Free A-Level Maths Practice Paper (PDF)

This free A-Level Maths practice paper mixes pure and applied topics in the style of a real exam sitting — surds, quadratics, coordinate geometry, binomial expansion, calculus and trigonometry, finishing with a statistics question and a short proof. Every question has a fully worked mark-scheme solution showing exactly where the method and accuracy marks come from, so you can mark your own attempt like an examiner. View it online below or download the print-ready PDF; no account needed.

1. Question 14 marks · Indices & surds
   Indices and surds.
   (a)

   Simplify $\frac{8^{\frac{2}{3}} \times 4^{-\frac{1}{2}}}{2^{-3}}$, giving your answer as an integer.

   [2]
   (b)

   Rationalise the denominator of $\frac{14}{3 - \sqrt{2}}$, giving your answer in the form $a + b\sqrt{2}$, where $a$ and $b$ are integers.

   [2]
2. Question 25 marks · Quadratics & the discriminant
   The function $f$ is defined by $f(x) = 2x^2 - 12x + 23$.
   $f(x) = 2x^2 - 12x + 23$
   (a)

   Express $f(x)$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are integers to be found.

   [2]
   (b)

   Hence write down the coordinates of the vertex of the curve $y = f(x)$.

   [1]
   (c)

   Using the discriminant, determine the number of real roots of the equation $f(x) = 0$.

   [2]
3. Question 36 marks · Coordinate geometry: circles
   The circle $C$ has centre $(3, -2)$ and passes through the point $P(7, 1)$.
   (a)

   Find an equation for $C$.

   [3]
   (b)

   Find an equation of the tangent to $C$ at the point $P$, giving your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers.

   [3]
4. Question 45 marks · Binomial expansion
   This question is about the binomial expansion of $(2 + 3x)^5$.
   $(2 + 3x)^5$
   (a)

   Find the first four terms, in ascending powers of $x$, of the binomial expansion of $(2 + 3x)^5$. Give each coefficient as an integer.

   [3]
   (b)

   Use your expansion with a suitable value of $x$ to estimate $2.03^5$, giving your answer to 4 decimal places.

   [2]
5. Question 57 marks · Differentiation & stationary points
   A curve has equation $y = 2x^3 - 9x^2 + 12x - 3$.
   $y = 2x^3 - 9x^2 + 12x - 3$
   (a)

   Find the coordinates of the stationary points of the curve.

   [4]
   (b)

   Determine the nature of each stationary point.

   [2]
   (c)

   State the range of values of $x$ for which $y$ is increasing.

   [1]
6. Question 66 marks · Integration & area
   The curve with equation $y = 4x - x^2$ and the line with equation $y = x$ enclose a finite region $R$. (For a sketch: the curve is an inverted parabola through the origin with maximum at $(2, 4)$; the line passes through the origin with gradient 1 and cuts the curve again to the right, with the curve above the line between the two intersections.)
   $y = 4x - x^2, \qquad y = x$
   (a)

   Find the coordinates of the points of intersection of the curve and the line.

   [2]
   (b)

   Find the exact area of the region $R$.

   [4]
7. Question 76 marks · Trigonometric equations
   Solve the equation $2\sin^2\theta + 3\cos\theta = 3$ for $0^\circ \leq \theta \leq 360^\circ$. Show each step of your working.
   $2\sin^2\theta + 3\cos\theta = 3, \qquad 0^\circ \leq \theta \leq 360^\circ$
8. Question 88 marks · Discrete random variables & the binomial distribution
   Statistics. The discrete random variable $X$ has the probability distribution $P(X = 1) = 0.1$, $P(X = 2) = 0.3$, $P(X = 3) = k$, $P(X = 4) = 0.2$, where $k$ is a constant.
   $P(X=1) = 0.1, \quad P(X=2) = 0.3, \quad P(X=3) = k, \quad P(X=4) = 0.2$
   (a)

   Find the value of $k$.

   [1]
   (b)

   Find $E(X)$ and $\text{Var}(X)$.

   [4]
   (c)

   Five independent observations of $X$ are taken. Find the probability that exactly two of them take the value 4.

   [3]
9. Question 94 marks · Proof
   Prove that, for every integer $n$, $(2n + 1)^2 - 1$ is divisible by 8.