Questions - AQA Paper 2 - A-Level Maths

The reaction force on the plank at $X$ is $4 g$ newtons.
17
1. Show that the mass of the plank is 9.6 kilograms.
  17
(ii) Find the reaction force, in terms of $g$, on the plank at $Y$
17
The support at $Y$ is moved so that the distance $Q Y = 1.4$ metres. The plank remains horizontally at rest in equilibrium.
It is claimed that the reaction force at $Y$ remains unchanged.
Explain, with a reason, whether this claim is correct.

AQA Paper 2 2023 June Q18

18 In this question $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors representing due east and due north respectively. A particle, $T$, is moving on a plane at a constant speed.
The path followed by $T$ makes the exact shape of a triangle $A B C$.
$T$ moves around $A B C$ in an anticlockwise direction as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-28_447_366_671_925} On its journey from $A$ to $B$ the velocity vector of $T$ is $( 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$
18

Find the speed of $T$ as it moves from $A$ to $B$ 18
On its journey from $B$ to $C$ the velocity vector of $T$ is $( - 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ Show that the acute angle $A B C = 60 ^ { \circ }$
18
It is given that $A B C$ is an equilateral triangle.
$T$ returns to its initial position after 9 seconds.
Vertex $B$ lies at position vector $\left[ \begin{array} { l } 1
0 \end{array} \right]$ metres with respect to a fixed origin $O$
Find the position vector of $C$

AQA Paper 2 2023 June Q19

19 A wooden toy comprises a train engine and a trailer connected to each other by a light, inextensible rod. The train engine has a mass of 1.5 kilograms.
The trailer has a mass 0.7 kilograms.
A string inclined at an angle of $40 ^ { \circ }$ above the horizontal is attached to the front of the train engine. The tension in the string is 2 newtons.
As a result the toy moves forward, from rest, in a straight line along a horizontal surface with acceleration $0.06 \mathrm {~ms} ^ { - 2 }$ as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-30_373_789_904_756} As it moves the train engine experiences a total resistance force of 0.8 N
19

Show that the total resistance force experienced by the trailer is approximately 0.6 N
19
At the instant that the toy reaches a speed of $0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the string breaks. As a result of this the train engine and trailer decelerate at a constant rate until they come to rest, having travelled a distance of $h$ metres. It can be assumed that the resistance forces remain unchanged.
19
1. Find the tension in the rod after the string has broken.
  
  19
  (ii) Find $h$
  Do not write outside the box
  
  \includegraphics[max width=\textwidth, alt={}]{de8a7d38-a665-4feb-854e-ac83f413d133-33_2488_1716_219_153}
  Nell and her pet dog Maia are visiting the beach.
  The beach surface can be assumed to be level and horizontal. Nell and Maia are initially standing next to each other.
  Nell throws a ball forward, from a height of 1.8 metres above the surface of the beach, at an angle of $60 ^ { \circ }$ above the horizontal with a speed of $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Exactly 0.2 seconds after the ball is thrown, Maia sets off from Nell and runs across the surface of the beach, in a straight line with a constant acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ Maia catches the ball when it is 0.3 metres above ground level as shown in the diagram below.
  \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-34_778_1287_1027_463}

AQA Paper 2 2023 June Q20

20 In this question use $g = 9.8 \mathrm {~m \mathrm {~s} ^ { - 2 }$} Find $a$

\includegraphics[max width=\textwidth, alt={}]{de8a7d38-a665-4feb-854e-ac83f413d133-36_2488_1719_219_150}

AQA Paper 2 2024 June Q1

1 One of the equations below is the equation of a circle. Identify this equation. Tick ( ✓ ) one box.
$( x + 1 ) ^ { 2 } - ( y + 2 ) ^ { 2 } = - 36$ □
$( x + 1 ) ^ { 2 } - ( y + 2 ) ^ { 2 } = 36$ □
$( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = - 36$ □
$( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36$ □

AQA Paper 2 2024 June Q2

2
The graph of $y = \mathrm { f } ( x )$ intersects the $x$-axis at ( $- 3,0$ ), ( 0,0 ) and ( 2,0 ) as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{28ee1134-b938-4c8f-b4ef-f3f9b210fdef-03_634_885_415_644} The shaded region $A$ has an area of 189 The shaded region $B$ has an area of 64
Find the value of $\int _ { - 3 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x$
Circle your answer.
-253
-125
125
253

AQA Paper 2 2024 June Q5

5 Given that
find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ $$y = \frac { x ^ { 3 } } { \sin x }$$

AQA Paper 2 2024 June Q6

6 It is given that $$( 2 \sin \theta + 3 \cos \theta ) ^ { 2 } + ( 6 \sin \theta - \cos \theta ) ^ { 2 } = 30$$ and that $\theta$ is obtuse. Find the exact value of $\sin \theta$. Fully justify your answer.

AQA Paper 2 2024 June Q7

7 On the first day of each month, Kate pays $\pounds 50$ into a savings account. Interest is paid on the total amount in the account on the last day of each month.
The interest rate is 0.2\% At the end of the $n$th month, the total amount of money in Kate's savings account is $\pounds T _ { n }$ Kate correctly calculates $T _ { 1 }$ and $T _ { 2 }$ as shown below: $$\begin{aligned} T _ { 1 } & = 50 \times 1.002 = 50.10
T _ { 2 } & = \left( T _ { 1 } + 50 \right) \times 1.002
& = ( ( 50 \times 1.002 ) + 50 ) \times 1.002
& = 50 \times 1.002 ^ { 2 } + 50 \times 1.002
& \approx 100.30 \end{aligned}$$ 7

Show that $T _ { 3 }$ is given by $$T _ { 3 } = 50 \times 1.002 ^ { 3 } + 50 \times 1.002 ^ { 2 } + 50 \times 1.002$$ 7
Kate uses her method to correctly calculate how much money she can expect to have in her savings account at the end of 10 years. 7
1. Find the amount of money Kate expects to have in her savings account at the end of 10 years.
  7
(ii) The amount of money in Kate's savings account at the end of 10 years may not be the amount she has correctly calculated. Explain why.

AQA Paper 2 2024 June Q8

2 marks

8 A zookeeper models the median mass of infant monkeys born at their zoo, up to the age of 2 years, by the formula $$y = a + b \log _ { 10 } x$$ where $y$ is the median mass in kilograms, $x$ is age in months and $a$ and $b$ are constants. The zookeeper uses the data shown below to determine the values of $a$ and $b$.

Age in months $( x )$	3	24
Median mass $( y )$	6.4	12

The zookeeper uses the data for monkeys aged 3 months to write the correct equation $$6.4 = a + b \log _ { 10 } 3$$ 8
1. Use the data for monkeys aged 24 months to write a second equation.
  8
(ii) Show that $$b = \frac { 5.6 } { \log _ { 10 } 8 }$$ 8
(iii) Find the value of $a$.
Give your answer to two decimal places.
\section*{Question 8 continues on the next question} 8
Use a suitable value for $x$ to determine whether the model can be used to predict the median mass of monkeys less than one week old.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}, center]{28ee1134-b938-4c8f-b4ef-f3f9b210fdef-13_2491_1757_173_121}

AQA Paper 2 2024 June Q9

2 marks

1. Find the binomial expansion of $( 1 + 3 x ) ^ { - 1 }$ up to and including the term in $x ^ { 2 }$
  9
(ii) Show that the first three terms in the binomial expansion of $$\frac { 1 } { 2 - 3 x }$$ form a geometric sequence and state the common ratio.
9
It is given that $$\frac { 36 x } { ( 1 + 3 x ) ( 2 - 3 x ) } \equiv \frac { P } { ( 2 - 3 x ) } + \frac { Q } { ( 1 + 3 x ) }$$ where $P$ and $Q$ are integers. Find the value of $P$ and the value of $Q$
9
1. Using your answers to parts (a) and (b), find the binomial expansion of $$\frac { 12 x } { ( 1 + 3 x ) ( 2 - 3 x ) }$$ up to and including the term in $x ^ { 2 }$
  [0pt] [2 marks]
  9
(ii) Find the range of values of $x$ for which the binomial expansion of $$\frac { 12 x } { ( 1 + 3 x ) ( 2 - 3 x ) }$$ is valid.

AQA Paper 2 2024 June Q10

10 The function f is defined by $$f ( x ) = x ^ { 2 } + 2 \cos x \text { for } - \pi \leq x \leq \pi$$ Determine whether the curve with equation $y = \mathrm { f } ( x )$ has a point of inflection at the point where $x = 0$ Fully justify your answer.
\includegraphics[max width=\textwidth, alt={}, center]{28ee1134-b938-4c8f-b4ef-f3f9b210fdef-19_2491_1757_173_121}

AQA Paper 2 2024 June Q11

A student states that 3 is the smallest value of $k$ in the interval $3 < k < 4$ Explain the error in the student's statement.
11
The student's teacher says there is no smallest value of $k$ in the interval $3 < k < 4$ The teacher gives the following correct proof: Step 1: Assume there is a smallest number in the interval $3 < k < 4$ and let this smallest number be $x$ Step 2: let $y = \frac { 3 + x } { 2 }$
Step 3: $3 < y < x$ which is a contradiction.
Step 4: Therefore, there is no smallest number in interval $3 < k < 4$ 11
1. Explain the contradiction stated in Step 3
  11
(ii) Prove that there is no largest value of $k$ in the interval $3 < k < 4$
\section*{END OF SECTION A TURN OVER FOR SECTION B}

AQA Paper 2 2024 June Q12

12 Two constant forces act on a particle, of mass 2 kilograms, so that it moves forward in a straight line. The two forces are:

a forward driving force of 10 newtons
a resistance force of 4 newtons.

Find the acceleration of the particle.
Circle your answer.
$2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
$3 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
$5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
$12 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

AQA Paper 2 2024 June Q13

13 A car starting from rest moves forward in a straight line. The motion of the car is modelled by the velocity-time graph below:
\includegraphics[max width=\textwidth, alt={}, center]{28ee1134-b938-4c8f-b4ef-f3f9b210fdef-23_476_738_459_715} One of the following assumptions about the motion of the car is implied by the graph. Identify this assumption. Tick ( ✓ ) one box. The car never accelerates. □ The acceleration of the car is always positive. □ The acceleration of the car can change instantaneously. □ The acceleration of the car is never constant. □

AQA Paper 2 2024 June Q14

3 marks

14 The displacement, $r$ metres, of a particle at time $t$ seconds is $$r = 6 t - 2 t ^ { 2 }$$ 14

Find the value of $r$ when $t = 4$
[0pt] [1 mark] 14
Determine the range of values of $t$ for which the displacement is positive.
[0pt] [2 marks]

AQA Paper 2 2024 June Q15

15 Two forces, $\mathbf { F } _ { \mathbf { 1 } }$ and $\mathbf { F } _ { \mathbf { 2 } }$, are acting on a particle of mass 3 kilograms. It is given that $$\mathbf { F } _ { \mathbf { 1 } } = \left[ \begin{array} { c } a
23 \end{array} \right] \text { newtons and } \mathbf { F } _ { \mathbf { 2 } } = \left[ \begin{array} { l } 4
b \end{array} \right] \text { newtons }$$ where $a$ and $b$ are constants. The particle has an acceleration of $\left[ \begin{array} { c } 4 b
a \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 2 }$ Find the value of $a$ and the value of $b$

AQA Paper 2 2024 June Q16

16 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ An apple tree stands on horizontal ground.
An apple hangs, at rest, from a branch of the tree.
A second apple also hangs, at rest, from a different branch of the tree.
The vertical distance between the two apples is $d$ centimetres.
At the same instant both apples begin to fall freely under gravity.
The first apple hits the ground after 0.5 seconds.
The second apple hits the ground 0.1 seconds later.
Show that $d$ is approximately 54

AQA Paper 2 2024 June Q17

17 A uniform rod is resting on two fixed supports at points $A$ and $B$.
$A$ lies at a distance $x$ metres from one end of the rod.
$B$ lies at a distance $( x + 0.1 )$ metres from the other end of the rod.
The rod has length $2 L$ metres and mass $m$ kilograms.
The rod lies horizontally in equilibrium as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{28ee1134-b938-4c8f-b4ef-f3f9b210fdef-28_332_880_726_644} The reaction force of the support on the rod at $B$ is twice the reaction force of the support on the rod at $A$. Show that $$L - x = k$$ where $k$ is a constant to be found.

AQA Paper 2 2024 June Q18

18 A particle is moving in a straight line through the origin $O$ The displacement of the particle, $r$ metres, from $O$, at time $t$ seconds is given by $$r = p + 2 t - q \mathrm { e } ^ { - 0.2 t }$$ where $p$ and $q$ are constants.
When $t = 3$, the acceleration of the particle is $- 1.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
18

Show that $q \approx 82$
18
The particle has an initial displacement of 5 metres. Find the value of $p$ Give your answer to two significant figures.
Turn over for the next question

AQA Paper 2 2024 June Q19

19 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A toy shoots balls upwards with an initial velocity of $7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
The advertisement for this toy claims the balls can reach a maximum height of 2.5 metres from the ground. 19

Suppose that the toy shoots the balls vertically upwards.
19
1. Verify the claim in the advertisement.
  19
(ii) State two modelling assumptions you have made in verifying this claim.
19
In fact the toy shoots the balls anywhere between 0 and 11 degrees from the vertical. The range of maximum heights, $h$ metres, above the ground which can be reached by the balls may be expressed as $$k < h \leq 2.5$$ Find the value of $k$

AQA Paper 2 2024 June Q20

1 marks

20 Two particles $P$ and $Q$ are moving in separate straight lines across a smooth horizontal surface.
$P$ moves with constant velocity $( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$
$Q$ moves from position vector $( 5 \mathbf { i } - 7 \mathbf { j } )$ metres to position vector $( 14 \mathbf { i } + 5 \mathbf { j } )$ metres during a 3 second period. 20

Show that $P$ and $Q$ move along parallel lines.
20
Stevie says
Q is also moving with a constant velocity of $( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$
Explain why Stevie may be incorrect.
[0pt] [1 mark] Question 20 continues on the next page 20
A third particle $R$ is moving with a constant speed of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, in a straight line, across the same surface.
$P$ and $R$ move along lines that intersect at a fixed point $X$
It is given that:
- $P$ passes through $X$ exactly 2 seconds after $R$ passes through $X$
- $P$ and $R$ are exactly 13 metres apart 3 seconds after $R$ passes through $X$
Show that $P$ and $R$ move along perpendicular lines.

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