Qualifications Scotland (SQA) Advanced Higher Mathematics Past Papers 2026

Structured + extended-response (non-calculator)

Duration: 1 hour

Structured + extended-response (calculator allowed)

Duration: 2 hours 30 minutes

Proof by induction — source sets, communication, and algebraic rigour

All three reports (2023, 2024, 2025) flag proof by induction as an area of consistent difficulty. In 2024, some candidates gave expressions for consecutive odd integers instead of consecutive integers, and few candidates gave a source set. Markers award separate marks for the source set, the closing statement, and the algebraic manipulation — losing any one of these costs the entire proof.

Paper 1 (Non-calculator), Paper 2 (Calculator)

Trigonometric exact values in a non-calculator setting

The 2023 and 2024 reports both explicitly flag this as a widespread error. Questions involving trigonometric exact values appear on Paper 1 every year (questions 6 and 9 in 2023; questions 7(a) and 7(b) in 2024). Errors here propagate into subsequent parts of multi-part questions, compounding the mark loss.

Paper 1 (Non-calculator)

Complex numbers — multiplying by the complex conjugate and Argand diagram representation

In 2025, some candidates did not attempt to multiply the numerator and denominator of a complex fraction by the complex conjugate of the denominator. This is a standard technique that eliminates the imaginary part of the denominator and must be automatic at this level.

Paper 1 (Non-calculator), Paper 2 (Calculator)

Matrices — identifying transformations, correct multiplication order, and inverse/power calculations

In 2024, many candidates did not determine the transformation associated with a given matrix, and many multiplied two matrices in the wrong order. In 2025, candidates were expected to find the inverse of a matrix where its square was given in terms of the matrix and the identity matrix — a more abstract application that most candidates could not complete.

Paper 1 (Non-calculator), Paper 2 (Calculator)

Differential equations — integrating factor technique, negative signs, and separation of variables

Integrating factor questions appear on Paper 2 every year and are consistently cited as areas of difficulty. In 2025, some candidates omitted the negative sign when determining the integrating factor. In 2024, many candidates did not correctly separate variables or integrate to give a negative logarithmic term. These errors invalidate the entire solution.

Paper 2 (Calculator)

Sequences and series — geometric vs arithmetic confusion, and sum to infinity conditions

In 2024, a few candidates attempted to process a geometric sequence as if it were arithmetic. In 2025, many candidates correctly identified the effect on the common ratio but did not accurately state the effect on the sum to infinity. The sum to infinity S = a/(1−r) and the condition |r| < 1 must be stated explicitly.

Paper 1 (Non-calculator), Paper 2 (Calculator)

Integration — by substitution, cyclic integration by parts, and volumes of revolution

In 2025, most candidates gave the correct form of integral for finding the volume of revolution but only a few produced the final exact value. In cyclic integration by parts, some candidates who noted the reappearance of the original integral stated that this meant no solution or an infinite solution — rather than recognising that the integral can be solved algebraically by collecting like terms.

Paper 2 (Calculator)

Points of inflection — the second-derivative test and common misconception

In 2024, a few candidates expressed the misconception that a zero second derivative combined with a non-zero first derivative is a sufficient condition for a point of inflection. Markers explicitly warned against this error. A zero second derivative is necessary but not sufficient; candidates must verify a change in concavity on either side.

Paper 1 (Non-calculator)