Questions - AQA - A-Level Maths

$\pounds 15$ per $\mathrm { m } ^ { 2 }$ for the base
$\pounds 8$ per $\mathrm { m } ^ { 2 }$ for the sides.

Modelling the sides and base of the fish tank as laminae, use calculus to find the height of the tank for which the overall cost of the materials has its minimum value. Fully justify your answer.
[0pt] [8 marks] 14
1. In reality, the thickness of the base and sides of the tank is 2.5 cm
  Briefly explain how you would refine your modelling to take account of the thickness of the sides and base of the tank of the tank.
  [0pt] [1 mark]
  LIH
  L
  LL 14
(ii) How would your refinement affect your answer to part (a)?
[0pt] [1 mark]

AQA Paper 1 Specimen Q15

8 marks Standard +0.3

15 The height $x$ metres, of a column of water in a fountain display satisfies the differential equation $\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 8 \sin 2 t } { 3 \sqrt { x } }$, where $t$ is the time in seconds after the display begins. 15

Solve the differential equation, given that initially the column of water has zero height.
Express your answer in the form $x = \mathrm { f } ( t )$
[0pt] [7 marks]
15
Find the maximum height of the column of water, giving your answer to the nearest cm .
[0pt] [1 mark]

AQA Paper 1 Specimen Q16

5 marks Challenging +1.8

16 A student argues that when a rational number is multiplied by an irrational number the result will always be an irrational number. 16

Identify the rational number for which the student's argument is not true. 16
Prove that the student is right for all rational numbers other than the one you have identified in part (a).
[0pt] [4 marks]
$17 \quad \mathrm { f } ( x ) = \sin x$
Using differentiation from first principles find the exact value of $f ^ { \prime } \left( \frac { \pi } { 6 } \right)$
Fully justify your answer.
[0pt] [6 marks] \section*{DO NOT WRITE ON THIS PAGE} ANSWER IN THE SPACES PROVIDED

AQA Paper 2 2018 June Q1

1 marks Easy -1.8

1 Which of these statements is correct? Tick one box. $$\begin{aligned} & x = 2 \Rightarrow x ^ { 2 } = 4 \\ & x ^ { 2 } = 4 \Rightarrow x = 2 \\ & x ^ { 2 } = 4 \Leftrightarrow x = 2 \\ & x ^ { 2 } = 4 \Rightarrow x = - 2 \end{aligned}$$

AQA Paper 2 2018 June Q2

1 marks Easy -1.8

2 Find the coefficient of $x ^ { 2 }$ in the expansion of $( 1 + 2 x ) ^ { 7 }$
Circle your answer. 4242184
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-03_1442_1168_219_365} Find the total shaded area. Circle your answer.
-68 60686128

AQA Paper 2 2018 June Q4

6 marks Moderate -0.3

4 A curve, $C$, has equation $y = x ^ { 2 } - 6 x + k$, where $k$ is a constant. The equation $x ^ { 2 } - 6 x + k = 0$ has two distinct positive roots. 4

Sketch $C$ on the axes below.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-04_1013_1016_534_513} 4
Find the range of possible values for $k$. Fully justify your answer.
\begin{center} \begin{tabular}{ | l | } \hline

AQA Paper 2 2018 June Q5

2 marks Easy -2.0

5 Prove that 23 is a prime number.
[0pt] [2 marks]
\end{tabular} \end{center}

AQA Paper 2 2018 June Q6

7 marks Standard +0.8

6 Find the coordinates of the stationary point of the curve with equation $$( x + y - 2 ) ^ { 2 } = \mathrm { e } ^ { y } - 1$$ $7 \quad$ A function f has domain $\mathbb { R }$ and range $\{ y \in \mathbb { R } : y \geq \mathrm { e } \}$ The graph of $y = \mathrm { f } ( x )$ is shown.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-08_922_1108_447_466} The gradient of the curve at the point $( x , y )$ is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x - 1 ) \mathrm { e } ^ { x }$
Find an expression for $\mathrm { f } ( x )$.
Fully justify your answer.

AQA Paper 2 2018 June Q8

10 marks Challenging +1.2

Determine a sequence of transformations which maps the graph of $y = \sin x$ onto the graph of $y = \sqrt { 3 } \sin x - 3 \cos x + 4$ Fully justify your answer.
8
(ii) Find the greatest value of $\frac { 1 } { \sqrt { 3 } \sin x - 3 \cos x + 4 }$

AQA Paper 2 2018 June Q9

22 marks Standard +0.3

9 A market trader notices that daily sales are dependent on two variables:
number of hours, $t$, after the stall opens
total sales, $x$, in pounds since the stall opened.
The trader models the rate of sales as directly proportional to $\frac { 8 - t } { x }$
After two hours the rate of sales is $\pounds 72$ per hour and total sales are $\pounds 336$
9

Show that $$x \frac { \mathrm {~d} x } { \mathrm {~d} t } = 4032 ( 8 - t )$$ 9
Hence, show that $$x ^ { 2 } = 4032 t ( 16 - t )$$ $\mathbf { 9 }$ (c) The stall opens at 09.30. 9
1. The trader closes the stall when the rate of sales falls below $\pounds 24$ per hour.
  Using the results in parts (a) and (b), calculate the earliest time that the trader closes the stall.
  9
(ii) Explain why the model used by the trader is not valid at 09.30.

AQA Paper 2 2018 June Q10

1 marks Easy -1.8

10 A garden snail moves in a straight line from rest to $1.28 \mathrm {~cm} \mathrm {~s} ^ { - 1 }$, with a constant acceleration in 1.8 seconds. Find the acceleration of the snail. Circle your answer.
$2.30 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
$0.71 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
$0.0071 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
$0.023 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

AQA Paper 2 2018 June Q11

1 marks Easy -1.8

11 A uniform rod, $A B$, has length 4 metres.
The rod is resting on a support at its midpoint $C$.
A particle of mass 4 kg is placed 0.6 metres to the left of $C$.
Another particle of mass 1.5 kg is placed $x$ metres to the right of $C$, as shown.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-16_277_908_1521_568} The rod is balanced in equilibrium at $C$.
Find $x$. Circle your answer.
[0pt] [1 mark]
$1.8 \mathrm {~m} \quad 1.5 \mathrm {~m} \quad 1.75 \mathrm {~m} \quad 1.6 \mathrm {~m}$

AQA Paper 2 2018 June Q12

5 marks Moderate -0.3

12 The graph below shows the velocity of an object moving in a straight line over a 20 second journey.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-17_579_1682_406_169} 12

Find the maximum magnitude of the acceleration of the object. 12
The object is at its starting position at times $0 , t _ { 1 }$ and $t _ { 2 }$ seconds.
Find $t _ { 1 }$ and $t _ { 2 }$

AQA Paper 2 2018 June Q13

8 marks Standard +0.3

13 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A boy attempts to move a wooden crate of mass 20 kg along horizontal ground. The coefficient of friction between the crate and the ground is 0.85 13

The boy applies a horizontal force of 150 N . Show that the crate remains stationary.
13
Instead, the boy uses a handle to pull the crate forward. He exerts a force of 150 N , at an angle of $15 ^ { \circ }$ above the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-19_244_915_408_561} Determine whether the crate remains stationary.
Fully justify your answer.

AQA Paper 2 2018 June Q14

6 marks Moderate -0.3

14 A quadrilateral has vertices $A , B , C$ and $D$ with position vectors given by $$\overrightarrow { O A } = \left[ \begin{array} { l } 3 \\ 5 \\ 1 \end{array} \right] , \overrightarrow { O B } = \left[ \begin{array} { r } - 1 \\ 2 \\ 7 \end{array} \right] , \overrightarrow { O C } = \left[ \begin{array} { l } 0 \\ 7 \\ 6 \end{array} \right] \text { and } \overrightarrow { O D } = \left[ \begin{array} { r } 4 \\ 10 \\ 0 \end{array} \right]$$ 14

Write down the vector $\overrightarrow { A B }$ 14
Show that $A B C D$ is a parallelogram, but not a rhombus.

AQA Paper 2 2018 June Q15

9 marks Standard +0.3

15 A driver is road-testing two minibuses, $A$ and $B$, for a taxi company. The performance of each minibus along a straight track is compared.
A flag is dropped to indicate the start of the test.
Each minibus starts from rest.
The acceleration in $\mathrm { ms } ^ { - 2 }$ of each minibus is modelled as a function of time, $t$ seconds, after the flag is dropped: The acceleration of $\mathrm { A } = 0.138 t ^ { 2 }$
The acceleration of $\mathrm { B } = 0.024 t ^ { 3 }$
15

Find the time taken for A to travel 100 metres.
Give your answer to four significant figures.
15
The company decides to buy the minibus which travels 100 metres in the shortest time. Determine which minibus should be bought.
15
The models assume that both minibuses start moving immediately when $t = 0$ In light of this, explain why the company may, in reality, make the wrong decision.
A particle is projected with an initial speed $u$, at an angle of $35 ^ { \circ }$ above the horizontal.
It lands at a point 10 metres vertically below its starting position.
The particle takes 1.5 seconds to reach the highest point of its trajectory.

AQA Paper 2 2018 June Q16

6 marks Moderate -0.3

$\quad$ Find $u$. 16 In this question use $g = 9.81 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ In this question use $\boldsymbol { g } = 9.81 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ 16
Find the total time that the particle is in flight.

AQA Paper 2 2018 June Q17

14 marks Moderate -0.3

17 A buggy is pulling a roller-skater, in a straight line along a horizontal road, by means of a connecting rope as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-24_240_1006_402_516} The combined mass of the buggy and driver is 410 kg
A driving force of 300 N and a total resistance force of 140 N act on the buggy.
The mass of the roller-skater is 72 kg
A total resistance force of $R$ newtons acts on the roller-skater.
The buggy and the roller-skater have an acceleration of $0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
17

1. Find $R$.
  
  17
  17
  (ii) Find the tension in the rope.
  17
2. The roller-skater releases the rope at a point $A$, when she reaches a speed of $6 \mathrm {~ms} ^ { - 1 }$ She continues to move forward, experiencing the same resistance force.
  The driver notices a change in motion of the buggy, and brings it to rest at a distance of 20 m from $A$. 17
3. 1. Determine whether the roller-skater will stop before reaching the stationary buggy.
    Fully justify your answer.
    17
4. (ii) Explain the change in motion that the driver noticed.

AQA Paper 2 2019 June Q1

1 marks Easy -1.8

1 Identify the graph of $y = 1 - | x + 2 |$ from the options below.
Tick ( $\checkmark$ ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_389_526_845_500}
B
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_362_442_1279_525}
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_113_116_977_1107}
C
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_496_704_1688_523}
D
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_474_686_2211_534}

AQA Paper 2 2019 June Q2

1 marks Easy -1.8

2 Simplify $\sqrt { a ^ { \frac { 2 } { 3 } } \times a ^ { \frac { 2 } { 5 } } }$
Circle your answer.
$a ^ { \frac { 2 } { 15 } }$
$a ^ { \frac { 4 } { 15 } }$
$a ^ { \frac { 8 } { 15 } }$
$a ^ { \frac { 16 } { 15 } }$

AQA Paper 2 2019 June Q3

1 marks Easy -1.8

3 Each of these functions has domain $x \in \mathbb { R }$
Which function does not have an inverse?
Circle your answer. $$\mathrm { f } ( x ) = x ^ { 3 } \quad \mathrm { f } ( x ) = 2 x + 1 \quad \mathrm { f } ( x ) = x ^ { 2 } \quad \mathrm { f } ( x ) = \mathrm { e } ^ { x }$$

AQA Paper 2 2019 June Q5

7 marks Moderate -0.3

5 Solve the differential equation $$\frac { \mathrm { d } t } { \mathrm {~d} x } = \frac { \ln x } { x ^ { 2 } t } \quad \text { for } x > 0$$ given $x = 1$ when $t = 2$
Write your answer in the form $t ^ { 2 } = \mathrm { f } ( x )$

AQA Paper 2 2019 June Q6

6 marks Standard +0.8

6 A curve has equation $$y = a \sin x + b \cos x$$ where $a$ and $b$ are constants. The maximum value of $y$ is 4 and the curve passes through the point $\left( \frac { \pi } { 3 } , 2 \sqrt { 3 } \right)$ as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-06_746_696_719_671} Find the exact values of $a$ and $b$.

AQA Paper 2 2019 June Q7

10 marks Standard +0.8

Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of $x ^ { 3 }$
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-08_997_1004_406_518} 7
The function $\mathrm { f } ( x )$ is defined by $$\mathrm { f } ( x ) = x ^ { 3 } + 3 p x ^ { 2 } + q$$ where $p$ and $q$ are constants and $p > 0$
7
1. Show that there is a turning point where the curve crosses the $y$-axis.
  7
(ii) The equation $\mathrm { f } ( x ) = 0$ has three distinct real roots. By considering the positions of the turning points find, in terms of $p$, the range of possible values of $q$.

AQA Paper 2 2019 June Q8

11 marks Standard +0.3

8 Theresa bought a house on 2 January 1970 for $\pounds 8000$. The house was valued by a local estate agent on the same date every 10 years up to 2010. The valuations are shown in the following table.

Year	1970	1980	1990	2000	2010
Valuation price	$\pounds 8000$	$\pounds 19000$	$\pounds 36000$	$\pounds 82000$	$\pounds 205000$

The valuation price of the house can be modelled by the equation $$V = p q ^ { t }$$ where $V$ pounds is the valuation price $t$ years after 2 January 1970 and $p$ and $q$ are constants. 8

Show that $V = p q ^ { t }$ can be written as $\log _ { 10 } V = \log _ { 10 } p + t \log _ { 10 } q$
8
The values in the table of $\log _ { 10 } V$ against $t$ have been plotted and a line of best fit has been drawn on the graph below.
$t$ 0 10 20 30 40
$\log _ { 10 } V$ 3.90 4.28 4.56 4.91 5.31

\includegraphics[max width=\textwidth, alt={}]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-11_1927_1207_625_338}
Using the given line of best fit, find estimates for the values of $p$ and $q$. Give your answers correct to three significant figures.
8
Determine the year in which Theresa's house will first be worth half a million pounds. 8
Explain whether your answer to part (c) is likely to be reliable.

Year	1970	1980	1990	2000	2010
Valuation price	\(\pounds 8000\)	\(\pounds 19000\)	\(\pounds 36000\)	\(\pounds 82000\)	\(\pounds 205000\)

\(t\)	0	10	20	30	40
\(\log _ { 10 } V\)	3.90	4.28	4.56	4.91	5.31

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