Questions M3 - 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

AQA M3 2009 June Q6

13 marks Standard +0.3

6 A smooth sphere $A$ of mass $m$ is moving with speed $5 u$ in a straight line on a smooth horizontal table. The sphere $A$ collides directly with a smooth sphere $B$ of mass $7 m$, having the same radius as $A$ and moving with speed $u$ in the same direction as $A$. The coefficient of restitution between $A$ and $B$ is $e$. \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-5_287_880_529_571}

Show that the speed of $B$ after the collision is $\frac { u } { 2 } ( e + 3 )$.
Given that the direction of motion of $A$ is reversed by the collision, show that $e > \frac { 3 } { 7 }$.
Subsequently, $B$ hits a wall fixed at right angles to the direction of motion of $A$ and $B$. The coefficient of restitution between $B$ and the wall is $\frac { 1 } { 2 }$. Given that after $B$ rebounds from the wall both spheres move in the same direction and collide again, show also that $e < \frac { 9 } { 13 }$.
(4 marks)

AQA M3 2009 June Q7

11 marks Challenging +1.2

7 A particle is projected from a point $O$ on a smooth plane which is inclined at $30 ^ { \circ }$ to the horizontal. The particle is projected down the plane with velocity $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $40 ^ { \circ }$ above the plane and first strikes it at a point $A$. The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-6_449_963_488_529}

Show that the time taken by the particle to travel from $O$ to $A$ is $$\frac { 20 \sin 40 ^ { \circ } } { g \cos 30 ^ { \circ } }$$
Find the components of the velocity of the particle parallel to and perpendicular to the slope as it hits the slope at $A$.
The coefficient of restitution between the slope and the particle is 0.5 . Find the speed of the particle as it rebounds from the slope.

Edexcel M3 2016 June Q1

8 marks Standard +0.3

A particle is attached to one end of a light inextensible string of length $l$. The other end of the string is attached to a fixed point $A$. The particle moves with constant angular speed $\omega$ in a horizontal circle. The centre of the circle is vertically below $A$ and the radius of the circle is $r$. Show that $\omega^2 = \frac{g}{\sqrt{l^2 - r^2}}$ [8]

Edexcel M3 2016 June Q2

9 marks Standard +0.3

A light elastic spring, of natural length $5a$ and modulus of elasticity $10mg$, has one end attached to a fixed point $A$ on a ceiling. A particle $P$ of mass $2m$ is attached to the other end of the spring and $P$ hangs freely in equilibrium at the point $O$.

Find the distance $AO$. [3]

The particle is now pulled vertically downwards a distance $\frac{1}{2}a$ from $O$ and released from rest.

Show that $P$ moves with simple harmonic motion. [4]
Find the period of the motion. [2]

Edexcel M3 2016 June Q3

7 marks Standard +0.8

A particle $P$ of mass $m$ is attached to one end of a light elastic string, of natural length $l$ and modulus of elasticity $4mg$. The other end of the string is attached to a fixed point $O$ on a rough horizontal plane. The coefficient of friction between $P$ and the plane is $\frac{2}{5}$. The particle is held at a point $A$ on the plane, where $OA = \frac{5}{4}l$, and released from rest. The particle comes to rest at the point $B$.

Show that $OB < l$ [4]
Find the distance $OB$. [3]

Edexcel M3 2016 June Q4

9 marks Standard +0.8

A particle $P$ of mass $m$ is fired vertically upwards from a point on the surface of the Earth and initially moves in a straight line directly away from the centre of the Earth. When $P$ is at a distance $x$ from the centre of the Earth, the gravitational force exerted by the Earth on $P$ is directed towards the centre of the Earth and has a magnitude which is inversely proportional to $x^2$. At the surface of the Earth the acceleration due to gravity is $g$. The Earth is modelled as a fixed sphere of radius $R$.

Show that the magnitude of the gravitational force acting on $P$ is $\frac{mgR^2}{x^2}$ [2]

The particle was fired with initial speed $U$ and the greatest height above the surface of the Earth reached by $P$ is $\frac{R}{20}$. Given that air resistance can be ignored,

find $U$ in terms of $g$ and $R$. [7]

Edexcel M3 2016 June Q5

11 marks Standard +0.8

A vertical ladder is fixed to a wall in a harbour. On a particular day the minimum depth of water in the harbour occurs at 0900 hours. The next time the water is at its minimum depth is 2115 hours on the same day. The bottom step of the ladder is 1 m above the lowest level of the water and 9 m below the highest level of the water. The rise and fall of the water level can be modelled as simple harmonic motion and the thickness of the step can be assumed to be negligible. Find

the speed, in metres per hour, at which the water level is moving when it reaches the bottom step of the ladder, [7]
the length of time, on this day, between the water reaching the bottom step of the ladder and the ladder being totally out of the water once more. [4]

Edexcel M3 2016 June Q6

14 marks Standard +0.8

\includegraphics{figure_1} A smooth solid hemisphere of radius 0.5 m is fixed with its plane face on a horizontal floor. The plane face has centre $O$ and the highest point of the surface of the hemisphere is $A$. A particle $P$ has mass 0.2 kg. The particle is projected horizontally with speed $u$ m s$^{-1}$ from $A$ and leaves the hemisphere at the point $B$, where $OB$ makes an angle $\theta$ with $OA$, as shown in Figure 1. The point $B$ is at a vertical distance of 0.1 m below the level of $A$. The speed of $P$ at $B$ is $v$ m s$^{-1}$

Show that $v^2 = u^2 + 1.96$ [3]
Find the value of $u$. [4]

The particle first strikes the floor at the point $C$.

Find the length of $OC$. [7]

Edexcel M3 2016 June Q7

17 marks Challenging +1.2

Use algebraic integration to show that the centre of mass of a uniform solid right circular cone of height $h$ is at a distance $\frac{3}{4}h$ from the vertex of the cone. [You may assume that the volume of a cone of height $h$ and base radius $r$ is $\frac{1}{3}\pi r^2 h$] [5]

\includegraphics{figure_2} A uniform solid $S$ consists of a right circular cone, of radius $r$ and height $5r$, fixed to a hemisphere of radius $r$. The centre of the plane face of the hemisphere is $O$ and this plane face coincides with the base of the cone, as shown in Figure 2.

Find the distance of the centre of mass of $S$ from $O$. [5]

The point $A$ lies on the circumference of the base of the cone. The solid is suspended by a string attached at $A$ and hangs freely in equilibrium.

Find the size of the angle between $OA$ and the vertical. [3]

The mass of the hemisphere is $M$. A particle of mass $kM$ is fixed to the surface of the hemisphere on the axis of symmetry of $S$. The solid is again suspended by the string attached at $A$ and hangs freely in equilibrium. The axis of symmetry of $S$ is now horizontal.

Find the value of $k$. [4]

Edexcel M3 Specimen Q1

7 marks Standard +0.3

\includegraphics{figure_1} A garden game is played with a small ball $B$ of mass $m$ attached to one end of a light inextensible string of length $13l$. The other end of the string is fixed to a point $A$ on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius $5l$ and centre $C$, where $C$ is vertically below $A$. Modelling the ball as a particle, find

the tension in the string, [3]
the speed of the ball. [4]

Edexcel M3 Specimen Q2

10 marks Standard +0.8

A particle $P$ of mass $m$ is above the surface of the Earth at distance $x$ from the centre of the Earth. The Earth exerts a gravitational force on $P$. The magnitude of this force is inversely proportional to $x^2$. At the surface of the Earth the acceleration due to gravity is $g$. The Earth is modelled as a sphere of radius $R$.

Prove that the magnitude of the gravitational force on $P$ is $\frac{mgR^2}{x^2}$. [3]

A particle is fired vertically upwards from the surface of the Earth with initial speed $3U$. At a height $R$ above the surface of the Earth the speed of the particle is $U$.

Find $U$ in terms of $g$ and $R$. [7]

Edexcel M3 Specimen Q3

9 marks Challenging +1.2

\includegraphics{figure_2} A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity $\lambda$ newtons. The other end of the spring is attached to a fixed point $O$ on a rough plane which is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac{3}{5}$. The coefficient of friction between the particle and the plane is 0.15. The particle is held on the plane at a point which is 1.5 m down the line of greatest slope from $O$, as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of $\lambda$. [9]

Edexcel M3 Specimen Q4

10 marks Standard +0.3

\includegraphics{figure_3} A container is formed by removing a right circular solid cone of height $4l$ from a uniform solid right circular cylinder of height $6l$. The centre $O$ of the plane face of the cone coincides with the centre of a plane face of the cylinder and the axis of the cone coincides with the axis of the cylinder, as shown in Figure 3. The cylinder has radius $2l$ and the base of the cone has radius $l$.

Find the distance of the centre of mass of the container from $O$. [6]

\includegraphics{figure_4} The container is placed on a plane which is inclined at an angle $\theta°$ to the horizontal. The open face is uppermost, as shown in Figure 4. The plane is sufficiently rough to prevent the container from sliding. The container is on the point of toppling.

Find the value of $\theta$. [4]

Edexcel M3 Specimen Q5

12 marks Standard +0.8

\includegraphics{figure_5} A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is fixed at the point $O$. The particle is initially held with $OP$ horizontal and the string taut. It is then projected vertically upwards with speed $u$, where $u^2 = 5ag$. When $OP$ has turned through an angle $\theta$ the speed of $P$ is $v$ and the tension in the string is $T$, as shown in Figure 5.

Find, in terms of $a$, $g$ and $\theta$, an expression for $v^2$. [3]
Find, in terms of $m$, $g$ and $\theta$, an expression for $T$. [4]
Prove that $P$ moves in a complete circle. [3]
Find the maximum speed of $P$. [2]

Edexcel M3 Specimen Q6

12 marks Standard +0.8

At time $t = 0$, a particle $P$ is at the origin $O$ moving with speed $2$ m s$^{-1}$ along the $x$-axis in the positive $x$-direction. At time $t$ seconds $(t > 0)$, the acceleration of $P$ has magnitude $\frac{3}{(t+1)^2}$ m s$^{-2}$ and is directed towards $O$.

Show that at time $t$ seconds the velocity of $P$ is $\left(\frac{3}{t+1} - 1\right)$ m s$^{-1}$. [5]
Find, to 3 significant figures, the distance of $P$ from $O$ when $P$ is instantaneously at rest. [7]

Edexcel M3 Specimen Q7

15 marks Standard +0.8

A light elastic string, of natural length $3a$ and modulus of elasticity $6mg$, has one end attached to a fixed point $A$. A particle $P$ of mass $2m$ is attached to the other end of the string and hangs in equilibrium at the point $O$, vertically below $A$.

Find the distance $AO$. [3]

The particle is now raised to point $C$ vertically below $A$, where $AC > 3a$, and is released from rest.

Show that $P$ moves with simple harmonic motion of period $2\pi\sqrt{\frac{a}{g}}$. [5]

It is given that $OC = \frac{1}{4}a$.

Find the greatest speed of $P$ during the motion. [3]

The point $D$ is vertically above $O$ and $OD = \frac{1}{8}a$. The string is cut as $P$ passes through $D$, moving upwards.

Find the greatest height of $P$ above $O$ in the subsequent motion. [4]

Edexcel M3 2002 January Q1

8 marks Standard +0.8

A particle $P$ of mass 0.2 kg moves away from the origin along the positive $x$-axis. It moves under the action of a force directed away from the origin $O$, of magnitude $\frac{5}{x+1}$ N, where $OP = x$ metres. Given that the speed of $P$ is 5 m s$^{-1}$ when $x = 0$, find the value of $x$, to 3 significant figures, when the speed of $P$ is 15 m s$^{-1}$. [8]

Edexcel M3 2002 January Q2

9 marks Standard +0.3

One end of a light elastic string, of natural length 2 m and modulus of elasticity 19.6 N, is attached to a fixed point $A$. A small ball $B$ of mass 0.5 kg is attached to the other end of the string. The ball is released from rest at $A$ and first comes to instantaneous rest at the point $C$, vertically below $A$.

Find the distance $AC$. [6]
Find the instantaneous acceleration of $B$ at $C$. [3]

Edexcel M3 2002 January Q3

10 marks Standard +0.3

\includegraphics{figure_1} A rod $AB$, of mass $2m$ and length $2a$, is suspended from a fixed point $C$ by two light strings $AC$ and $BC$. The rod rests horizontally in equilibrium with $AC$ making an angle $\alpha$ with the rod, where $\tan \alpha = \frac{3}{4}$, and with $AC$ perpendicular to $BC$, as shown in Fig. 1.

Give a reason why the rod cannot be uniform. [1]
Show that the tension in $BC$ is $\frac{4}{5}mg$ and find the tension in $AC$. [5]

The string $BC$ is elastic, with natural length $a$ and modulus of elasticity $kmg$, where $k$ is constant.

Find the value of $k$. [4]

Edexcel M3 2002 January Q4

10 marks Standard +0.3

\includegraphics{figure_2} Figure 2 shows the region $R$ bounded by the curve with equation $y^2 = rx$, where $r$ is a positive constant, the $x$-axis and the line $x = r$. A uniform solid of revolution $S$ is formed by rotating $R$ through one complete revolution about the $x$-axis.

Show that the distance of the centre of mass of $S$ from $O$ is $\frac{4}{5}r$. [6]

The solid is placed with its plane face on a plane which is inclined at an angle $\alpha$ to the horizontal. The plane is sufficiently rough to prevent $S$ from sliding. Given that $S$ does not topple,

find, to the nearest degree, the maximum value of $\alpha$. [4]

Edexcel M3 2002 January Q5

10 marks Challenging +1.2

A cyclist is travelling around a circular track which is banked at 25° to the horizontal. The coefficient of friction between the cycle's tyres and the track is 0.6. The cyclist moves with constant speed in a horizontal circle of radius 40 m, without the tyres slipping. Find the maximum speed of the cyclist. [10]

Edexcel M3 2002 January Q6

13 marks Standard +0.3

The points $O$, $A$, $B$ and $C$ lie in a straight line, in that order, where $OA = 0.6$ m, $OB = 0.8$ m and $OC = 1.2$ m. A particle $P$, moving along this straight line, has a speed of $\left(\frac{1}{10}\sqrt{5}\right)$ m s$^{-1}$ at $A$, $\left(\frac{1}{5}\sqrt{5}\right)$ m s$^{-1}$ at $B$ and is instantaneously at rest at $C$.

Show that this information is consistent with $P$ performing simple harmonic motion with centre $O$. [5]

Given that $P$ is performing simple harmonic motion with centre $O$,

show that the speed of $P$ at $O$ is 0.6 m s$^{-1}$, [2]
find the magnitude of the acceleration of $P$ as it passes $A$, [2]
find, to 3 significant figures, the time taken for $P$ to move directly from $A$ to $B$. [4]