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100+ Free Advanced Higher Mathematics Practice Questions

Pass your Scottish Advanced Higher Mathematics (C847 77) exam on the first try — instant access, no signup required.

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Evaluate the integral of 1/((x - 1)(x + 2)) dx using partial fractions.

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Key Facts: Advanced Higher Mathematics Exam

A-D

Pass grades on Advanced Higher

Qualifications Scotland

100 marks

Question paper total

AH Mathematics course specification C847 77

3 hours

Question paper duration

Qualifications Scotland

100

Free practice questions here

OpenExamPrep

Qualifications Scotland Advanced Higher Mathematics is a one-year course assessed through a 3-hour, 100-mark written question paper plus a project assignment. Content spans complex numbers, advanced calculus, proof, matrices and 3D vectors, with grades A to D counting as a pass on the 2026 specification.

Sample Advanced Higher Mathematics Practice Questions

Try these sample questions to test your Advanced Higher Mathematics exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Express (3x + 5) / ((x + 1)(x + 2)) in partial fractions.
A.2/(x + 1) + 1/(x + 2)
B.1/(x + 1) + 2/(x + 2)
C.3/(x + 1) + 2/(x + 2)
D.2/(x + 1) - 1/(x + 2)
Explanation: Write (3x + 5)/((x+1)(x+2)) = A/(x+1) + B/(x+2). Then 3x + 5 = A(x+2) + B(x+1). x = -1 gives 2 = A so A = 2; x = -2 gives -1 = -B so B = 1. The decomposition is 2/(x+1) + 1/(x+2).
2Express (x^2 + 2)/((x - 1)(x^2 + 1)) in partial fractions.
A.3/(2(x - 1)) - (x + 1)/(2(x^2 + 1))
B.1/(x - 1) + 1/(x^2 + 1)
C.2/(x - 1) + x/(x^2 + 1)
D.3/(x - 1) + (x + 1)/(x^2 + 1)
Explanation: Write A/(x - 1) + (Bx + C)/(x^2 + 1). Then x^2 + 2 = A(x^2 + 1) + (Bx + C)(x - 1). x = 1 gives 3 = 2A so A = 3/2. Compare x^2 coefficients: 1 = A + B so B = -1/2. Compare constants: 2 = A - C so C = A - 2 = -1/2. Result: 3/(2(x - 1)) - (x + 1)/(2(x^2 + 1)).
3Express the improper fraction (x^3 + x + 1)/(x^2 + 1) in the form quotient + remainder/(x^2 + 1).
A.x + 1/(x^2 + 1)
B.x + x/(x^2 + 1)
C.x^2 + 1/(x^2 + 1)
D.1 + x/(x^2 + 1)
Explanation: Polynomial divide x^3 + x + 1 by x^2 + 1: x * (x^2 + 1) = x^3 + x. Subtract: (x^3 + x + 1) - (x^3 + x) = 1. So the quotient is x and remainder is 1, giving x + 1/(x^2 + 1).
4Find the term independent of x in the binomial expansion of (2x + 1/x^2)^6.
A.60
B.15
C.240
D.20
Explanation: General term: C(6, r) (2x)^(6-r) (1/x^2)^r = C(6, r) 2^(6-r) x^(6 - 3r). Set 6 - 3r = 0 so r = 2. Constant term = C(6, 2) * 2^4 = 15 * 16 = 240.
5Find the coefficient of x^3 in the binomial expansion of (1 + 2x)^5.
A.80
B.40
C.10
D.32
Explanation: Term in x^3 is C(5,3)(2x)^3 = 10 * 8x^3 = 80x^3. The coefficient is 80.
6Using the binomial series for negative indices, the expansion of (1 + x)^(-2) up to the term in x^2 is:
A.1 - 2x + 3x^2
B.1 + 2x + 3x^2
C.1 - 2x - 3x^2
D.1 - x + x^2
Explanation: (1 + x)^n with n = -2 gives 1 + nx + n(n-1)/2! x^2 = 1 + (-2)x + (-2)(-3)/2 x^2 = 1 - 2x + 3x^2.
7Expand (1 + x)^(1/2) as a Maclaurin/binomial series up to the term in x^2.
A.1 + x/2 - x^2/8
B.1 + x/2 + x^2/8
C.1 - x/2 + x^2/8
D.1 + x/2 - x^2/4
Explanation: n = 1/2 gives 1 + (1/2)x + (1/2)(-1/2)/2 x^2 = 1 + x/2 - x^2/8.
8If z = 3 + 4i, find |z| and arg(z) in radians (give arg to 2 d.p.).
A.|z| = 5, arg(z) = 0.93
B.|z| = 7, arg(z) = 0.93
C.|z| = 5, arg(z) = 1.33
D.|z| = 25, arg(z) = 0.75
Explanation: |z| = sqrt(3^2 + 4^2) = sqrt(25) = 5. arg(z) = arctan(4/3) which is approximately 0.93 radians (about 53.13 degrees).
9Express z = -1 + i in polar form r(cos theta + i sin theta) with theta in radians.
A.sqrt(2)(cos(3pi/4) + i sin(3pi/4))
B.sqrt(2)(cos(pi/4) + i sin(pi/4))
C.2(cos(3pi/4) + i sin(3pi/4))
D.sqrt(2)(cos(-pi/4) + i sin(-pi/4))
Explanation: r = sqrt((-1)^2 + 1^2) = sqrt(2). The point (-1, 1) is in the second quadrant, so arg = pi - arctan(1/1) = pi - pi/4 = 3pi/4. Polar form: sqrt(2)(cos(3pi/4) + i sin(3pi/4)).
10Using De Moivre's theorem, find (cos(pi/6) + i sin(pi/6))^6.
A.-1
B.1
C.i
D.-i
Explanation: De Moivre's theorem: (cos theta + i sin theta)^n = cos(n theta) + i sin(n theta). With theta = pi/6 and n = 6: cos(pi) + i sin(pi) = -1 + 0i = -1.

About the Advanced Higher Mathematics Exam

Advanced Higher Mathematics (course code C847 77) is offered by Qualifications Scotland as the most advanced Scottish school mathematics qualification. The course covers methods in algebra and calculus, applications of algebra and calculus, and geometry, proof and systems of equations across three units, assessed by a 100-mark written question paper and a separately marked project.

Questions

100 scored questions

Time Limit

3 hours for the question paper plus 25 hours for the project

Passing Score

Grade A is the highest, A-D count as a pass (A-B-C-D), No Award is a fail

Exam Fee

Funded by Scottish Government for school candidates; private candidate entry fee approx GBP 49.10 per subject (Qualifications Scotland (formerly SQA))

Advanced Higher Mathematics Exam Content Outline

Core

Methods in Algebra: Partial Fractions and Binomial Theorem

Partial fractions including improper fractions and irreducible quadratic factors, binomial theorem with negative and fractional indices

Core

Complex Numbers

Argand diagram, modulus and argument, polar and exponential form, De Moivre's theorem, nth roots, loci in the complex plane

Core

Differentiation Methods

First principles, standard derivatives, chain product and quotient rules, implicit and parametric differentiation, logarithmic differentiation, stationary points and inflection

Core

Integration Methods

Standard integrals, substitution, integration by parts, partial fractions, trigonometric identities, areas, volumes of revolution, arc length

Core

Differential Equations

Separable variables and first-order linear differential equations using the integrating factor method

Core

Sequences, Series and Maclaurin Expansions

Arithmetic and geometric series, sum to infinity, standard summation formulae, convergence, Maclaurin series, recurrence relations

Core

Proof Techniques

Direct proof, proof by contradiction, counter-example, proof by mathematical induction, disproof at Advanced Higher rigour

Core

Matrices and Number Theory

Matrix operations, 2x2 and 3x3 inverses, Gaussian elimination, transformations, Euclidean algorithm and elementary congruences

Core

Vectors and Geometry in 3D

Scalar and vector products, equations of lines and planes in 3D, intersections, angle between line and plane

How to Pass the Advanced Higher Mathematics Exam

What You Need to Know

  • Passing score: Grade A is the highest, A-D count as a pass (A-B-C-D), No Award is a fail
  • Exam length: 100 questions
  • Time limit: 3 hours for the question paper plus 25 hours for the project
  • Exam fee: Funded by Scottish Government for school candidates; private candidate entry fee approx GBP 49.10 per subject

Keys to Passing

  • Complete 500+ practice questions
  • Score 80%+ consistently before scheduling
  • Focus on highest-weighted sections
  • Use our AI tutor for tough concepts

Advanced Higher Mathematics Study Tips from Top Performers

1Work through past papers from Qualifications Scotland and SQA in the May exam style — patterns and topic emphasis repeat year on year
2Drill proof by induction and proof by contradiction structures until you can write them with no hesitation
3Memorise standard Maclaurin series and standard integrals; they appear in nearly every paper
4Use the official marking instructions to learn how method marks (M), strategy marks (S), and proof marks are awarded

Frequently Asked Questions

Who awards Advanced Higher Mathematics?

Advanced Higher Mathematics is awarded by Qualifications Scotland, the awarding body formed from the Scottish Qualifications Authority (SQA) on 1 February 2026. The course specification and grading framework are unchanged from the previous SQA syllabus.

When is the Advanced Higher Mathematics exam sat?

The question paper is sat in the May exam diet at the end of S6 (or post-school). The project component is internally completed across the year and externally marked by Qualifications Scotland alongside the written paper.

How is Advanced Higher Mathematics graded?

Advanced Higher courses are graded A, B, C, D, or No Award. Grades A through D count as a pass; the project assignment contributes alongside the 100-mark question paper to determine the final overall grade.

How does Advanced Higher Maths compare to A-Level Maths?

Advanced Higher Maths is widely regarded as more demanding than single A-Level Mathematics and broadly comparable to A-Level Further Mathematics. It covers complex numbers, matrices, proof by induction, and differential equations not always required at A-Level.