Questions - Edexcel - A-Level Maths

Explain why it is reasonable to assume that the tensions in each half of the string are equal.
Find the mass of the object.
Find the elastic potential energy of the string when the object is suspended from it.

Edexcel M3 Q6

13 marks Challenging +1.2

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-4_455_540_201_660} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Figure 3 shows part of the curve $y = x ^ { 2 } + 1$. The shaded region enclosed by the curve, the coordinate axes and the line $x = 1$ is rotated through $360 ^ { \circ }$ about the $x$-axis.

Find the coordinates of the centre of mass of the solid obtained. The solid is suspended from a point on its larger circular rim and hangs in equilibrium.
Find, correct to the nearest degree, the acute angle which the plane surfaces of the solid make with the vertical.
(3 marks)

Edexcel M3 Q7

20 marks Standard +0.8

7. A particle of mass 0.5 kg is hanging vertically at one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point. The particle is given an initial horizontal speed of $u \mathrm {~ms} ^ { - 1 }$.

Show that the particle will perform complete circles if $u \geq \sqrt { 3 g }$. Given that $u = 5$,
find, correct to the nearest degree, the angle through which the string turns before it becomes slack,
find, correct to the nearest centimetre, the greatest height the particle reaches above its position when the string becomes slack.

Edexcel M3 Q1

8 marks Moderate -0.3

A particle $P$ of mass 1.5 kg moves from rest at the origin such that at time $t$ seconds it is subject to a single force of magnitude $( 4 t + 3 ) \mathrm { N }$ in the direction of the positive $x$-axis.

Find the magnitude of the impulse exerted by the force during the interval $1 \leq t \leq 4$. Given that at time $T$ seconds, $P$ has a speed of $22 \mathrm {~ms} ^ { - 1 }$,
find the value of $T$ correct to 3 significant figures.

Edexcel M3 Q2

8 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-2_469_465_776_680} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A particle $P$ of mass 0.5 kg is at rest at the highest point $A$ of a smooth sphere, centre $O$, of radius 1.25 m which is fixed to a horizontal surface. When $P$ is slightly disturbed it slides along the surface of the sphere. Whilst $P$ is in contact with the sphere it has speed $v \mathrm {~ms} ^ { - 1 }$ when $\angle A O P = \theta$ as shown in Figure 1.

Show that $v ^ { 2 } = 24.5 ( 1 - \cos \theta )$.
Find the value of $\cos \theta$ when $P$ leaves the surface of the sphere.

Edexcel M3 Q3

12 marks Standard +0.8

3. A car starts from rest at the point $O$ and moves along a straight line. The car accelerates to a maximum velocity, $V \mathrm {~ms} ^ { - 1 }$, before decelerating and coming to rest again at the point $A$. The acceleration of the car during this journey, $a \mathrm {~ms} ^ { - 2 }$, is modelled by the formula $$a = \frac { 500 - k x } { 150 }$$ where $x$ is the distance in metres of the car from $O$.
Using this model and given that the car is travelling at $16 \mathrm {~ms} ^ { - 1 }$ when it is 40 m from $O$,

find $k$,
show that $V = 41$, correct to 2 significant figures,
find the distance $O A$.

Edexcel M3 Q4

12 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-3_316_536_1087_639} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A particle $P$ of mass 2 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity $\lambda$. The other end of the string is fixed to a point $A$ on a rough plane inclined at an angle of $30 ^ { \circ }$ to the horizontal as shown in Figure 2. The coefficient of friction between $P$ and the plane is $\frac { 1 } { 6 } \sqrt { 3 }$. $P$ is held at rest at $A$ and then released. It first comes to instantaneous rest at the point $B , 2.2 \mathrm {~m}$ from $A$. For the motion of $P$ from $A$ to $B$,

show that the work done against friction is 10.78 J ,
find the change in the gravitational potential energy of $P$. By using the work-energy principle, or otherwise,
find $\lambda$.

Edexcel M3 Q5

16 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-4_693_554_196_717} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} A flask is modelled as a uniform solid formed by removing a cylinder of radius $r$ and height $h$ from a cylinder of radius $\frac { 4 } { 3 } r$ and height $\frac { 3 } { 2 } h$ with the same axis of symmetry and a common plane as shown in Figure 3.

Show that the centre of mass of the flask is a distance of $\frac { 9 } { 10 } h$ from the open end of the flask. The flask is made from a material of density $\rho$ and is filled to the level of the open plane face with a liquid of density $k \rho$. Given that the centre of mass of the flask and liquid together is a distance of $\frac { 15 } { 22 } h$ from the open end of the flask,
find the value of $k$.
Explain why it may be advantageous to make the base of the flask from a more dense material.
(2 marks)

Edexcel M3 Q6

19 marks Standard +0.8

6. A particle $P$ of mass 2.5 kg is moving with simple harmonic motion in a straight line between two points $A$ and $B$ on a smooth horizontal table. When $P$ is 3 m from $O$, the centre of the oscillations, its speed is $6 \mathrm {~ms} ^ { - 1 }$. When $P$ is 2.25 m from $O$, its speed is $8 \mathrm {~ms} ^ { - 1 }$.

Show that $A B = 7.5 \mathrm {~m}$.
Find the period of the motion.
Find the kinetic energy of $P$ when it is 2.7 m from $A$.
Show that the time taken by $P$ to travel directly from $A$ to the midpoint of $O B$ is $\frac { \pi } { 4 }$ seconds.

Edexcel M4 Q3

8 marks Challenging +1.8

3. A smooth uniform sphere $P$ of mass $m$ is falling vertically and strikes a fixed smooth inclined plane with speed $u$. The plane is inclined at an angle $\theta , \theta < 45 ^ { \circ }$, to the horizontal. The coefficient of restitution between $P$ and the inclined plane is $e$. Immediately after $P$ strikes the plane, $P$ moves horizontally.

Show that $e = \tan ^ { 2 } \theta$.

Edexcel M4 2002 June Q1

9 marks Standard +0.8

1. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c68c85a1-9d80-4ced-bfb6-c7b5347e9bb8-2_450_1417_391_339}

\end{figure} Two smooth uniform spheres $A$ and $B$, of equal radius, are moving on a smooth horizontal plane. Sphere $A$ has mass 2 kg and sphere $B$ has mass 3 kg . The spheres collide and at the instant of collision the line joining their centres is parallel to $\mathbf { i }$. Before the collision $A$ has velocity ( $3 \mathbf { i } - \mathbf { j }$ ) $\mathrm { m } \mathrm { s } ^ { - 1 }$ and after the collision it has velocity $( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. Before the collision the velocity of $B$ makes an angle $\alpha$ with the line of centres, as shown in Fig. 1, where $\tan \alpha = 2$. The coefficient of restitution between the spheres is $\frac { 1 } { 2 }$. Find, in terms of $\mathbf { i }$ and $\mathbf { j }$, the velocity of $B$ before the collision.
(9)

Edexcel M4 2002 June Q2

10 marks Challenging +1.2

2. Ship $A$ is steaming on a bearing of $060 ^ { \circ }$ at $30 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ and at 9 a.m. it is 20 km due west of a second ship $B$. Ship $B$ steams in a straight line.

Find the least speed of $B$ if it is to intercept $A$. Given that the speed of $B$ is $24 \mathrm {~km} \mathrm {~h} ^ { - 1 }$,
find the earliest time at which it can intercept $A$.

Edexcel M4 2002 June Q3

12 marks Standard +0.8

3. The engine of a car of mass 800 kg works at a constant rate of 32 kW . The car travels along a straight horizontal road and the resistance to motion of the car is proportional to the speed of the car. The car starts from rest and $t$ seconds later it has a speed of $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $$800 v \frac { \mathrm {~d} v } { \mathrm {~d} t } = 32000 - k v ^ { 2 } , \text { where } k \text { is a positive constant. }$$ Given that the limiting speed of the car is $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find
the value of $k$,
$v$ in terms of $t$.

Edexcel M4 2002 June Q4

13 marks Challenging +1.8

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c68c85a1-9d80-4ced-bfb6-c7b5347e9bb8-3_424_422_1181_844}

\end{figure} Four identical uniform rods, each of mass $m$ and length $2 a$, are freely jointed to form a rhombus $A B C D$. The rhombus is suspended from $A$ and is prevented from collapsing by an elastic string which joins $A$ to $C$, with $\angle B A D = 2 \theta , 0 \leq \theta \leq \frac { 1 } { 3 } \pi$, as shown in Fig. 2. The natural length of the elastic string is $2 a$ and its modulus of elasticity is $4 m g$.

Show that the potential energy, $V$, of the system is given by $$V = 4 m g a \left[ ( 2 \cos \theta - 1 ) ^ { 2 } - 2 \cos \theta \right] + \text { constant } .$$
Hence find the non-zero value of $\theta$ for which the system is in equilibrium.
Determine whether this position of equilibrium is stable or unstable.

Edexcel M4 2002 June Q5

14 marks Challenging +1.2

5. At time $t = 0$ particles $P$ and $Q$ start simultaneously from points which have position vectors $( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }$ and $( - \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }$ respectively, relative to a fixed origin $O$. The velocities of $P$ and $Q$ are $( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and $( 2 \mathbf { i } + \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ respectively.

Show that $P$ and $Q$ collide and find the position vector of the point at which they collide. A third particle $R$ moves in such a way that its velocity relative to $P$ is parallel to the vector ( $- 5 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$ ) and its velocity relative to $Q$ is parallel to the vector $( - 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )$. Given that all three particles collide simultaneously, find
1. the velocity of $R$,
2. the position vector of $R$ at time $t = 0$.

Edexcel M4 2002 June Q6

17 marks Challenging +1.3

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{c68c85a1-9d80-4ced-bfb6-c7b5347e9bb8-4_244_1264_1314_382}

\end{figure} A particle $P$ of mass 2 kg is attached to the mid-point of a light elastic spring of natural length 2 m and modulus of elasticity 4 N . One end $A$ of the elastic spring is attached to a fixed point on a smooth horizontal table. The spring is then stretched until its length is 4 m and its other end $B$ is held at a point on the table where $A B = 4 \mathrm {~m}$. At time $t = 0 , P$ is at rest on the table at the point $O$ where $A O = 2 \mathrm {~m}$, as shown in Fig. 3. The end $B$ is now moved on the table in such a way that $A O B$ remains a straight line. At time $t$ seconds, $A B = \left( 4 + \frac { 1 } { 2 } \sin 4 t \right) \mathrm { m }$ and $A P = ( 2 + x ) \mathrm { m }$.

Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 x = \sin 4 t$$
Hence find the time when $P$ first comes to instantaneous rest. END

Edexcel M4 2003 June Q1

8 marks Standard +0.8

A wooden ball of mass 0.01 kg falls vertically into a pond of water. The speed of the ball as it enters the water is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When the ball is $x$ metres below the surface of the water and moving downwards with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the water provides a resistance of magnitude $0.02 v ^ { 2 } \mathrm {~N}$ and an upward buoyancy force of magnitude 0.158 N .

Show that, while the ball is moving downwards, $$- 2 v ^ { 2 } - 6 = v \frac { \mathrm {~d} v } { \mathrm {~d} x }$$
Hence find, to 3 significant figures, the greatest distance below the surface of the water reached by the ball.

Edexcel M4 2003 June Q2

8 marks Standard +0.3

2. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{47e1d96b-4582-4324-a946-66989a2c66fc-2_455_1084_1112_487}

\end{figure} A man, who rows at a speed $v$ through still water, rows across a river which flows at a speed $u$. The man sets off from the point $A$ on one bank and wishes to land at the point $B$ on the opposite bank, where $A B$ is perpendicular to both banks, as shown in Fig. 1.

Show that, for this to be possible, $v > u$. Given that $v < u$ and that he rows from $A$ so as to reach a point $C$, on the opposite bank, which is as close to $B$ as possible,
find, in terms of $u$ and $v$, the ratio of $B C$ to the width of the river.
(5)

Edexcel M4 2003 June Q3

9 marks Challenging +1.2

3. A man walks due north at a constant speed $u$ and the wind seems to him to be blowing from the direction $30 ^ { \circ }$ east of north. On his return journey, when he is walking at the same speed $u$ due south, the wind seems to him to be blowing from the direction $30 ^ { \circ }$ south of east. Assuming that the velocity, $\mathbf { w }$, of the wind relative to the earth is constant, find

the magnitude of $\mathbf { w }$, in terms of $u$,
the direction of $\mathbf { w }$.

Edexcel M4 2003 June Q4

15 marks Challenging +1.2

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{47e1d96b-4582-4324-a946-66989a2c66fc-3_581_826_801_648}

\end{figure} A uniform rod $A B$, of length $2 a$ and mass $8 m$, is free to rotate in a vertical plane about a fixed smooth horizontal axis through $A$. One end of a light elastic string, of natural length $a$ and modulus of elasticity $\frac { 4 } { 5 } \mathrm { mg }$, is fixed to $B$. The other end of the string is attached to a small ring which is free to slide on a smooth straight horizontal wire which is fixed in the same vertical plane as $A B$ at a height 7a vertically above $A$. The rod $A B$ makes an angle $\theta$ with the upward vertical at $A$, as shown in Fig. 2.

Show that the potential energy $V$ of the system is given by $$V = \frac { 8 } { 5 } m g a \left( \cos ^ { 2 } \theta - \cos \theta \right) + \text { constant. }$$
Hence find the values of $\theta , 0 \leq \theta \leq \pi$, for which the system is in equilibrium.
Determine the nature of these positions of equilibrium.

Edexcel M4 2003 June Q5

17 marks Challenging +1.2

5. A light elastic string, of natural length $2 a$ and modulus of elasticity $m g$, has a particle $P$ of mass $m$ attached to its mid-point. One end of the string is attached to a fixed point $A$ and the other end is attached to a fixed point $B$ which is at a distance $4 a$ vertically below $A$.

Show that $P$ hangs in equilibrium at the point $E$ where $A E = \frac { 5 } { 2 } a$. The particle $P$ is held at a distance $3 a$ vertically below $A$ and is released from rest at time $t = 0$. When the speed of the particle is $v$, there is a resistance to motion of magnitude $2 m k v$, where $k = \sqrt { } \left( \frac { g } { a } \right)$. At time $t$ the particle is at a distance $\left( \frac { 5 } { 2 } a + x \right)$ from $A$.
Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 k \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 k ^ { 2 } x = 0$$
Hence find $x$ in terms of $t$.

Edexcel M4 2003 June Q6

18 marks Challenging +1.2

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{47e1d96b-4582-4324-a946-66989a2c66fc-5_652_725_296_620}

\end{figure} A small smooth uniform sphere $S$ is at rest on a smooth horizontal floor at a distance $d$ from a straight vertical wall. An identical sphere $T$ is projected along the floor with speed $U$ towards $S$ and in a direction which is perpendicular to the wall. At the instant when $T$ strikes $S$ the line joining their centres makes an angle $\alpha$ with the wall, as shown in Fig. 3. Each sphere is modelled as having negligible diameter in comparison with $d$. The coefficient of restitution between the spheres is $e$.

Show that the components of the velocity of $T$ after the impact, parallel and perpendicular to the line of centres, are $\frac { 1 } { 2 } U ( 1 - e ) \sin \alpha$ and $U \cos \alpha$ respectively.
Show that the components of the velocity of $T$ after the impact, parallel and perpendicular to the wall, are $\frac { 1 } { 2 } U ( 1 + e ) \cos \alpha \sin \alpha$ and $\frac { 1 } { 2 } U \left[ 2 - ( 1 + e ) \sin ^ { 2 } \alpha \right]$ respectively. The spheres $S$ and $T$ strike the wall at the points $A$ and $B$ respectively.
Given that $e = \frac { 2 } { 3 }$ and $\tan \alpha = \frac { 3 } { 4 }$,
find, in terms of $d$, the distance $A B$. \section*{END}

Edexcel M4 2004 June Q1

6 marks Standard +0.3

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively.]

An aeroplane makes a journey from a point $P$ to a point $Q$ which is due east of $P$. The wind velocity is $w ( \cos \theta \mathbf { i } + \sin \theta \mathbf { j } )$, where $w$ is a positive constant. The velocity of the aeroplane relative to the wind is $v ( \cos \phi \mathbf { i } - \sin \phi \mathbf { j } )$, where $v$ is a constant and $v > w$. Given that $\theta$ and $\phi$ are both acute angles,

show that $v \sin \phi = w \sin \theta$,
find, in terms of $v , w$ and $\theta$, the speed of the aeroplane relative to the ground.

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