Questions - AQA - A-Level Maths

the motion is due to gravitational force only
the acceleration due to gravity remains constant throughout.

Show that the time taken for the ball to travel from $M$ to $N$ is $\frac { 2 u } { g }$ seconds.
[0pt] [2 marks] 13
Point $M$ is $h$ metres above the Earth. Show that $h > \frac { 4 u ^ { 2 } } { g }$
Fully justify your answer.
The car is moving in a straight line.
The acceleration $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ of the car at time $t$ seconds is given by $$a = 3 k t ^ { 2 } - 2 k t + 1$$ where $k$ is a constant.
When $t = 3$ the car has a velocity of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Show that $k = \frac { 1 } { 3 }$

AQA Paper 2 2023 June Q14

Moderate -0.8

14 A car has an initial velocity of $1 \mathrm {~ms} ^ { - 1 }$ A particle, $Q$, moves in a straight line across a rough horizontal surface.
A horizontal driving force of magnitude $D$ newtons acts on $Q$
$Q$ moves with a constant acceleration of $0.91 \mathrm {~ms} ^ { - 2 }$
$Q$ has a weight of 0.65 N
The only resistance force acting on $Q$ is due to friction.
The coefficient of friction between $Q$ and the surface is 0.4 Find $D$

AQA Paper 2 2023 June Q15

Standard +0.3

15 In this question use $g = 9.8 \mathrm {~ms} ^ { - 2 }$ In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$

AQA Paper 2 2023 June Q16

Moderate -0.8

16 A particle moves under the action of two forces, $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ It is given that $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 1.6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 2 } = ( k \mathbf { i } + 5 k \mathbf { j } ) \mathrm { N } \end{aligned}$$ where $k$ is a constant.
The acceleration of the particle is $( 3.2 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$
Find $k$
\includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-25_2488_1716_219_153}

AQA Paper 2 2023 June Q17

Standard +0.3

17 A uniform plank $P Q$, of length 7 metres, lies horizontally at rest, in equilibrium, on two fixed supports at points $X$ and $Y$ The distance $P X$ is 1.4 metres and the distance $Q Y$ is 2 metres as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-26_56_689_534_762}
\includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-26_225_830_607_694} 17

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

Moderate -0.3

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

Moderate -0.3

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

Moderate -0.8

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

Easy -2.0

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

Moderate -0.8

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

Standard +0.3

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

AQA Paper 2 2024 June Q6

Standard +0.8

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

Moderate -0.8

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 Standard +0.3

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

Standard +0.3

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

Moderate -0.8

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}