Higher Mathematics: Mixed Practice Across the Course

Ten questions across the Higher course: two on differentiation, two on integration, and one each on logarithms, the circle, vectors in three dimensions, the remainder theorem, a trigonometric equation and the limit of a recurrence relation.

The wrong options are where the teaching is. In the derivative of 3x^-3, the tempting answer adds one to the power instead of subtracting, because going from minus three to minus four feels like moving the wrong way — and that single instinct accounts for most lost marks on negative indices. In the area question, one option is what you get by integrating the constant term and quietly leaving the x-squared behind, which is easy to do because 4x evaluates cleanly and looks finished. In the circle question, one option changes the signs of the coefficients but forgets to halve them.

Two of the questions are about knowing which situation you are in rather than about algebra at all. The remainder theorem needs P(2) and not P(-2) when the divisor is x minus 2, and a sine equation solved over a full turn has a second solution in the second quadrant, not the third — both are decisions taken before any working starts, and both are where a confident candidate can lose a mark without noticing.

Each explanation sets the working out line by line and names the error behind the closest wrong answer, because Higher credits method as well as the final number.

Every question here was written for this set. Past papers and marking instructions are published by Qualifications Scotland and are their copyright; nothing is reproduced from them.

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  • Differentiate a term with a negative index without moving the power the wrong way
  • Find a stationary point and classify it using the second derivative
  • Evaluate a definite integral and find the area under a curve, integrating every term
  • Solve an exponential equation by writing both sides to the same base
  • Find the centre and radius of a circle from its general equation, halving and changing signs correctly
  • Calculate a scalar product in three dimensions
  • Apply the remainder theorem with the right value substituted
  • Solve a trigonometric equation over a full turn, finding both solutions
  • Find the limit of a recurrence relation from the equation L = mL + c

Find the derivative of f(x) = 3x^(-3) + 2x^2. — one of ten questions written for this set, alongside a stationary point, a definite integral, an area under a curve, an exponential equation, a circle, a scalar product, the remainder theorem, a trigonometric equation and a recurrence limit.

Sample question

Graph of y = 4 − x² for x from 0 to 2.5, with the region under the curve and above the x-axis between x = 0 and x = 2 shaded.

Find the area enclosed between the curve $y = 4 - x^2$ and the x-axis from $x = 0$ to $x = 2$.

See the answer

$16/3$

Integrate and then evaluate between the limits: $\int_0^2 (4 - x^2)\,dx = [4x - x^3/3]_0^2 = (8 - 8/3) - 0 = 16/3$. Each wrong option comes from a specific slip. $8$ is what you get if only the constant term is integrated and the $x^2$ is left behind — the commonest error here, because $4x$ evaluates cleanly and looks finished. $8/3$ is the second term on its own, with the $4x$ dropped. And $4$ is simply the height of the curve at $x = 0$, which is a value of $y$ rather than an area at all.

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← Higher Mathematics

↑ National 5 and Higher (Scotland)

National 5 and Higher qualifications are awarded by Qualifications Scotland, which took over the awarding functions of the Scottish Qualifications Authority (SQA) in February 2026. Zestly is not affiliated with or endorsed by Qualifications Scotland or the SQA. These materials are our own and reproduce no past-paper content.