Questions - A-Level Maths

AQA Further AS Paper 2 Mechanics 2020 June Q3

3 marks Easy -1.2

3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of $3.84 \times 10 ^ { 8 }$ metres.
Find the approximate speed of the moon relative to Earth, in metres per second.

AQA Further AS Paper 2 Mechanics 2020 June Q4

4 marks Moderate -0.3

4 A particle $P$, of mass $m \mathrm {~kg}$, collides with a particle $Q$, of mass 2 kg Immediately before the collision the velocity of $P$ is $\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ is $\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ As a result of the collision the particles coalesce into a single particle which moves with velocity $\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$, where $k$ is a constant. Find the value of $k$.

AQA Further AS Paper 2 Mechanics 2020 June Q5

5 marks Moderate -0.3

5 A train consisting of an engine and eight carriages moves on a straight horizontal track. A constant resistive force of 2400 N acts on the engine.
A constant resistive force of 300 N acts on each of the eight carriages.
The maximum speed of the train on the track is $120 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ Find the maximum power output of the engine.
Fully justify your answer.

AQA Further AS Paper 2 Mechanics 2020 June Q6

5 marks Standard +0.3

6 The magnitude of the gravitational force $F$ between two planets of masses $m _ { 1 }$ and $m _ { 2 }$ with centres at a distance $d$ apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where $G$ is a constant.
6

Show that $G$ must have dimensions $L ^ { 3 } M ^ { - 1 } T ^ { - 2 }$, where $L$ represents length, $M$ represents mass and $T$ represents time.
6
The lifetime $t$ of a planet is thought to depend on its mass $m$, its radius $r$, the constant $G$ and a dimensionless constant $k$ such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where $a , b$ and $c$ are constants.
Determine the values of $a , b$ and $c$.

AQA Further AS Paper 2 Mechanics 2020 June Q7

9 marks Standard +0.3

7 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ As part of a competition, Jo-Jo makes a small pop-up rocket.
It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket. The rocket is released from rest and moves vertically upwards.
The mass of the rocket is 18 grams and the stiffness constant of the spring is $60 \mathrm { Nm } ^ { - 1 }$ Initially the spring is compressed by 3 cm
7

Find the speed of the rocket when the spring first reaches its natural length.
7
By considering energy find the distance that the rocket rises. 7
In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position. With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not. Fully justify your answer.

AQA Further AS Paper 2 Mechanics 2020 June Q14

4 marks Moderate -0.5

14 J
18J
42 J 3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of $3.84 \times 10 ^ { 8 }$ metres.
Find the approximate speed of the moon relative to Earth, in metres per second.
4 A particle $P$, of mass $m \mathrm {~kg}$, collides with a particle $Q$, of mass 2 kg Immediately before the collision the velocity of $P$ is $\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ is $\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ As a result of the collision the particles coalesce into a single particle which moves with velocity $\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$, where $k$ is a constant. Find the value of $k$.
5 A train consisting of an engine and eight carriages moves on a straight horizontal track. A constant resistive force of 2400 N acts on the engine.
A constant resistive force of 300 N acts on each of the eight carriages.
The maximum speed of the train on the track is $120 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ Find the maximum power output of the engine.
Fully justify your answer.
6 The magnitude of the gravitational force $F$ between two planets of masses $m _ { 1 }$ and $m _ { 2 }$ with centres at a distance $d$ apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where $G$ is a constant.
6

Show that $G$ must have dimensions $L ^ { 3 } M ^ { - 1 } T ^ { - 2 }$, where $L$ represents length, $M$ represents mass and $T$ represents time.
6

The lifetime $t$ of a planet is thought to depend on its mass $m$, its radius $r$, the constant $G$ and a dimensionless constant $k$ such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where $a , b$ and $c$ are constants.
Determine the values of $a , b$ and $c$.
7 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ As part of a competition, Jo-Jo makes a small pop-up rocket.
It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket. The rocket is released from rest and moves vertically upwards.
The mass of the rocket is 18 grams and the stiffness constant of the spring is $60 \mathrm { Nm } ^ { - 1 }$ Initially the spring is compressed by 3 cm
7

Find the speed of the rocket when the spring first reaches its natural length.
7
By considering energy find the distance that the rocket rises. 7
In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position. With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not. Fully justify your answer.
8 Two smooth spheres $A$ and $B$ have the same radius and are free to move on a smooth horizontal surface. The masses of $A$ and $B$ are $2 m$ and $m$ respectively.
Both $A$ and $B$ are initially at rest.
The sphere $A$ is set in motion directly towards $B$ with speed $3 u$ and at the same time $B$ is set in motion directly towards $A$ with speed $2 u$. Subsequently $A$ and $B$ collide directly. $A$ The coefficient of restitution between the spheres is $e$.
8
Show that the speed of $B$ after the collision is given by $$\frac { 2 u ( 2 + 5 e ) } { 3 }$$ \section*{Question 8 continues on the next page} 8
Given that the direction of the velocity of $A$ is reversed during the collision, find the range of possible values of $e$. Fully justify your answer.
[0pt] [4 marks]
8

Given that the magnitude of the impulse that $A$ exerts on $B$ is $\frac { 19 m u } { 3 }$, find the value of $e$.

Question number	Additional page, if required. Write the question numbers in the left-hand margin.
	$\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$

AQA Further AS Paper 2 Mechanics Specimen Q1

1 marks Easy -1.2

1 A child, of mass 40 kg , moves at constant speed of $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a fairground ride.
The path of the child is a circle of radius 4 metres.
Find the magnitude of the resultant force acting on the child.
Circle your answer.
[0pt] [1 mark]
6.3 N
50 N
130 N
250 N

AQA Further AS Paper 2 Mechanics Specimen Q2

1 marks Easy -1.2

2 The graph shows how a force, $F$, varies with time during a period of 0.8 seconds. \includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-03_440_960_568_516} Find the magnitude of the impulse of $F$ during the 0.8 seconds.
Circle your answer.
[0pt] [1 mark]
1.0 Ns
1.6 Ns
2.2 Ns
3.2 Ns Turn over for the next question

AQA Further AS Paper 2 Mechanics Specimen Q3

4 marks Moderate -0.3

3 A tank full of liquid has a hole made in its base.
Two students, Sarah and David, propose two different models for the speed, $v$, at which liquid exits the tank. David thinks that $v$ will depend on the height of the liquid in the tank, $h$, the acceleration due to gravity, $g$, and the density of the liquid, $\rho$, such that $v \propto g ^ { a } h ^ { b } \rho ^ { c }$ where $a$, $b$ and $c$ are constants. Sarah thinks that $v$ will not depend on the density of the liquid and suggests the model $v \propto g ^ { a } h ^ { b }$ 3

By considering dimensions, explain which student's model should be rejected.
[0pt] [2 marks]
3
Find the values of the constants in order for the model that you did not reject in part (a) to be dimensionally consistent.
[0pt] [2 marks]

AQA Further AS Paper 2 Mechanics Specimen Q4

5 marks Moderate -0.3

4 A cricket ball of mass 156 grams is thrown from a point which is 1.5 metres above the ground, with a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ A tennis ball of mass 58 grams is thrown from the same point, with the same speed.
Prove that both balls hit the ground with the same speed.
Clearly state any assumptions you have made and how you have used them.
[0pt] [5 marks]

AQA Further AS Paper 2 Mechanics Specimen Q5

4 marks Standard +0.3

5 Two small smooth discs, $C$ and $D$, have equal radii and masses of 2 kg and 3 kg respectively. The discs are sliding on a smooth horizontal surface towards each other and collide directly. Disc $C$ has speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and disc $D$ has speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ as they collide. The coefficient of restitution between $C$ and $D$ is 0.6 The diagram shows the discs, viewed from above, before the collision. \includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-06_343_712_868_753} 5

Show that the speed of $D$ immediately after the collision is $1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 2 significant figures.
5
Find the speed of $C$ immediately after the collision.
[0pt] [2 marks]
5
In fact the horizontal surface on which the discs are sliding is not smooth.
Explain how the introduction of friction will affect your answer to part (b).
[0pt] [2 marks]
Turn over for the next question

AQA Further AS Paper 2 Mechanics Specimen Q6

4 marks Standard +0.3

6 A car, of mass 1200 kg , moves on a straight horizontal road where it has a maximum speed of $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ When the car travels at a speed of $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ it experiences a resistance force which can be modelled as being of magnitude 30 v newtons. 6

Show that the power output of the car is 48000 W , when it is travelling at its maximum speed. 6
Find the maximum acceleration of the car when it is travelling at a speed of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ [0pt] [4 marks]

AQA Further AS Paper 2 Mechanics Specimen Q7

3 marks Standard +0.3

7 A disc, of mass 0.15 kg , slides across a smooth horizontal table and collides with a vertical wall which is perpendicular to the path of the disc. The disc is in contact with the wall for 0.02 seconds and then rebounds.
A possible model for the force, $F$ newtons, exerted on the disc by the wall, whilst in contact, is given by $$F = k t ^ { 2 } ( t - b ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.020$$ where $k$ and $b$ are constants.
The force is initially zero and becomes zero again as the disc loses contact with the wall. 7

State the value of $b$.
7
Find the magnitude of the impulse on the disc, giving your answer in terms of $k$.
7
The disc is travelling at $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it hits the wall.
The disc rebounds with a speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Find $k$.
[0pt] [3 marks]

AQA Further AS Paper 2 Mechanics Specimen Q8

6 marks Challenging +1.2

8 In this question use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
A particle, of mass 2 kg , is attached to one end of a light elastic string of natural length 0.2 metres. The other end of the string is attached to a fixed point $O$.
The particle is pulled down and released from rest at a point 0.6 metres directly below $O$.
The particle then moves vertically and next comes to rest when it is 0.1 metres below $O$.
Assume that no air resistance acts on the particle.
8

Find the maximum speed of the particle.
[0pt] [6 marks]
8
Describe one way in which the model you have used could be refined.

OCR Further Pure Core AS 2018 June Q1

5 marks Moderate -0.3

1

Find a vector which is perpendicular to both $\left( \begin{array} { r } 1 \\ 3 \\ - 2 \end{array} \right)$ and $\left( \begin{array} { r } - 3 \\ - 6 \\ 4 \end{array} \right)$.
The cartesian equation of a line is $\frac { x } { 2 } = y - 3 = 2 z + 4$. Express the equation of this line in vector form.

OCR Further Pure Core AS 2018 June Q2

3 marks Standard +0.3

2 In this question you must show detailed reasoning.
The cubic equation $2 x ^ { 3 } + 3 x ^ { 2 } - 5 x + 4 = 0$ has roots $\alpha , \beta$ and $\gamma$. By making an appropriate substitution, or otherwise, find a cubic equation with integer coefficients whose roots are $\frac { 1 } { \alpha } , \frac { 1 } { \beta }$ and $\frac { 1 } { \gamma }$.

OCR Further Pure Core AS 2018 June Q3

9 marks Moderate -0.3

3 In this question you must show detailed reasoning.
The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by $z _ { 1 } = 2 - 3 i$ and $z _ { 2 } = a + 4 i$ where $a$ is a real number.

Express $z _ { 1 }$ in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures.
Find $z _ { 1 } z _ { 2 }$ in terms of $a$, writing your answer in the form $c + \mathrm { id }$.
The real and imaginary parts of a complex number on an Argand diagram are $x$ and $y$ respectively. Given that the point representing $z _ { 1 } z _ { 2 }$ lies on the line $y = x$, find the value of $a$.
Given instead that $z _ { 1 } z _ { 2 } = \left( z _ { 1 } z _ { 2 } \right) ^ { * }$ find the value of $a$.

OCR Further Pure Core AS 2018 June Q4

7 marks Standard +0.3

4 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r r } 2 & 1 & 2 \\ 1 & - 1 & 1 \\ 2 & 2 & a \end{array} \right)$.

Show that $\operatorname { det } \mathbf { A } = 6 - 3 a$.
State the value of $a$ for which $\mathbf { A }$ is singular.
Given that $\mathbf { A }$ is non-singular find $\mathbf { A } ^ { - 1 }$ in terms of $a$.

OCR Further Pure Core AS 2018 June Q5

10 marks Moderate -0.3

5 In this question you must show detailed reasoning.

Express $( 2 + 3 \mathrm { i } ) ^ { 3 }$ in the form $a + \mathrm { i } b$.
Hence verify that $2 + 3$ i is a root of the equation $3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52 = 0$.
Express $3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52$ as the product of a linear factor and a quadratic factor with real coefficients.

OCR Further Pure Core AS 2018 June Q6

7 marks Moderate -0.5

6 The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { r r } t & 6 \\ t & - 2 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { r r } 2 t & 4 \\ t & - 2 \end{array} \right)$ where $t$ is a constant.

Show that $| \mathbf { A } | = | \mathbf { B } |$.
Verify that $| \mathbf { A B } | = | \mathbf { A } \| \mathbf { B } |$.
Given that $| \mathbf { A B } | = - 1$ explain what this means about the constant $t$.

OCR Further Pure Core AS 2018 June Q7

6 marks Moderate -0.3

7 Prove by induction that $2 ^ { n + 1 } + 5 \times 9 ^ { n }$ is divisible by 7 for all integers $n \geqslant 1$.

OCR Further Pure Core AS 2018 June Q8

13 marks Standard +0.8

8 The $2 \times 2$ matrix A represents a transformation T which has the following properties.

The image of the point $( 0,1 )$ is the point $( 3,4 )$.
An object shape whose area is 7 is transformed to an image shape whose area is 35 .
T has a line of invariant points.
1. Find a possible matrix for $\mathbf { A }$.

The transformation S is represented by the matrix $\mathbf { B }$ where $\mathbf { B } = \left( \begin{array} { l l } 3 & 1 \\ 2 & 2 \end{array} \right)$.

Find the equation of the line of invariant points of S .

Show that any line of the form $y = x + c$ is an invariant line of S .

OCR Further Pure Core AS 2022 June Q1

8 marks Moderate -0.8

1

Determine whether the point $( 19 , - 12,17 )$ lies on the line $\mathbf { r } = \left( \begin{array} { r } 4 \\ - 2 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right)$. Vectors $\mathbf { a }$ and $\mathbf { b }$ are given by $\mathbf { a } = \left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)$ and $\mathbf { b } = \left( \begin{array} { r } - 3 \\ 6 \\ 2 \end{array} \right)$.
1. Find, in degrees, the angle between $\mathbf { a }$ and $\mathbf { b }$.
2. Find a vector which is perpendicular to both $\mathbf { a }$ and $\mathbf { b }$.

OCR Further Pure Core AS 2022 June Q2

7 marks Moderate -0.3

2 Matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { r r } a & 1 \\ - 1 & 3 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } - 2 & 5 \\ - 1 & 0 \end{array} \right)$ where $a$ is a constant.

Find the following matrices.
- $\mathbf { A } + \mathbf { B }$
- AB
- $\mathbf { A } ^ { 2 }$
- 1. Given that the determinant of $\mathbf { A }$ is 25 find the value of $a$.
  2. You are given instead that the following system of equations does not have a unique solution.
$$\begin{array} { r } a x + y = - 2 \\ - x + 3 y = - 6 \end{array}$$ Determine the value of $a$.

OCR Further Pure Core AS 2022 June Q4

5 marks Standard +0.3

4 Prove that $3 ^ { n } > 10 n$ for all integers $n \geqslant 4$.

Questions (33218 questions)

Browse by module

Browse by board

AQA Further AS Paper 2 Mechanics 2020 June Q3

AQA Further AS Paper 2 Mechanics 2020 June Q4

AQA Further AS Paper 2 Mechanics 2020 June Q5

AQA Further AS Paper 2 Mechanics 2020 June Q6

AQA Further AS Paper 2 Mechanics 2020 June Q7

AQA Further AS Paper 2 Mechanics 2020 June Q14

AQA Further AS Paper 2 Mechanics Specimen Q1

AQA Further AS Paper 2 Mechanics Specimen Q2

AQA Further AS Paper 2 Mechanics Specimen Q3

AQA Further AS Paper 2 Mechanics Specimen Q4

AQA Further AS Paper 2 Mechanics Specimen Q5

AQA Further AS Paper 2 Mechanics Specimen Q6

AQA Further AS Paper 2 Mechanics Specimen Q7

AQA Further AS Paper 2 Mechanics Specimen Q8

OCR Further Pure Core AS 2018 June Q1

OCR Further Pure Core AS 2018 June Q2

OCR Further Pure Core AS 2018 June Q3

OCR Further Pure Core AS 2018 June Q4

OCR Further Pure Core AS 2018 June Q5

OCR Further Pure Core AS 2018 June Q6

OCR Further Pure Core AS 2018 June Q7

OCR Further Pure Core AS 2018 June Q8

OCR Further Pure Core AS 2022 June Q1

OCR Further Pure Core AS 2022 June Q2

OCR Further Pure Core AS 2022 June Q4

Question number	Additional page, if required. Write the question numbers in the left-hand margin.
	\(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)