Questions - Edexcel M2 - A-Level Maths

the velocity of the particle at time $t$ seconds,
the displacement of the particle from the origin at time $t$ seconds,
the values of $t$ at which the particle is instantaneously at rest.

Edexcel M2 2011 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-06_365_776_264_584} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A box of mass 30 kg is held at rest at point $A$ on a rough inclined plane. The plane is inclined at $20 ^ { \circ }$ to the horizontal. Point $B$ is 50 m from $A$ up a line of greatest slope of the plane, as shown in Figure 1. The box is dragged from $A$ to $B$ by a force acting parallel to $A B$ and then held at rest at $B$. The coefficient of friction between the box and the plane is $\frac { 1 } { 4 }$. Friction is the only non-gravitational resistive force acting on the box. Modelling the box as a particle,

find the work done in dragging the box from $A$ to $B$. The box is released from rest at the point $B$ and slides down the slope. Using the workenergy principle, or otherwise,
find the speed of the box as it reaches $A$.
January 2011

Edexcel M2 2011 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-10_823_908_269_513} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The uniform L-shaped lamina $A B C D E F$, shown in Figure 2, has sides $A B$ and $F E$ parallel, and sides $B C$ and $E D$ parallel. The pairs of parallel sides are 9 cm apart. The points $A , F$, $D$ and $C$ lie on a straight line.
$A B = B C = 36 \mathrm {~cm} , F E = E D = 18 \mathrm {~cm} . \angle A B C = \angle F E D = 90 ^ { \circ }$, and $\angle B C D = \angle E D F = \angle E F D = \angle B A C = 45 ^ { \circ }$.

Find the distance of the centre of mass of the lamina from
1. side $A B$,
2. side $B C$. The lamina is freely suspended from $A$ and hangs in equilibrium.
Find, to the nearest degree, the size of the angle between $A B$ and the vertical.

Edexcel M2 2011 January Q6

\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]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-12_689_1042_360_459} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} At time $t = 0$, a particle $P$ is projected from the point $A$ which has position vector 10j metres with respect to a fixed origin $O$ at ground level. The ground is horizontal. The velocity of projection of $P$ is $( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, as shown in Figure 3. The particle moves freely under gravity and reaches the ground after $T$ seconds.

For $0 \leqslant t \leqslant T$, show that, with respect to $O$, the position vector, $\mathbf { r }$ metres, of $P$ at time $t$ seconds is given by $$\mathbf { r } = 3 t \mathbf { i } + \left( 10 + 5 t - 4.9 t ^ { 2 } \right) \mathbf { j }$$
Find the value of $T$.
Find the velocity of $P$ at time $t$ seconds $( 0 \leqslant t \leqslant T )$. When $P$ is at the point $B$, the direction of motion of $P$ is $45 ^ { \circ }$ below the horizontal.
Find the time taken for $P$ to move from $A$ to $B$.
Find the speed of $P$ as it passes through $B$.

Edexcel M2 2011 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-14_442_986_264_479} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A uniform plank $A B$, of weight 100 N and length 4 m , rests in equilibrium with the end $A$ on rough horizontal ground. The plank rests on a smooth cylindrical drum. The drum is fixed to the ground and cannot move. The point of contact between the plank and the drum is $C$, where $A C = 3 \mathrm {~m}$, as shown in Figure 4. The plank is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 3 }$. The coefficient of friction between the plank and the ground is $\mu$. Modelling the plank as a rod, find the least possible value of $\mu$.

Edexcel M2 2011 January Q8

A particle $P$ of mass $m \mathrm {~kg}$ is moving with speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a straight line on a smooth horizontal floor. The particle strikes a fixed smooth vertical wall at right angles and rebounds. The kinetic energy lost in the impact is 64 J . The coefficient of restitution between $P$ and the wall is $\frac { 1 } { 3 }$.
1. Show that $m = 4$.
  (6)
After rebounding from the wall, $P$ collides directly with a particle $Q$ which is moving towards $P$ with speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The mass of $Q$ is 2 kg and the coefficient of restitution between $P$ and $Q$ is $\frac { 1 } { 3 }$.
Show that there will be a second collision between $P$ and the wall.

Edexcel M2 2012 January Q1

A tennis ball of mass 0.1 kg is hit by a racquet. Immediately before being hit, the ball has velocity $30 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The racquet exerts an impulse of $( - 2 \mathbf { i } - 4 \mathbf { j } ) \mathrm { Ns }$ on the ball. By modelling the ball as a particle, find the velocity of the ball immediately after being hit.
A particle $P$ is moving in a plane. At time $t$ seconds, $P$ is moving with velocity $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$, where $\mathbf { v } = 2 t \mathbf { i } - 3 t ^ { 2 } \mathbf { j }$.

Find

the speed of $P$ when $t = 4$
the acceleration of $P$ when $t = 4$ Given that $P$ is at the point with position vector $( - 4 \mathbf { i } + \mathbf { j } ) \mathrm { m }$ when $t = 1$,
find the position vector of $P$ when $t = 4$

Edexcel M2 2012 January Q3

3. A cyclist and her cycle have a combined mass of 75 kg . The cyclist is cycling up a straight road inclined at $5 ^ { \circ }$ to the horizontal. The resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude 20 N . At the instant when the cyclist has a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, she is decelerating at $0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Find the rate at which the cyclist is working at this instant. When the cyclist passes the point $A$ her speed is $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At $A$ she stops working but does not apply the brakes. She comes to rest at the point $B$. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 20 N .
Use the work-energy principle to find the distance $A B$.

Edexcel M2 2012 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{717c6949-db0f-4c2b-87a6-a7adf8c30a9e-06_415_981_237_475} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The trapezium $A B C D$ is a uniform lamina with $A B = 4 \mathrm {~m}$ and $B C = C D = D A = 2 \mathrm {~m}$, as shown in Figure 1.

Show that the centre of mass of the lamina is $\frac { 4 \sqrt { } 3 } { 9 } \mathrm {~m}$ from $A B$. The lamina is freely suspended from $D$ and hangs in equilibrium.
Find the angle between $D C$ and the vertical through $D$.

Edexcel M2 2012 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{717c6949-db0f-4c2b-87a6-a7adf8c30a9e-08_597_981_217_461} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A uniform rod $A B$ has mass 4 kg and length 1.4 m . The end $A$ is resting on rough horizontal ground. A light string $B C$ has one end attached to $B$ and the other end attached to a fixed point $C$. The string is perpendicular to the rod and lies in the same vertical plane as the rod. The rod is in equilibrium, inclined at $20 ^ { \circ }$ to the ground, as shown in Figure 2.

Find the tension in the string. Given that the rod is about to slip,
find the coefficient of friction between the rod and the ground.

Edexcel M2 2012 January Q6

6. Three identical particles, $A , B$ and $C$, lie at rest in a straight line on a smooth horizontal table with $B$ between $A$ and $C$. The mass of each particle is $m$. Particle $A$ is projected towards $B$ with speed $u$ and collides directly with $B$. The coefficient of restitution between each pair of particles is $\frac { 2 } { 3 }$.

Find, in terms of $u$,
1. the speed of $A$ after this collision,
2. the speed of $B$ after this collision.
Show that the kinetic energy lost in this collision is $\frac { 5 } { 36 } m u ^ { 2 }$ After the collision between $A$ and $B$, particle $B$ collides directly with $C$.
Find, in terms of $u$, the speed of $C$ immediately after this collision between $B$ and $C$.

Edexcel M2 2012 January Q7

7. [In this question, the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are horizontal and vertical respectively.] \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{717c6949-db0f-4c2b-87a6-a7adf8c30a9e-12_414_1234_338_354} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The point $O$ is a fixed point on a horizontal plane. A ball is projected from $O$ with velocity $( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, and passes through the point $A$ at time $t$ seconds after projection. The point $B$ is on the horizontal plane vertically below $A$, as shown in Figure 3. It is given that $O B = 2 A B$. Find

the value of $t$,
the speed, $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$, of the ball at the instant when it passes through $A$. At another point $C$ on the path the speed of the ball is also $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the time taken for the ball to travel from $O$ to $C$.

Edexcel M2 2013 January Q1

Two uniform rods $A B$ and $B C$ are rigidly joined at $B$ so that $\angle A B C = 90 ^ { \circ }$. Rod $A B$ has length 0.5 m and mass 2 kg . Rod $B C$ has length 2 m and mass 3 kg . The centre of mass of the framework of the two rods is at $G$.
1. Find the distance of $G$ from $B C$.
The distance of $G$ from $A B$ is 0.6 m .
The framework is suspended from $A$ and hangs freely in equilibrium.
Find the angle between $A B$ and the downward vertical at $A$.

Edexcel M2 2013 January Q2

2. A lorry of mass 1800 kg travels along a straight horizontal road. The lorry's engine is working at a constant rate of 30 kW . When the lorry's speed is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, its acceleration is $0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The magnitude of the resistance to the motion of the lorry is $R$ newtons.

Find the value of $R$. The lorry now travels up a straight road which is inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 1 } { 12 }$. The magnitude of the non-gravitational resistance to motion is $R$ newtons. The lorry travels at a constant speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the new rate of working of the lorry's engine.

Edexcel M2 2013 January Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ad18c22c-2fc5-4844-99b8-492f758bb24e-05_876_757_125_589} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A ladder, of length 5 m and mass 18 kg , has one end $A$ resting on rough horizontal ground and its other end $B$ resting against a smooth vertical wall. The ladder lies in a vertical plane perpendicular to the wall and makes an angle $\alpha$ with the horizontal ground, where $\tan \alpha = \frac { 4 } { 3 }$, as shown in Figure 1. The coefficient of friction between the ladder and the ground is $\mu$. A woman of mass 60 kg stands on the ladder at the point $C$, where $A C = 3 \mathrm {~m}$. The ladder is on the point of slipping. The ladder is modelled as a uniform rod and the woman as a particle. Find the value of $\mu$.

Edexcel M2 2013 January Q4

4. At time $t$ seconds the velocity of a particle $P$ is $[ ( 4 t - 5 ) \mathbf { i } + 3 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }$. When $t = 0$, the position vector of $P$ is $( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }$, relative to a fixed origin $O$.

Find the value of $t$ when the velocity of $P$ is parallel to the vector $\mathbf { j }$.
Find an expression for the position vector of $P$ at time $t$ seconds. A second particle $Q$ moves with constant velocity $( - 2 \mathbf { i } + c \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. When $t = 0$, the position vector of $Q$ is $( 11 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m }$. The particles $P$ and $Q$ collide at the point with position vector ( $d \mathbf { i } + 14 \mathbf { j }$ ) m.
Find
1. the value of $c$,
2. the value of $d$.

Edexcel M2 2013 January Q5

5. The point $A$ lies on a rough plane inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 24 } { 25 }$. A particle $P$ is projected from $A$, up a line of greatest slope of the plane, with speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The mass of $P$ is 2 kg and the coefficient of friction between $P$ and the plane is $\frac { 5 } { 12 }$. The particle comes to instantaneous rest at the point $B$ on the plane, where $A B = 1.5 \mathrm {~m}$. It then moves back down the plane to $A$.

Find the work done against friction as $P$ moves from $A$ to $B$.
Use the work-energy principle to find the value of $U$.
Find the speed of $P$ when it returns to $A$.

Edexcel M2 2013 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ad18c22c-2fc5-4844-99b8-492f758bb24e-11_531_931_230_520} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A ball is thrown from a point $O$, which is 6 m above horizontal ground. The ball is projected with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\theta$ above the horizontal. There is a thin vertical post which is 4 m high and 8 m horizontally away from the vertical through $O$, as shown in Figure 2. The ball passes just above the top of the post 2 s after projection. The ball is modelled as a particle.

Show that $\tan \theta = 2.2$
Find the value of $u$. The ball hits the ground $T$ seconds after projection.
Find the value of $T$. Immediately before the ball hits the ground the direction of motion of the ball makes an angle $\alpha$ with the horizontal.
Find $\alpha$.

Edexcel M2 2013 January Q7

7. A particle $A$ of mass $m$ is moving with speed $u$ on a smooth horizontal floor when it collides directly with another particle $B$, of mass $3 m$, which is at rest on the floor. The coefficient of restitution between the particles is $e$. The direction of motion of $A$ is reversed by the collision.

Find, in terms of $e$ and $u$,
1. the speed of $A$ immediately after the collision,
2. the speed of $B$ immediately after the collision. After being struck by $A$ the particle $B$ collides directly with another particle $C$, of mass $4 m$, which is at rest on the floor. The coefficient of restitution between $B$ and $C$ is $2 e$. Given that the direction of motion of $B$ is reversed by this collision,
find the range of possible values of $e$,
determine whether there will be a second collision between $A$ and $B$.

Edexcel M2 2014 January Q2

2. $$y = 2 x ^ { 2 } - \frac { 4 } { \sqrt { } x } + 1 , \quad x > 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving each term in its simplest form.
Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$, giving each term in its simplest form.
\includegraphics[max width=\textwidth, alt={}, center]{e6a4beaa-2c1f-4a98-bc63-4ddb8611db45-05_104_97_2613_1784}

Edexcel M2 2014 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e6a4beaa-2c1f-4a98-bc63-4ddb8611db45-08_835_777_118_596} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of a curve with equation $y = \mathrm { f } ( x )$. The curve crosses the $y$-axis at $( 0,3 )$ and has a minimum at $P ( 4,2 )$. On separate diagrams, sketch the curve with equation

$y = \mathrm { f } ( x + 4 )$,
$y = 2 \mathrm { f } ( x )$. On each diagram, show clearly the coordinates of the minimum point and any point of intersection with the $y$-axis.

Edexcel M2 2014 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e6a4beaa-2c1f-4a98-bc63-4ddb8611db45-12_650_885_255_603} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The straight line $l _ { 1 }$ has equation $2 y = 3 x + 7$
The line $l _ { 1 }$ crosses the $y$-axis at the point $A$ as shown in Figure 2.

1. State the gradient of $l _ { 1 }$
2. Write down the coordinates of the point $A$. Another straight line $l _ { 2 }$ intersects $l _ { 1 }$ at the point $B ( 1,5 )$ and crosses the $x$-axis at the point $C$, as shown in Figure 2. Given that $\angle A B C = 90 ^ { \circ }$,
find an equation of $l _ { 2 }$ in the form $a x + b y + c = 0$, where $a$, b and $c$ are integers. The rectangle $A B C D$, shown shaded in Figure 2, has vertices at the points $A , B , C$ and $D$.
Find the exact area of rectangle $A B C D$.

Edexcel M2 2001 June Q1

At time $t$ seconds, a particle $P$ has position vector $r$ metres relative to a fixed origin $O$, where

$$\mathbf { r } = \left( t ^ { 2 } + 2 t \right) \mathbf { i } + \left( t - 2 t ^ { 2 } \right) \mathbf { j } .$$ Show that the acceleration of $P$ is constant and find its magnitude. \section*{2.} \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{7670376b-a7b6-4f8c-83b5-bad3f5ff5162-2_861_588_837_807}

\end{figure} Figure 1 shows a decoration which is made by cutting 2 circular discs from a sheet of uniform card. The discs are joined so that they touch at a point $D$ on the circumference of both discs. The discs are coplanar and have centres $A$ and $B$ with radii 10 cm and 20 cm respectively.

Find the distance of the centre of mass of the decoration from B. The point $C$ lies on the circumference of the smaller disc and $\angle C A B$ is a right angle. The decoration is freely suspended from C and hangs in equilibrium.
Find, in degrees to one decimal place, the angle between AB and the vertical.

Edexcel M2 2001 June Q3

3. A uniform ladder $A B$, of mass $m$ and length $2 a$, has one end $A$ on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.5 . The other end $B$ of the ladder rests against a smooth vertical wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, and makes an angle of $30 ^ { \circ }$ with the wall. A man of mass $5 m$ stands on the ladder which remains in equilibrium. The ladder is modelled as a uniform rod and the man as a particle. The greatest possible distance of the man from $A$ is $k a$. Find the value of $k$.

Edexcel M2 2001 June Q4

4. The unit vectors $\mathbf { i }$ and $\mathbf { j }$ lie in a vertical plane, $\mathbf { i }$ being horizontal and $\mathbf { j }$ vertical. A ball of mass 0.1 kg is hit by a bat which gives it an impulse of ( $3.5 \mathbf { i } + 3 \mathbf { j }$ ) Ns. The velocity of the ball immediately after being hit is $( 10 \mathbf { i } + 25 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.

Find the velocity of the ball immediately before it is hit. In the subsequent motion the ball is modelled as a particle moving freely under gravity. When it is hit the ball is 1 m above horizontal ground.
Find the greatest height of the ball above the ground in the subsequent motion. The ball is caught when it is again 1 m above the ground.
Find the distance from the point where the ball is hit to the point where it is caught.

Questions — Edexcel M2 (551 questions)

Browse by module

Browse by board

Edexcel M2 2011 January Q3

Edexcel M2 2011 January Q4

Edexcel M2 2011 January Q5

Edexcel M2 2011 January Q6

Edexcel M2 2011 January Q7

Edexcel M2 2011 January Q8

Edexcel M2 2012 January Q1

Edexcel M2 2012 January Q3

Edexcel M2 2012 January Q4

Edexcel M2 2012 January Q5

Edexcel M2 2012 January Q6

Edexcel M2 2012 January Q7

Edexcel M2 2013 January Q1

Edexcel M2 2013 January Q2

Edexcel M2 2013 January Q3

Edexcel M2 2013 January Q4

Edexcel M2 2013 January Q5

Edexcel M2 2013 January Q6

Edexcel M2 2013 January Q7

Edexcel M2 2014 January Q2

Edexcel M2 2014 January Q4

Edexcel M2 2014 January Q6

Edexcel M2 2001 June Q1

Edexcel M2 2001 June Q3

Edexcel M2 2001 June Q4