Questions - Edexcel - 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 M3 2019 January Q2

12 marks Challenging +1.2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-04_573_456_264_712} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A small ball $P$ of mass $m$ is attached to the midpoint of a light inextensible string of length $2 a$. The ends of the string are attached to fixed points $A$ and $B$, where $A$ is vertically above $B$ and $A B = a$, as shown in Figure 1. The system rotates about the line $A B$ with constant angular speed $\omega$. The ball moves in a horizontal circle with both parts of the string taut. The tension in the string must be less than $3 m g$ otherwise the string will break. Given that the time taken by the ball to complete one revolution is $S$, show that $$\pi \sqrt { \frac { a } { g } } < S < \pi \sqrt { \frac { k a } { g } }$$ stating the value of the constant $k$.

Edexcel M3 2019 January Q3

12 marks Standard +0.3

3. A particle $P$ is moving in a straight line with simple harmonic motion between two points $A$ and $B$, where $A B$ is $2 a$ metres. The point $C$ lies on the line $A B$ and $A C = \frac { 1 } { 2 } a$ metres. The particle passes through $C$ with speed $\frac { 3 a \sqrt { 3 } } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the period of the motion. The maximum magnitude of the acceleration of $P$ is $45 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Find
the value of $a$,
the maximum speed of $P$. The point $D$ lies on $A B$ and $P$ takes a quarter of one period to travel directly from $C$ to $D$.
Find the distance CD.

Edexcel M3 2019 January Q4

13 marks Challenging +1.2

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-12_364_718_278_612} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The ends of a light elastic string, of natural length $4 l$ and modulus of elasticity $\lambda$, are attached to two fixed points $A$ and $B$, where $A B$ is horizontal and $A B = 4 l$. A particle $P$ of mass $2 m$ is attached to the midpoint of the string. The particle hangs freely in equilibrium at a distance $\frac { 3 } { 2 } l$ vertically below the midpoint of $A B$, as shown in Figure 2.

Show that $\lambda = \frac { 20 } { 3 } m g$. The particle is pulled vertically downwards from its equilibrium position until the total length of the string is 6l. The particle is then released from rest.
Show that $P$ comes to instantaneous rest before reaching the line $A B$.

Edexcel M3 2019 January Q5

16 marks Standard +0.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-16_492_442_237_744} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The region $R$, shown shaded in Figure 3, is bounded by the circle with centre $O$ and radius $r$, the line with equation $x = \frac { 3 } { 5 } r$ and the $x$-axis. The region is rotated through one complete revolution about the $x$-axis to form a uniform solid $S$.

Use algebraic integration to show that the $x$ coordinate of the centre of mass of $S$ is $\frac { 48 } { 65 } r$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-16_394_643_1311_653} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A bowl is made from a uniform solid hemisphere of radius 6 cm by removing a hemisphere of radius 5 cm . Both hemispheres have the same centre $A$ and the same axis of symmetry. The bowl is fixed with its open plane face uppermost and horizontal. Liquid is poured into the bowl. The depth of the liquid is 2 cm , as shown in Figure 4. The mass of the empty bowl is $5 M \mathrm {~kg}$ and the mass of the liquid is $2 M \mathrm {~kg}$.
Find, to 3 significant figures, the distance from $A$ to the centre of mass of the bowl with its liquid.

Edexcel M3 2019 January Q6

16 marks Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-20_497_643_237_653} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a hollow sphere, with centre $O$ and internal radius $a$, which is fixed to a horizontal surface. A particle $P$ of mass $m$ is projected horizontally with speed $\sqrt { \frac { 7 a g } { 2 } }$ from the lowest point $A$ of the inner surface of the sphere. The particle moves in a vertical circle with centre $O$ on the smooth inner surface of the sphere. The particle passes through the point $B$, on the inner surface of the sphere, where $O B$ is horizontal.

Find, in terms of $m$ and $g$, the normal reaction exerted on $P$ by the surface of the sphere when $P$ is at $B$. The particle leaves the inner surface of the sphere at the point $C$, where $O C$ makes an angle $\theta , \theta > 0$, with the upward vertical.
Show that, after leaving the surface of the sphere at $C$, the particle is next in contact with the surface at $A$.
END

Edexcel M3 2021 January Q1

8 marks Standard +0.8

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-02_469_758_251_593} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The region $R$, shown shaded in Figure 1, is bounded by the curve with equation $y = \frac { 1 } { x }$, the line with equation $x = 1$, the positive $x$-axis and the line with equation $x = a$ where $a > 1$ A uniform solid $S$ is formed by rotating $R$ through $2 \pi$ radians about the $x$-axis.

Show that the volume of $S$ is $$\pi \left( 1 - \frac { 1 } { a } \right)$$
Find the $x$ coordinate of the centre of mass of $S$.

Edexcel M3 2021 January Q2

10 marks Standard +0.8

2. A particle $P$ of mass $m$ is at a distance $x$ above the surface of the Earth. The Earth exerts a gravitational force on $P$. This force is directed towards the centre of the Earth. The magnitude of this force is inversely proportional to the square of the distance of $P$ from the centre of the Earth. 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 on $P$ is $\frac { m g R ^ { 2 } } { ( x + R ) ^ { 2 } }$ A particle is released from rest from a point above the surface of the Earth. When the particle is at a distance $R$ above the surface of the Earth, the particle has speed $U$. Air resistance is modelled as being negligible.
Find, in terms of $U , g$ and $R$, the speed of the particle when it strikes the surface of the Earth.

VIAV SIHI NI III IM I ON OC VIAV SIMI NI III M M O N OO VIUV SIMI NI JIIYM ION OC

Edexcel M3 2021 January Q3

11 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-08_506_527_251_712} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A fairground ride consists of a cabin $C$ that travels in a horizontal circle with a constant angular speed about a fixed vertical central axis. The cabin is attached to one end of each of two rigid arms, each of length 5 m . The other end of the top arm is attached to the fixed point $A$ at the top of the central axis of the ride. The other end of the lower arm is attached to the fixed point $B$ on the central axis, where $A B$ is 8 m , as shown in Figure 2. Both arms are free to rotate about the central axis. The arms are modelled as light inextensible rods. The cabin, together with the people inside, is modelled as a particle. The cabin completes one revolution every 2 seconds. Given that the combined mass of the cabin and the people is 600 kg ,

find
1. the tension in the upper arm of the ride,
2. the tension in the lower arm of the ride. In a refined model, it is assumed that both arms stretch to a length of 5.1 m .
State how this would affect the sum of the tensions in the two arms, justifying your answer.

Edexcel M3 2021 January Q4

9 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-12_442_506_251_721} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform right solid cone $C$ has diameter $6 a$ and height $8 a$, as shown in Figure 3.
The solid $S$ is formed by removing a cone of height $4 a$ from the top of $C$ and then removing an identical, inverted cone. The vertex of the removed cone is at the point $O$ in the centre of the base of $C$, as shown in Figure 4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-12_236_502_1126_721} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure}

Find the distance of the centre of mass of $S$ from $O$.
(5) The point $A$ lies on the circumference of the base of $S$ and the point $B$ lies on the circumference of the top of $S$. The points $O$, $A$ and $B$ all lie in the same vertical plane, as shown in Figure 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-12_248_449_1845_749} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The solid $S$ is freely suspended from the point $B$ and hangs in equilibrium.
Find the size of the angle that $A B$ makes with the downward vertical.

Edexcel M3 2021 January Q5

13 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-16_720_232_251_858} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} The fixed points, $A$ and $B$, are a distance $10 a$ apart, with $B$ vertically above $A$. One end of a light elastic string, of natural length $2 a$ and modulus of elasticity $2 m g$, is attached to a particle $P$ of mass $m$ and the other end is attached to $A$. One end of another light elastic string, of natural length $4 a$ and modulus of elasticity $6 m g$, is attached to $P$ and the other end is attached to $B$. The particle $P$ rests in equilibrium at the point $C$, as shown in Figure 6.

Show that each string has an extension of $2 a$.
(5) The particle $P$ is now pulled down vertically, so that it is a distance $a$ below $C$ and then released from rest.
Show that in the subsequent motion, $P$ performs simple harmonic motion.
Find, in terms of $a$ and $g$, the speed of $P$ when it is a distance $\frac { 7 } { 2 } a$ above $A$.

Edexcel M3 2021 January Q6

13 marks Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-20_789_858_121_536} \captionsetup{labelformat=empty} \caption{Figure 7}

\end{figure} A particle of mass $m$ is attached to one end of a light inextensible string of length $2 a$. The other end of the string is attached to a fixed point $O$. The particle is initially held at the point $A$ with the string taut and $O A$ making an angle of $60 ^ { \circ }$ with the downward vertical. The particle is then projected upwards with a speed of $3 \sqrt { a g }$, perpendicular to $O A$, in the vertical plane containing $O A$, as shown in Figure 7. In an initial model of the motion of the particle, it is assumed that the string does not break. Using this model,

show that the particle performs complete vertical circles. In a refined model it is assumed that the string will break if the tension in it exceeds 7 mg . Using this refined model,
show that the particle still performs complete vertical circles. \includegraphics[max width=\textwidth, alt={}, center]{8a687d17-ec7e-463f-84dd-605f5c230db1-20_2249_50_314_1982}

Edexcel M3 2021 January Q7

11 marks Challenging +1.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-24_394_1027_248_461} \captionsetup{labelformat=empty} \caption{Figure 8}

\end{figure} A particle $P$ of mass 0.5 kg is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity 3 N . The other end of the string is attached to a fixed point $O$ on a rough plane. The plane is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 2 } { 7 }$ The coefficient of friction between $P$ and the plane is $\frac { \sqrt { 5 } } { 5 }$ The particle $P$ is initially at rest at the point $O$, as shown in Figure 8. The particle $P$ then receives an impulse of magnitude 4 Ns, directed up a line of greatest slope of the plane. The particle $P$ moves up the plane and comes to rest at the point $A$.

Find the extension of the elastic string when $P$ is at $A$.
Show that the particle does not remain at rest at $A$.

Edexcel M3 2022 January Q1

6 marks Standard +0.3

A light elastic string $A B$ has natural length $11 a$ and modulus of elasticity $6 m g$

A particle of mass $4 m$ is attached to the point $C$ on the string where $A C = 8 a$ and a particle of mass $2 m$ is attached to the end $B$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-02_581_202_429_957} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The end $A$ of the string is attached to a fixed point and the string hangs vertically below $A$ with the particle of mass $4 m$ in equilibrium at the point $P$ and the particle of mass $2 m$ in equilibrium at the point $Q$, as shown in Figure 1.

Find the length $A P$
Find the length $P Q$

Edexcel M3 2022 January Q2

7 marks Standard +0.8

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-04_479_853_246_607} \captionsetup{labelformat=empty} \caption{Figure 2}

\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 $A$ which lies above a smooth horizontal table. The particle $P$ moves in a horizontal circle on the table with the string taut. The centre of the circle is the point $O$ on the table, where $A O$ is vertical and the string makes a constant angle $\theta ^ { \circ }$ with $A O$, as shown in Figure 2. Given that $P$ moves with constant angular speed $\sqrt { \frac { 2 g } { a } }$, find the range of possible values of $\theta$

Edexcel M3 2022 January Q3

8 marks Challenging +1.2

A particle $P$ is moving along the $x$-axis. At time $t$ seconds, where $t \geqslant 0 , P$ is $x$ metres from the origin $O$ and is moving with speed $v \mathrm {~ms} ^ { - 1 }$

The acceleration of $P$ has magnitude $\frac { 2 } { ( 2 x + 1 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }$ and is directed towards $O$ When $t = 0 , P$ passes through $O$ in the positive $x$ direction with speed $1 \mathrm {~ms} ^ { - 1 }$

Find $v$ in terms of $x$
Show that $x = \frac { 1 } { 2 } ( \sqrt { ( 4 t + 1 ) } - 1 )$

Edexcel M3 2022 January Q4

11 marks Standard +0.8

A uniform solid hemisphere $H$ has radius $r$ and centre $O$

Show that the centre of mass of $H$ is $\frac { 3 r } { 8 }$ from $O$ $$\left[ \text { You may assume that the volume of } H \text { is } \frac { 2 \pi r ^ { 3 } } { 3 } \right]$$ A uniform solid $S$, shown below in Figure 3, is formed by attaching a uniform solid right circular cylinder of height $h$ and radius $r$ to $H$, so that one end of the cylinder coincides with the plane face of $H$. The point $A$ is the point on $H$ such that $O A = r$ and $O A$ is perpendicular to the plane face of $H$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-12_592_791_909_660} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
Show that the distance of the centre of mass of $S$ from $A$ is $$\frac { 5 r ^ { 2 } + 12 r h + 6 h ^ { 2 } } { 8 r + 12 h }$$ The solid $S$ can rest in equilibrium on a horizontal plane with any point of the curved surface of the hemisphere in contact with the plane.
Find $r$ in terms of $h$.

Edexcel M3 2022 January Q5

13 marks Challenging +1.2

5. \begin{figure}[h]

[diagram]

\captionsetup{labelformat=empty} \caption{Figure 4}

\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 fixed point $O$. The particle $P$ is held at rest vertically below $O$, with the string taut, as shown in Figure 4. The particle is then projected horizontally with speed $u$, where $u > \sqrt { 2 a g }$ Air resistance is modelled as being negligible.
At the instant when the string makes an angle $\theta$ with the upward vertical through $O$, the speed of $P$ is $v$ and the string goes slack.

Show that $3 v ^ { 2 } = u ^ { 2 } - 2 a g$ From the instant when the string goes slack to the instant when $O P$ is next horizontal, $P$ moves as a projectile. The time from the instant when the string goes slack to the instant when $O P$ is next horizontal is $T$ Given that $\theta = 30 ^ { \circ }$
show that $T = \frac { 2 v } { g }$
Hence, show that the string goes taut again when it is next horizontal.

Edexcel M3 2022 January Q6

15 marks Challenging +1.2

A particle $P$ of mass $m$ is attached to one end of a light elastic spring of natural length 2l. The other end of the spring is attached to a fixed point $A$. The particle $P$ hangs in equilibrium vertically below $A$, at the point $E$ where $A E = 6 l$. The particle $P$ is then raised a vertical distance $2 l$ and released from rest.

Air resistance is modelled as being negligible.

Show that $P$ moves with simple harmonic motion of period $T$ where $$T = 4 \pi \sqrt { \frac { l } { g } }$$
Find, in terms of $m , l$ and $g$, the kinetic energy of $P$ as it passes through $E$
Find, in terms of $T$, the exact time from the instant when $P$ is released to the instant when $P$ has moved a distance 31 .

Edexcel M3 2022 January Q7

15 marks Challenging +1.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-24_396_992_246_539} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point $O$ on a rough plane which is inclined to the horizontal at an angle $\alpha$ The string lies along a line of greatest slope of the plane.
The particle $P$ is held at rest on the plane at the point $A$, where $O A = a$, as shown in Figure 5. The particle $P$ is released from $A$ and slides down the plane, coming to rest at the point $B$. The coefficient of friction between $P$ and the plane is $\mu$, where $\mu < \tan \alpha$ Air resistance is modelled as being negligible.

Show that $A B = a ( \sin \alpha - \mu \cos \alpha )$. Given that $\tan \alpha = \frac { 3 } { 4 }$ and $\mu = \frac { 1 } { 2 }$
find, in terms of $a$ and $g$, the maximum speed of $P$ as it moves from $A$ to $B$
Describe the motion of $P$ after it reaches $B$, justifying your answer.

A particle $P$ of mass $m \mathrm {~kg}$ is initially held at rest at the point $O$ on a smooth inclined plane. The plane is inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 2 } { 5 }$

The particle is released from rest and slides down the plane against a force which acts towards $O$. The force has magnitude $\frac { 1 } { 3 } m x ^ { 2 } \mathrm {~N}$, where $x$ metres is the distance of $P$ from $O$.

Find the speed of $P$ when $x = 2$ The particle first comes to instantaneous rest at the point $A$.
Find the distance $O A$.

Edexcel M3 2022 January Q4

8 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bd1e2b07-4a87-49d6-addd-c9f67467ef2f-12_659_513_246_774} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A thin uniform right hollow cylinder, of radius $2 a$ and height $k a$, has a base but no top. A thin uniform hemispherical shell, also of radius $2 a$, is made of the same material as the cylinder. The hemispherical shell is attached to the end of the cylinder forming a container $C$. The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of $C$ is $O$, as shown in Figure 3.

Show that the distance from $O$ to the centre of mass of $C$ is $$\frac { \left( k ^ { 2 } + 4 k + 4 \right) } { 2 ( k + 3 ) } a$$ The container is placed with its circular base on a plane which is inclined at $30 ^ { \circ }$ to the horizontal. The plane is sufficiently rough to prevent $C$ from sliding. The container is on the point of toppling.
Find the value of $k$.

Edexcel M3 2022 January Q5

12 marks Moderate -0.3

A particle $P$ is moving along the $x$-axis. At time $t$ seconds the displacement of $P$ from the origin $O$ is $x$ metres, where $x = 4 \cos \left( \frac { 1 } { 5 } \pi t \right)$

Prove that $P$ is moving with simple harmonic motion.
Find the period of the motion.
State the amplitude of the motion.
Find, in terms of $\pi$, the maximum speed of $P$ The points $A$ and $B$ lie on the $x$-axis, on opposite sides of $O$, with $O A = 1.5 \mathrm {~m}$ and $O B = 2.5 \mathrm {~m}$.
Find the time taken by $P$ to move directly from $A$ to $B$.

Edexcel M3 2022 January Q6

15 marks Standard +0.8

A particle $P$ of mass 1.2 kg is attached to the midpoint of a light elastic string of natural length 0.5 m and modulus of elasticity $\lambda$ newtons.

The fixed points $A$ and $B$ are 0.8 m apart on a horizontal ceiling. One end of the string is attached to $A$ and the other end of the string is attached to $B$. Initially $P$ is held at rest at the midpoint $M$ of the line $A B$ and the tension in the string is 30 N .

Show that $\lambda = 50$ The particle is now held at rest at the point $C$, where $C$ is 0.3 m vertically below $M$. The particle is released from rest.
Find the magnitude of the initial acceleration of $P$
Find the speed of $P$ at the instant immediately before it hits the ceiling.

Questions — Edexcel (10514 questions)

Browse by module

Browse by board

Edexcel M3 2019 January Q2

Edexcel M3 2019 January Q3

Edexcel M3 2019 January Q4

Edexcel M3 2019 January Q5

Edexcel M3 2019 January Q6

Edexcel M3 2021 January Q1

Edexcel M3 2021 January Q2

Edexcel M3 2021 January Q3

Edexcel M3 2021 January Q4

Edexcel M3 2021 January Q5

Edexcel M3 2021 January Q6

Edexcel M3 2021 January Q7

Edexcel M3 2022 January Q1

Edexcel M3 2022 January Q2

Edexcel M3 2022 January Q3

Edexcel M3 2022 January Q4

Edexcel M3 2022 January Q5

Edexcel M3 2022 January Q6

Edexcel M3 2022 January Q7

Edexcel M3 2022 January Q1

Edexcel M3 2022 January Q2

Edexcel M3 2022 January Q3

Edexcel M3 2022 January Q4

Edexcel M3 2022 January Q5

Edexcel M3 2022 January Q6