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100+ Free Higher Mathematics Practice Questions

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Differentiate y = 3x^4 - 2x + 5 with respect to x.

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Key Facts: Higher Mathematics Exam

C847 76

Course code

Qualifications Scotland

130 marks

Total marks across both papers

Qualifications Scotland Higher Maths spec

SCQF 6

Qualification level

SCQF framework

100

Free practice questions here

OpenExamPrep

Qualifications Scotland Higher Mathematics is assessed through two end-of-course papers worth 130 marks. Coverage spans vectors, functions, polynomials, trigonometry, calculus, the straight line, the circle, and sequences, graded A-D.

Sample Higher Mathematics Practice Questions

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

1Vectors u = (2, -1, 3) and v = (1, 4, -2) are given in component form. Find u + v.
A.(3, 3, 1)
B.(3, -3, 1)
C.(1, 3, 1)
D.(3, 3, -1)
Explanation: Add corresponding components: (2+1, -1+4, 3+(-2)) = (3, 3, 1). Vector addition is component-wise.
2Find the magnitude of the vector a = (2, -2, 1).
A.3
B.5
C.sqrt(5)
D.sqrt(7)
Explanation: |a| = sqrt(2^2 + (-2)^2 + 1^2) = sqrt(4 + 4 + 1) = sqrt(9) = 3. The magnitude is the square root of the sum of squared components.
3Find a unit vector in the direction of b = (6, 0, -8).
A.(0.6, 0, -0.8)
B.(6, 0, -8)
C.(0.5, 0, -0.5)
D.(0.75, 0, -1)
Explanation: |b| = sqrt(36 + 0 + 64) = sqrt(100) = 10. Divide each component by 10: (6/10, 0/10, -8/10) = (0.6, 0, -0.8).
4Vectors a = (1, 2, -1) and b = (3, -1, 2). Compute the scalar product a.b.
A.-1
B.1
C.5
D.-5
Explanation: a.b = (1)(3) + (2)(-1) + (-1)(2) = 3 - 2 - 2 = -1. The scalar (dot) product sums the products of corresponding components.
5For which value of k are the vectors (2, k, 3) and (4, -1, 2) perpendicular?
A.14
B.-14
C.7
D.-7
Explanation: Perpendicular vectors satisfy a.b = 0: (2)(4) + (k)(-1) + (3)(2) = 0, so 8 - k + 6 = 0, giving k = 14.
6Find the angle (to the nearest degree) between vectors p = (1, 0, 1) and q = (1, 1, 0).
A.60
B.45
C.90
D.30
Explanation: p.q = 1; |p| = sqrt(2); |q| = sqrt(2). cos theta = 1 / (sqrt(2) * sqrt(2)) = 1/2, so theta = 60 degrees.
7A, B and C have position vectors a = (1, 2, 3), b = (3, 4, 7) and c = (5, 6, 11). Determine whether they are collinear.
A.Yes, collinear; AC = 2 AB
B.No, not collinear
C.Yes, but AB and AC are anti-parallel
D.Cannot determine without further information
Explanation: AB = b - a = (2, 2, 4); AC = c - a = (4, 4, 8) = 2(2, 2, 4) = 2 AB. Since AC is a scalar multiple of AB and they share point A, the points are collinear.
8P divides AB in the ratio 2:1, where A = (1, 0, 2) and B = (7, 3, -1). Find the position vector of P.
A.(5, 2, 0)
B.(3, 1, 1)
C.(4, 1.5, 0.5)
D.(9, 4, -1)
Explanation: Using the section formula, P = A + (2/3)(B - A) = (1, 0, 2) + (2/3)(6, 3, -3) = (1, 0, 2) + (4, 2, -2) = (5, 2, 0).
9Given a = (3, -2, 6), find the scalar magnitude |a|.
A.7
B.5
C.sqrt(11)
D.11
Explanation: |a| = sqrt(9 + 4 + 36) = sqrt(49) = 7. Square each component, sum, then take the square root.
10If a = (1, 2, 2) and b = (2, 1, -2), what is 3a - 2b?
A.(-1, 4, 10)
B.(7, 4, 10)
C.(-1, 8, 2)
D.(1, 4, 10)
Explanation: 3a = (3, 6, 6); 2b = (4, 2, -4); 3a - 2b = (3 - 4, 6 - 2, 6 - (-4)) = (-1, 4, 10).

About the Higher Mathematics Exam

Scottish Higher Mathematics (course code C847 76) is a SCQF Level 6 qualification offered by Qualifications Scotland. The course is assessed by two written question papers — Paper 1 non-calculator (60 marks) and Paper 2 calculator (70 marks) — covering three areas: Expressions and Functions, Relationships and Calculus, and Applications.

Questions

100 scored questions

Time Limit

3 hours 15 minutes total (Paper 1: 1h 30min, Paper 2: 1h 45min)

Passing Score

Grade C (50%) is the minimum pass, Grades A-D awarded (A is the highest)

Exam Fee

Entry fees set by presenting centre, typically £15-£40 per subject (Qualifications Scotland (formerly SQA))

Higher Mathematics Exam Content Outline

Core

Expressions and Functions

Vectors in 3D, composite and inverse functions, exponential and logarithmic functions, log laws and graph transformations

Core

Relationships and Calculus

Polynomial functions, factor theorem, quadratic theory, trigonometric equations, addition and double-angle formulae, differentiation and integration

Core

Applications

Straight line geometry, circle equations, sequences and recurrence relations, applications of differentiation and integration

How to Pass the Higher Mathematics Exam

What You Need to Know

  • Passing score: Grade C (50%) is the minimum pass, Grades A-D awarded (A is the highest)
  • Exam length: 100 questions
  • Time limit: 3 hours 15 minutes total (Paper 1: 1h 30min, Paper 2: 1h 45min)
  • Exam fee: Entry fees set by presenting centre, typically £15-£40 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

Higher Mathematics Study Tips from Top Performers

1Practise both Paper 1 (non-calculator) and Paper 2 (calculator) past papers from Qualifications Scotland
2Master the exact values for sin, cos and tan at radians pi/6, pi/4, pi/3 — they are tested every year on Paper 1
3Learn the standard formats: completing the square, circle equation, R sin(theta + alpha), and the synthetic-division layout
4Review the SQA marker reports each autumn — common errors (sign slips, wrong chain-rule constant) repeat year on year

Frequently Asked Questions

Who administers Scottish Higher Mathematics?

Higher Mathematics is set and awarded by Qualifications Scotland (formerly SQA). As of 1 February 2026 the awarding body uses the Qualifications Scotland name; the course code C847 76 and exam structure are unchanged.

How is Higher Mathematics assessed?

Two written question papers in May: Paper 1 is non-calculator (60 marks, 1 hour 30 minutes) and Paper 2 is calculator (70 marks, 1 hour 45 minutes). The papers contribute 100% of the course award.

What grade is needed to pass Higher Mathematics?

Grades A-D are awarded with Grade C (around 50%) being the standard pass. Many Scottish universities require Grade B or A in Higher Mathematics for STEM courses.

What topics are covered in Higher Mathematics?

Three areas: Expressions and Functions (vectors, log/exp, transformations), Relationships and Calculus (polynomials, trigonometry, differentiation, integration) and Applications (straight line, circle, sequences, optimisation).