Questions M5 - A-Level Maths

Edexcel M5 2014 June Q5

A particle moves in a plane so that its position vector $\mathbf { r }$ metres at time $t$ seconds satisfies the differential equation

$$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + ( \tan t ) \mathbf { r } = \left( \cos ^ { 2 } t \right) \mathbf { i } - ( 3 \cos t ) \mathbf { j } , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ When $t = 0$, the particle is at the point with position vector $4 \mathbf { j } \mathrm {~m}$. Find $\mathbf { r }$ in terms of $t$.

Edexcel M5 2014 June Q6

6. Three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ act on a rigid body at the points with position vectors, $\mathbf { r } _ { 1 } , \mathbf { r } _ { 2 }$ and $\mathbf { r } _ { 3 }$ respectively, where
$\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { N }$
$\mathbf { F } _ { 2 } = ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }$
$\mathbf { F } _ { 3 } = ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }$
$\mathbf { r } _ { 1 } = ( \mathbf { i } - \mathbf { k } ) \mathrm { m }$
$\mathbf { r } _ { 2 } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }$
$\mathbf { r } _ { 3 } = ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { m }$ The system of the three forces is equivalent to a single force $\mathbf { R }$ acting at the point with position vector ( $\mathbf { 3 i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }$, together with a couple of moment $\mathbf { G }$.

Find $\mathbf { R }$.
Find $\mathbf { G }$.

Edexcel M5 2014 June Q7

7. A raindrop absorbs water as it falls vertically under gravity through a cloud. In a model of the motion the cloud is assumed to consist of stationary water particles. At time $t$, the mass of the raindrop is $m$ and the speed of the raindrop is $v$. At time $t = 0$, the raindrop is at rest. The rate of increase of the mass of the raindrop with respect to time is modelled as being $m k v$, where $k$ is a positive constant.

Ignoring air resistance, show from first principles, that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = g - k v ^ { 2 }$$
Find the time taken for the raindrop to reach a speed of $\frac { 1 } { 2 } \sqrt { } \left( \frac { g } { k } \right)$

Edexcel M5 2014 June Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{57b98cdd-4121-4495-b500-185cbf3ff1a8-13_739_739_276_607} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform circular disc of radius $2 a$ has centre $O$. The points $P , Q , R$ and $S$ on the disc are the vertices of a square with centre $O$ and $O P = a$. Four circular holes, each of radius $\frac { a } { 2 }$, and with centres $P , Q , R$ and $S$, are drilled in the disc to produce the lamina $L$, shown shaded in Figure 1. The mass of $L$ is $M$.

Show that the moment of inertia of $L$ about an axis through $O$, and perpendicular to the plane of $L$, is $\frac { 55 M a ^ { 2 } } { 24 }$ The lamina $L$ is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to $L$ and which passes through a point $A$ on the circumference of $L$. At time $t , A O$ makes an angle $\theta$ with the downward vertical through $A$.
Show that $\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { 48 g } { 151 a } \sin \theta$
Hence find the period of small oscillations of $L$ about its position of stable equilibrium. The magnitude of the component, in a direction perpendicular to $A O$, of the force exerted on $L$ by the axis is $X$.
Find $X$ in terms of $M , g$ and $\theta$. \includegraphics[max width=\textwidth, alt={}, center]{57b98cdd-4121-4495-b500-185cbf3ff1a8-14_159_1662_2416_173}

Edexcel M5 2014 June Q1

A small bead is threaded on a smooth, straight horizontal wire which passes through the point $A ( - 3,1 )$ and the point $B ( 2,5 )$ in the $x - y$ plane. The bead moves under the action of a horizontal force $\mathbf { F }$ of magnitude 8.5 N whose line of action is parallel to the line with equation $15 x - 8 y + 4 = 0$. The unit on both the $x$ and $y$ axes has length one metre. Find the work done by $\mathbf { F }$ as it moves the bead from $A$ to $B$.
(8)
A particle $P$ moves in a plane so that its position vector, $\mathbf { r }$ metres at time $t$ seconds, satisfies the differential equation

$$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + \mathbf { r } = t \mathbf { i } + \mathrm { e } ^ { - t } \mathbf { j }$$ When $t = 0$ the particle is at the point with position vector $( \mathbf { i } + \mathbf { j } ) \mathrm { m }$. Find $\mathbf { r }$ in terms of $t$.

Edexcel M5 2014 June Q3

3. Three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ act on a rigid body at the points with position vectors $\mathbf { r } _ { 1 } , \mathbf { r } _ { 2 }$ and $\mathbf { r } _ { 3 }$ respectively.
$\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } ) \mathrm { N }$ and $\mathbf { r } _ { 1 } = ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) \mathrm { m }$,
$\mathbf { F } _ { 2 } = ( \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }$ and $\mathbf { r } _ { 2 } = ( 3 \mathbf { i } - \mathbf { j } - \mathbf { k } ) \mathrm { m }$,
$\mathbf { F } _ { 3 } = ( - 3 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \mathrm { N }$ and $\mathbf { r } _ { 3 } = ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }$.
Show that the system is equivalent to a couple and find the magnitude of the vector moment of this couple.

Edexcel M5 2014 June Q4

4. A spacecraft is travelling in a straight line in deep space where all external forces can be assumed to be negligible. The spacecraft decelerates by ejecting fuel at a constant speed $k$ relative to the spacecraft, in the direction of motion of the spacecraft. At time $t$, the spacecraft has speed $v$ and mass $m$.

Show, from first principles, that while the spacecraft is ejecting fuel, $$\frac { \mathrm { d } v } { \mathrm {~d} m } - \frac { k } { m } = 0$$ At time $t = 0$, the spacecraft has speed $U$ and mass $M$.
Find the mass of the spacecraft when it comes to rest. Given that $m = M \mathrm { e } ^ { - \alpha t ^ { 2 } }$, where $\alpha$ is a positive constant, and that the spacecraft comes to rest at time $t = T$,
find, in terms of $U$ and $T$ only, the distance travelled by the spacecraft in decelerating from speed $U$ to rest.

Edexcel M5 2014 June Q5

A uniform rod $A B$, of mass $m$ and length $2 a$, is free to rotate in a vertical plane about a fixed smooth horizontal axis $L$. The axis $L$ is perpendicular to the rod and passes through the point $P$ of the rod, where $A P = \frac { 2 } { 3 } a$.
1. Find the moment of inertia of the rod about $L$.
The rod is held at rest with $B$ vertically above $P$ and is slightly displaced.
Find the angular speed of the rod when $P B$ makes an angle $\theta$ with the upward vertical.
Find the magnitude of the angular acceleration of the rod when $P B$ makes an angle $\theta$ with the upward vertical.
Find, in terms of $g$ and $a$ only, the angular speed of the rod when the force acting on the rod at $P$ is perpendicular to the rod.

Edexcel M5 2014 June Q6

6. (a) Prove, using integration, that the moment of inertia of a uniform circular disc, of mass $m$ and radius $a$, about an axis through the centre of the disc and perpendicular to the plane of the disc is $\frac { 1 } { 2 } m a ^ { 2 }$.
[0pt] [You may assume without proof that the moment of inertia of a uniform hoop of mass $m$ and radius $r$ about an axis through its centre and perpendicular to its plane is $m r ^ { 2 }$.] \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f1b010d7-14fb-4d23-88eb-d8183e9da3c7-09_528_528_635_708} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform plane shape $S$ of mass $M$ is formed by removing a uniform circular disc with centre $O$ and radius $a$ from a uniform circular disc with centre $O$ and radius $2 a$, as shown in Figure 1. The shape $S$ is free to rotate about a fixed smooth axis $L$, which passes through $O$ and lies in the plane of the shape.
(b) Show that the moment of inertia of $S$ about $L$ is $\frac { 5 } { 4 } M a ^ { 2 }$. The shape $S$ is at rest in a horizontal plane and is free to rotate about the axis $L$. A particle of mass $M$ falls vertically and strikes $S$ at the point $A$, where $O A = \frac { 3 } { 2 } a$ and $O A$ is perpendicular to $L$. The particle adheres to $S$ at $A$. Immediately before the particle strikes $S$ the speed of the particle is $u$.
(c) Find, in terms of $M$ and $u$, the loss in kinetic energy due to the impact.

Edexcel M5 2015 June Q1

A particle $P$ moves from the point $A$, with position vector ( $2 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k }$ ) m , where $a$ is a positive constant, to the point $B$, with position vector ( $- \mathbf { i } + a \mathbf { j } - \mathbf { k }$ ) m , under the action of a constant force $\mathbf { F } = ( 2 \mathbf { i } + a \mathbf { j } - 3 \mathbf { k } )$ N. The work done by $\mathbf { F }$, as it moves the particle $P$ from $A$ to $B$, is 3 J . Find the value of $a$.
(6)

Edexcel M5 2015 June Q2

2. A particle $P$ moves so that its position vector, $\mathbf { r }$ metres, at time $t$ seconds, where $0 \leqslant t < \frac { \pi } { 2 }$, satisfies the differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } - ( \tan t ) \mathbf { r } = ( \sin t ) \mathbf { i }$$ When $t = 0 , \mathbf { r } = - \frac { 1 } { 2 } \mathbf { i }$.
Find $\mathbf { r }$ in terms of $t$.

Edexcel M5 2015 June Q3

A rigid body is in equilibrium under the action of three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 } \mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ act at the points with position vectors $\mathbf { r } _ { 1 }$ and $\mathbf { r } _ { 2 }$ respectively, where $\mathbf { F } _ { 1 } = ( 2 \mathbf { j } + \mathbf { k } ) \mathrm { N } \quad \mathbf { r } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \mathrm { m } \mathbf { F } _ { 2 } = ( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { N } \quad \mathbf { r } _ { 2 } = ( - \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }$
1. Find the magnitude of $\mathbf { F } _ { 3 }$
2. Find a vector equation of the line of action of $\mathbf { F } _ { 3 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$, where $\mathbf { a }$ and $\mathbf { b }$ are constant vectors and $t$ is a scalar parameter.
\includegraphics[max width=\textwidth, alt={}, center]{cac4dd38-796c-414b-9b80-fe39ab12d41b-11_62_49_2643_1886}

Edexcel M5 2015 June Q4

4. A particle $P$, whose initial mass is $m _ { 0 }$, is projected vertically upwards from the ground at time $t = 0$ with speed $\frac { g } { k }$, where $k$ is a constant. As the particle moves upwards it gains mass by picking up small droplets of moisture from the atmosphere. The droplets are at rest before they are picked up. At time $t$ the speed of $P$ is $v$ and its mass has increased to $m _ { 0 } \mathrm { e } ^ { k t }$. Assuming that, during the motion, the acceleration due to gravity is constant,

show that, while $P$ is moving upwards, $$k v + \frac { \mathrm { d } v } { \mathrm {~d} t } = - g$$
find, in terms of $m _ { 0 }$, the mass of $P$ when it reaches its greatest height above the ground.
(6)

Edexcel M5 2015 June Q5

A uniform circular disc, of mass $m$ and radius $a$, is free to rotate about a fixed smooth horizontal axis $L$. The axis $L$ is a tangent to the disc at the point $A$. The centre $O$ of the disc moves in a vertical plane that is perpendicular to $L$.

The disc is held at rest with its plane horizontal and released.

Find the angular acceleration of the disc when it has turned through an angle of $\frac { \pi } { 3 }$
Find the magnitude of the component, in a direction perpendicular to the disc, of the force of the axis $L$ acting on the disc at $A$, when the disc has turned through an angle of $\frac { \pi } { 3 }$

Edexcel M5 2015 June Q6

A pendulum is modelled as a uniform rod $A B$, of mass $3 m$ and length $2 a$, which has a particle of mass $2 m$ attached at $B$. The pendulum is free to rotate in a vertical plane about a fixed smooth horizontal axis $L$ which passes through $A$. The vertical plane is perpendicular to the axis $L$.
1. Find the period of small oscillations of the pendulum about its position of stable equilibrium.
The pendulum is hanging at rest in a vertical position, with $B$ below $A$, when it is given a horizontal impulse of magnitude $J$. The impulse acts at $B$ in a vertical plane which is perpendicular to the axis $L$. Given that the pendulum turns through an angle of $60 ^ { \circ }$ before first coming to instantaneous rest,
find $J$.

Edexcel M5 2015 June Q7

7. (a) Find, using integration, the moment of inertia of a uniform solid hemisphere, of mass $m$ and radius $a$, about a diameter of its plane face.
[0pt] [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass $m$ and radius $r$, about a diameter is $\frac { 1 } { 4 } m r ^ { 2 }$.]
(b) Hence find the moment of inertia of a uniform solid sphere, of mass $M$ and radius $a$, about a diameter.

Edexcel M5 2016 June Q1

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

A bead $P$ of mass 0.4 kg is threaded on a smooth straight horizontal wire. The wire lies along the line with vector equation $\mathbf { r } = ( \mathbf { i } + 2 \mathbf { j } ) + \lambda ( - 2 \mathbf { i } + 3 \mathbf { j } )$. The bead is initially at rest at the point $A$ with position vector $( - \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }$. A constant horizontal force $( 0.5 \mathbf { i } + \mathbf { j } ) \mathrm { N }$ acts on $P$ and moves it along the wire to the point $B$. At $B$ the speed of $P$ is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the position vector of $B$.

Edexcel M5 2016 June Q2

A particle $P$ is moving in a plane. At time $t$ seconds the position vector of $P$ is $\mathbf { r }$ metres and the velocity of $P$ is $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$. When $t = \frac { \pi } { 2 } , P$ is instantaneously at rest at the point with position vector $( \mathbf { i } - \mathbf { j } ) \mathrm { m }$.

Given that $\mathbf { r }$ satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 4 \mathbf { r } = ( 3 \sin t ) \mathbf { i }$$ find $\mathbf { v }$ in terms of $t$.
(13)

Edexcel M5 2016 June Q3

Three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ act on a rigid body at the points with position vectors $\mathbf { r } _ { 1 } , \mathbf { r } _ { 2 }$ and $\mathbf { r } _ { 3 }$ respectively, where
$\mathbf { F } _ { 1 } = ( 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }$
$\mathbf { F } _ { 3 } = ( \mathbf { i } + \mathbf { j } ) \mathrm { N }$
$\mathbf { r } _ { 1 } = ( 4 \mathbf { j } - \mathbf { k } ) \mathrm { m }$
$\mathbf { r } _ { 3 } = ( 3 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }$
$\mathbf { F } _ { 1 } = ( 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }$
$\mathbf { r } _ { 1 } = ( 4 \mathbf { j } - \mathbf { k } ) \mathrm { m }$

$$\begin{aligned} & \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { k } ) \mathrm { N }
& \mathbf { r } _ { 2 } = ( 2 \mathbf { i } + \mathbf { k } ) \mathrm { m } \end{aligned}$$ j The system of the three forces is equivalent to a single force $\mathbf { R }$ acting through the point with position vector $( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }$, together with a couple of moment $\mathbf { G }$.

Find $\mathbf { R }$.
Find $\mathbf { G }$. respectively, where The

Edexcel M5 2016 June Q4

4. Find, using integration, the moment of inertia of a uniform cylindrical shell of radius $r$, height $h$ and mass $M$, about a diameter of one end.
(10)

Edexcel M5 2016 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f932d7cb-1299-41d1-8248-cfbf639795ed-08_613_649_221_644} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform piece of wire $A B C$, of mass $2 m$ and length $4 a$, is bent into two straight equal portions, $A B$ and $B C$, which are at right angles to each other, as shown in Figure 1. The wire rotates freely in a vertical plane about a fixed smooth horizontal axis $L$ which passes through $A$ and is perpendicular to the plane of the wire.

Show that the moment of inertia of the wire about $L$ is $\frac { 20 m a ^ { 2 } } { 3 }$
By writing down an equation of rotational motion for the wire as it rotates about $L$, find the period of small oscillations of the wire about its position of stable equilibrium.

Edexcel M5 2016 June Q6

6. A firework rocket, excluding its fuel, has mass $m _ { 0 } \mathrm {~kg}$. The rocket moves vertically upwards by ejecting burnt fuel vertically downwards with constant speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 } , u > 24.5$, relative to the rocket. The rocket starts from rest on the ground at time $t = 0$. At time $t$ seconds, $t \leqslant 2$, the speed of the rocket is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the mass of the rocket including its fuel is $m _ { 0 } ( 5 - 2 t ) \mathrm { kg }$. It is assumed that air resistance is negligible and the acceleration due to gravity is constant.

Show that, for $t \leqslant 2$ $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 2 u } { 5 - 2 t } - 9.8$$
Find the speed of the rocket at the instant when all of its fuel has been burnt.

Edexcel M5 2016 June Q7

7. A uniform square lamina $P Q R S$, of mass $m$ and side $2 a$, is free to rotate about a fixed smooth horizontal axis which passes through $P$ and $Q$. The lamina hangs at rest in a vertical plane with $S R$ below $P Q$ and is given a horizontal impulse of magnitude $J$ at the midpoint of $S R$. The impulse is perpendicular to $S R$.

Find the initial angular speed of the lamina.
Find the magnitude of the angular deceleration of the lamina at the instant when the lamina has turned through $\frac { \pi } { 6 }$ radians.
Find the magnitude of the component of the force exerted on the lamina by the axis, in a direction perpendicular to the lamina, at the instant when the lamina has turned through $\frac { \pi } { 6 }$ radians.
\includegraphics[max width=\textwidth, alt={}, center]{f932d7cb-1299-41d1-8248-cfbf639795ed-12_2255_50_315_1978}

Edexcel M5 2017 June Q1

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

A bead $P$ of mass 0.08 kg is threaded on a smooth straight horizontal wire which lies along the line with equation $y = 2 x - 1$. The unit of length on both axes is the metre. Initially the bead is at rest at the point $( a , b )$. A force $( 6 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }$ acts on $P$ and moves it along the wire so that $P$ passes through the point $( 5,9 )$ with speed $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the value of $a$ and the value of $b$.

Questions M5 (158 questions)

Browse by module

Browse by board

Edexcel M5 2014 June Q4

Edexcel M5 2014 June Q5

Edexcel M5 2014 June Q6

Edexcel M5 2014 June Q7

Edexcel M5 2014 June Q8

Edexcel M5 2014 June Q1

Edexcel M5 2014 June Q3

Edexcel M5 2014 June Q4

Edexcel M5 2014 June Q5

Edexcel M5 2014 June Q6

Edexcel M5 2015 June Q1

Edexcel M5 2015 June Q2

Edexcel M5 2015 June Q3

Edexcel M5 2015 June Q4

Edexcel M5 2015 June Q5

Edexcel M5 2015 June Q6

Edexcel M5 2015 June Q7

Edexcel M5 2016 June Q1

Edexcel M5 2016 June Q2

Edexcel M5 2016 June Q3

Edexcel M5 2016 June Q4

Edexcel M5 2016 June Q5

Edexcel M5 2016 June Q6

Edexcel M5 2016 June Q7

Edexcel M5 2017 June Q1