SQA Higher Mathematics splits into 12 main topics, from straight lines and functions to vectors, circles, the wave function, differentiation and integration. For each one below you get what you need to know, a link to the formulae, and an exam-style question with a worked answer. Then use SQA’s free past papers to practise the same topic in real exam questions.
Revising one topic at a time, then mixing them in full past papers, is one of the most effective ways to prepare for prelims in January and February and for the SQA exam later in the spring.
SQA HigherScotlandS5 and S6
How to revise Higher Maths by topic
Pick one topic from the list below and read what you need to know.
Try the exam-style question without notes, then check the worked answer.
Find that topic in real past papers. SQA publishes Higher Mathematics papers and marking instructions free on its past papers page. Most topics appear in every year’s papers, so search a few years for questions on your topic.
Mark your answers with the marking instructions, and note the method marks you missed.
Finish with full papers under timed conditions once each topic feels secure: Paper 1 is non-calculator and Paper 2 allows a calculator.
Keep the Higher Maths formula sheet beside you: it shows which formulae SQA prints in the exam and which you must learn.
1. Straight lines
What you need to know: Gradient, m = tan θ, parallel and perpendicular lines, y − b = m(x − a), midpoints, perpendicular bisectors, medians and altitudes, and points of intersection.
A is (2, 1) and B is (6, 9). Find the equation of the perpendicular bisector of AB.
Show the worked answer
Midpoint of AB = (4, 5). Gradient of AB = (9 − 1)/(6 − 2) = 2, so the perpendicular gradient is −½. y − 5 = −½(x − 4), which gives x + 2y = 14.
What you need to know: Domain and range, composite functions f(g(x)), inverse functions, and sketching related graphs such as y = f(x) + a, y = f(x + a), y = −f(x) and y = kf(x).
f(x) = 2x + 3 and g(x) = x². Find f(g(x)), g(f(x)) and the inverse function f⁻¹(x).
Show the worked answer
f(g(x)) = f(x²) = 2x² + 3. g(f(x)) = (2x + 3)² = 4x² + 12x + 9. For the inverse, swap x and y in y = 2x + 3: x = 2y + 3, so f⁻¹(x) = (x − 3)/2.
What you need to know: Laws of logarithms, solving log and exponential equations, ln and e, exponential growth and decay, and finding a and b from straight-line graphs of log y against x or log x.
Solve log₂(x + 1) + log₂(x − 1) = 3, where x > 1.
Show the worked answer
Combine: log₂((x + 1)(x − 1)) = 3, so x² − 1 = 2³ = 8 and x² = 9. Since x > 1, x = 3.
What you need to know: Synthetic division, the factor and remainder theorems, factorising cubics and quartics, solving polynomial equations, and finding where curves meet.
Show that (x − 2) is a factor of f(x) = x³ − 3x² − 4x + 12, and factorise f(x) fully.
Show the worked answer
f(2) = 8 − 12 − 8 + 12 = 0, so (x − 2) is a factor. Synthetic division gives the quotient x² − x − 6 = (x − 3)(x + 2), so f(x) = (x − 2)(x − 3)(x + 2).
What you need to know: Completing the square, the nature of roots using b² − 4ac, quadratic inequalities, and showing that a line is a tangent to a curve.
Find the values of k for which x² + kx + 9 = 0 has equal roots.
Show the worked answer
Equal roots when b² − 4ac = 0: k² − 4 × 1 × 9 = 0, so k² = 36 and k = 6 or k = −6.
What you need to know: Components and magnitude, collinearity, dividing a line in a ratio, the scalar product, angles between vectors and perpendicular vectors.
a = (2, −1, 2) and b = (1, 4, k) as column vectors. Find k so that a and b are perpendicular.
Show the worked answer
Perpendicular when a·b = 0: 2 × 1 + (−1) × 4 + 2k = 0, so 2k − 2 = 0 and k = 1.
What you need to know: Centre and radius from either form of the equation, the equation of a tangent, whether a line meets a circle, and whether circles touch.
The circle x² + y² − 6x + 4y − 12 = 0 passes through P(6, 2). Find the centre, the radius and the equation of the tangent at P.
Show the worked answer
Centre (−g, −f) = (3, −2). Radius = √(9 + 4 + 12) = 5. Gradient of the radius to P = (2 − (−2))/(6 − 3) = 4/3, so the tangent gradient is −3/4. y − 2 = −¾(x − 6), which gives 3x + 4y = 26.
What you need to know: Radians and exact values, the addition and double-angle formulae, and solving trig equations, including ones that become quadratics in sin x or cos x.
Solve cos 2x + cos x = 0 for 0 ≤ x < 360°.
Show the worked answer
Use cos 2x = 2cos²x − 1: 2cos²x + cos x − 1 = 0, so (2cos x − 1)(cos x + 1) = 0. cos x = ½ gives x = 60° or 300°; cos x = −1 gives x = 180°. So x = 60°, 180°, 300°.
What you need to know: Writing a cos x + b sin x as k cos(x ± α) or k sin(x ± α), finding maximum and minimum values, and solving equations with it.
Write 3cos x + 4sin x in the form k cos(x − α), where k > 0 and 0 ≤ α < 360°.
Show the worked answer
k cos(x − α) = k cos x cos α + k sin x sin α, so k cos α = 3 and k sin α = 4. k = √(3² + 4²) = 5. Both are positive, so α is in the first quadrant: tan α = 4/3, α = 53.1°. So 3cos x + 4sin x = 5cos(x − 53.1)°.
What you need to know: Differentiating powers, the chain rule, sin and cos, tangents, stationary points and their nature, increasing and decreasing functions, optimisation and rates of change.
Find the stationary points of y = x³ − 6x² + 9x + 1 and determine their nature.
Show the worked answer
dy/dx = 3x² − 12x + 9 = 3(x − 1)(x − 3) = 0, so x = 1 or x = 3. When x = 1, y = 5; when x = 3, y = 1. d²y/dx² = 6x − 12 is −6 at x = 1 and 6 at x = 3, so (1, 5) is a maximum and (3, 1) is a minimum.
What you need to know: Integrating powers, (ax + b)ⁿ, sin and cos, definite integrals, area under a curve and between curves, and solving differential equations of the form dy/dx = f(x).
Find the area enclosed by y = 2x and y = x².
Show the worked answer
The graphs meet where x² = 2x, so x = 0 and x = 2. Between them 2x is above x². Area = ∫₀² (2x − x²) dx = [x² − x³/3] from 0 to 2 = 4 − 8/3 = 4/3 square units.
MathsDaily follows the curriculum in England, so it has no SQA Higher level, but the A-level daily set practises much of the same calculus, algebra, logs and trigonometry, with ten fresh questions every morning.
Free. No sign-up. The closest daily level to SQA Higher.
Common questions
Where can I find Higher Maths past papers by topic?
SQA publishes full Higher Mathematics past papers and marking instructions free on its past papers page. They are organised by year, not by topic, so use the topic list on this page to know what to look for, then search several years of papers for questions on one topic.
What topics are in Higher Maths?
The main topics are straight lines, functions and graphs, logarithms and exponentials, polynomials, quadratics, recurrence relations, vectors, circles, trigonometric formulae and equations, the wave function, differentiation and integration.
Is there a formula sheet in Higher Maths?
Yes, but it is short: circles, the scalar product, the addition and double-angle formulae, and the derivatives and integrals of sin ax and cos ax. See the Higher Maths formula sheet for what is given and what to learn.
Are these real SQA questions?
No. The questions on this page are original exam-style questions written by MathsDaily, with checked worked answers. For real exam questions, use the SQA past papers.
When are the Higher Maths prelims?
Most schools in Scotland hold prelims in January or February, before the SQA exam diet in the spring. Check the dates with your school.