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Broad General Education

Senior Phase

Senior Phase · S6

Advanced Higher Mathematics

University-level mathematics, taught in school.

  • SCQF Level 7
  • 3 units
  • 9 practice questions
  • 6 key dates
Start practising View key dates

Overview

About this course

Advanced Higher Mathematics extends pupils well beyond Higher into the mathematics of first-year university courses – advanced calculus, complex numbers, matrices, proof and sequences. It is highly regarded by universities and provides an exceptional foundation for mathematics, physics, engineering, computing, economics and actuarial degrees.

Course structure

What your child will learn

The course is organised into 3 units. Topics are usually taught in the order shown, though teachers may adapt the sequence for their class. Select a unit to see its topics.

1 Methods in Algebra & Calculus5 topics
  • Partial fractions
  • Differentiation – product, quotient, chain, implicit, parametric
  • Integration – substitution, by parts, partial fractions
  • First-order differential equations – separable, integrating factor
  • Second-order differential equations
2 Applications of Algebra & Calculus5 topics
  • Binomial theorem
  • Complex numbers – polar form, de Moivre, roots
  • Sequences and series – arithmetic, geometric, Maclaurin
  • Summation formulae
  • Applications of calculus – volumes of revolution, related rates
3 Geometry, Proof & Systems of Equations5 topics
  • Matrices – determinants, inverses, transformations
  • Gaussian elimination
  • Vectors – lines and planes in 3D
  • Number theory – Euclidean algorithm
  • Methods of proof – direct, contrapositive, contradiction, induction

Assessment

How the course is assessed

01

Paper 1 (Non-calculator)

35 marks · 1 hour.

02

Paper 2 (Calculator)

80 marks · 2 hours 30 minutes.

03

Grading

Graded A–D on the combined total of 115 marks.

Assessment practice

Practise for forthcoming assessments

9 questions worth 29 marks, matched to the assessments on this course. Try each question on paper first, then select it to reveal the worked solution. You can also print a blank set for revision.

Set 1 · November

Methods in Algebra & Calculus Assessment

4 questions · 11 marks

Print this set

  1. Q1

    Differentiate y = x² e^(3x).

    2 marks
    Show solution

    Solution

    dy/dx = 2x e^(3x) + 3x² e^(3x) = x e^(3x)(2 + 3x).

  2. Q2

    Find ∫ x cos x dx.

    3 marks
    Show solution

    Solution

    By parts: x sin x + cos x + c.

  3. Q3

    Express (3x + 1) ⁄ ((x − 1)(x + 2)) in partial fractions.

    3 marks
    Show solution

    Solution

    4⁄3(x − 1) + 5⁄3(x + 2).

  4. Q4

    Given x² + xy + y² = 7, find dy/dx.

    3 marks
    Show solution

    Solution

    2x + y + x(dy/dx) + 2y(dy/dx) = 0 → dy/dx = −(2x + y)⁄(x + 2y).

Set 2 · February

Prelim

5 questions · 18 marks

Print this set

  1. Q1

    Express z = 1 + √3 i in polar form and hence find z⁶.

    4 marks
    Show solution

    Solution

    |z| = 2, arg z = π⁄3. z⁶ = 64(cos 2π + i sin 2π) = 64.

  2. Q2

    Prove by induction that Σᵣ₌₁ⁿ r = n(n + 1)⁄2.

    4 marks
    Show solution

    Solution

    True for n = 1. Assume true for k; then sum to k + 1 = k(k + 1)⁄2 + (k + 1) = (k + 1)(k + 2)⁄2. Hence true for all n ≥ 1.

  3. Q3

    Find the inverse of the matrix A = [[2, 1], [5, 3]].

    2 marks
    Show solution

    Solution

    det = 1; A⁻¹ = [[3, −1], [−5, 2]].

  4. Q4

    Solve dy/dx + 2y = e^(−x), given y = 1 when x = 0.

    5 marks
    Show solution

    Solution

    Integrating factor e^(2x): y = e^(−x) + c e^(−2x); c = 0 → y = e^(−x).

  5. Q5

    Find the Maclaurin series for ln(1 + 2x) up to the term in x³.

    3 marks
    Show solution

    Solution

    2x − 2x² + (8⁄3)x³.

Key dates

Important dates for this course

Session 2025–26. Dates marked * are indicative pending the official SQA timetable; items marked TBC will be updated as soon as they are confirmed. Select “+ cal” on any date to add it to your own calendar.

  • Prelim
  • SQA Exam
  • Parents' Evening
  • Assessment
  • Information

Thu 6 Nov 2025

S5/S6 Parents' Evening

Parents' Evening + cal

w/c 10 Nov 2025

Methods in Algebra & Calculus assessment

Assessment + cal

Thu 5 Feb 2026

Advanced Higher Prelim (Papers 1 & 2)

Prelim + cal

Mar 2026

Easter revision school (dates TBC)

Information

May 2026

SQA Exam – Papers 1 & 2 (date TBC)*

SQA Exam

Tue 4 Aug 2026

SQA results day

Information + cal

Progression

Where this course leads

Degree-level study in mathematics, physics, engineering, computing, economics or related disciplines.

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