Questions Paper 2 - A-Level Maths

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AQA Paper 2 2018 June Q16

$\quad$ 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

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 Q7

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

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.

AQA Paper 2 2019 June Q9

2 marks

Show that the first two terms of the binomial expansion of $\sqrt { 4 - 2 x ^ { 2 } }$ are $$2 - \frac { x ^ { 2 } } { 2 }$$ 9
State the range of values of $x$ for which the expansion found in part (a) is valid.
[0pt] [2 marks]
9
Hence, find an approximation for $$\int _ { 0 } ^ { 0.4 } \sqrt { \cos x } \mathrm {~d} x$$ giving your answer to five decimal places.
Fully justify your answer.
9
A student decides to use this method to find an approximation for $$\int _ { 0 } ^ { 1.4 } \sqrt { \cos x } \mathrm {~d} x$$ Explain why this may not be a suitable method.

AQA Paper 2 2019 June Q10

10 The diagram below shows a velocity-time graph for a particle moving with velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t$ seconds.
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-16_684_1653_847_182} Which statement is correct? Tick ( $\checkmark$ ) one box. The particle was stationary for $9 \leq t \leq 12$ □ The particle was decelerating for $12 \leq t \leq 20$ □ The particle had a displacement of zero when $t = 6$ □ The particle's speed when $t = 4$ was $- 12 \mathrm {~ms} ^ { - 1 }$ □

AQA Paper 2 2019 June Q11

11 A wooden crate rests on a rough horizontal surface.
The coefficient of friction between the crate and the surface is 0.6 A forward force acts on the crate, parallel to the surface.
When this force is 600 N , the crate is on the point of moving.
Find the weight of the crate.
Circle your answer. $1000 \mathrm {~N} 360 \mathrm {~N} \quad 100 \mathrm {~kg} \quad 36 \mathrm {~kg}$

AQA Paper 2 2019 June Q12

12 A particle, under the action of two constant forces, is moving across a perfectly smooth horizontal surface at a constant speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ The first force acting on the particle is $( 400 \mathbf { i } + 180 \mathbf { j } ) \mathbf { N }$.
The second force acting on the particle is $( p \mathbf { i } - 180 \mathbf { j } ) \mathrm { N }$.
Find the value of $p$.
Circle your answer. $- 400 - 390390400$

AQA Paper 2 2019 June Q13

13 In a school experiment, a particle, of mass $m$ kilograms, is released from rest at a point $h$ metres above the ground. At the instant it reaches the ground, the particle has velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$
13

Show that $$v = \sqrt { 2 g h }$$ 13
A student correctly used $h = 18$ and measured $v$ as 20
The student's teacher claims that the machine measuring the velocity must have been faulty. Determine if the teacher's claim is correct. Fully justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-19_2488_1716_219_153}

AQA Paper 2 2019 June Q14

14 A metal rod, of mass $m$ kilograms and length 20 cm , lies at rest on a horizontal shelf. end of the rod, $B$, extends 6 cm beyond the edge of the shelf, $A$, as shown in the The end of the rod, $B$, extends 6 cm beyond the edge of the shelf, $A$, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-20_337_757_488_639} 14

The rod is in equilibrium when an object of mass 0.28 kilograms hangs from the midpoint of $A B$. Show that $m = 0.21$ 14
The object of mass 0.28 kilograms is removed. A number, $n$, of identical objects, each of mass 0.048 kg , are hung from the rod all at a distance of 1 cm from $B$. Find the maximum value of $n$ such that the rod remains horizontal.
14
State one assumption you have made about the rod.

AQA Paper 2 2019 June Q15

15 Four buoys on the surface of a large, calm lake are located at $A , B , C$ and $D$ with position vectors given by $$\overrightarrow { O A } = \left[ \begin{array} { l } 410
710 \end{array} \right] , \overrightarrow { O B } = \left[ \begin{array} { r } - 210
530 \end{array} \right] , \overrightarrow { O C } = \left[ \begin{array} { l } - 340
- 310 \end{array} \right] \text { and } \overrightarrow { O D } = \left[ \begin{array} { c } 590
- 40 \end{array} \right]$$ All values are in metres.
15

Prove that the quadrilateral $A B C D$ is a trapezium but not a parallelogram.
15
A speed boat travels directly from $B$ to $C$ at a constant speed in 50 seconds. Find the speed of the boat between $B$ and $C$.

AQA Paper 2 2019 June Q16

6 marks

16 An elite athlete runs in a straight line to complete a 100-metre race. During the race, the athlete's velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, may be modelled by $$v = 11.71 - 11.68 \mathrm { e } ^ { - 0.9 t } - 0.03 \mathrm { e } ^ { 0.3 t }$$ where $t$ is the time, in seconds, after the starting pistol is fired.
16

Find the maximum value of $v$, giving your answer to one decimal place.
Fully justify your answer.
16
Find an expression for the distance run in terms of $t$.
[0pt] [6 marks]
16
The athlete's actual time for this race is 9.8 seconds. Comment on the accuracy of the model.
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-27_2488_1719_219_150}

AQA Paper 2 2019 June Q17

17 Lizzie is sat securely on a wooden sledge.
The combined mass of Lizzie and the sledge is $M$ kilograms.
The sledge is being pulled forward in a straight line along a horizontal surface by means of a light inextensible rope, which is attached to the front of the sledge. This rope stays inclined at an acute angle $\theta$ above the horizontal and remains taut as the sledge moves forward.
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-28_282_786_726_756} The sledge remains in contact with the surface throughout.
The coefficient of friction between the sledge and the surface is $\mu$ and there are no other resistance forces. Lizzie and the sledge move forward with constant acceleration, $a \mathrm {~ms} ^ { - 2 }$
The tension in the rope is a constant $T$ Newtons.
17

Show that $$T = \frac { M ( a + \mu g ) } { \cos \theta + \mu \sin \theta }$$ 17
It is known that when $M = 30 , \theta = 30 ^ { \circ }$, and $T = 40$, the sledge remains at rest.
Lizzie uses these values with the relationship formed in part (a) to find the value for $\mu$ Explain why her value for $\mu$ may be incorrect.
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-30_2488_1719_219_150}
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-31_2488_1719_219_150}
\includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-32_2496_1721_214_148}

AQA Paper 2 2020 June Q1

1 marks

1 Which one of these functions is decreasing for all real values of $x$ ?
Circle your answer.
[0pt] [1 mark] $$\mathrm { f } ( x ) = \mathrm { e } ^ { x } \quad \mathrm { f } ( x ) = - \mathrm { e } ^ { 1 - x } \quad \mathrm { f } ( x ) = - \mathrm { e } ^ { x - 1 } \quad \mathrm { f } ( x ) = - \mathrm { e } ^ { - x }$$

AQA Paper 2 2020 June Q2

1 marks

2 Which one of the following equations has no real solutions?
Tick ( $\checkmark$ ) one box.
[0pt] [1 mark] $$\begin{aligned} & \cot x = 0
& \ln x = 0
& | x + 1 | = 0
& \sec x = 0 \end{aligned}$$ □
□
□
□

AQA Paper 2 2020 June Q3

3 Find the coefficient of $x ^ { 2 }$ in the binomial expansion of $\left( 2 x - \frac { 3 } { x } \right) ^ { 8 }$

AQA Paper 2 2020 June Q4

4 Using small angle approximations, show that for small, non-zero, values of $x$ $$\frac { x \tan 5 x } { \cos 4 x - 1 } \approx A$$ where $A$ is a constant to be determined.
\includegraphics[max width=\textwidth, alt={}, center]{27339c29-c4a1-480c-b882-930f8dacc7af-05_2488_1716_219_153}

AQA Paper 2 2020 June Q5

6 marks

5 Use integration by substitution to show that $$\int _ { - \frac { 1 } { 4 } } ^ { 6 } x \sqrt { 4 x + 1 } \mathrm {~d} x = \frac { 875 } { 12 }$$ Fully justify your answer.
[0pt] [6 marks]

AQA Paper 2 2020 June Q6

6 The line $L$ has equation $$5 y + 12 x = 298$$ A circle, $C$, has centre $( 7,9 )$
$L$ is a tangent to $C$.
6

Find the coordinates of the point of intersection of $L$ and $C$.
Fully justify your answer.
Find the equation of $C$. 6
Find the equation of $C$.
$7 \quad a$ and $b$ are two positive irrational numbers. The sum of $a$ and $b$ is rational. The product of $a$ and $b$ is rational.
Caroline is trying to prove $\frac { 1 } { a } + \frac { 1 } { b }$ is rational.
Here is her proof:
Step $1 \quad \frac { 1 } { a } + \frac { 1 } { b } = \frac { 2 } { a + b }$
Step 22 is rational and $a + b$ is non-zero and rational.
Step 3 Therefore $\frac { 2 } { a + b }$ is rational.
Step 4 Hence $\frac { 1 } { a } + \frac { 1 } { b }$ is rational.

AQA Paper 2 2020 June Q7

4 marks

1. Identify Caroline's mistake.
  7

(ii) Write down a correct version of the proof.

Prove by contradiction that the difference of any rational number and any irrational number is irrational.

[4 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

\includegraphics[max width=\textwidth, alt={}]{27339c29-c4a1-480c-b882-930f8dacc7af-12_2492_1722_217_150}

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

Questions Paper 2 (402 questions)

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