Questions - AQA - A-Level Maths

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

1 marks Easy -1.8

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

1 marks Easy -1.8

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

1 marks Easy -1.2

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

5 marks Easy -1.2

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

8 marks Moderate -0.3

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

9 marks Easy -1.2

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

16 marks Standard +0.8

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

9 marks Standard +0.3

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 Easy -1.8

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 Easy -2.0

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 marks Standard +0.3

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 marks Standard +0.3

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 Standard +0.3

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

8 marks Standard +0.3

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

7 marks Standard +0.3

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}

AQA Paper 2 2020 June Q8

10 marks Standard +0.3

8 The curve defined by the parametric equations $$x = t ^ { 2 } \text { and } y = 2 t \quad - \sqrt { 2 } \leq t \leq \sqrt { 2 }$$ is shown in Figure 1 below. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{27339c29-c4a1-480c-b882-930f8dacc7af-13_1063_1022_607_507}

\end{figure} 8

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

10 marks Standard +0.8

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 Easy -1.8

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

1 marks Easy -1.2

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

3 marks Moderate -0.8

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

7 marks Standard +0.3

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

5 marks Moderate -0.8

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

5 marks Standard +0.3

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

6 marks Standard +0.3

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 $$\sin 2 \theta \geq \frac { d g } { u ^ { 2 } }$$

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