Questions - A-Level Maths

Browse by module

All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4

Browse by board

AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4

Sort by: Default | Easiest first | Hardest first

OCR MEI C3 Q10

4 marks Easy -1.2

Verify the following statement: $$\text{'} 2^p - 1 \text{ is a prime number for all prime numbers } p \text{ less than 11'.} [2]
Calculate $23 \times 89$, and hence disprove this statement: $$\text{'} 2^p - 1 \text{ is a prime number for all prime numbers } p \text{'.} [2]

Given that $t$ is an integer greater than 1, show that $2t$, $t^2 - 1$ and $t^2 + 1$ form a Pythagorean triple. [3]
The two smallest integers of a Pythagorean triple are 20 and 21. Find the third integer. Use this triple to show that not all Pythagorean triples can be expressed in the form $2t$, $t^2 - 1$ and $t^2 + 1$. [3]

AQA C4 2010 June Q1

7 marks Easy -1.2

The polynomial $f(x)$ is defined by $f(x) = 8x^3 + 6x^2 - 14x - 1$. Find the remainder when $f(x)$ is divided by $(4x - 1)$. [2 marks]
The polynomial $g(x)$ is defined by $g(x) = 8x^3 + 6x^2 - 14x + d$.
1. Given that $(4x - 1)$ is a factor of $g(x)$, find the value of the constant $d$. [2 marks]
2. Given that $g(x) = (4x - 1)(ax^2 + bx + c)$, find the values of the integers $a$, $b$ and $c$. [3 marks]

AQA C4 2010 June Q2

9 marks Moderate -0.3

A curve is defined by the parametric equations $$x = 1 - 3t, \quad y = 1 + 2t^3$$

Find $\frac{dy}{dx}$ in terms of $t$. [3 marks]
Find an equation of the normal to the curve at the point where $t = 1$. [4 marks]
Find a cartesian equation of the curve. [2 marks]

AQA C4 2010 June Q3

8 marks Moderate -0.3

1. Express $\frac{7x - 3}{(x + 1)(3x - 2)}$ in the form $\frac{A}{x + 1} + \frac{B}{3x - 2}$. [3 marks]
2. Hence find $\int \frac{7x - 3}{(x + 1)(3x - 2)} dx$. [2 marks]
Express $\frac{6x^2 + x + 2}{2x^2 - x + 1}$ in the form $P + \frac{Qx + R}{2x^2 - x + 1}$. [3 marks]

AQA C4 2010 June Q4

7 marks Moderate -0.3

1. Find the binomial expansion of $(1 + x)^{\frac{3}{2}}$ up to and including the term in $x^2$. [2 marks]
2. Find the binomial expansion of $(16 + 9x)^{\frac{3}{2}}$ up to and including the term in $x^2$. [3 marks]
Use your answer to part (a)(ii) to show that $13^{\frac{3}{2}} \approx 46 + \frac{a}{b}$, stating the values of the integers $a$ and $b$. [2 marks]

AQA C4 2010 June Q5

11 marks Standard +0.3

1. Show that the equation $3\cos 2x + 2\sin x + 1 = 0$ can be written in the form $$3\sin^2 x - \sin x - 2 = 0$$ [3 marks]
2. Hence, given that $3\cos 2x + 2\sin x + 1 = 0$, find the possible values of $\sin x$. [2 marks]
1. Express $3\cos 2x + 2\sin 2x$ in the form $R\cos(2x - \alpha)$, where $R > 0$ and $0° < \alpha < 90°$, giving $\alpha$ to the nearest $0.1°$. [3 marks]
2. Hence solve the equation $$3\cos 2x + 2\sin 2x + 1 = 0$$ for all solutions in the interval $0° < x < 180°$, giving $x$ to the nearest $0.1°$. [3 marks]

AQA C4 2010 June Q6

7 marks Standard +0.3

A curve has equation $x^3 y + \cos(\pi y) = 7$.

Find the exact value of the $x$-coordinate at the point on the curve where $y = 1$. [2 marks]
Find the gradient of the curve at the point where $y = 1$. [5 marks]

AQA C4 2010 June Q7

12 marks Standard +0.3

The point $A$ has coordinates $(4, -3, 2)$. The line $l_1$ passes through $A$ and has equation $\mathbf{r} = \begin{bmatrix} 4 \\ -3 \\ 2 \end{bmatrix} + \lambda \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}$. The line $l_2$ has equation $\mathbf{r} = \begin{bmatrix} -1 \\ 3 \\ 4 \end{bmatrix} + \mu \begin{bmatrix} 1 \\ -2 \\ -1 \end{bmatrix}$. The point $B$ lies on $l_2$ where $\mu = 2$.

Find the vector $\overrightarrow{AB}$. [3 marks]
1. Show that the lines $l_1$ and $l_2$ intersect. [4 marks]
2. The lines $l_1$ and $l_2$ intersect at the point $P$. Find the coordinates of $P$. [1 mark]
The point $C$ lies on a line which is parallel to $l_1$ and which passes through the point $B$. The points $A$, $B$, $C$ and $P$ are the vertices of a parallelogram. Find the coordinates of the two possible positions of the point $C$. [4 marks]

AQA C4 2010 June Q8

14 marks Standard +0.3

Solve the differential equation $$\frac{dx}{dt} = -\frac{1}{5}(x + 1)^{\frac{1}{2}}$$ given that $x = 80$ when $t = 0$. Give your answer in the form $x = f(t)$. [6 marks]
A fungus is spreading on the surface of a wall. The proportion of the wall that is unaffected after time $t$ hours is $x\%$. The rate of change of $x$ is modelled by the differential equation $$\frac{dx}{dt} = -\frac{1}{5}(x + 1)^{\frac{1}{2}}$$ At $t = 0$, the proportion of the wall that is unaffected is 80%. Find the proportion of the wall that will still be unaffected after 60 hours. [2 marks]
A biologist proposes an alternative model for the rate at which the fungus is spreading on the wall. The total surface area of the wall is $9\text{ m}^2$. The surface area that is affected at time $t$ hours is $A\text{ m}^2$. The biologist proposes that the rate of change of $A$ is proportional to the product of the surface area that is affected and the surface area that is unaffected.
1. Write down a differential equation for this model. (You are not required to solve your differential equation.) [2 marks]
2. A solution of the differential equation for this model is given by $$A = \frac{9}{1 + 4e^{-0.09t}}$$ Find the time taken for 50% of the area of the wall to be affected. Give your answer in hours to three significant figures. [4 marks]

AQA C4 2016 June Q1

11 marks Moderate -0.3

Express $\frac{19x - 3}{(1 + 2x)(3 - 4x)}$ in the form $\frac{A}{1 + 2x} + \frac{B}{3 - 4x}$. [3 marks]
1. Find the binomial expansion of $\frac{19x - 3}{(1 + 2x)(3 - 4x)}$ up to and including the term in $x^2$. [7 marks]
2. State the range of values of $x$ for which this expansion is valid. [1 mark]

AQA C4 2016 June Q2

5 marks Standard +0.3

By forming and solving a suitable quadratic equation, find the solutions of the equation $$3 \cos 2\theta - 5 \cos \theta + 2 = 0$$ in the interval $0° < \theta < 360°$, giving your answers to the nearest $0.1°$. [5 marks]

AQA C4 2016 June Q3

8 marks Standard +0.3

Express $\frac{3 + 13x - 6x^2}{2x - 3}$ in the form $Ax + B + \frac{C}{2x - 3}$. [4 marks]
Show that $\int_3^6 \frac{3 + 13x - 6x^2}{2x - 3} \, dx = p + q \ln 3$, where $p$ and $q$ are rational numbers. [4 marks]

AQA C4 2016 June Q4

7 marks Moderate -0.3

The mass of radioactive atoms in a substance can be modelled by the equation $$m = m_0 k^t$$ where $m_0$ grams is the initial mass, $m$ grams is the mass after $t$ days and $k$ is a constant. The value of $k$ differs from one substance to another.

1. A sample of radioactive iodine reduced in mass from 24 grams to 12 grams in 8 days. Show that the value of the constant $k$ for this substance is 0.917004, correct to six decimal places. [1 mark]
2. A similar sample of radioactive iodine reduced in mass to 1 gram after 60 days. Calculate the initial mass of this sample, giving your answer to the nearest gram. [2 marks]
The half-life of a radioactive substance is the time it takes for a mass of $m_0$ to reduce to a mass of $\frac{1}{2}m_0$. A sample of radioactive vanadium reduced in mass from exactly 10 grams to 8.106 grams in 100 days. Find the half-life of radioactive vanadium, giving your answer to the nearest day. [4 marks]

AQA C4 2016 June Q5

10 marks Standard +0.3

It is given that $\sin A = \frac{\sqrt{5}}{3}$ and $\sin B = \frac{1}{\sqrt{5}}$, where the angles $A$ and $B$ are both acute.

1. Show that the exact value of $\cos B = \frac{2}{\sqrt{5}}$. [1 mark]
2. Hence show that the exact value of $\sin 2B$ is $\frac{4}{5}$. [2 marks]
1. Show that the exact value of $\sin(A - B)$ can be written as $p(5 - \sqrt{5})$, where $p$ is a rational number. [4 marks]
2. Find the exact value of $\cos(A - B)$ in the form $r + s\sqrt{5}$, where $r$ and $s$ are rational numbers. [3 marks]

AQA C4 2016 June Q6

15 marks Standard +0.3

The line $l_1$ passes through the point $A(0, 6, 9)$ and the point $B(4, -6, -11)$. The line $l_2$ has equation $\mathbf{r} = \begin{bmatrix} -1 \\ 5 \\ -2 \end{bmatrix} + \lambda \begin{bmatrix} 3 \\ -5 \\ 1 \end{bmatrix}$.

The acute angle between the lines $l_1$ and $l_2$ is $\theta$. Find the value of $\cos \theta$ as a fraction in its lowest terms. [5 marks]
Show that the lines $l_1$ and $l_2$ intersect and find the coordinates of the point of intersection. [5 marks]
The points $C$ and $D$ lie on line $l_2$ such that $ACBD$ is a parallelogram. \includegraphics{figure_6} The length of $AB$ is three times the length of $CD$. Find the coordinates of the points $C$ and $D$. [5 marks]

AQA C4 2016 June Q7

9 marks Standard +0.8

A curve $C$ is defined by the parametric equations $$x = \frac{4 - e^{-6t}}{4}, \quad y = \frac{e^{3t}}{3t}, \quad t \neq 0$$

Find the exact value of $\frac{dy}{dx}$ at the point on $C$ where $t = \frac{2}{3}$. [5 marks]
Show that $x = \frac{4 - e^{-6t}}{4}$ can be rearranged into the form $e^{3t} = \frac{e}{2\sqrt{(1-x)}}$. [2 marks]
Hence find the Cartesian equation of $C$, giving your answer in the form $$y = \frac{e}{f(x)[1 - \ln(f(x))]}$$ [2 marks]

AQA C4 2016 June Q8

10 marks Standard +0.8

It is given that $\theta = \tan^{-1}\left(\frac{3x}{2}\right)$.

By writing $\theta = \tan^{-1}\left(\frac{3x}{2}\right)$ as $2\tan\theta = 3x$, use implicit differentiation to show that $$\frac{d\theta}{dx} = \frac{k}{4 + 9x^2}$$, where $k$ is an integer. [3 marks]
Hence solve the differential equation $$9y(4 + 9x^2)\frac{dy}{dx} = \cosec 3y$$ given that $x = 0$ when $y = \frac{\pi}{3}$. Give your answer in the form $\mathbf{g}(y) = \mathbf{h}(x)$. [7 marks]

Edexcel C4 Q1

6 marks Standard +0.3

Use integration by parts to find the exact value of $\int_1^3 x^2 \ln x \, dx$. [6]

Edexcel C4 Q2

12 marks Moderate -0.3

Fluid flows out of a cylindrical tank with constant cross section. At time $t$ minutes, $t \geq 0$, the volume of fluid remaining in the tank is $V$ m$^3$. The rate at which the fluid flows, in m$^3$ min$^{-1}$, is proportional to the square root of $V$.

Show that the depth $h$ metres of fluid in the tank satisfies the differential equation $$\frac{dh}{dt} = -k\sqrt{h}, \quad \text{where } k \text{ is a positive constant.}$$ [3]
Show that the general solution of the differential equation may be written as $$h = (A - Bt)^2, \quad \text{where } A \text{ and } B \text{ are constants.}$$ [4] Given that at time $t = 0$ the depth of fluid in the tank is 1 m, and that 5 minutes later the depth of fluid has reduced to 0.5 m,
find the time, $T$ minutes, which it takes for the tank to empty. [3]
Find the depth of water in the tank at time $0.5T$ minutes. [2]

Edexcel C4 Q3

14 marks Standard +0.3

Use the identity for $\cos(A + B)$ to prove that $\cos 2A = 2\cos^2 A - 1$. [2]
Use the substitution $x = 2\sqrt{2} \sin \theta$ to prove that $$\int_2^{\sqrt{6}} \sqrt{(8 - x^2)} \, dx = \frac{1}{3}(\pi + 3\sqrt{3} - 6).$$ [7]

A curve is given by the parametric equations $$x = \sec \theta, \quad y = \ln(1 + \cos 2\theta), \quad 0 \leq \theta < \frac{\pi}{2}.$$