Questions M2 - A-Level Maths

Show that the distance of $G$ from the side $A B$ is $\frac { 7 } { 3 } a$.
Find the distance of $G$ from the side $A E$. The mass of $L$ is $M$. A particle of mass $k M$ is attached to $L$ at the point $E$. The lamina, with the particle attached, is freely suspended from $A$ and hangs in equilibrium with the diagonal $A C$ vertical.
Find the value of $k$.

Edexcel M2 2017 June Q7

Standard +0.3

7. Three particles $A , B$ and $C$ lie at rest in a straight line on a smooth horizontal surface, with $B$ between $A$ and $C$. The particles $A$, $B$ and $C$ have mass $6 m$, 4 $m$ and $m$ respectively. Particle $A$ is projected towards $B$ with speed $3 u$ and $A$ collides directly with $B$. Immediately after this collision, the speed of $B$ is $w$. The coefficient of restitution between $A$ and $B$ is $\frac { 1 } { 6 }$.

Show that $w = \frac { 21 } { 10 } u$.
Express the total kinetic energy of $A$ and $B$ lost in the collision as a fraction of the total kinetic energy of $A$ and $B$ immediately before the collision. After being struck by $A$, the particle $B$ collides directly with $C$. The coefficient of restitution between $B$ and $C$ is $e$. After the collision between $B$ and $C$, there are no further collisions between the particles.
Find the range of possible values of $e$.

Edexcel M2 2018 June Q1

Moderate -0.8

A particle $P$ of mass 0.7 kg is moving with velocity ( $\mathbf { i } - 2 \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$ when it receives an impulse. Immediately after receiving the impulse, $P$ is moving with velocity $( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
1. Find the impulse.
2. Find, in degrees, the size of the angle between the direction of the impulse and the direction of motion of $P$ immediately before receiving the impulse.
  (3)

Edexcel M2 2018 June Q2

Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a3f28425-4acf-4878-b0e3-15b5bc8a92d7-04_494_1116_226_415} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform rod $A B$, of mass 6 kg and length 1.6 m , rests with its end $A$ on rough horizontal ground. The rod is held in equilibrium at $30 ^ { \circ }$ to the horizontal by a light string attached to the rod at $B$. The string is at $40 ^ { \circ }$ to the horizontal and lies in the same vertical plane as the rod, as shown in Figure 1. The tension in the string is $T$ newtons. The coefficient of friction between the ground and the rod is $\mu$.

Show that, to 3 significant figures, $T = 27.1$
Find the set of values of $\mu$ for which equilibrium is possible. \includegraphics[max width=\textwidth, alt={}, center]{a3f28425-4acf-4878-b0e3-15b5bc8a92d7-07_27_40_2802_1893}

Edexcel M2 2018 June Q3

Standard +0.3

3. A cyclist and his bicycle, with a combined mass of 75 kg , move along a straight horizontal road. The cyclist is working at a constant rate of 180 W . There is a constant resistance to the motion of the cyclist and his bicycle of magnitude $R$ newtons. At the instant when the speed of the cyclist is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, his acceleration is $0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Find the value of $R$. Later, the cyclist moves up a straight road with a constant speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The road is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 1 } { 21 }$. The cyclist is working at a rate of 180 W and the resistance to the motion of the cyclist and his bicycle from non-gravitational forces is again the same constant force of magnitude $R$ newtons.
Find the value of $v$.

Edexcel M2 2018 June Q4

Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a3f28425-4acf-4878-b0e3-15b5bc8a92d7-12_702_1182_226_379} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The uniform lamina $A B C$ is an isosceles triangle with $A B = B C , A C = 6 a$ and the distance from $B$ to $A C$ is $3 a$. The uniform lamina $M N C$ is an isosceles triangle with $M N = N C$ and $M C = 3 a$. Triangles $A B C$ and $M N C$ are similar and are made of the same material. The lamina $L$ is formed by fixing triangle $M N C$ on top of triangle $A B C$, as shown in Figure 2.

Show that the distance of the centre of mass of $L$ from $A C$ is $\frac { 9 } { 10 } a$ The lamina $L$ is freely suspended from $B$ and hangs in equilibrium.
Find, to the nearest degree, the size of the angle between $A B$ and the downward vertical.

Edexcel M2 2018 June Q5

Standard +0.3

5. A particle $P$ of mass 0.3 kg moves under the action of a single force $\mathbf { F }$ newtons. At time $t$ seconds $( t \geqslant 0 ) , P$ has velocity $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$, where $$\mathbf { v } = \left( 3 t ^ { 2 } - 4 t \right) \mathbf { i } + \left( 3 t ^ { 2 } - 8 t + 4 \right) \mathbf { j }$$

Find $\mathbf { F }$ when $t = 4$ At the instants when $P$ is at the points $A$ and $B$, particle $P$ is moving parallel to the vector i.
Find the distance $A B$.

Edexcel M2 2018 June Q6

Standard +0.3

6. A particle $P$ is projected from a fixed point $A$ with speed $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\alpha$ above the horizontal and moves freely under gravity. As $P$ passes through the point $B$ on its path, $P$ is moving with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\beta$ below the horizontal.

By considering energy, find the vertical distance between $A$ and $B$. Particle $P$ takes 1.5 seconds to travel from $A$ to $B$.
Find the size of angle $\alpha$.
Find the size of angle $\beta$.
Find the length of time for which the speed of $P$ is less than $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Edexcel M2 2018 June Q7

Standard +0.8

7. Three particles $A$, $B$ and $C$ have masses $2 m , 3 m$ and $4 m$ respectively. The particles lie at rest in a straight line on a smooth horizontal surface, with $B$ between $A$ and $C$. Particle $A$ is projected towards $B$ with speed $u$ and collides directly with $B$. The coefficient of restitution between $A$ and $B$ is $e$. The kinetic energy of $A$ immediately after the collision is one ninth of the kinetic energy of $A$ immediately before the collision. Given that the direction of motion of $A$ is unchanged by the collision,

find the value of $e$. After the collision between $A$ and $B$ there is a direct collision between $B$ and $C$. The coefficient of restitution between $B$ and $C$ is $f$, where $f < \frac { 3 } { 4 }$. The speed of $B$ immediately after the collision with $C$ is $V$.
1. Express $V$ in terms of $f$ and $u$.
2. Hence show that there will be a second collision between $A$ and $B$.

Edexcel M2 2020 June Q1

Moderate -0.3

A particle of mass 2 kg is moving with velocity $( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ when it receives an impulse $\mathbf { I N }$ s, such that $\mathbf { I } = a \mathbf { i } + b \mathbf { j }$

Immediately after receiving the impulse, the particle is moving with velocity $\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }$, where $\lambda$ is a constant. Given that the magnitude of $\mathbf { I }$ is $\sqrt { 40 }$, find the two possible impulses.
(5)

Edexcel M2 2020 June Q2

Standard +0.3

A truck of weight 9000 N is travelling up a hill on a straight road that is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 1 } { 15 }$

When the truck travels up the hill with the engine working at $3 P$ watts, the truck is moving at a constant speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Later on, the truck travels down the hill along the same road, with the engine working at $P$ watts. At the instant when the speed of the truck is $12 \mathrm {~ms} ^ { - 1 }$, the acceleration of the truck is $\frac { g } { 20 }$ The resistance to motion of the truck from non-gravitational forces is a constant force of magnitude $R$ newtons in all circumstances. Find (i) the value of $P$,
(ii) the value of $R$.
WIHW SIHI NIT INHM ION OC
WIIV SIHI NI III IM I ON OC
VAYV SIHI NI JLIUM ION OO

Edexcel M2 2020 June Q3

Standard +0.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1dea68fe-7916-41ed-894e-6b48f8d989bb-08_476_725_251_605} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform rod $A B$, of mass 25 kg and length 3 m , has end $A$ resting on rough horizontal ground. The end $B$ rests against a rough vertical wall. The rod is in a vertical plane perpendicular to the wall.
The coefficient of friction between the rod and the ground is $\frac { 4 } { 5 }$
The coefficient of friction between the rod and the wall is $\frac { 3 } { 5 }$
The rod rests in limiting equilibrium.
The rod is at an angle of $\theta$ to the ground, as shown in Figure 1. Find the exact value of $\tan \theta$.
DO NOT WRITEIN THIS AREA

VIXV SIHIANI III IM IONOO

VIAV SIHI NI JYHAM ION OO

VI4V SIHI NI JLIYM ION OO

Edexcel M2 2020 June Q4

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

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

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

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

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

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

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

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

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

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

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

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

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}

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