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Broad General Education

Senior Phase

Senior Phase · S5 / S6

Higher Mathematics

Calculus, proof and problem solving at the highest level.

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

Overview

About this course

Higher Mathematics is one of the most respected and widely required qualifications in Scotland. Pupils study differentiation, integration, logarithms, trigonometry, vectors and the geometry of lines and circles. The course demands consistent effort from day one – pupils are expected to complete regular homework and attend supported study as needed. It is essential for many STEM, economics, computing and business 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 Expressions & Functions6 topics
  • Logarithms and exponentials
  • Trig – radians, exact values, addition formulae, double angle
  • Wave function a cos x + b sin x
  • Composite and inverse functions
  • Graphs of related functions
  • Vectors – 3D, scalar product, collinearity
2 Relationships & Calculus6 topics
  • Polynomials – remainder theorem, factorising
  • Quadratic theory and the discriminant
  • Solving trig equations
  • Differentiation – rules, chain rule, trig
  • Integration – indefinite and definite
  • Differential equations
3 Applications6 topics
  • Straight line – medians, altitudes, perpendicular bisectors
  • The circle – equation, tangents, intersections
  • Recurrence relations
  • Optimisation
  • Area between curves
  • Rates of change

Assessment

How the course is assessed

01

Paper 1 (Non-calculator)

70 marks · 1 hour 30 minutes.

02

Paper 2 (Calculator)

80 marks · 1 hour 45 minutes.

03

Grading

Graded A–D on the combined total of 150 marks.

Assessment practice

Practise for forthcoming assessments

10 questions worth 33 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 · January

Prelim – Paper 1 (Non-calculator)

5 questions · 17 marks

Print this set

  1. Q1

    Find the equation of the tangent to y = x³ − 4x at the point where x = 2.

    4 marks
    Show solution

    Solution

    dy/dx = 3x² − 4 = 8 at x = 2; y = 0; tangent y = 8x − 16.

  2. Q2

    Given f(x) = 2x + 1 and g(x) = x² − 3, find g(f(x)).

    2 marks
    Show solution

    Solution

    g(2x + 1) = (2x + 1)² − 3 = 4x² + 4x − 2.

  3. Q3

    Show that (x − 2) is a factor of x³ − 3x² + 4 and factorise fully.

    4 marks
    Show solution

    Solution

    Synthetic division gives remainder 0: (x − 2)(x² − x − 2) = (x − 2)(x − 2)(x + 1) = (x − 2)²(x + 1).

  4. Q4

    Evaluate ∫₁⁴ (3√x) dx.

    3 marks
    Show solution

    Solution

    ∫3x^½ dx = 2x^{3⁄2}; [2(8) − 2(1)] = 14.

  5. Q5

    Solve 2 cos 2x° − 1 = 0 for 0 ≤ x < 360.

    4 marks
    Show solution

    Solution

    cos 2x = ½ → 2x = 60, 300, 420, 660 → x = 30, 150, 210, 330.

Set 2 · January

Prelim – Paper 2 (Calculator)

5 questions · 16 marks

Print this set

  1. Q1

    A circle has equation x² + y² − 6x + 4y − 12 = 0. Find its centre and radius.

    2 marks
    Show solution

    Solution

    Centre (3, −2); r = √(9 + 4 + 12) = 5.

  2. Q2

    A recurrence relation is uₙ₊₁ = 0.8uₙ + 12, u₀ = 40. Find the limit and explain why it exists.

    3 marks
    Show solution

    Solution

    A limit exists because −1 < 0.8 < 1. L = 12⁄(1 − 0.8) = 60.

  3. Q3

    Express 3 cos x° − 4 sin x° in the form k cos(x + a)°, k > 0, 0 ≤ a < 360.

    4 marks
    Show solution

    Solution

    k = 5, tan a = 4⁄3, a in Q1 → a = 53.1°. So 5 cos(x + 53.1)°.

  4. Q4

    Points A(1, 2, −1), B(3, 0, 1) and C(7, −4, 5). Show that A, B and C are collinear.

    3 marks
    Show solution

    Solution

    AB = (2, −2, 2), BC = (4, −4, 4) = 2AB; parallel with B common, so collinear.

  5. Q5

    Solve log₃(x + 2) + log₃(x − 4) = 3.

    4 marks
    Show solution

    Solution

    (x + 2)(x − 4) = 27 → x² − 2x − 35 = 0 → (x − 7)(x + 5) = 0 → x = 7 (x = −5 rejected).

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

w/c 29 Sep 2025

Unit 1 class assessment

Assessment + cal

Thu 6 Nov 2025

S5/S6 Parents' Evening

Parents' Evening + cal

w/c 1 Dec 2025

Unit 2 class assessment

Assessment + cal

Mon 19 Jan 2026

Higher Prelim (Papers 1 & 2)

Prelim + cal

Mar 2026

Easter revision school (dates TBC)

Information

Thu 7 May 2026

SQA Exam – Paper 1 (9:00) & Paper 2 (10:50)*

SQA Exam + cal

Tue 4 Aug 2026

SQA results day

Information + cal

Progression

Where this course leads

Advanced Higher Mathematics and/or Advanced Higher Mechanics in S6.

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