Questions - AQA Paper 2 - A-Level Maths

Sketch the graph of $y = \frac { 1 } { x ^ { 2 } }$
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-12_1273_1083_404_482} 8
The graph of $y = \frac { 1 } { x ^ { 2 } }$ can be transformed onto the graph of $y = \frac { 9 } { x ^ { 2 } }$ using a stretch in one direction. Beth thinks the stretch should be in the $y$-direction.
Paul thinks the stretch should be in the $x$-direction.
State, giving reasons for your answer, whether Beth is correct, Paul is correct, both are correct or neither is correct.
[0pt] [3 marks]

AQA Paper 2 2022 June Q9

9 Given that $$\log _ { 2 } x ^ { 3 } - \log _ { 2 } y ^ { 2 } = 9$$ show that $$x = A y ^ { p }$$ where $A$ is an integer and $p$ is a rational number.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-15_2488_1716_219_153}

AQA Paper 2 2022 June Q10

2 marks

10 A gardener has a greenhouse containing 900 tomato plants. The gardener notices that some of the tomato plants are damaged by insects.
Initially there are 25 damaged tomato plants.
The number of tomato plants damaged by insects is increasing by $32 \%$ each day.
10

The total number of plants damaged by insects, $x$, is modelled by $$x = A \times B ^ { t }$$ where $A$ and $B$ are constants and $t$ is the number of days after the gardener first noticed the damaged plants. 10
1. Use this model to find the total number of plants damaged by insects 5 days after the gardener noticed the damaged plants.
  10
(ii) Explain why this model is not realistic in the long term.
10
A refined model assumes the rate of increase of the number of plants damaged by insects is given by $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x ( 900 - x ) } { 2700 }$$ 10

Show that $$\int \left( \frac { A } { x } + \frac { B } { 900 - x } \right) \mathrm { d } x = \int \mathrm { d } t$$ where $A$ and $B$ are positive integers to be found.

(iii) Hence, find the number of days it takes from when the damage is first noticed until half of the plants are damaged by the insects.

[2 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

AQA Paper 2 2022 June Q11

1 marks

11 A moon vehicle has a mass of 212 kg and a length of 3 metres.
On the moon the vehicle has a weight of 345 N
Calculate a value for acceleration due to gravity on the moon.
Circle your answer.
[0pt] [1 mark] $$0.614 \mathrm {~m} \mathrm {~s} ^ { - 2 } \quad 1.63 \mathrm {~m} \mathrm {~s} ^ { - 2 } \quad 1.84 \mathrm {~m} \mathrm {~s} ^ { - 2 } \quad 4.89 \mathrm {~m} \mathrm {~s} ^ { - 2 }$$

AQA Paper 2 2022 June Q12

1 marks

12 A car is travelling along a straight horizontal road with initial velocity $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$
The car begins to accelerate at a constant rate $a \mathrm {~ms} ^ { - 2 }$ for 5 seconds, to reach a final velocity of $4 u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Express $a$ in terms of $u$.
Circle your answer.
[0pt] [1 mark]
$a = 0.2 u$
$a = 0.4 u$
$a = 0.6 u$
$a = 0.8 u$

AQA Paper 2 2022 June Q13

Show that $$h = 2.5 \sin ^ { 2 } \theta$$ 13 In this question use $g = 9.8 \mathrm {~ms} ^ { - 2 }$ 13
Hence, given that $0 ^ { \circ } \leq \theta \leq 60 ^ { \circ }$, find the maximum value of $h$.
13
Nisha claims that the larger the size of the ball, the greater the maximum vertical height will be. State whether Nisha is correct, giving a reason for your answer.

AQA Paper 2 2022 June Q14

14 A $\pounds 2$ coin has a diameter of 28 mm and a mass of 12 grams. A uniform rod $A B$ of length 160 mm and a fixed load of mass $m$ grams are used to check that a $\pounds 2$ coin has the correct mass. The rod rests with its midpoint on a support.
A $\pounds 2$ coin is placed face down on the rod with part of its curved edge directly above $A$. The fixed load is hung by a light inextensible string from a point directly below the other end of the rod at $B$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-22_195_766_854_639} 14

Given that the rod is horizontal and rests in equilibrium, find $m$.
14
State an assumption you have made about the $\pounds 2$ coin to answer part (a).

AQA Paper 2 2022 June Q15

4 marks

15 A car is moving in a straight line along a horizontal road. The graph below shows how the car's velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ changes with time, $t$ seconds.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-23_509_746_456_648} Over the period $0 \leq t \leq 15$ the car has a total displacement of - 7 metres.
Initially the car has velocity $0 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Find the next time when the velocity of the car is $0 \mathrm {~ms} ^ { - 1 }$
[0pt] [4 marks]

AQA Paper 2 2022 June Q16

16 Two particles, $P$ and $Q$, move in the same horizontal plane. Particle $P$ is initially at rest at the point with position vector $( - 4 \mathbf { i } + 5 \mathbf { j } )$ metres and moves with constant acceleration $( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 2 }$ Particle $Q$ moves in a straight line, passing through the points with position vectors $( \mathbf { i } - \mathbf { j } )$ metres and $( 10 \mathbf { i } + c \mathbf { j } )$ metres.
$P$ and $Q$ are moving along parallel paths.
16

Show that $c = - 13$
16
1. Find an expression for the position vector of $P$ at time $t$ seconds.
  16
(ii) Hence, prove that the paths of $P$ and $Q$ are not collinear.

AQA Paper 2 2022 June Q17

17 A particle is moving such that its position vector, $\mathbf { r }$ metres, at time $t$ seconds, is given by $$\mathbf { r } = \mathrm { e } ^ { t } \cos t \mathbf { i } + \mathrm { e } ^ { t } \sin t \mathbf { j }$$ Show that the magnitude of the acceleration of the particle, $a \mathrm {~ms} ^ { - 2 }$, is given by $$a = 2 \mathrm { e } ^ { t }$$ Fully justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-27_2490_1728_217_141}

AQA Paper 2 2022 June Q18

18 An object, $O$, of mass $m$ kilograms is hanging from a ceiling by two light, inelastic strings of different lengths. The shorter string, of length 0.8 metres, is fixed to the ceiling at $A$.
The longer string, of length 1.2 metres, is fixed to the ceiling at $B$.
This object hangs 0.6 metres directly below the ceiling as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-28_252_940_667_552} 18

Show that the tension in the shorter string is over $30 \%$ more than the tension in the longer string.
18
The tension in the longer string is known to be $2 g$ newtons. Find the value of $m$.
A rough wooden ramp is 10 metres long and is inclined at an angle of $25 ^ { \circ }$ above the horizontal. The bottom of the ramp is at the point $O$. A crate of mass 20 kg is at rest at the point $A$ on the ramp.
The crate is pulled up the ramp using a rope attached to the crate.
Once in motion, the rope remains taut and parallel to the line of greatest slope of the ramp.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-30_252_842_804_598}

AQA Paper 2 2022 June Q19

3 marks

The tension in the rope is 230 N
The crate accelerates up the ramp at $1.2 \mathrm {~ms} ^ { - 2 }$
Find the coefficient of friction between the crate and the ramp.
19
1. The crate takes 3.8 seconds to reach the top of the ramp.
  Find the distance $O A$.
  [0pt] [3 marks]
  19
(ii) Other than air resistance, state one assumption you have made about the crate in answering part (b)(i).
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-32_2492_1721_217_150}

AQA Paper 2 2023 June Q1

1 The graph of $y = a x ^ { 2 } + b x + c$ has roots $x = 2$ and $x = 5$, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-02_905_963_717_625} State the set of values of $x$ which satisfy $$a x ^ { 2 } + b x + c > 0$$ Tick ( $\checkmark$ ) one box. $$\begin{aligned} & \{ x : x < 2 \} \cup \{ x : x > 5 \}
& \{ x : 0 < x < 2 \} \cap \{ x : x > 5 \}
& \{ x : 2 < x < 5 \}
& \{ x : 2 > x > 5 \} \end{aligned}$$ \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-02_118_115_1950_1087}
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AQA Paper 2 2023 June Q2

2 It is given that $$\int _ { 0 } ^ { 6 } \mathrm { f } ( x ) \mathrm { d } x = 20 \text { and } \int _ { 3 } ^ { 6 } \mathrm { f } ( x ) \mathrm { d } x = - 10$$ Find the value of $\int _ { 0 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x$
Circle your answer. $- 30 - 101030$

AQA Paper 2 2023 June Q3

3 A circle has equation $$( x - 5 ) ^ { 2 } + ( y - 13 ) ^ { 2 } = 16$$ Find the radius of the circle. Circle your answer. 41216256

AQA Paper 2 2023 June Q4

4 A curve has equation $$y = \frac { x ^ { 2 } } { 8 } + 4 \sqrt { x }$$ 4

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$
4
The point $P$ with coordinates $( 4,10 )$ lies on the curve.
Find an equation of the tangent to the curve at the point $P$
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4
Show that the curve has no stationary points.

AQA Paper 2 2023 June Q5

5 Ziad is training to become a long-distance swimmer. He trains every day by swimming lengths at his local pool.
The length of the pool is 25 metres.
Each day he increases the number of lengths that he swims by four.
On his first day of training, Ziad swims 10 lengths of the pool.
5

Write down an expression for the number of lengths Ziad will swim on his $n$th day of training. 5
1. Ziad's target is to be able to swim at least 3000 metres in one day.
  Determine the minimum number of days he will need to train to reach his target.
  5
(ii) Ziad's coach claims that when he reaches his target he will have covered a total distance of over 50000 metres. Determine if Ziad's coach is correct.

AQA Paper 2 2023 June Q6

1 marks

6 Victoria, a market researcher, believes the average weekly value, $\pounds V$ million, of online grocery sales in the UK has grown exponentially since 2009. Victoria models the incomplete data, shown in the table, using the formula $$V = a \times b ^ { N }$$ where $N$ is the number of years since 2009 and $a$ and $b$ are constants.

Victoria wishes to determine the values of $a$ and $b$ in her formula.
To do this she plots a graph of $\log _ { 10 } V$ against $N$ and then draws a line of best fit as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-08_757_1040_1169_589} The equation of Victoria's line of best fit is $$\log _ { 10 } V = 0.057 N + 1.76$$ 6
1. Use the equation of Victoria's line of best fit to show that, correct to three significant figures, $a = 57.5$
  [0pt] [1 mark]
  6
(ii) Use the equation of Victoria's line of best fit to find the value of $b$
Give your answer to three significant figures. 6
According to Victoria's model, state the yearly percentage increase in the average weekly value of online grocery sales. 6
1. Use Victoria's model to predict the average weekly value of online grocery sales in 2025.
  6
(ii) Explain why the prediction made in part (c)(i) may be unreliable.

AQA Paper 2 2023 June Q7

7 The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = \sqrt { 10 - 2 x } \text { for } \quad x \leq 5
& \mathrm {~g} ( x ) = \frac { 1 } { x } \quad \text { for } \quad x \neq 0 \end{aligned}$$ The function $h$ has maximum possible domain and is defined by $$\mathrm { h } ( x ) = \operatorname { gf } ( x )$$ 7

Find an expression for $\mathrm { h } ( x )$
7
Find the domain of h
7
Show that $\mathrm { h } ^ { - 1 } ( x ) = 5 - \frac { 1 } { 2 x ^ { 2 } }$
\includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-11_2488_1716_219_153}

AQA Paper 2 2023 June Q8

Given that $\cos \theta \neq \pm 1$, prove the identity $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } \equiv 2 \operatorname { cosec } ^ { 2 } \theta$$ 8
Hence, find the set of values of $A$ for which the equation $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } = A$$ has real solutions.
Fully justify your answer.
8
Given that $\theta$ is obtuse and $$\frac { 1 } { 1 - \cos \theta } + \frac { 1 } { 1 + \cos \theta } = 16$$ find the exact value of $\cot \theta$

AQA Paper 2 2023 June Q9

1 marks

Find the first three terms, in ascending powers of $x$, of the binomial expansion of $$( 1 + x ) ^ { - \frac { 1 } { 2 } }$$ 9
A student substitutes $x = 2$ into the expansion of $( 1 + x ) ^ { - \frac { 1 } { 2 } }$ to find an approximation for $\frac { 1 } { \sqrt { 3 } }$ Explain the mistake in the student's approach.
[0pt] [1 mark] 9
By substituting $x = - \frac { 1 } { 4 }$ in your expansion for $( 1 + x ) ^ { - \frac { 1 } { 2 } }$ find an approximation for $\frac { 1 } { \sqrt { 3 } }$ Give your answer to three significant figures.

AQA Paper 2 2023 June Q10

10
\end{tabular} & Do not write outside the box
\hline \end{tabular} \end{center} 10
Given that $a$ and $b$ are distinct positive numbers, use proof by contradiction to prove that $$\frac { a } { b } + \frac { b } { a } > 2$$ \section*{END OF SECTION A
TURN OVER FOR SECTION B}

AQA Paper 2 2023 June Q11

11 A decoration is hanging freely from a fixed point on a ceiling.
The decoration has a mass of 0.2 kilograms.
The decoration is hanging by a light, inextensible wire.
The wire is 0.1 metres long.
Find the tension in the wire. Circle your answer.
0.02 N
0.02 g N
0.2 N
0.2 g N

AQA Paper 2 2023 June Q12

12 A particle moves in a straight line.
After the first 4 seconds of its motion, the displacement of the particle from its initial position is 0 metres. One of the graphs on the opposite page shows the velocity $v \mathrm {~ms} ^ { - 1 }$ of the particle after time $t$ seconds of its motion. Identify the correct graph.
Tick ( $\checkmark$ ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-19_2249_896_260_484}

AQA Paper 2 2023 June Q13

2 marks

13 A ball falls freely towards the Earth.
The ball passes through two different fixed points $M$ and $N$ before reaching the Earth's surface. At $M$ the ball has velocity $u \mathrm {~ms} ^ { - 1 }$
At $N$ the ball has velocity $3 u \mathrm {~m} \mathrm {~s} ^ { - 1 }$
It can be assumed that:

the motion is due to gravitational force only
the acceleration due to gravity remains constant throughout.

Show that the time taken for the ball to travel from $M$ to $N$ is $\frac { 2 u } { g }$ seconds.
[0pt] [2 marks] 13
Point $M$ is $h$ metres above the Earth. Show that $h > \frac { 4 u ^ { 2 } } { g }$
Fully justify your answer.
The car is moving in a straight line.
The acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ of the car at time $t$ seconds is given by $$a = 3 k t ^ { 2 } - 2 k t + 1$$ where $k$ is a constant.
When $t = 3$ the car has a velocity of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Show that $k = \frac { 1 } { 3 }$

Questions — AQA Paper 2 (140 questions)

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