Questions - Edexcel M3 - A-Level Maths

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Edexcel M3 2018 Specimen Q6

17 marks Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-20_442_723_237_605} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} The shaded region $R$ is bounded by part of the curve with equation $y = x ^ { 2 } + 3$, the $x$-axis, the $y$-axis and the line with equation $x = 2$, as shown in Figure 4. The unit of length on each axis is one centimetre. The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a uniform solid $S$.
Using algebraic integration,

show that the volume of $S$ is $\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }$,
show that, to 2 decimal places, the centre of mass of $S$ is 1.30 cm from $O$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-20_483_469_1402_767} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A uniform right circular solid cone, of base radius 7 cm and height 6 cm , is joined to $S$ to form a solid $T$. The base of the cone coincides with the larger plane face of $S$, as shown in Figure 5. The vertex of the cone is $V$.
The mass per unit volume of $S$ is twice the mass per unit volume of the cone.
Find the distance from $V$ to the centre of mass of $T$. The point $A$ lies on the circumference of the base of the cone. The solid $T$ is suspended from $A$ and hangs freely in equilibrium.
Find the size of the angle between $V A$ and the vertical.
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Q6

VIIIV SIHI NI JAIIM ION OC VIIIV SIHI NI JIHMM ION OO VI4V SIHI NI JIIYM IONOO

Edexcel M3 Q2

Challenging +1.2

2. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-004_513_399_303_785}

\end{figure} A uniform solid right circular cone has base radius $a$ and semi-vertical angle $\alpha$, where $\tan \alpha = \frac { 1 } { 3 }$. The cone is freely suspended by a string attached at a point $A$ on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of $\theta ^ { \circ }$ with the upward vertical, as shown in Figure 1. Find, to one decimal place, the value of $\theta$.

Edexcel M3 Q4

Challenging +1.2

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-006_574_510_324_726}

\end{figure} 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 attached to a point $O$. The point $A$ is vertically below $O$, and $O A = a$. The particle is projected horizontally from $A$ with speed $\sqrt { } ( 3 a g )$. When $O P$ makes an angle $\theta$ with the upward vertical through $O$ and the string is still taut, the tension in the string is $T$ and the speed of $P$ is $v$, as shown in Figure 2.

Find, in terms of $a , g$ and $\theta$, an expression for $v ^ { 2 }$.
Show that $T = ( 1 - 3 \cos \theta ) m g$. The string becomes slack when $P$ is at the point $B$.
Find, in terms of $a$, the vertical height of $B$ above $A$. After the string becomes slack, the highest point reached by $P$ is $C$.
Find, in terms of $a$, the vertical height of $C$ above $B$.

Edexcel M3 Q5

Standard +0.8

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-008_531_691_299_657}

\end{figure} One end of a light inextensible string is attached to a fixed point $A$. The other end of the string is attached to a fixed point $B$, vertically below $A$, where $A B = h$. A small smooth ring $R$ of mass $m$ is threaded on the string. The ring $R$ moves in a horizontal circle with centre $B$, as shown in Figure 3. The upper section of the string makes a constant angle $\theta$ with the downward vertical and $R$ moves with constant angular speed $\omega$. The ring is modelled as a particle.

Show that $\omega ^ { 2 } = \frac { g } { h } \left( \frac { 1 + \sin \theta } { \sin \theta } \right)$.
Deduce that $\omega > \sqrt { \frac { 2 g } { h } }$. Given that $\omega = \sqrt { \frac { 3 g } { h } }$,
find, in terms of $m$ and $g$, the tension in the string.

Edexcel M3 Q6

Standard +0.3

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-010_515_1015_319_477}

\end{figure} The shaded region $R$ is bounded by the curve with equation $y = \frac { 1 } { 2 x ^ { 2 } }$, the $x$-axis and the lines $x = 1$ and $x = 2$, as shown in Figure 4. The unit of length on each axis is 1 m . A uniform solid $S$ has the shape made by rotating $R$ through $360 ^ { \circ }$ about the $x$-axis.

Show that the centre of mass of $S$ is $\frac { 2 } { 7 } \mathrm {~m}$ from its larger plane face. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{ab85ec29-b1fc-45a9-9343-09feb33ab6c5-010_616_431_1420_778}
\end{figure} A sporting trophy $T$ is a uniform solid hemisphere $H$ joined to the solid $S$. The hemisphere has radius $\frac { 1 } { 2 } \mathrm {~m}$ and its plane face coincides with the larger plane face of $S$, as shown in Figure 5. Both $H$ and $S$ are made of the same material.
Find the distance of the centre of mass of $T$ from its plane face.

Edexcel M3 2003 January Q1

5 marks Standard +0.3

1. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{044c5866-0a12-4309-8ced-b463e1615fb0-2_383_789_335_681}

\end{figure} A particle of mass 5 kg is attached to one end of two light elastic strings. The other ends of the strings are attached to a hook on a beam. The particle hangs in equilibrium at a distance 120 cm below the hook with both strings vertical, as shown in Fig. 1. One string has natural length 100 cm and modulus of elasticity 175 N . The other string has natural length 90 cm and modulus of elasticity $\lambda$ newtons. Find the value of $\lambda$.
(5)

Edexcel M3 2003 January Q2

9 marks Standard +0.3

2. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{044c5866-0a12-4309-8ced-b463e1615fb0-2_389_601_1362_693}

\end{figure} A light inextensible string of length $8 l$ has its ends fixed to two points $A$ and $B$, where $A$ is vertically above $B$. A small smooth ring of mass $m$ is threaded on the string. The ring is moving with constant speed in a horizontal circle with centre $B$ and radius 3l, as shown in Fig. 2. Find

the tension in the string,
the speed of the ring.
State briefly in what way your solution might no longer be valid if the ring were firmly attached to the string.
(1) \section*{3.} \section*{Figure 3}
\includegraphics[max width=\textwidth, alt={}]{044c5866-0a12-4309-8ced-b463e1615fb0-3_564_1051_438_541}
A child's toy consists of a uniform solid hemisphere attached to a uniform solid cylinder. The plane face of the hemisphere coincides with the plane face of the cylinder, as shown in Fig. 3. The cylinder and the hemisphere each have radius $r$, and the height of the cylinder is $h$. The material of the hemisphere is 6 times as dense as the material of the cylinder. The toy rests in equilibrium on a horizontal plane with the cylinder above the hemisphere and the axis of the cylinder vertical.

Edexcel M3 2003 January Q4

11 marks Standard +0.3

4. A piston $P$ in a machine moves in a straight line with simple harmonic motion about a point $O$, which is the centre of the oscillations. The period of the oscillations is $\pi \mathrm { s }$. When $P$ is 0.5 m from $O$, its speed is $2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find

the amplitude of the motion,
the maximum speed of $P$ during the motion,
the maximum magnitude of the acceleration of $P$ during the motion,
the total time, in s to 2 decimal places, in each complete oscillation for which the speed of $P$ is greater than $2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Edexcel M3 2003 January Q5

12 marks Standard +0.8

5. A car of mass 800 kg moves along a horizontal straight road. At time $t$ seconds, the resultant force acting on the car has magnitude $\frac { 48000 } { ( t + 2 ) ^ { 2 } }$ newtons, acting in the direction of the motion of the car. When $t = 0$, the car is at rest.

Show that the speed of the car approaches a limiting value as $t$ increases and find this value.
Find the distance moved by the car in the first 6 seconds of its motion.

Edexcel M3 2003 January Q6

12 marks Standard +0.8

6. A light elastic string has natural length 4 m and modulus of elasticity 58.8 N . A particle $P$ of mass 0.5 kg is attached to one end of the string. The other end of the string is attached to a vertical point $A$. The particle is released from rest at $A$ and falls vertically.

Find the distance travelled by $P$ before it immediately comes to instantaneous rest for the first time. The particle is now held at a point 7 m vertically below $A$ and released from rest.
Find the speed of the particle when the string first becomes slack.

Edexcel M3 2003 January Q7

16 marks Challenging +1.2

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{044c5866-0a12-4309-8ced-b463e1615fb0-5_604_596_391_760}

\end{figure} Part of a hollow spherical shell, centre $O$ and radius $a$, is removed to form a bowl with a plane circular rim. The bowl is fixed with the circular rim uppermost and horizontal. The point $A$ is the lowest point of the bowl. The point $B$ is on the rim of the bowl and $\angle A O B = 120 ^ { \circ }$, as shown in Fig. 4. A smooth small marble of mass $m$ is placed inside the bowl at $A$ and given an initial horizontal speed $u$. The direction of motion of the marble lies in the vertical plane $A O B$. The marble stays in contact with the bowl until it reaches $B$. When the marble reaches $B$, its speed is $v$.

Find an expression for $v ^ { 2 }$.
For the case when $u ^ { 2 } = 6 g a$, find the normal reaction of the bowl on the marble as the marble reaches $B$.
Find the least possible value of $u$ for the marble to reach $B$. The point $C$ is the other point on the rim of the bowl lying in the vertical plane $O A B$.
Find the value of $u$ which will enable the marble to leave the bowl at $B$ and meet it again at the point $C$.

Edexcel M3 2004 January Q2

9 marks Moderate -0.3

2. A particle $P$ moves along the $x$-axis. At time $t$ seconds its acceleration is $\left( - 4 \mathrm { e } ^ { - 2 t } \right) \mathrm { m } \mathrm { s } ^ { - 2 }$ in the direction of $x$ increasing. When $t = 0 , P$ is at the origin $O$ and is moving with speed $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the direction of $x$ increasing.

Find an expression for the velocity of $P$ at time $t$.
Find the distance of $P$ from $O$ when $P$ comes to instantaneous rest.
(6)

Edexcel M3 2004 January Q3

10 marks Standard +0.8

3. Above the earth's surface, the magnitude of the force on a particle due to the earth's gravity is inversely proportional to the square of the distance of the particle from the centre of the earth. Assuming that the earth is a sphere of radius $R$, and taking $g$ as the acceleration due to gravity at the surface of the earth,

prove that the magnitude of the gravitational force on a particle of mass $m$ when it is a distance $x ( x \geq R )$ from the centre of the earth is $\frac { m g R ^ { 2 } } { x ^ { 2 } }$. A particle is fired vertically upwards from the surface of the earth with initial speed $u$, where $u ^ { 2 } = \frac { 3 } { 2 } g R$. Ignoring air resistance,
find, in terms of $g$ and $R$, the speed of the particle when it is at a height $2 R$ above the surface of the earth.

Edexcel M3 2004 January Q4

11 marks Challenging +1.2

4. A particle $P$ of mass $m$ is attached to one end of a light elastic string of length $a$ and modulus of elasticity $\frac { 1 } { 2 } m g$. The other end of the string is fixed at the point $A$ which is at a height $2 a$ above a smooth horizontal table. The particle is held on the table with the string making an angle $\beta$ with the horizontal, where $\tan \beta = \frac { 3 } { 4 }$.

Find the elastic energy stored in the string in this position. The particle is now released. Assuming that $P$ remains on the table,
find the speed of $P$ when the string is vertical. By finding the vertical component of the tension in the string when $P$ is on the table and $A P$ makes an angle $\theta$ with the horizontal,
show that the assumption that $P$ remains in contact with the table is justified.

Edexcel M3 2004 January Q5

12 marks Standard +0.3

5. A piston in a machine is modelled as a particle of mass 0.2 kg attached to one end $A$ of a light elastic spring, of natural length 0.6 m and modulus of elasticity 48 N . The other end $B$ of the spring is fixed and the piston is free to move in a horizontal tube which is assumed to be smooth. The piston is released from rest when $A B = 0.9 \mathrm {~m}$.

Prove that the motion of the piston is simple harmonic with period $\frac { \pi } { 10 } \mathrm {~s}$.
(5)
Find the maximum speed of the piston.
(2)
Find, in terms of $\pi$, the length of time during each oscillation for which the length of the spring is less than 0.75 m .
(5)

Edexcel M3 2004 January Q6

12 marks Standard +0.8

6. Figure 2 \includegraphics[max width=\textwidth, alt={}, center]{c4b453e7-8a32-458b-8041-58c9e4ef9533-5_691_1067_241_584} A uniform solid cylinder has radius $2 a$ and height $\frac { 3 } { 2 } a$. A hemisphere of radius $a$ is removed from the cylinder. The plane face of the hemisphere coincides with the upper plane face of the cylinder, and the centre $O$ of the hemisphere is also the centre of this plane face, as shown in Fig. 2. The remaining solid is $S$.

Find the distance of the centre of mass of $S$ from $O$.
(6) The lower plane face of $S$ rests in equilibrium on a desk lid which is inclined at an angle $\theta$ to the horizontal. Assuming that the lid is sufficiently rough to prevent $S$ from slipping, and that $S$ is on the point of toppling when $\theta = \alpha$,
find the value of $\alpha$.
(3) Given instead that the coefficient of friction between $S$ and the lid is 0.8 , and that $S$ is on the point of sliding down the lid when $\theta = \beta$,
find the value of $\beta$.
(3)

Edexcel M3 2004 January Q7

14 marks Challenging +1.2

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{c4b453e7-8a32-458b-8041-58c9e4ef9533-6_710_729_172_672}

\end{figure} 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 a point $O$. The particle is held with the string taut and $O P$ horizontal. It is then projected vertically downwards with speed $u$, where $u ^ { 2 } = \frac { 3 } { 2 } g a$. When $O P$ has turned through an angle $\theta$ and the string is still taut, the speed of $P$ is $v$ and the tension in the string is $T$, as shown in Fig. 3.

Find an expression for $v ^ { 2 }$ in terms of $a , g$ and $\theta$.
Find an expression for $T$ in terms of $m , g$ and $\theta$.
Prove that the string becomes slack when $\theta = 210 ^ { \circ }$.
State, with a reason, whether $P$ would complete a vertical circle if the string were replaced by a light rod. After the string becomes slack, $P$ moves freely under gravity and is at the same level as $O$ when it is at the point $A$.
Explain briefly why the speed of $P$ at $A$ is $\sqrt { } \left( \frac { 3 } { 2 } g a \right)$. The direction of motion of $P$ at $A$ makes an angle $\varphi$ with the horizontal.
Find $\varphi$.

Edexcel M3 2006 January Q1

8 marks Moderate -0.3

1. \section*{Figure 1}

\includegraphics[max width=\textwidth, alt={}]{67a9cf74-833f-4b4a-9fde-3c62dcc08e8c-2_515_1157_276_516}

A particle $P$ of mass 0.8 kg is attached to one end of a light inelastic string, of natural length 1.2 m and modulus of elasticity 24 N . The other end of the string is attached to a fixed point $A$. A horizontal force of magnitude $F$ newtons is applied to $P$. The particle $P$ in in equilibrium with the string making an angle $60 ^ { \circ }$ with the downward vertical, as shown in Figure 1. Calculate

the value of $F$,
the extension of the string,
the elasticity stored in the string.

Edexcel M3 2006 January Q2

8 marks Moderate -0.3

2. A particle $P$ moves along the $x$-axis. At time $t$ seconds the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and its acceleration is $2 \sin \frac { 1 } { 2 } t \mathrm {~m} \mathrm {~s} ^ { - 2 }$, both measured in the direction of $O x$. Given that $v = 4$ when $t = 0$,

find $v$ in terms of $t$,
calculate the distance travelled by $P$ between the times $t = 0$ and $t = \frac { \pi } { 2 }$.

Edexcel M3 2006 January Q3

8 marks Standard +0.3

3. A rocket is fired vertically upwards with speed $U$ from a point on the Earth's surface. The rocket is modelled as a particle $P$ of constant mass $m$, and the Earth as a fixed sphere of radius $R$. At a distance $x$ from the centre of the Earth, the speed of $P$ is $v$. The only force acting on $P$ is directed towards the centre of the Earth and has magnitude $\frac { c m } { x ^ { 2 } }$, where $c$ is a constant.

Show that $v ^ { 2 } = U ^ { 2 } + 2 c \left( \frac { 1 } { x } - \frac { 1 } { R } \right)$. The kinetic energy of $P$ at $x = 2 R$ is half of its kinetic energy at $x = R$.
Find $c$ in terms of $U$ and $R$.
(3)

Edexcel M3 2006 January Q4

9 marks Standard +0.3

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{67a9cf74-833f-4b4a-9fde-3c62dcc08e8c-3_531_387_1226_845}

\end{figure} A body consists of a uniform solid circular cylinder $C$, together with a uniform solid hemisphere $H$ which is attached to $C$. The plane face of $H$ coincides with the upper plane face of $C$, as shown in Figure 2. The cylinder $C$ has base radius $r$, height $h$ and mass 3M. The mass of $H$ is $2 M$. The point $O$ is the centre of the base of $C$.

Show that the distance of the centre of mass of the body from $O$ is $$\frac { 14 h + 3 r } { 20 } .$$ The body is placed with its plane face on a rough plane which is inclined at an angle $\alpha$ to the horizontal, where tan $\alpha = \frac { 4 } { 3 }$. The plane is sufficiently rough to prevent slipping. Given that the body is on the point of toppling,
find $h$ in terms of $r$.

Edexcel M3 2006 January Q5

13 marks Standard +0.8

5. A light elastic string of natural length $l$ has one end attached to a fixed point $A$. A particle $P$ of mass $m$ is attached tot he other end of the string and hangs in equilibrium at the point $O$, where $A O = \frac { 5 } { 4 } l$.

Find the modulus of the elasticity of the string. The particle $P$ is then pulled down and released from rest. At time $t$ the length of the string is $\frac { 5 l } { 4 } + x$.
Prove that, while the string is taut, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g x } { l }$$ When $P$ is released, $A P = \frac { 7 } { 4 } l$. The point $B$ is a distance $l$ vertically below $A$.
Find the speed of $P$ at $B$.
Describe briefly the motion of $P$ after it has passed through $B$ for the first time until it next passes through $O$.

Edexcel M3 2006 January Q6

14 marks Standard +0.8

6. One end of a light inextensible string of length $l$ is attached to a fixed point $A$. The other end is attached to a particle $P$ of mass $m$ which is hanging freely at rest at point $B$. The particle $P$ is projected horizontally from $B$ with speed $\sqrt { } ( 3 g l )$. When $A P$ makes an angle $\theta$ with the downward vertical and the string remains taut, the tension in the string is $T$.

Show that $T = m g ( 1 + 3 \cos \theta )$.
Find the speed of $P$ at the instant when the string becomes slack.
Find the maximum height above the level of $B$ reached by $P$.

Edexcel M3 2006 January Q7

15 marks Standard +0.3

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{67a9cf74-833f-4b4a-9fde-3c62dcc08e8c-5_625_1141_319_424}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $2 a$. The other end of the string is fixed to a point $A$ which is vertically above the point $O$ on a smooth horizontal table. The particle $P$ remains in contact with the surface of the table and moves in a circle with centre $O$ and with angular speed $\sqrt { \left( \frac { k g } { 3 a } \right) }$, where $k$ is a constant. Throughout the motion the string remains taut and $\angle A P O = 30 ^ { \circ }$, as shown in Figure 3.

Show that the tension in the string is $\frac { 2 k m g } { 3 }$.
Find, in terms of $m , g$ and $k$, the normal reaction between $P$ and the table.
Deduce the range of possible values of $k$. The angular speed of $P$ is changed to $\sqrt { \left( \frac { 2 g } { a } \right) }$. The particle $P$ now moves in a horizontal circle above the table. The centre of this circle is $X$.
Show that $X$ is the mid-point of $O A$.

Edexcel M3 2007 January Q1

6 marks Moderate -0.3

A particle $P$ moves along the $x$-axis. At time $t = 0 , P$ passes through the origin $O$, moving in the positive $x$-direction. At time $t$ seconds, the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $O P = x$ metres. The acceleration of $P$ is $\frac { 1 } { 12 } ( 30 - x ) \mathrm { m } \mathrm { s } ^ { - 2 }$, measured in the positive $x$-direction.
1. Give a reason why the maximum speed of $P$ occurs when $x = 30$.
Given that the maximum speed of $P$ is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$,
find an expression for $v ^ { 2 }$ in terms of $x$.

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Questions — Edexcel M3 (510 questions)

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Edexcel M3 2018 Specimen Q6

Edexcel M3 Q2

Edexcel M3 Q4

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Edexcel M3 2003 January Q1

Edexcel M3 2003 January Q2

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