Questions M5 - A-Level Maths

Browse by module

All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4

Browse by board

AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4

Sort by: Default | Easiest first | Hardest first

Edexcel M5 Q7

16 marks Challenging +1.8

A uniform lamina of mass $m$ is in the shape of a triangle $ABC$. The perpendicular distance of $C$ from the line $AB$ is $h$. Prove, using integration, that the moment of inertia of the lamina about $AB$ is $\frac{1}{6}mh^2$. [7]

Deduce the radius of gyration of a uniform square lamina of side $2a$, about a diagonal. [3]

The points $X$ and $Y$ are the mid-points of the sides $RQ$ and $RS$ respectively of a square $PQRS$ of side $2a$. A uniform lamina of mass $M$ is in the shape of $PQXYS$.

Show that the moment of inertia of this lamina about $XY$ is $\frac{79}{84}Ma^2$. [6]

Edexcel M5 Q1

7 marks Standard +0.8

A particle moves in a plane in such a way that its position vector $\mathbf{r}$ metres at time $t$ seconds satisfies the differential equation $$\frac{d^2\mathbf{r}}{dt^2} - 2\frac{d\mathbf{r}}{dt} = 0$$ When $t = 0$, the particle is at the origin and is moving with velocity $(4i + 2j)$ m s$^{-1}$. Find $\mathbf{r}$ in terms of $t$. [7]

Edexcel M5 Q2

11 marks Challenging +1.3

Three forces $\mathbf{F}_1 = (3i - j + k)$ N, $\mathbf{F}_2 = (2i - k)$ N, and $\mathbf{F}_3$ act on a rigid body. The force $\mathbf{F}_1$ acts through the point with position vector $(i + 2j + k)$ m, the force $\mathbf{F}_2$ acts through the point with position vector $(i - 2j)$ m and the force $\mathbf{F}_3$ acts through the point with position vector $(i + j + k)$ m. Given that the system $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ reduces to a couple $\mathbf{G}$,

find $\mathbf{G}$. [6]

The line of action of $\mathbf{F}_3$ is changed so that the system $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ now reduces to a couple $(6i + 8j + 2k)$ N m.

Find an equation of the new 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. [5]

Edexcel M5 2006 June Q1

6 marks Standard +0.3

Prove, using integration, that the moment of inertia of a uniform rod, of mass $m$ and length $2a$, about an axis perpendicular to the rod through one end is $\frac{4}{3}ma^2$. [3]
Hence, or otherwise, find the moment of inertia of a uniform square lamina, of mass $M$ and side $2a$, about an axis through one corner and perpendicular to the plane of the lamina. [3]

Edexcel M5 2006 June Q2

9 marks Standard +0.8

A particle of mass 0.5 kg is at rest at the point with position vector $(2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})$ m. The particle is then acted upon by two constant forces $\mathbf{F}_1$ and $\mathbf{F}_2$. These are the only two forces acting on the particle. Subsequently, the particle passes through the point with position vector $(4\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})$ m with speed 12 m s$^{-1}$. Given that $\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j} - \mathbf{k})$ N, find $\mathbf{F}_2$. [9]

Edexcel M5 2006 June Q3

10 marks Challenging +1.2

A particle $P$ moves in the $x$-$y$ plane and has position vector $\mathbf{r}$ metres at time $t$ seconds. It is given that $\mathbf{r}$ satisfies the differential equation $$\frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2} = 2\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}.$$ When $t = 0$, $P$ is at the point with position vector $3\mathbf{i}$ metres and is moving with velocity $\mathbf{j}$ m s$^{-1}$.

Find $\mathbf{r}$ in terms of $t$. [8]
Describe the path of $P$, giving its cartesian equation. [2]

Edexcel M5 2006 June Q4

12 marks Challenging +1.2

A force system consists of three forces $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ acting on a rigid body. $\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j})$ N and acts at the point with position vector $(-\mathbf{i} + 4\mathbf{j})$ m. $\mathbf{F}_2 = (-\mathbf{j} + \mathbf{k})$ N and acts at the point with position vector $(2\mathbf{i} + \mathbf{j} + \mathbf{k})$ m. $\mathbf{F}_3 = (3\mathbf{i} - \mathbf{j} + \mathbf{k})$ N and acts at the point with position vector $(\mathbf{i} - \mathbf{j} + 2\mathbf{k})$ m. It is given that this system can be reduced to a single force $\mathbf{R}$.

Find $\mathbf{R}$. [2]
Find a vector equation of the line of action of $\mathbf{R}$, giving your answer in the form $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$, where $\mathbf{a}$ and $\mathbf{b}$ are constant vectors and $\lambda$ is a parameter. [10]

Edexcel M5 2006 June Q5

12 marks Challenging +1.8

A space-ship is moving in a straight line in deep space and needs to reduce its speed from $U$ to $V$. This is done by ejecting fuel from the front of the space-ship at a constant speed $k$ relative to the space-ship. When the speed of the space-ship is $v$, its mass is $m$.

Show that, while the space-ship is ejecting fuel, $\frac{\mathrm{d}m}{\mathrm{d}v} = -\frac{m}{k}$. [6]

The initial mass of the space-ship is $M$.

Find, in terms of $U$, $V$, $k$ and $M$, the amount of fuel which needs to be used to reduce the speed of the space-ship from $U$ to $V$. [6]

Edexcel M5 2006 June Q6

12 marks Challenging +1.3

A uniform circular disc, of mass $m$, radius $a$ and centre $O$, is free to rotate in a vertical plane about a fixed smooth horizontal axis. The axis passes through the mid-point $A$ of a radius of the disc.

Find an equation of motion for the disc when the line $AO$ makes an angle $\theta$ with the downward vertical through $A$. [5]
Hence find the period of small oscillations of the disc about its position of stable equilibrium. [2]

When the line $AO$ makes an angle $\theta$ with the downward vertical through $A$, the force acting on the disc at $A$ is $\mathbf{F}$.

Find the magnitude of the component of $\mathbf{F}$ perpendicular to $AO$. [5]

Edexcel M5 2006 June Q7

14 marks Challenging +1.8

Particles $P$ and $Q$ have mass $3m$ and $m$ respectively. Particle $P$ is attached to one end of a light inextensible string and $Q$ is attached to the other end. The string passes over a circular pulley which can freely rotate in a vertical plane about a fixed horizontal axis through its centre $O$. The pulley is modelled as a uniform circular disc of mass $2m$ and radius $a$. The pulley is sufficiently rough to prevent the string slipping. The system is at rest with the string taut. A third particle $R$ of mass $m$ falls freely under gravity from rest for a distance $a$ before striking and adhering to $Q$. Immediately before $R$ strikes $Q$, particles $P$ and $Q$ are at rest with the string taut.

Show that, immediately after $R$ strikes $Q$, the angular speed of the pulley is $\frac{1}{3}\sqrt{\frac{g}{2a}}$. [5]

When $R$ strikes $Q$, there is an impulse in the string attached to $Q$.

Find the magnitude of this impulse. [3]

Given that $P$ does not hit the pulley,

find the distance that $P$ moves upwards before first coming to instantaneous rest. [6]

$\mathbf{R}$, [2]
$\mathbf{G}$. [4]

A third force $\mathbf{F}_3$ is now added to the system. The force $\mathbf{F}_3$ acts at the point with position vector $(2\mathbf{i} - \mathbf{k})$ m and the three forces $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ are equivalent to a couple.

Find the magnitude of the couple. [6]

Edexcel M5 2011 June Q6

7 marks Challenging +1.8

A uniform rod $AB$ of mass $4m$ is free to rotate in a vertical plane about a fixed smooth horizontal axis, $L$, through $A$. The rod is hanging vertically at rest when it is struck at its end $B$ by a particle of mass $m$. The particle is moving with speed $u$, in a direction which is horizontal and perpendicular to $L$, and after striking the rod it rebounds in the opposite direction with speed $v$. The coefficient of restitution between the particle and the rod is $1$. Show that $u = 7v$. [7]

Edexcel M5 2011 June Q8

17 marks Challenging +1.3

A pendulum consists of a uniform rod $PQ$, of mass $3m$ and length $2a$, which is rigidly fixed at its end $Q$ to the centre of a uniform circular disc of mass $m$ and radius $a$. The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis $L$ which passes through the end $P$ of the rod and is perpendicular to the rod.

Show that the moment of inertia of the pendulum about $L$ is $\frac{33}{4}ma^2$. [5]

The pendulum is released from rest in the position where $PQ$ makes an angle $\alpha$ with the downward vertical. At time $t$, $PQ$ makes an angle $\theta$ with the downward vertical.

Show that the angular speed, $\dot{\theta}$, of the pendulum satisfies $$\dot{\theta}^2 = \frac{40g(\cos\theta - \cos\alpha)}{33a}$$ [4]
Hence, or otherwise, find the angular acceleration of the pendulum. [3]

Given that $\alpha = \frac{\pi}{20}$ and that $PQ$ has length $\frac{8}{33}$ m,

find, to 3 significant figures, an approximate value for the angular speed of the pendulum $0.2$ s after it has been released from rest. [5]

Edexcel M5 2012 June Q1

9 marks Challenging +1.3

A particle $P$ moves in a plane such that its position vector $\mathbf{r}$ metres at time $t$ seconds $(t > 0)$ satisfies the differential equation $$\frac{d\mathbf{r}}{dt} - \frac{2}{t}\mathbf{r} = 4\mathbf{i}$$ When $t = 1$, the particle is at the point with position vector $(\mathbf{i} + \mathbf{j})$ m. Find $\mathbf{r}$ in terms of $t$. [9]

Edexcel M5 2012 June Q2

10 marks Challenging +1.2

A rocket, with initial mass 1500 kg, including 600 kg of fuel, is launched vertically upwards from rest. The rocket burns fuel at a rate of 15 kg s$^{-1}$ and the burnt fuel is ejected vertically downwards with a speed of 1000 m s$^{-1}$ relative to the rocket. At time $t$ seconds after launch $(t \leqslant 40)$ the rocket has mass $m$ kg and velocity $v$ m s$^{-1}$.

Show that $$\frac{dv}{dt} + \frac{1000}{m}\frac{dm}{dt} = -9.8$$ [5]
Find $v$ at time $t$, $0 \leqslant t \leqslant 40$ [5]

Edexcel M5 2012 June Q3

12 marks Challenging +1.8

A uniform rod $PQ$ of mass $m$ and length $3a$, is free to rotate about a fixed smooth horizontal axis $L$, which passes through the end $P$ of the rod and is perpendicular to the rod. The rod hangs at rest in equilibrium with $Q$ vertically below $P$. One end of a light inextensible string of length $2a$ is attached to the rod at $P$ and the other end is attached to a particle of mass $3m$. The particle is held with the string taut, and horizontal and perpendicular to $L$, and is then released. After colliding, the particle sticks to the rod forming a body $B$.

Show that the moment of inertia of $B$ about $L$ is $15ma^2$. [2]
Show that $B$ first comes to instantaneous rest after it has turned through an angle $\arccos\left(\frac{9}{25}\right)$. [10]

Edexcel M5 2012 June Q4

6 marks Challenging +1.8

A body consists of a uniform plane circular disc, of radius $r$ and mass $2m$, with a particle of mass $3m$ attached to the circumference of the disc at the point $P$. The line $PQ$ is a diameter of the disc. The body is free to rotate in a vertical plane about a fixed smooth horizontal axis, $L$, which is perpendicular to the plane of the disc and passes through $Q$. The body is held with $QP$ making an angle $\beta$ with the downward vertical through $Q$, where $\sin \beta = 0.25$, and released from rest. Find the magnitude of the component, perpendicular to $PQ$, of the force acting on the body at $Q$ at the instant when it is released. [You may assume that the moment of inertia of the body about $L$ is $15mr^2$.] [6]

Edexcel M5 2012 June Q5

10 marks Standard +0.8

The points $P$ and $Q$ have position vectors $4\mathbf{i} - 6\mathbf{j} - 12\mathbf{k}$ and $2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k}$ respectively, relative to a fixed origin $O$. Three forces, $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$, act along $\overrightarrow{OP}$, $\overrightarrow{OQ}$ and $\overrightarrow{QP}$ respectively, and have magnitudes 7 N, 3 N and $3\sqrt{10}$ N respectively.

Express $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ in vector form. [3]
Show that the resultant of $\mathbf{F}_1$, $\mathbf{F}_2$ and $\mathbf{F}_3$ is $(2\mathbf{i} - 10\mathbf{j} - 16\mathbf{k})$ N. [2]
Find a vector equation of the line of action of this resultant, giving your answer in the form $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$, where $\mathbf{a}$ and $\mathbf{b}$ are constant vectors and $\lambda$ is a parameter. [5]

Edexcel M5 2012 June Q7

16 marks Challenging +1.2

A uniform lamina of mass $m$ is in the shape of a triangle $ABC$. The perpendicular distance of $C$ from the line $AB$ is $h$. Prove, using integration, that the moment of inertia of the lamina about $AB$ is $\frac{1}{6}mh^2$. [7]
Deduce the radius of gyration of a uniform square lamina of side $2a$, about a diagonal. [3]

The points $X$ and $Y$ are the mid-points of the sides $RQ$ and $RS$ respectively of a square $PQRS$ of side $2a$. A uniform lamina of mass $M$ is in the shape of $PQXYS$.