Advanced Higher Maths: the 2027 exam, both papers and practice questions

The Advanced Higher Maths exam in 2027

The times are from the 2027 exam timetable, and the papers match the course specification. The course is worth 32 SCQF credit points and is graded on the two papers alone. Maths wasn't one of the courses given a shorter exam for 2026 to 2027, so the format is unchanged.

The formulae list gives standard derivatives and integrals (including the inverse trigonometric functions), summation formulae, the binomial theorem, the Maclaurin expansion, de Moivre's theorem, the vector product and matrix transformations.

What the course covers

• Calculus: differentiating products, quotients, exponentials and logarithms, inverse trigonometric functions, and implicit and parametric equations; integration by substitution and by parts; volumes of revolution; first and second order differential equations.
• Algebra, proof and number theory: partial fractions, asymptotes and graphs of rational functions, the binomial theorem, arithmetic and geometric series, Maclaurin expansions, proof by induction, direct and indirect proof, and Euclid's algorithm.
• Matrices, vectors and complex numbers: Gaussian elimination, determinants and inverses, transformation matrices, the vector product, lines and planes in three dimensions, and complex numbers including de Moivre's theorem.

Advanced Higher Maths grade boundaries

In 2026, out of 115, an A needed 81, a B 68, a C 56 and a D 43. Of the 4,784 candidates, 41.3% got an A. The 2026 course report says Paper 1 was less demanding than intended and Paper 2 more demanding, and both were allowed for when the boundaries were set. Boundaries move every year.

Advanced Higher Maths practice questions

We wrote these for this guide. They suit Paper 1, so try them without a calculator.

1. Partial fractions

Question: split (4x + 7) ÷ ((x + 1)(x + 2)) into partial fractions. Answer: set it equal to A ÷ (x + 1) + B ÷ (x + 2) and multiply through: 4x + 7 = A(x + 2) + B(x + 1). Substituting x = −1 leaves 3 = A. Substituting x = −2 leaves −1 = −B, so B = 1. The answer is 3 ÷ (x + 1) + 1 ÷ (x + 2). Check it with the Partial Fractions Calculator.

2. Integration by parts

Question: find the integral of xe^2x with respect to x. Answer: let u = x and dv/dx = e^2x, so du/dx = 1 and v = ½e^2x. Then the integral is ½xe^2x − ∫½e^2x dx = ½xe^2x − ¼e^2x + c. Differentiate your answer to check it, and don't forget the constant.

3. De Moivre's theorem

Question: prove, with de Moivre's theorem, that cos 3θ = 4cos³θ − 3cos θ. Answer: (cos θ + i sin θ)³ = cos 3θ + i sin 3θ. Expanding the left side with the binomial theorem, the real part is cos³θ − 3cos θ sin²θ. Replace sin²θ with 1 − cos²θ: cos³θ − 3cos θ + 3cos³θ = 4cos³θ − 3cos θ. Equate real parts and you're done. Warm up on trigonometric identities in our Trigonometry course and press Mark.

4. Proof by induction

Question: prove that 1 + 3 + 5 + ... + (2n − 1) = n² for all positive integers n. Answer: when n = 1, both sides are 1, so it's true. Suppose the result holds when n = k, meaning the first k odd numbers add up to k². Add the next term, 2(k + 1) − 1 = 2k + 1: the sum becomes k² + 2k + 1 = (k + 1)², which is the statement for n = k + 1. The statement holds for n = 1, and whenever it holds for k it also holds for k + 1, so by induction it holds for every positive integer n. Write that last sentence in full: it carries a mark.

5. Matrices

Question: the matrix M has rows (4, 3) and (5, 4). Find M⁻¹. Answer: the determinant is 4 × 4 − 3 × 5 = 1. Swap the leading diagonal, negate the off-diagonal pair, then scale by one over the determinant: M⁻¹ has rows (4, −3) and (−5, 4). Multiply it by the original to check you get the identity matrix, or use the Matrix Calculator.

Common Advanced Higher Maths mistakes

These are from the 2026 course report:

• Not simplifying final answers, including numerical fractions.
• Basic algebra slips in longer questions, and missing brackets when raising a common ratio to a power.
• Ignoring the context in related rates: a volume that is falling has a negative rate of change.
• Mixing up normal and direction vectors when working with lines and planes.
• Loose proof language: stating the contrapositive wrongly, or not saying that the numerator and denominator of a rational number are integers.

Use a daily paper as a short timed prelim

Today's Advanced Higher Maths paper is written at this level, but it doesn't follow the Scottish course exactly. It covers complex numbers and Argand diagrams, matrices and transformations, Maclaurin series, proof by induction, second order differential equations and volumes of revolution, all of which are in the course. It also includes hyperbolic functions, polar coordinates and roots of polynomials, which aren't in Advanced Higher Maths. It doesn't cover partial fractions, the main differentiation and integration techniques, series, Euclid's algorithm, Gaussian elimination, or lines and planes, so practise those from the official past papers. For prelim dates, see our prelims guide.

Read next

Scotland · Advanced HigherAdvanced Higher Mechanics: 2027 Exam Guide and PracticeAdvanced Higher Mechanics explained: the 2027 exam date, the single 100-mark paper, the formulae list, 2026 grade boundaries and worked practice questions.