Questions - Edexcel M2 - A-Level Maths

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Edexcel M2 2020 June Q4

9 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1dea68fe-7916-41ed-894e-6b48f8d989bb-12_662_716_255_614} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The uniform lamina $L$, shown shaded in Figure 2, is formed by removing the square $P Q R V$, of side $2 a$, and the square $R S T U$, of side $4 a$, from a uniform square lamina $A B C D$, of side $8 a$. The lines $Q R U$ and $V R S$ are straight. The side $A D$ is parallel to $P V$ and the side $A B$ is parallel to $P Q$. The distance between $A D$ and $P V$ is $a$ and the distance between $A B$ and $P Q$ is $a$. The centre of mass of $L$ is at the point $G$.

Show that the distance of $G$ from the side $A D$ is $\frac { 42 } { 11 } a$ The mass of $L$ is $M$. A particle of mass $k M$ is attached to $L$ at $C$. The lamina, with the attached particle, is freely suspended from $B$ and hangs in equilibrium with $B C$ making an angle of $45 ^ { \circ }$ with the horizontal.
Find the value of $k$.

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel M2 2020 June Q5

10 marks Moderate -0.3

5. At time $t$ seconds ( $t \geqslant 0$ ), a particle $P$ has velocity $\mathbf { v m ~ s } ^ { - 1 }$, where $$\mathbf { v } = \left( 3 t ^ { 2 } - 9 t + 6 \right) \mathbf { i } + \left( t ^ { 2 } + t - 6 \right) \mathbf { j }$$

Find the acceleration of $P$ when $t = 3$ When $t = 0 , P$ is at the fixed point $O$.
The particle comes to instantaneous rest at the point $A$.
Find the distance $O A$.

Edexcel M2 2020 June Q6

10 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1dea68fe-7916-41ed-894e-6b48f8d989bb-20_273_1058_246_443} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A rough straight ramp is fixed to horizontal ground. The ramp has length 15 m and is inclined at an angle $\alpha$ to the ground, where $\tan \alpha = \frac { 5 } { 12 }$. The line $A B$ is a line of greatest slope of the ramp, where $A$ is at the bottom of the ramp, and $B$ is at the top of the ramp, as shown in Figure 3. A particle $P$ of mass 6 kg is projected up the ramp with speed $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from $A$ in a straight line towards $B$. The coefficient of friction between $P$ and the ramp is 0.25

Find the work done against friction as $P$ moves from $A$ to $B$. At the instant $P$ reaches $B$, the speed of $P$ is $v \mathrm {~ms} ^ { - 1 }$. After leaving the ramp at $B$, the particle $P$ moves freely under gravity until it hits the horizontal ground at the point $C$. Immediately before hitting the ground at $C$, the speed of $P$ is $w \mathrm {~ms} ^ { - 1 }$
Use the work-energy principle to find
1. the value of $v$,
2. the value of $w$.
  
  \includegraphics[max width=\textwidth, alt={}, center]{1dea68fe-7916-41ed-894e-6b48f8d989bb-23_86_49_2617_1884}

Edexcel M2 2020 June Q7

10 marks Standard +0.3

7. Particle $A$ of mass $3 m$ is moving in a straight line with speed $2 u$ on a smooth horizontal surface. Particle $A$ collides directly with particle $B$ of mass $m$, which is moving along the same straight line and in the same direction as $A$. Immediately before the collision, the speed of $B$ is $u$.
As a result of the collision, the direction of motion of $B$ is unchanged and the kinetic energy gained by $B$ is $\frac { 48 } { 25 } m u ^ { 2 }$

Find the coefficient of restitution between $A$ and $B$.
(8) After the collision, $B$ hits a smooth fixed vertical wall that is perpendicular to the direction of motion of $B$. The coefficient of restitution between $B$ and the wall is $f$. Given that the speed of $B$ immediately after first hitting the wall is equal to the speed of $A$ immediately after its first collision with $B$,
find the value of $f$.

Edexcel M2 2020 June Q8

13 marks Standard +0.8

8. [In this question, the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are in a vertical plane, with $\mathbf { i }$ being horizontal and $\mathbf { j }$ being vertically upwards.] \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1dea68fe-7916-41ed-894e-6b48f8d989bb-28_426_1145_347_338} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} At time $t = 0$, a small ball is projected from a fixed point $O$ on horizontal ground. The ball is projected from $O$ with velocity ( $p \mathbf { i } + q \mathbf { j }$ ) $\mathrm { ms } ^ { - 1 }$, where $p$ and $q$ are positive constants. The ball moves freely under gravity. At time $t = 3$ seconds, the ball passes through the point $A$ with velocity ( $8 \mathbf { i } - 12 \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$, as shown in Figure 4.

Find the speed of the ball at the instant it is projected from $O$. For an interval of $T$ seconds the speed, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of the ball is such that $v \leqslant 10$
Find the value of $T$. At the point $B$ on the path of the ball, the direction of motion of the ball is perpendicular to the direction of motion of the ball at $A$.
Find the vertical height of $B$ above $A$.

Edexcel M2 2021 June Q1

5 marks Moderate -0.3

A motorcyclist and his motorcycle have a combined mass of 480 kg .

The motorcyclist drives down a straight road that is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 1 } { 12 }$, with the engine of the motorcycle working at 3.5 kW . The motorcycle is moving at a constant speed of $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resistance to the motion of the motorcycle is modelled as a constant force with magnitude 20 V newtons. Find the value of $V$.
(5)

Edexcel M2 2021 June Q2

8 marks Standard +0.3

2. A particle $P$ of mass 1.5 kg moves under the action of a single force $\mathbf { F }$ newtons. At time $t$ seconds, $t \geqslant 0 , P$ has velocity $\mathbf { v } \mathrm { ms } ^ { - 1 }$, where $$\mathbf { v } = \left( 5 t ^ { 2 } - t ^ { 3 } \right) \mathbf { i } + \left( 2 t ^ { 3 } - 8 t \right) \mathbf { j }$$

Find $\mathbf { F }$ when $t = 2$ At time $t = 0 , P$ is at the origin $O$.
Find the position vector of $P$ relative to $O$ at the instant when $P$ is moving in the direction of the vector $\mathbf { j }$

Edexcel M2 2021 June Q3

8 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-06_645_684_260_639} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The uniform lamina $A B C D$ is a square of side $6 a$. The template $T$, shown shaded in Figure 1, is formed by removing the right-angled triangle $E F G$ and the circle, centre $H$ and radius $a$, from the square lamina. Triangle $E F G$ has $E F = E G = 4 a$, with $E F$ parallel to $A B$ and $E G$ parallel to $A D$. The distance between $A B$ and $E F$ is $a$ and the distance between $A D$ and $E G$ is $a$. The point $H$ lies on $A C$ and the distance of $H$ from $B C$ is $2 a$.

Show that the centre of mass of $T$ is a distance $\frac { 4 ( 67 - 3 \pi ) } { 3 ( 28 - \pi ) } a$ from $A D$. The template $T$ is suspended from the ceiling by two light inextensible vertical strings. One string is attached to $T$ at $A$ and the other string is attached to $T$ at $B$ so that $T$ hangs in equilibrium with $A B$ horizontal. The weight of $T$ is $W$. The tension in the string attached to $T$ at $B$ is $k W$, where $k$ is a constant.
Find the value of $k$, giving your answer to 2 decimal places.

Edexcel M2 2021 June Q4

6 marks Moderate -0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-10_410_369_251_790} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass 0.3 kg is moving with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along a straight line on a smooth horizontal plane. The particle receives a horizontal impulse of magnitude $J$ Ns. The speed of $P$ immediately after receiving the impulse is $8 \mathrm {~ms} ^ { - 1 }$. The angle between the direction of motion of $P$ before it receives the impulse and the direction of the impulse is $60 ^ { \circ }$, as shown in Figure 2. Find the value of $J$.
(6)

Edexcel M2 2021 June Q5

9 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-12_638_595_251_676} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform rod, of length $8 a$ and mass $M$, has one end freely hinged to a fixed point $A$ on a vertical wall. One end of a light inextensible string is attached to the rod at the point $B$, where $A B = 5 a$. The other end of the string is attached to the wall at the point $C$, where $A C = 5 a$ and $C$ is vertically above $A$. The rod rests in equilibrium in a vertical plane perpendicular to the wall with angle $B A C = 70 ^ { \circ }$, as shown in Figure 3.

Find, in terms of $M$ and $g$, the tension in the string. The magnitude of the force acting on the rod at $A$ is $\lambda M g$, where $\lambda$ is a constant.
Find, to 2 significant figures, the value of $\lambda$.

Edexcel M2 2021 June Q6

15 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-16_273_819_260_566} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Two particles, $A$ and $B$, of mass 2 kg and 3 kg respectively, are connected by a light inextensible string. Particle $A$ is held at rest at the point $X$ on a fixed rough ramp that is inclined at an angle $\theta$ to the horizontal, where $\tan \theta = \frac { 5 } { 12 }$. The string passes over a small smooth pulley $P$ that is fixed at the top of the ramp. Particle $B$ hangs vertically below $P$, 2 m above the ground, as shown in Figure 4. The particles are released from rest with the string taut so that $A$ moves up the ramp and the section of the string from $A$ to $P$ is parallel to a line of greatest slope of the ramp. The coefficient of friction between $A$ and the ramp is $\frac { 3 } { 8 }$ Air resistance is ignored.

Find the potential energy lost by the system as $A$ moves 2 m up the ramp.
Find the work done against friction as $A$ moves 2 m up the ramp. When $B$ hits the ground, $B$ is brought to rest by the impact and does not rebound and $A$ continues to move up the ramp.
Use the work-energy principle to find the speed of $B$ at the instant before it hits the ground. Particle $A$ comes to instantaneous rest at the point $Y$ on the ramp, where $X Y = ( 2 + d ) \mathrm { m }$.
Use the work-energy principle to find the value of $d$.

Edexcel M2 2021 June Q7

11 marks Standard +0.3

\hspace{0pt} [In this question, the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are in a vertical plane, $\mathbf { i }$ being horizontal and $\mathbf { j }$ being vertically upwards.]

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-20_289_837_347_486} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A small ball is projected with velocity $( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ from a fixed point $A$ on horizontal ground. The ball hits the ground at the point $B$, as shown in Figure 5. The motion of the ball is modelled as a particle moving freely under gravity.

Find the distance $A B$. When the height of the ball above the ground is more than $h$ metres, the speed of the ball is less than $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
Find the smallest possible value of $h$. When the ball is at the point $C$ on its path, the direction of motion of the ball is perpendicular to the direction of motion of the ball at the instant before it hits the ground at $B$.
Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the velocity of the ball when it is at $C$.

Edexcel M2 2021 June Q8

13 marks Standard +0.8

Particles $A , B$ and $C$, of masses $2 m , m$ and $3 m$ respectively, lie at rest in a straight line on a smooth horizontal plane with $B$ between $A$ and $C$. Particle $A$ is projected towards particle $B$ with speed $2 u$ and collides directly with $B$.

The coefficient of restitution between each pair of particles is $e$.

1. Show that the speed of $B$ immediately after the collision with $A$ is $\frac { 4 } { 3 } u ( 1 + e )$
2. Find the speed of $A$ immediately after the collision with $B$. At the instant when $A$ collides with $B$, particle $C$ is projected with speed $u$ towards $B$ so that $B$ and $C$ collide directly.
Show that there will be a second collision between $A$ and $B$. \includegraphics[max width=\textwidth, alt={}, center]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-27_2644_1840_118_111}

Edexcel M2 2022 June Q1

8 marks Standard +0.3

At time $t$ seconds, $t \geqslant 0$, a particle $P$ has position vector $\mathbf { r }$ metres with respect to a fixed origin $O$, where

$$\mathbf { r } = \left( t ^ { 3 } - 8 t \right) \mathbf { i } + \left( \frac { 1 } { 3 } t ^ { 3 } - t ^ { 2 } + 2 t \right) \mathbf { j }$$

Find the acceleration of $P$ when $t = 4$ At time $T$ seconds, $T \geqslant 0 , P$ is moving in the direction of ( $2 \mathbf { i } + \mathbf { j }$ )
Find the value of $T$

Edexcel M2 2022 June Q2

6 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7eedd755-0dfd-4506-b7fd-23b9def4ebc8-04_508_780_258_584} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The point $A$ lies on a smooth horizontal floor between two fixed smooth parallel vertical walls $W X$ and $Y Z$, as shown in the plan view in Figure 1.
The distance between $W X$ and $Y Z$ is $3 d$.
The distance of $A$ from $Y Z$ is $d$.
A particle is projected from $A$ along the floor with speed $u$ towards $Y Z$ in a direction perpendicular to $Y Z$. The coefficient of restitution between the particle and each wall is $\frac { 2 } { 3 }$ The time taken for the particle to move from $A$, bounce off each wall once and return to A for the first time is $T _ { 1 }$

Find $T _ { 1 }$ in terms of $d$ and $u$. The ball returns to $A$ for the first time after bouncing off each wall once. The further time taken for the particle to move from $A$, bounce off each wall once and return to $A$ for the second time is $T _ { 2 }$
Find $T _ { 2 }$ in terms of $d$ and $u$.

show that $T = 20$ Given also that the rod is in limiting equilibrium,
find the value of the coefficient of friction between the rod and the ground.

Edexcel M2 2022 June Q6

12 marks Standard +0.3

6. Two particles, $P$ and $Q$, are moving in opposite directions along the same straight line on a smooth horizontal surface so that the particles collide directly.
The mass of $P$ is $k m$ and the mass of $Q$ is $m$.
Immediately before the collision, the speed of $P$ is $x$ and the speed of $Q$ is $y$. Immediately after the collision, $P$ and $Q$ are moving in the same direction, the speed of $P$ is $v$ and the speed of $Q$ is $2 v$. The coefficient of restitution between $P$ and $Q$ is $\frac { 1 } { 5 }$ The magnitude of the impulse received by $Q$ in the collision is $5 m v$

Find (i) $y$ in terms of $v$ (ii) $x$ in terms of $v$ (iii) the value of $k$
Find, in terms of $m$ and $v$, the total kinetic energy lost in the collision between $P$ and $Q$.

Edexcel M2 2022 June Q7

12 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7eedd755-0dfd-4506-b7fd-23b9def4ebc8-20_679_695_260_628} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The template shown in Figure 3 is formed by joining together three separate laminas. All three laminas lie in the same plane.

PQUV is a uniform square lamina with sides of length $3 a$
URST is a uniform square lamina with sides of length $6 a$
$Q R U$ is a uniform triangular lamina with $U Q = 3 a , U R = 6 a$ and angle $Q U R = 90 ^ { \circ }$

The mass per unit area of $P Q U V$ is $k$, where $k$ is a constant.
The mass per unit area of URST is $k$.
The mass per unit area of $Q R U$ is $2 k$.
The distance of the centre of mass of the template from $Q T$ is $d$.

Show that $d = \frac { 29 } { 14 } a$ The template is freely suspended from the point $Q$ and hangs in equilibrium with $Q R$ at $\theta ^ { \circ }$ to the downward vertical.
Find the value of $\theta$

Edexcel M2 2022 June Q8

14 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7eedd755-0dfd-4506-b7fd-23b9def4ebc8-24_259_1045_255_456} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a rough ramp fixed to horizontal ground.
The ramp is inclined at angle $\alpha$ to the ground, where $\tan \alpha = \frac { 1 } { 6 }$ The point $A$ is on the ground at the bottom of the ramp.
The point $B$ is at the top of the ramp.
The line $A B$ is a line of greatest slope of the ramp and $A B = 4 \mathrm {~m}$.
A particle $P$ of mass 3 kg is projected with speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from $A$ directly towards $B$.
The coefficient of friction between the particle and the ramp is $\frac { 3 } { 4 }$

Find the work done against friction as $P$ moves from $A$ to $B$. Given that at the instant $P$ reaches the point $B$, the speed of $P$ is $5 \mathrm {~ms} ^ { - 1 }$
use the work-energy principle to find the value of $U$. The particle leaves the ramp at $B$, and moves freely under gravity until it hits the ground at the point $C$.
Find the horizontal distance from $B$ to $C$.

Edexcel M2 2023 June Q1

7 marks Moderate -0.3

A particle $P$ of mass 0.3 kg is moving with velocity $5 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$

The particle receives an impulse I Ns.
Immediately after receiving the impulse, the velocity of $P$ is $( 7 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$

Find the magnitude of $\mathbf { I }$
Find the angle between the direction of $\mathbf { I }$ and the direction of motion of $P$ immediately before receiving the impulse.

Edexcel M2 2023 June Q2

10 marks Moderate -0.3

\hspace{0pt} [In this question, the perpendicular unit vectors $\mathbf { i }$ and $\mathbf { j }$ are in a horizontal plane.]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A particle $P$ is moving on a smooth horizontal plane.
At time $t$ seconds $( t \geqslant 0 )$, the position vector of $P$, relative to a fixed point $O$, is $\mathbf { r }$ metres and the velocity of $P$ is $\mathbf { v } \mathrm { ms } ^ { - 1 }$ where $$\mathbf { v } = \left( 4 t ^ { 2 } - 5 t \right) \mathbf { i } + ( - 10 t - 12 ) \mathbf { j }$$ When $t = 0 , \mathbf { r } = 2 \mathbf { i } + 6 \mathbf { j }$

Find $\mathbf { r }$ when $t = 2$ When $t = T$ particle $P$ is moving in the direction of the vector $\mathbf { i } - 2 \mathbf { j }$
Find the value of $T$
Find the exact magnitude of the acceleration of $P$ when $t = 2.5$

Edexcel M2 2023 June Q3

8 marks Challenging +1.2

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{52966963-2e62-4361-bcd5-a76322f8621e-08_1141_810_287_148} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{52966963-2e62-4361-bcd5-a76322f8621e-08_752_803_484_1114} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The uniform triangular lamina $A B C$, shown in Figure 1, has height $9 y$, base $B C = 6 x$, and $A B = A C$ The points $P$ and $Q$ are such that $A P : P C = A Q : Q B = 2 : 1$ The lamina is folded along $P Q$ to form the folded lamina $F$ The distance of the centre of mass of $F$ from $P Q$ is $d$

Show that $d = \frac { 16 } { 9 } y$ The folded lamina is suspended from $P$ and hangs freely in equilibrium with $P Q$ at an angle $\alpha$ to the downward vertical.
Given that $\tan \alpha = \frac { 64 } { 81 }$
find $x$ in terms of $y$

Edexcel M2 2023 June Q4

12 marks Standard +0.3

A particle $P$ of mass $3 m$ and a particle $Q$ of mass $5 m$ are moving towards each other along the same straight line on a smooth horizontal surface. The particles collide directly.

Immediately before the collision, the speed of $P$ is $u$ and the speed of $Q$ is $k u$.
Immediately after the collision, the speed of $P$ is $2 v$ and the speed of $Q$ is $v$.
The direction of motion of each particle is reversed by the collision.
In the collision, $P$ receives an impulse of magnitude $15 m v$.

Show that $u = 3 v$.
Find the value of $k$. The coefficient of restitution between $P$ and $Q$ is $e$.
Find the value of $e$. The total kinetic energy lost in the collision is $\lambda m v ^ { 2 }$
Find the value of $\lambda$.

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