the 2027 exam

The Higher Maths exam in 2027

The timings are from the 2027 exam timetable, and they match the papers set in the course specification, last amended in May 2023. The course is graded on the exam alone. Each paper begins with a formulae list: the equations of a circle, the scalar product, trigonometric formulae including the double angle formulae, and the derivatives and integrals of sin ax and cos ax.

Higher Maths grade boundaries

In 2026, out of 120, an A needed 83, a B 70, a C 58 and a D 45, and 37.6% of the 19,792 candidates got an A (course report 2026). The report says some questions were slightly more demanding than intended, which was allowed for when the boundaries were set. Boundaries move every year.

Higher Maths practice questions

We wrote these questions for this guide. Try each one before reading the answer, and show working: you earn marks for method.

1. The straight line

Question: a line passes through A(2, −1) and B(6, 7). Find its equation. Answer: gradient = (7 − (−1)) ÷ (6 − 2) = 2. Using y − b = m(x − a) with A: y + 1 = 2(x − 2), so y = 2x − 5.

2. The circle

Question: find the centre and radius of x² + y² − 6x + 4y − 12 = 0. Answer: compare with the formulae list: 2g = −6 and 2f = 4, so g = −3 and f = 2. The centre is (−g, −f) = (3, −2), and the radius is √(9 + 4 + 12) = 5.

3. Differentiation

Question: differentiate y = 3x⁴ − 2/x. Answer: rewrite 2/x as 2x⁻¹ first. Then dy/dx = 12x³ + 2x⁻², which you can write as 12x³ + 2/x². Not rewriting the fraction as a power is the usual slip.

4. Integration

Question: evaluate the integral of 3√x from x = 1 to x = 4. Answer: write 3√x as 3x^(1/2), which integrates to 2x^(3/2). Then 2 × 4^(3/2) − 2 × 1 = 16 − 2 = 14.

5. Logarithms

Question: solve log₂ x + log₂(x − 2) = 3. Answer: combine the logs: log₂[x(x − 2)] = 3, so x(x − 2) = 2³ = 8. Then x² − 2x − 8 = 0 gives (x − 4)(x + 2) = 0. Reject x = −2, because you can't take the log of a negative number, so x = 4. Check with the Logarithm Calculator.

6. The wave function

Question: write 3 cos x° + 4 sin x° in the form k cos(x − a)°, where k > 0 and 0 ≤ a < 360. Answer: expanding gives k cos a = 3 and k sin a = 4. So k = √(3² + 4²) = 5, and tan a = 4/3 with both values positive, so a = 53.1. The answer is 5 cos(x − 53.1)°. Practise trigonometric identities in our Trigonometry course and press Mark.

7. Recurrence relations

Question: a sequence is defined by uₙ₊₁ = 0.6uₙ + 8. Explain why it has a limit, and find it. Answer: it has a limit because the multiplier, 0.6, lies between −1 and 1. At the limit L = 0.6L + 8, so 0.4L = 8 and L = 20. State the reason in words: it carries a mark.

Common Higher Maths mistakes

• Unclear working. The 2026 course report noted that some candidates set out working inconsistently or made transcription errors. Copy numbers carefully from line to line.
• Not rewriting roots and fractions as powers before differentiating or integrating.
• Wrong quadrant in the wave function. Use the signs of k sin a and k cos a.
• Keeping invalid answers to log equations.
• No reason given for a recurrence relation's limit.

Use a daily paper as a short timed prelim

For more Higher Maths practice questions, sit today's Higher Maths papers in Algebra and Functions and Calculus. For prelim dates and how prelims are used, see the prelims guide.

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