Questions Paper 2 - A-Level Maths

Find a Cartesian equation of the curve in the form $y ^ { 2 } = \mathrm { f } ( x )$
8
The point $A$ lies on the curve where $t = a$ The tangent to the curve at $A$ is at an angle $\theta$ to a line through $A$ parallel to the $x$-axis. The point $B$ has coordinates $( 1,0 )$
The line $A B$ is at an angle $\phi$ to the $x$-axis.
\includegraphics[max width=\textwidth, alt={}, center]{27339c29-c4a1-480c-b882-930f8dacc7af-14_846_936_678_552} 8
1. By considering the gradient of the curve, show that $$\tan \theta = \frac { 1 } { a }$$ 8
(ii) Find $\tan \phi$ in terms of $a$. 8
(iii) Show that $\tan 2 \theta = \tan \phi$

AQA Paper 2 2020 June Q9

9 A cylinder is to be cut out of the circular face of a solid hemisphere. The cylinder and the hemisphere have the same axis of symmetry.
The cylinder has height $h$ and the hemisphere has a radius of $R$.
\includegraphics[max width=\textwidth, alt={}, center]{27339c29-c4a1-480c-b882-930f8dacc7af-16_467_792_534_625} 9

Show that the volume, $V$, of the cylinder is given by $$V = \pi R ^ { 2 } h - \pi h ^ { 3 }$$ 9
Find the maximum volume of the cylinder in terms of $R$. Fully justify your answer.

AQA Paper 2 2020 June Q11

1 marks

11 A number of forces act on a particle such that the resultant force is $\binom { 6 } { - 3 } \mathrm {~N}$
One of the forces acting on the particle is $\binom { 8 } { - 5 } \mathrm {~N}$
Calculate the total of the other forces acting on the particle.
Circle your answer.
[0pt] [1 mark] $$\binom { 2 } { - 2 } \mathrm {~N} \quad \binom { 14 } { - 8 } \mathrm {~N} \quad \binom { - 2 } { 2 } \mathrm {~N} \quad \binom { - 14 } { 8 } \mathrm {~N}$$

AQA Paper 2 2020 June Q12

12 A particle, $P$, is moving with constant velocity $8 \mathbf { i } - 12 \mathbf { j }$ A second particle, $Q$, is moving with constant velocity $a \mathbf { i } + 9 \mathbf { j }$
$Q$ travels in a direction which is parallel to the motion of $P$.
Find $a$.
Circle your answer.
-6
-5
5
6

AQA Paper 2 2020 June Q13

2 marks

13 A uniform rod, $A B$, has length 7 metres and mass 4 kilograms. The rod rests on a single fixed pivot point, $C$, where $A C = 2$ metres.
A particle of weight $W$ newtons is fixed at $A$, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{27339c29-c4a1-480c-b882-930f8dacc7af-20_321_898_532_571} The system is in equilibrium with the rod resting horizontally.
13

Find $W$, giving your answer in terms of $g$.
[0pt] [2 marks] 13
Explain how you have used the fact that the rod is uniform in part (a).

AQA Paper 2 2020 June Q14

4 marks

14 At time $t$ seconds a particle, $P$, has position vector $\mathbf { r }$ metres, with respect to a fixed origin, such that $$\mathbf { r } = \left( t ^ { 3 } - 5 t ^ { 2 } \right) \mathbf { i } + \left( 8 t - t ^ { 2 } \right) \mathbf { j }$$ 14

Find the exact speed of $P$ when $t = 2$
[0pt] [4 marks]
14
Bella claims that the magnitude of acceleration of $P$ will never be zero.
Determine whether Bella's claim is correct.
Fully justify your answer.

AQA Paper 2 2020 June Q15

1 marks

15 A particle is moving in a straight line with velocity $\mathrm { vm } \mathrm { s } ^ { - 1 }$ at time $t$ seconds as shown by the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{27339c29-c4a1-480c-b882-930f8dacc7af-22_1155_1189_424_424} 15

Use the trapezium rule with four strips to estimate the distance travelled by the particle during the time period $20 \leq t \leq 100$
15
Over the same time period, the curve can be very closely modelled by a particular quadratic. Explain how you could find an alternative estimate using this quadratic.
[0pt] [1 mark]

AQA Paper 2 2020 June Q16

16 Two particles $A$ and $B$ are released from rest from different starting points above a horizontal surface.
$A$ is released from a height of $h$ metres.
$B$ is released at a time $t$ seconds after $A$ from a height of $k h$ metres, where $0 < k < 1$ Both particles land on the surface 5 seconds after $A$ was released. Assuming any resistance forces may be ignored, prove that $$t = 5 ( 1 - \sqrt { k } )$$ Fully justify your answer.

AQA Paper 2 2020 June Q17

17 A ball is projected forward from a fixed point, $P$, on a horizontal surface with an initial speed $u \mathrm {~ms} ^ { - 1 }$, at an acute angle $\theta$ above the horizontal. The ball needs to first land at a point at least $d$ metres away from $P$.
You may assume the ball may be modelled as a particle and that air resistance may be ignored. Show that $$\sin 2 \theta \geq \frac { d g } { u ^ { 2 } }$$

AQA Paper 2 2020 June Q18

18 Block $A$, of mass 0.2 kg , lies at rest on a rough plane. The plane is inclined at an angle $\theta$ to the horizontal, such that $\tan \theta = \frac { 7 } { 24 }$
A light inextensible string is attached to $A$ and runs parallel to the line of greatest slope until it passes over a smooth fixed pulley at the top of the slope. The other end of this string is attached to particle $B$, of mass 2 kg , which is held at rest so that the string is taut, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{27339c29-c4a1-480c-b882-930f8dacc7af-28_400_1022_731_511} 18

$B$ is released from rest so that it begins to move vertically downwards with an acceleration of $\frac { 543 } { 625 } \mathrm {~g} \mathrm {~m} \mathrm {~s} ^ { - 2 }$ Show that the coefficient of friction between $A$ and the surface of the inclined plane is 0.17
18
In this question use $g = 9.81 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ When $A$ reaches a speed of $0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the string breaks.
18
1. Find the distance travelled by $A$ after the string breaks until first coming to rest.
  18
(ii) State an assumption that could affect the validity of your answer to part (b)(i).
\includegraphics[max width=\textwidth, alt={}, center]{27339c29-c4a1-480c-b882-930f8dacc7af-31_2488_1716_219_153}

AQA Paper 2 2020 June Q19

19 A particle moves so that its acceleration, $a \mathrm {~ms} ^ { - 2 }$, at time $t$ seconds may be modelled in terms of its velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, as $$a = - 0.1 v ^ { 2 }$$ The initial velocity of the particle is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
19

By first forming a suitable differential equation, show that $$v = \frac { 20 } { 5 + 2 t }$$ 19
Find the acceleration of the particle when $t = 5.5$

AQA Paper 2 2021 June Q1

1 Four possible sketches of $y = a x ^ { 2 } + b x + c$ are shown below.
Given $b ^ { 2 } - 4 a c = 0$ and $a , b$ and $c$ are non-zero constants, which sketch is the only one that could possibly be correct? Tick ( $\checkmark$ ) one box. A
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_383_303_995_550}
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_113_111_1128_1009} B
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_113_111_1562_1009} C
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_223_300_1868_548}
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_108_109_2001_1009} D
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-02_388_301_2305_549}
□

AQA Paper 2 2021 June Q2

2 A curve has equation $y = \mathrm { f } ( x )$ The curve has a point of inflection at $x = 7$
It is given that $\mathrm { f } ^ { \prime } ( 7 ) = a$ and $\mathrm { f } ^ { \prime \prime } ( 7 ) = b$, where $a$ and $b$ are real numbers. Identify which one of the statements below must be true.
Circle your answer.
$\mathrm { f } ^ { \prime } ( 7 ) \neq 0$
$\mathrm { f } ^ { \prime } ( 7 ) = 0$
$\mathrm { f } ^ { \prime \prime } ( 7 ) \neq 0$
$\mathrm { f } ^ { \prime \prime } ( 7 ) = 0$

AQA Paper 2 2021 June Q3

3 A sequence is defined by $$u _ { 1 } = a \text { and } u _ { n + 1 } = - 1 \times u _ { n }$$ Find $\sum _ { n = 1 } ^ { 95 } u _ { n }$
Circle your answer.
$- a$
0
$a$
95a

AQA Paper 2 2021 June Q4

On Figure 1 add a sketch of the graph of $$y = | 3 x - 6 |$$ 4
Find the coordinates of the points of intersection of the two graphs.
Fully justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-05_2488_1716_219_153}

AQA Paper 2 2021 June Q5

5 Express $$\frac { 5 ( x - 3 ) } { ( 2 x - 11 ) ( 4 - 3 x ) }$$ in the form $$\frac { A } { ( 2 x - 11 ) } + \frac { B } { ( 4 - 3 x ) }$$ where $A$ and $B$ are integers.

AQA Paper 2 2021 June Q6

6 Show that the solution of the equation $$5 ^ { x } = 3 ^ { x + 4 }$$ can be written as $$x = \frac { \ln 81 } { \ln 5 - \ln 3 }$$ Fully justify your answer.

AQA Paper 2 2021 June Q7

7 A circle has equation $$x ^ { 2 } + y ^ { 2 } - 6 x - 8 y = p$$ 7

1. State the coordinates of the centre of the circle.
  7
(ii) Find the radius of the circle in terms of $p$.
7
The circle intersects the coordinate axes at exactly three points. Find the two possible values of $p$.

AQA Paper 2 2021 June Q8

8 Kai is proving that $n ^ { 3 } - n$ is a multiple of 3 for all positive integer values of $n$. Kai begins a proof by exhaustion.
Step 1 $$n ^ { 3 } - n = n \left( n ^ { 2 } - 1 \right)$$ Step 2 When $n = 3 m$, where $m$ is a $n ^ { 3 } - n = 3 m \left( 9 m ^ { 2 } - 1 \right)$ non-negative integer which is a multiple of 3 Step 3 When $n = 3 m + 1$, $$\begin{aligned} & n ^ { 3 } - n = ( 3 m + 1 ) \left( ( 3 m + 1 ) ^ { 2 } - 1 \right)
& = ( 3 m + 1 ) \left( 9 m ^ { 2 } \right)
& = 3 ( 3 m + 1 ) \left( 3 m ^ { 2 } \right) \end{aligned}$$ Step 5 Therefore $n ^ { 3 } - n$ is a multiple of 3 for all positive integer values of $n$ 8

Explain the two mistakes that Kai has made after Step 3. Step 4 \section*{which is a multiple of 3
which is a multiple of 3}
\includegraphics[max width=\textwidth, alt={}]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-10_67_134_964_230}
别
all positive integer values of $n$ \section*{a} \includegraphics[max width=\textwidth, alt={}]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-10_58_49_1037_370} 墐 pount $\_\_\_\_$
$\_\_\_\_$ " $\_\_\_\_$
8
Correct Kai's argument from Step 4 onwards.

AQA Paper 2 2021 June Q9

9 A robotic arm which is attached to a flat surface at the origin $O$, is used to draw a graphic design. The arm is made from two rods $O P$ and $P Q$, each of length $d$, which are joined at $P$.
A pen is attached to the arm at $Q$.
The coordinates of the pen are controlled by adjusting the angle $O P Q$ and the angle $\theta$ between $O P$ and the $x$-axis. For this particular design the pen is made to move so that the two angles are always equal to each other with $0 \leq \theta \leq \frac { \pi } { 2 }$ as shown in Figure 2. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-12_805_867_989_584}

\end{figure} 9

Show that the $x$-coordinate of the pen can be modelled by the equation $$x = d \left( \cos \theta + \sin \left( 2 \theta - \frac { \pi } { 2 } \right) \right)$$ 9
Hence, show that $$x = d \left( 1 + \cos \theta - 2 \cos ^ { 2 } \theta \right)$$ 9
It can be shown that $$x = \frac { 9 d } { 8 } - d \left( \cos \theta - \frac { 1 } { 4 } \right) ^ { 2 }$$ State the greatest possible value of $x$ and the corresponding value of $\cos \theta$
9
Figure 3 below shows the arm when the $x$-coordinate is at its greatest possible value. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-14_570_773_456_630}
\end{figure} Find, in terms of $d$, the exact distance $O Q$.
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-15_2488_1716_219_153}

AQA Paper 2 2021 June Q10

8 marks

10 The function h is defined by $$\mathrm { h } ( x ) = \frac { \sqrt { x } } { x - 3 }$$ where $h$ has its maximum possible domain.
10

Find the domain of h .
Give your answer using set notation. 10
Alice correctly calculates $$h ( 1 ) = - 0.5 \text { and } h ( 4 ) = 2$$ She then argues that since there is a change of sign there must be a value of $x$ in the interval $1 < x < 4$ that gives $\mathrm { h } ( x ) = 0$ Explain the error in Alice's argument.
[0pt] [2 marks]
10
By considering any turning points of h , determine whether h has an inverse function. Fully justify your answer.
[0pt] [6 marks]

AQA Paper 2 2021 June Q11

11 A particle's displacement, $r$ metres, with respect to time, $t$ seconds, is defined by the equation $$r = 3 \mathrm { e } ^ { 0.5 t }$$ Find an expression for the velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of the particle at time $t$ seconds.
Circle your answer.
$v = 1.5 \mathrm { e } ^ { 0.5 t }$
$v = 6 \mathrm { e } ^ { 0.5 t }$
$v = 1.5 t \mathrm { e } ^ { 0.5 t }$
$v = 6 t e ^ { 0.5 t }$

AQA Paper 2 2021 June Q12

1 marks

12 A particle has a speed of $6 \mathrm {~ms} ^ { - 1 }$ in a direction relative to unit vectors $\mathbf { i }$ and $\mathbf { j }$ as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-18_307_542_1528_749} The velocity of this particle can be expressed as a vector $\left[ \begin{array} { l } v _ { 1 }
v _ { 2 } \end{array} \right] \mathrm { ms } ^ { - 1 }$
Find the correct expression for $v _ { 2 }$
Circle your answer.
[0pt] [1 mark]
$v _ { 2 } = 6 \cos 30 ^ { \circ }$
$v _ { 2 } = 6 \sin 30 ^ { \circ }$
$v _ { 2 } = - 6 \sin 30 ^ { \circ }$
$v _ { 2 } = - 6 \cos 30 ^ { \circ }$

AQA Paper 2 2021 June Q13

13 A vehicle, of total mass 1200 kg , is travelling along a straight, horizontal road at a constant speed of $13 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ This vehicle begins to accelerate at a constant rate.
After 40 metres it reaches a speed of $17 \mathrm {~ms} ^ { - 1 }$
Find the resultant force acting on the vehicle during the period of acceleration.

AQA Paper 2 2021 June Q14

14 A motorised scooter is travelling along a straight path with velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ over time $t$ seconds as shown by the following graph.
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-20_1120_1134_420_452} Noosha says that, in the period $\mathbf { 1 2 } \leq \boldsymbol { t } \leq \mathbf { 3 6 }$, the scooter travels approximately 130 metres. Determine if Noosha is correct, showing clearly any calculations you have used.

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