Questions - AQA - A-Level Maths

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = a \sin \theta + b \operatorname { cosec } \theta$$ where $a$ and $b$ are integers.
2. Find the gradient of the normal to the curve at the point where $\theta = \frac { \pi } { 6 }$.
Show that the cartesian equation of the curve can be expressed as $$y ^ { 2 } = p x ^ { 2 } \left( 4 - x ^ { 2 } \right)$$ where $p$ is a rational number.

AQA C4 2012 June Q6

8 marks Standard +0.8

6 A curve is defined by the equation $9 x ^ { 2 } - 6 x y + 4 y ^ { 2 } = 3$. Find the coordinates of the two stationary points of this curve.

AQA C4 2012 June Q7

12 marks Standard +0.3

$\mathbf { 7 } \quad$ The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 0 \\ - 2 \\ q \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right]$, where $q$ is an integer. The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { l } 8 \\ 3 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { l } 2 \\ 5 \\ 4 \end{array} \right]$. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $P$.

Show that $q = 4$ and find the coordinates of $P$.
Show that $l _ { 1 }$ and $l _ { 2 }$ are perpendicular.
The point $A$ lies on the line $l _ { 1 }$ where $\lambda = 1$.
1. Find $A P ^ { 2 }$.
2. The point $B$ lies on the line $l _ { 2 }$ so that the right-angled triangle $A P B$ is isosceles. Find the coordinates of the two possible positions of $B$.

AQA C4 2012 June Q8

12 marks Standard +0.3

A water tank has a height of 2 metres. The depth of the water in the tank is $h$ metres at time $t$ minutes after water begins to enter the tank. The rate at which the depth of the water in the tank increases is proportional to the difference between the height of the tank and the depth of the water. Write down a differential equation in the variables $h$ and $t$ and a positive constant $k$.
(You are not required to solve your differential equation.)
1. Another water tank is filling in such a way that $t$ minutes after the water is turned on, the depth of the water, $x$ metres, increases according to the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 15 x \sqrt { 2 x - 1 } }$$ The depth of the water is 1 metre when the water is first turned on.
  Solve this differential equation to find $t$ as a function of $x$.
2. Calculate the time taken for the depth of the water in the tank to reach 2 metres, giving your answer to the nearest 0.1 of a minute.
  (l mark)

AQA C4 2013 June Q1

10 marks Standard +0.3

1. Express $\frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) }$ in the form $\frac { A } { 2 + x } + \frac { B } { 1 - 3 x }$, where $A$ and $B$ are integers.
  (3 marks)
2. Hence show that $\int _ { - 1 } ^ { 0 } \frac { 5 - 8 x } { ( 2 + x ) ( 1 - 3 x ) } \mathrm { d } x = p \ln 2$, where $p$ is rational.
  (4 marks)
1. Given that $\frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }$ can be written as $C + \frac { 5 - 8 x } { 2 - 5 x - 3 x ^ { 2 } }$, find the value of $C$.
  (1 mark)
2. Hence find the exact value of the area of the region bounded by the curve $y = \frac { 9 - 18 x - 6 x ^ { 2 } } { 2 - 5 x - 3 x ^ { 2 } }$, the $x$-axis and the lines $x = - 1$ and $x = 0$. You may assume that $y > 0$ when $- 1 \leqslant x \leqslant 0$.

AQA C4 2013 June Q2

8 marks Moderate -0.3

2 The acute angles $\alpha$ and $\beta$ are given by $\tan \alpha = \frac { 2 } { \sqrt { 5 } }$ and $\tan \beta = \frac { 1 } { 2 }$.

1. Show that $\sin \alpha = \frac { 2 } { 3 }$, and find the exact value of $\cos \alpha$.
2. Hence find the exact value of $\sin 2 \alpha$.
Show that the exact value of $\cos ( \alpha - \beta )$ can be expressed as $\frac { 2 } { 15 } ( k + \sqrt { 5 } )$, where $k$ is an integer.

AQA C4 2013 June Q3

7 marks Standard +0.3

Find the binomial expansion of $( 1 + 6 x ) ^ { - \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
1. Find the binomial expansion of $( 27 + 6 x ) ^ { - \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$, simplifying the coefficients.
2. Given that $\sqrt [ 3 ] { \frac { 2 } { 7 } } = \frac { 2 } { \sqrt [ 3 ] { 28 } }$, use your binomial expansion from part (b)(i) to obtain an approximation to $\sqrt [ 3 ] { \frac { 2 } { 7 } }$, giving your answer to six decimal places.
  (2 marks)

AQA C4 2013 June Q4

12 marks Standard +0.3

4 A curve is defined by the parametric equations $x = 8 \mathrm { e } ^ { - 2 t } - 4 , y = 2 \mathrm { e } ^ { 2 t } + 4$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
The point $P$, where $t = \ln 2$, lies on the curve.
1. Find the gradient of the curve at $P$.
2. Find the coordinates of $P$.
3. The normal at $P$ crosses the $x$-axis at the point $Q$. Find the coordinates of $Q$.
Find the Cartesian equation of the curve in the form $x y + 4 y - 4 x = k$, where $k$ is an integer.
(3 marks)

AQA C4 2013 June Q5

11 marks Standard +0.3

5 The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 4 x ^ { 3 } - 11 x - 3$.

Use the Factor Theorem to show that ( $2 x + 3$ ) is a factor of $\mathrm { f } ( x )$.
Write $\mathrm { f } ( x )$ in the form $( 2 x + 3 ) \left( a x ^ { 2 } + b x + c \right)$, where $a , b$ and $c$ are integers.
1. Show that the equation $2 \cos 2 \theta \sin \theta + 9 \sin \theta + 3 = 0$ can be written as $4 x ^ { 3 } - 11 x - 3 = 0$, where $x = \sin \theta$.
2. Hence find all solutions of the equation $2 \cos 2 \theta \sin \theta + 9 \sin \theta + 3 = 0$ in the interval $0 ^ { \circ } < \theta < 360 ^ { \circ }$, giving your solutions to the nearest degree.

AQA C4 2013 June Q6

14 marks Standard +0.8

6 The points $A , B$ and $C$ have coordinates $( 3 , - 2,4 ) , ( 1 , - 5,6 )$ and $( - 4,5 , - 1 )$ respectively. The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { r } 7 \\ - 7 \\ 5 \end{array} \right]$.

Show that the point $C$ lies on the line $l$.
Find a vector equation of the line that passes through points $A$ and $B$.
The point $D$ lies on the line through $A$ and $B$ such that the angle $C D A$ is a right angle. Find the coordinates of $D$.
The point $E$ lies on the line through $A$ and $B$ such that the area of triangle $A C E$ is three times the area of triangle $A C D$. Find the coordinates of the two possible positions of $E$.

AQA C4 2013 June Q7

3 marks Moderate -0.8

7 The height of the tide in a certain harbour is $h$ metres at time $t$ hours. Successive high tides occur every 12 hours. The rate of change of the height of the tide can be modelled by a function of the form $a \cos ( k t )$, where $a$ and $k$ are constants. The largest value of this rate of change is 1.3 metres per hour. Write down a differential equation in the variables $h$ and $t$. State the values of the constants $a$ and $k$.

AQA C4 2013 June Q8

10 marks Standard +0.8

$\quad$ Find $\int t \cos \left( \frac { \pi } { 4 } t \right) \mathrm { d } t$.
The platform of a theme park ride oscillates vertically. For the first 75 seconds of the ride, $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { t \cos \left( \frac { \pi } { 4 } t \right) } { 32 x }$$ where $x$ metres is the height of the platform above the ground after time $t$ seconds.
At $t = 0$, the height of the platform above the ground is 4 metres.
Find the height of the platform after 45 seconds, giving your answer to the nearest centimetre.
(6 marks)

AQA C4 2014 June Q1

5 marks Moderate -0.3

1 A curve is defined by the parametric equations $x = \frac { t ^ { 2 } } { 2 } + 1 , y = \frac { 4 } { t } - 1$.

Find the gradient at the point on the curve where $t = 2$.
Find a Cartesian equation of the curve.
\includegraphics[max width=\textwidth, alt={}]{9f03a5f3-7fea-4fb7-b3bd-b4c0cdf662a2-02_1730_1709_977_153}

AQA C4 2014 June Q2

7 marks Moderate -0.3

Given that $\frac { 4 x ^ { 3 } - 2 x ^ { 2 } + 16 x - 3 } { 2 x ^ { 2 } - x + 2 }$ can be expressed as $A x + \frac { B ( 4 x - 1 ) } { 2 x ^ { 2 } - x + 2 }$, find the values of the constants $A$ and $B$.
The gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x ^ { 3 } - 2 x ^ { 2 } + 16 x - 3 } { 2 x ^ { 2 } - x + 2 }$$ The point $( - 1,2 )$ lies on the curve. Find the equation of the curve.
[0pt] [4 marks]

AQA C4 2014 June Q3

7 marks Moderate -0.3

Find the binomial expansion of $( 1 - 4 x ) ^ { \frac { 1 } { 4 } }$ up to and including the term in $x ^ { 2 }$.
[0pt] [2 marks]
Find the binomial expansion of $( 2 + 3 x ) ^ { - 3 }$ up to and including the term in $x ^ { 2 }$.
Hence find the binomial expansion of $\frac { ( 1 - 4 x ) ^ { \frac { 1 } { 4 } } } { ( 2 + 3 x ) ^ { 3 } }$ up to and including the term in $x ^ { 2 }$.
[0pt] [2 marks]

AQA C4 2014 June Q4

11 marks Moderate -0.3

4 A painting was valued on 1 April 2001 at $\pounds 5000$.
The value of this painting is modelled by $$V = A p ^ { t }$$ where $\pounds V$ is the value $t$ years after 1 April 2001, and $A$ and $p$ are constants.

Write down the value of $A$.
According to the model, the value of this painting on 1 April 2011 was $\pounds 25000$. Using this model:
1. show that $p ^ { 10 } = 5$;
2. use logarithms to find the year in which the painting will be valued at $\pounds 75000$.
A painting by another artist was valued at $\pounds 2500$ on 1 April 1991. The value of this painting is modelled by $$W = 2500 q ^ { t }$$ where $\pounds W$ is the value $t$ years after 1 April 1991, and $q$ is a constant.
1. Show that, according to the two models, the value of the two paintings will be the same $T$ years after 1 April 1991, $$\text { where } T = \frac { \ln \left( \frac { 5 } { 2 } \right) } { \ln \left( \frac { p } { q } \right) }$$
2. Given that $p = 1.029 q$, find the year in which the two paintings will have the same value.
  [0pt] [1 mark]

AQA C4 2014 June Q5

15 marks Standard +0.3

1. Express $3 \sin x + 4 \cos x$ in the form $R \sin ( x + \alpha )$ where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.
2. Hence solve the equation $3 \sin 2 \theta + 4 \cos 2 \theta = 5$ in the interval $0 ^ { \circ } < \theta < 360 ^ { \circ }$, giving your solutions to the nearest $0.1 ^ { \circ }$.
1. Show that the equation $\tan 2 \theta \tan \theta = 2$ can be written as $2 \tan ^ { 2 } \theta = 1$.
2. Hence solve the equation $\tan 2 \theta \tan \theta = 2$ in the interval $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$, giving your solutions to the nearest $0.1 ^ { \circ }$.
1. Use the Factor Theorem to show that $2 x - 1$ is a factor of $8 x ^ { 3 } - 4 x + 1$.
2. Show that $4 \cos 2 \theta \cos \theta + 1$ can be written as $8 x ^ { 3 } - 4 x + 1$ where $x = \cos \theta$.
3. Given that $\theta = 72 ^ { \circ }$ is a solution of $4 \cos 2 \theta \cos \theta + 1 = 0$, use the results from parts (c)(i) and (c)(ii) to show that the exact value of $\cos 72 ^ { \circ }$ is $\frac { ( \sqrt { 5 } - 1 ) } { p }$ where $p$ is an integer.
  [0pt] [3 marks]

AQA C4 2014 June Q6

10 marks Moderate -0.3

6 The line $l _ { 1 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 4 \\ - 5 \\ 3 \end{array} \right] + \lambda \left[ \begin{array} { r } - 1 \\ 3 \\ 1 \end{array} \right]$.
The line $l _ { 2 }$ has equation $\mathbf { r } = \left[ \begin{array} { r } 7 \\ - 8 \\ 6 \end{array} \right] + \mu \left[ \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right]$.
The point $P$ lies on $l _ { 1 }$ where $\lambda = - 1$. The point $Q$ lies on $l _ { 2 }$ where $\mu = 2$.

Show that the vector $\overrightarrow { P Q }$ is parallel to $\left[ \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right]$.
The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $R ( 3 , b , c )$.
1. Show that $b = - 2$ and find the value of $c$.
2. The point $S$ lies on a line through $P$ that is parallel to $l _ { 2 }$. The line $R S$ is perpendicular to the line $P Q$.
  \includegraphics[max width=\textwidth, alt={}, center]{9f03a5f3-7fea-4fb7-b3bd-b4c0cdf662a2-16_887_1159_1320_443} Find the coordinates of $S$.
  $7 \quad$ A curve has equation $\cos 2 y + y \mathrm { e } ^ { 3 x } = 2 \pi$.
  The point $A \left( \ln 2 , \frac { \pi } { 4 } \right)$ lies on this curve.

AQA C4 2014 June Q8

11 marks Standard +0.3

Express $\frac { 16 x } { ( 1 - 3 x ) ( 1 + x ) ^ { 2 } }$ in the form $\frac { A } { 1 - 3 x } + \frac { B } { 1 + x } + \frac { C } { ( 1 + x ) ^ { 2 } }$.
Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 16 x \mathrm { e } ^ { 2 y } } { ( 1 - 3 x ) ( 1 + x ) ^ { 2 } }$$ where $y = 0$ when $x = 0$.
Give your answer in the form $\mathrm { f } ( y ) = \mathrm { g } ( x )$.
[0pt] [7 marks]

AQA C4 2015 June Q1

9 marks Moderate -0.8

1 It is given that $\mathrm { f } ( x ) = \frac { 19 x - 2 } { ( 5 - x ) ( 1 + 6 x ) }$ can be expressed as $\frac { A } { 5 - x } + \frac { B } { 1 + 6 x }$, where $A$ and $B$ are integers.

Find the values of $A$ and $B$.
Hence show that $\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = k \ln 5$, where $k$ is a rational number.
[0pt] [6 marks]

AQA C4 2015 June Q2

8 marks Standard +0.3

Express $2 \cos x - 5 \sin x$ in the form $R \cos ( x + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$, giving your value of $\alpha$, in radians, to three significant figures.
1. Hence find the value of $x$ in the interval $0 < x < 2 \pi$ for which $2 \cos x - 5 \sin x$ has its maximum value. Give your value of $x$ to three significant figures.
2. Use your answer to part (a) to solve the equation $2 \cos x - 5 \sin x + 1 = 0$ in the interval $0 < x < 2 \pi$, giving your solutions to three significant figures.
  [0pt] [3 marks]

AQA C4 2015 June Q3

9 marks Moderate -0.3

The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + d$, where $d$ is a constant. When $\mathrm { f } ( x )$ is divided by ( $2 x + 1$ ), the remainder is - 2 . Use the Remainder Theorem to find the value of $d$.
The polynomial $\mathrm { g } ( x )$ is defined by $\mathrm { g } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + 3$.
1. Given that $x = - \frac { 1 } { 2 }$ is a solution of the equation $\mathrm { g } ( x ) = 0$, write $\mathrm { g } ( x )$ as a product of three linear factors.
2. The function h is defined by $\mathrm { h } ( x ) = \frac { 4 x ^ { 2 } - 1 } { \mathrm {~g} ( x ) }$ for $x > 2$. Simplify $\mathrm { h } ( x )$, and hence show that h is a decreasing function.
  [0pt] [4 marks]

AQA C4 2015 June Q4

7 marks Standard +0.3

Find the binomial expansion of $( 1 + 5 x ) ^ { \frac { 1 } { 5 } }$ up to and including the term in $x ^ { 2 }$.
1. Find the binomial expansion of $( 8 + 3 x ) ^ { - \frac { 2 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
2. Use your expansion from part (b)(i) to find an estimate for $\sqrt [ 3 ] { \frac { 1 } { 81 } }$, giving your answer to four decimal places.
  [0pt] [2 marks]

AQA C4 2015 June Q5

11 marks Standard +0.8

5 A curve is defined by the parametric equations $x = \cos 2 t , y = \sin t$.
The point $P$ on the curve is where $t = \frac { \pi } { 6 }$.

Find the gradient at $P$.
Find the equation of the normal to the curve at $P$ in the form $y = m x + c$.
The normal at $P$ intersects the curve again at the point $Q ( \cos 2 q , \sin q )$. Use the equation of the normal to form a quadratic equation in $\sin q$ and hence find the $x$-coordinate of $Q$.
[0pt] [5 marks]

AQA C4 2015 June Q6

12 marks Challenging +1.2

6 The points $A$ and $B$ have coordinates $( 3,2,10 )$ and $( 5 , - 2,4 )$ respectively.
The line $l$ passes through $A$ and has equation $\mathbf { r } = \left[ \begin{array} { r } 3 \\ 2 \\ 10 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right]$.

Find the acute angle between $l$ and the line $A B$.
The point $C$ lies on $l$ such that angle $A B C$ is $90 ^ { \circ }$.
\includegraphics[max width=\textwidth, alt={}, center]{fdd3905e-11f7-4b20-adfe-4c686018a221-12_360_339_762_852} Find the coordinates of $C$.
The point $D$ is such that $B D$ is parallel to $A C$ and angle $B C D$ is $90 ^ { \circ }$. The point $E$ lies on the line through $B$ and $D$ and is such that the length of $D E$ is half that of $A C$. Find the coordinates of the two possible positions of $E$.
[0pt] [4 marks]

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