A-Level Maths Questions

Looking for A-Level maths questions to practise on — samples to see the standard, or a steady supply for revision? This page gives you both: five original exam-style questions with full worked answers below, and a free question generator that produces unlimited questions like them, by topic and difficulty. Everything targets H2 Mathematics (9758); the pure-maths examples are equally good practice for any A-Level syllabus.

Sample question 1 — Vectors

The line $l$ has equation $\mathbf{r} = \begin{pmatrix}1\\2\\3\end{pmatrix} + \lambda\begin{pmatrix}2\\-1\\2\end{pmatrix}$ and the plane $p$ has equation $x + 2y - 2z = 5$.

(a) Find the coordinates of the point where $l$ meets $p$. [3]

(b) Find the acute angle between $l$ and $p$. [3]

Worked answer.

(a) A point on $l$ has coordinates $(1+2\lambda,\; 2-\lambda,\; 3+2\lambda)$. Substituting into the plane equation:

So the point of intersection is $\left(-2,\; \tfrac{7}{2},\; 0\right)$.

(b) With direction $\mathbf{d} = (2, -1, 2)$ and normal $\mathbf{n} = (1, 2, -2)$:

More on this topic: vectors notes and examples.

Sample question 2 — Integration

(a) Find $\displaystyle\int x e^{2x}\,dx$. [3]

(b) Hence find the exact value of $\displaystyle\int_0^1 x e^{2x}\,dx$. [2]

Worked answer.

(a) Integrating by parts with $u = x$, $\dfrac{dv}{dx} = e^{2x}$:

(b) Evaluating from $0$ to $1$:

More on this topic: integration techniques.

Sample question 3 — Sequences and series

The first two terms of a geometric progression have sum $24$, and the sum to infinity is $32$. All terms are positive.

(a) Find the first term and the common ratio. [4]

(b) Find the least value of $n$ for which the sum of the first $n$ terms exceeds $31.5$. [3]

Worked answer.

(a) The conditions give $a(1+r) = 24$ and $\dfrac{a}{1-r} = 32$, so $a = 32(1-r)$. Substituting:

Since all terms are positive, $r = \tfrac{1}{2}$ and $a = 16$. (Check: $16 + 8 = 24$; $\tfrac{16}{1/2} = 32$.)

(b) $S_n = 32\left(1 - \left(\tfrac{1}{2}\right)^n\right) > 31.5 \implies \left(\tfrac{1}{2}\right)^n < \tfrac{1}{64} \implies 2^n > 64$, so $n > 6$ and the least value is $n = 7$.

More on this topic: sequences and series.

Sample question 4 — Normal distribution

The scores in a national test are modelled by $X \sim N(70, 8^2)$.

(a) Find the probability that a randomly chosen student scores more than $80$. [2]

(b) The top $10\%$ of students receive a distinction. Find the minimum distinction score. [3]

Worked answer.

(a) Standardising, $z = \dfrac{80-70}{8} = 1.25$, so $P(X > 80) = 1 - \Phi(1.25) = 0.106$ (3 s.f.) — or read directly from the GC.

(b) We need $k$ with $P(X < k) = 0.9$. The GC (invNorm) gives $z = 1.2816$, so

More on this topic: the normal distribution.

Sample question 5 — Differentiation

A spherical balloon is inflated so that its volume increases at a constant rate of $24\ \text{cm}^3\,\text{s}^{-1}$. Find the rate of increase of its radius at the instant when the radius is $2$ cm. [4]

Worked answer.

With $V = \tfrac{4}{3}\pi r^3$, we have $\dfrac{dV}{dr} = 4\pi r^2 = 16\pi$ when $r = 2$. By the chain rule:

More on this topic: differentiation and its applications.

More example questions, topic by topic

Every one of our topic guides contains further worked examples in the same exam style — pick the topic you are revising:

Functions and graphs · Equations and inequalities · Sequences and series · Vectors · Complex numbers · Differentiation · Maclaurin series · Integration · Differential equations · P&C · Probability · Discrete random variables · Normal distribution · Sampling · Hypothesis testing · Correlation and regression

Need more than samples? Generate unlimited questions

Five samples show you the standard — but revision needs volume. MathChat’s free A-Level maths question generator creates a fresh, original exam-style question every time: choose from six H2 topics, set the difficulty from foundation to distinction level, or match the prelim style of 19 junior colleges. Every generated question comes with a verified answer, and the built-in AI tutor explains the method step by step when you are stuck.

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For the full picture of how generated practice fits into revision, see the H2 maths practice questions guide.

Frequently asked questions

Where can I get free A-Level maths questions with answers? Every sample question on this page comes with a full worked answer, and the free question generator creates unlimited A-Level style H2 Maths questions, each with a verified answer.

Are these real past-year A-Level questions? No — they are original questions written in the same style and at the same difficulty as A-Level and JC prelim questions, so you can practise freely without copyright-restricted papers.

Do these questions cover the H2 Maths (9758) syllabus? Yes. The samples and the generator target the Singapore H2 Mathematics 9758 syllabus — see the full syllabus guide. Pure maths topics such as vectors, calculus and sequences also overlap heavily with other A-Level maths syllabuses.

Can I generate more questions like these? Yes — the free generator produces unlimited exam-style questions by topic, difficulty and JC prelim style, with a worked answer for every question.

How are the answers checked? Generated answers are verified with a computer algebra system before they are shown, and the free AI tutor can walk you through the full method step by step if you get stuck.