Questions — Edexcel (10514 questions)

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Edexcel M3 2021 June Q2
9 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b99b3eb0-9bca-42e3-bea9-3b0454a872db-04_374_1084_246_493} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a fairground ride that consists of a chair of mass \(m\) attached to one end of a rigid arm of length \(\frac { 5 a } { 4 }\). The other end of the arm is freely hinged to the rim of a thin horizontal circular disc of radius \(a\). The disc rotates with constant angular speed \(\omega\) about a vertical axis through the centre of the disc. As the ride rotates the arm remains in a vertical plane through the centre of the disc. The arm makes a constant angle \(\theta\) with the vertical, where \(\tan \theta = \frac { 3 } { 4 }\) The chair is modelled as a particle and the arm is modelled as a light rod.
  1. Find the tension in the arm in terms of \(m\) and \(g\)
  2. Find \(\omega\) in terms of \(a\) and \(g\)
Edexcel M3 2021 June Q3
9 marks Standard +0.8
  1. The finite region enclosed by the curve with equation \(y = 3 - \sqrt { x }\) and the lines \(x = 0\) and \(y = 0\) is rotated through \(2 \pi\) radians about the \(x\)-axis, to form a uniform solid \(S\).
Use algebraic integration to
  1. show that the volume of \(S\) is \(\frac { 27 } { 2 } \pi\)
  2. find the \(x\) coordinate of the centre of mass of \(S\).
Edexcel M3 2021 June Q4
9 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b99b3eb0-9bca-42e3-bea9-3b0454a872db-12_483_848_306_589} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A circus performer has mass \(m\). She is attached to one end of a cable of length \(l\). The other end of the cable is attached to a fixed point \(O\) Initially she is held at rest at point \(A\) with the cable taut and at an angle of \(30 ^ { \circ }\) below the horizontal, as shown in Figure 3. The circus performer is released from \(A\) and she moves on a vertical circular path with centre \(O\) The circus performer is modelled as a particle and the cable is modelled as light and inextensible.
  1. Find, in terms of \(m\) and \(g\), the tension in the cable at the instant immediately after the circus performer is released.
  2. Show that, during the motion following her release, the greatest tension in the cable is 4 times the least tension in the cable.
Edexcel M3 2021 June Q5
11 marks Standard +0.8
  1. A particle \(P\) of mass 0.5 kg moves on the \(x\)-axis under the action of a single force.
At time \(t\) seconds, \(t \geqslant 0\)
  • \(O P = x\) metres, \(0 \leqslant x < \frac { \pi } { 2 }\)
  • the force has magnitude \(\sin 2 x \mathrm {~N}\) and is directed towards the origin \(O\)
  • \(P\) is moving in the positive \(x\) direction with speed \(v \mathrm {~ms} ^ { - 1 }\)
At time \(t = 0 , P\) passes through the origin with speed \(2 \mathrm {~ms} ^ { - 1 }\)
  1. Show that \(v = 2 \cos x\)
  2. Show that \(t = \frac { 1 } { 2 } \ln ( \sqrt { 2 } + 1 )\) when \(x = \frac { \pi } { 4 }\)
Edexcel M3 2021 June Q6
14 marks Standard +0.8
  1. A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string, of natural length 0.8 m and modulus of elasticity 0.6 N . The other end of the string is fixed to a point \(A\) on a rough horizontal table. The coefficient of friction between \(P\) and the table is \(\frac { 1 } { 7 }\)
The particle \(P\) is projected from \(A\), with speed \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along the surface of the table.
After travelling 0.8 m from \(A\), the particle passes through the point \(B\) on the table.
  1. Find the speed of \(P\) at the instant it passes through \(B\). The particle \(P\) comes to rest at the point \(C\) on the table, where \(A B C\) is a straight line.
  2. Find the total distance travelled by \(P\) as it moves directly from \(A\) to \(C\).
  3. Show that \(P\) remains at rest at \(C\).
Edexcel M3 2021 June Q7
17 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b99b3eb0-9bca-42e3-bea9-3b0454a872db-24_177_876_260_593} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The fixed points \(A\) and \(B\) are 7 m apart on a smooth horizontal surface.
A light elastic string has natural length 2 m and modulus of elasticity 4 N . One end of the string is attached to a particle \(P\) of mass 2 kg and the other end is attached to \(A\) Another light elastic string has natural length 3 m and modulus of elasticity 2 N . One end of this string is attached to \(P\) and the other end is attached to \(B\) The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 4.
  1. Show that \(O A = 2.5 \mathrm {~m}\). The particle \(P\) now receives an impulse of magnitude 6Ns in the direction \(O B\)
    1. Show that \(P\) initially moves with simple harmonic motion with centre \(O\)
    2. Determine the amplitude of this simple harmonic motion. The point \(C\) lies on \(O B\). As \(P\) passes through \(C\) the string attached to \(B\) becomes slack.
  3. Find the speed of \(P\) as it passes through \(C\)
  4. Find the time taken for \(P\) to travel directly from \(O\) to \(C\)
Edexcel M3 2022 June Q1
6 marks Standard +0.3
  1. A particle \(P\) moves in a straight line with simple harmonic motion between two fixed points \(A\) and \(B\). The particle performs 2 complete oscillations per second. The midpoint of \(A B\) is \(O\) and the midpoint of \(O A\) is \(C\)
The length of \(A B\) is 0.6 m .
  1. Find the maximum speed of \(P\)
  2. Find the time taken by \(P\) to move directly from \(O\) to \(C\)
Edexcel M3 2022 June Q2
9 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e837bb9-4ada-4f0f-8b21-2730611335f2-04_390_515_246_772} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hemispherical bowl of internal radius \(6 r\) is fixed with its circular rim horizontal. The centre of the circular rim is \(O\) and the point \(A\) on the surface of the bowl is vertically below \(O\). A particle \(P\) moves in a horizontal circle, with centre \(C\), on the smooth inner surface of the bowl. The particle moves with constant angular speed \(\sqrt { \frac { g } { 4 r } }\). The point \(C\) lies on \(O A\), as shown in Figure 1. Find, in terms of \(r\), the distance \(O C\)
Edexcel M3 2022 June Q3
10 marks Standard +0.8
  1. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.} A particle \(P\) is moving along a straight line. At time \(t\) seconds, \(P\) is a distance \(x\) metres from a fixed point \(O\) on the line and is moving away from \(O\) with speed \(\frac { 50 } { 2 x + 3 } \mathrm {~ms} ^ { - 1 }\)
  1. Find the deceleration of \(P\) when \(x = 12\) Given that \(x = 4\) when \(t = 1\)
  2. find the value of \(t\) when \(x = 12\)
Edexcel M3 2022 June Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e837bb9-4ada-4f0f-8b21-2730611335f2-12_357_737_260_664} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\lambda\), is fixed to a point \(A\) on a smooth plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. A small ball \(B\) of mass \(m\) is attached to the other end of the elastic string. Initially, \(B\) is held at rest at the point \(C\) on the plane with the elastic string lying along a line of greatest slope of the plane. The point \(C\) is below \(A\) and \(A C = l\), as shown in Figure 2 . The ball is released and comes to instantaneous rest at a point \(D\) on the plane.
The points \(A , C\) and \(D\) all lie along a line of greatest slope of the plane and \(A D = \frac { 5 l } { 4 }\) The ball is modelled as a particle and air resistance is modelled as being negligible.
Using the model,
  1. show that \(\lambda = 4 \mathrm { mg }\)
  2. find, in terms of \(g\) and \(l\), the greatest speed of \(B\) as it moves from \(C\) to \(D\)
Edexcel M3 2022 June Q5
10 marks Challenging +1.2
  1. Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac { 3 } { 8 } r\) from the centre of its plane face.
    [0pt] [You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ] \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2e837bb9-4ada-4f0f-8b21-2730611335f2-16_355_574_571_749} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A uniform solid hemisphere of radius \(r\) is joined to a uniform solid right circular cone made of the same material to form a toy. The cone has base radius \(r\) and height \(k r\). The centre of the base of the cone is \(O\). The plane face of the cone coincides with the plane face of the hemisphere, as shown in Figure 3. The toy can rest in equilibrium on a horizontal plane with any point of the curved surface of the hemisphere in contact with the plane.
  2. Find the exact value of \(k\)
Edexcel M3 2022 June Q6
13 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e837bb9-4ada-4f0f-8b21-2730611335f2-20_499_748_244_653} \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 is held at the point \(A\), where \(O A = a\) and \(O A\) is horizontal, as shown in Figure 4. The particle is projected vertically downwards with speed \(\sqrt { \frac { 9 a g } { 5 } }\) When the string makes an angle \(\theta\) with the downward vertical through \(O\) and the string is still taut, the tension in the string is \(S\).
  1. Show that \(S = \frac { 3 } { 5 } m g ( 5 \cos \theta + 3 )\) At the instant when the string becomes slack, the speed of \(P\) is \(v\)
  2. Show that \(v = \sqrt { \frac { 3 a g } { 5 } }\)
  3. Find the maximum height of \(P\) above the horizontal level of \(O\)
Edexcel M3 2022 June Q7
16 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e837bb9-4ada-4f0f-8b21-2730611335f2-24_165_1392_258_338} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows two fixed points, \(A\) and \(B\), which are 5 m apart on a smooth horizontal floor. A particle \(P\) of mass 1.25 kg is attached to one end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to \(A\) A second light elastic string, of natural length 1.2 m and modulus of elasticity \(\lambda\) newtons, has one end attached to \(P\) and the other end attached to \(B\) Initially \(P\) rests in equilibrium at the point \(O\), where \(A O = 3 \mathrm {~m}\)
  1. Show that \(\lambda = 15\) The particle is now projected along the floor towards \(B\) At time \(t\) seconds, \(P\) is a displacement \(x\) metres from \(O\) in the direction \(O B\)
  2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion where \(\ddot { x } = - 18 x\) The initial speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  3. Find the speed of \(P\) at the instant when the string \(P B\) becomes slack. Both strings are taut for \(T\) seconds during one complete oscillation.
  4. Find the value of \(T\)
Edexcel M3 2023 June Q1
5 marks Standard +0.8
  1. In this question you must show all stages in your working.
Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dceb2432-117c-40fe-bf3d-782beeb42e41-02_579_1059_386_502} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The finite region \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis, the line with equation \(x = 3\), the curve with equation \(y = \sqrt { ( x + 1 ) }\) and the \(y\)-axis.
Find the \(\boldsymbol { y }\) coordinate of the centre of mass of a uniform lamina in the shape of \(R\).
Edexcel M3 2023 June Q2
6 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dceb2432-117c-40fe-bf3d-782beeb42e41-04_424_510_246_767} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light elastic string \(A B\) has modulus of elasticity \(2 m g\) and natural length \(k a\), where \(k\) is a constant.
The end \(A\) of the elastic string is attached to a fixed point. The other end \(B\) is attached to a particle of mass \(m\). The particle is held in equilibrium, with the elastic string taut, by a force that acts in a direction that is perpendicular to the string. The line of action of the force and the elastic string lie in the same vertical plane. The string makes an angle \(\theta\) with the downward vertical at \(A\), as shown in Figure 2. Given that the length \(A B = \frac { 21 } { 10 } a\) and \(\tan \theta = \frac { 3 } { 4 }\), find the value of \(k\).
Edexcel M3 2023 June Q3
8 marks Challenging +1.2
  1. A uniform solid right circular cone \(C\) has base radius \(r\), height \(H\) and vertex \(V\). A uniform solid \(S\), shown in Figure 3, is formed by removing from \(C\) a uniform solid right circular cone of height \(h ( h < H )\) that has the same base and axis of symmetry as \(C\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dceb2432-117c-40fe-bf3d-782beeb42e41-08_725_1152_422_461} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
  1. Show that the distance of the centre of mass of \(S\) from \(V\) is $$\frac { 1 } { 4 } ( 3 H - h )$$ The solid \(S\) is suspended by two vertical light strings. The first string is attached to \(S\) at \(V\) and the second string is attached to \(S\) at a point on the circumference of the circular base of \(S\).
    The solid \(S\) hangs freely in equilibrium with its axis of symmetry horizontal.
    The tension in the first string is \(T _ { 1 }\) and the tension in the second string is \(T _ { 2 }\)
  2. Find \(\frac { T _ { 1 } } { T _ { 2 } }\), giving your answer in terms of \(H\) and \(h\), in its simplest form.
Edexcel M3 2023 June Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dceb2432-117c-40fe-bf3d-782beeb42e41-12_360_1004_246_534} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A car is travelling round a circular track. The track is banked at an angle \(\alpha\) to the horizontal, as shown in Figure 4. The car and driver are modelled as a particle.
The car moves round the track with constant speed in a horizontal circle of radius \(r\).
When the car is moving with speed \(\frac { 1 } { 2 } \sqrt { g r }\) round the circle, there is no sideways friction between the tyres of the car and the track.
  1. Show that \(\tan \alpha = \frac { 1 } { 4 }\) The sideways friction between the tyres of the car and the track has coefficient of friction \(\mu\), where \(\mu < 4\) The maximum speed at which the car can move round the circle without slipping sideways is \(V\).
  2. Find \(V\) in terms of \(\mu , r\) and \(g\).
Edexcel M3 2023 June Q5
12 marks Standard +0.8
  1. The centre of the Earth is the point \(O\) and the Earth is modelled as a fixed sphere of radius \(R\).
    At time \(t = 0\), a particle \(P\) is projected vertically upwards with speed \(U\) from a point \(A\) on the surface of the Earth.
At time \(t\) seconds, where \(t \geqslant 0\)
  • \(\quad P\) is a distance \(x\) from \(O\)
  • \(P\) is moving with speed \(v\)
  • \(P\) has acceleration of magnitude \(\frac { g R ^ { 2 } } { x ^ { 2 } }\) directed towards \(O\)
Air resistance is modelled as being negligible.
  1. Show that \(v ^ { 2 } = \frac { 2 g R ^ { 2 } } { x } + U ^ { 2 } - 2 g R\) Particle \(P\) is first moving with speed \(\frac { 1 } { 2 } \sqrt { g R }\) at the point \(B\).
  2. Given that \(U = \sqrt { g R }\) find, in terms of \(R\), the distance \(A B\).
  3. Find, in terms of \(g\) and \(R\), the smallest value of \(U\) that would ensure that \(P\) never comes to rest, explaining your reasoning.
Edexcel M3 2023 June Q6
16 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dceb2432-117c-40fe-bf3d-782beeb42e41-20_134_653_243_707} \captionsetup{labelformat=empty} \caption{Figure 5}
\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 with the string taut and horizontal and is then projected vertically downwards with speed \(u\), as shown in Figure 5. Air resistance is modelled as being negligible.
At the instant when the string has turned through an angle \(\theta\) and the string is taut, the tension in the string is \(T\).
  1. Show that \(T = \frac { m u ^ { 2 } } { a } + 3 m g \sin \theta\) Given that \(u = 2 \sqrt { \frac { 3 a g } { 5 } }\)
  2. find, in terms of \(a\) and \(g\), the speed of \(P\) at the instant when the string goes slack.
  3. Hence find, in terms of \(a\), the maximum height of \(P\) above \(O\) in the subsequent motion.
Edexcel M3 2023 June Q7
16 marks Challenging +1.8
  1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\). The other end of the string is attached to a fixed point on a ceiling. The particle \(P\) hangs in equilibrium at a distance \(D\) below the ceiling.
The particle \(P\) is now pulled vertically downwards until it is a distance \(3 l\) below the ceiling and released from rest. Given that \(P\) comes to instantaneous rest just before it reaches the ceiling,
  1. show that \(D = \frac { 5 l } { 3 }\)
  2. Show that, while the elastic string is stretched, \(P\) moves with simple harmonic motion, with period \(2 \pi \sqrt { \frac { 2 l } { 3 g } }\)
  3. Find, in terms of \(g\) and \(l\), the exact time from the instant when \(P\) is released to the instant when the elastic string first goes slack.
Edexcel M3 2024 June Q1
8 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{176ae8f8-7de9-4993-825a-6067614526ae-02_314_677_296_696} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A light elastic string \(A B\) has natural length \(4 a\) and modulus of elasticity \(\lambda\). The end \(A\) is attached to a fixed point and the end \(B\) is attached to a particle of mass \(m\). The particle is held in equilibrium, with the string stretched, by a horizontal force of magnitude \(k m g\).
The line of action of the horizontal force lies in the vertical plane containing the elastic string.
The string \(A B\) makes an angle \(\alpha\) with the vertical, where \(\tan \alpha = \frac { 4 } { 3 }\) With the particle in this position, \(A B = 5 a\), as shown in Figure 1.
  1. Show that \(\lambda = \frac { 20 m g } { 3 }\)
  2. Find the value of \(k\).
Edexcel M3 2024 June Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{176ae8f8-7de9-4993-825a-6067614526ae-04_351_563_296_751} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A thin hemispherical shell, with centre \(O\) and radius \(a\), is fixed with its open end uppermost and horizontal. A particle \(P\) of mass \(m\) moves in a horizontal circle on the smooth inner surface of the shell. The vertical distance of \(P\) below the level of \(O\) is \(d\), as shown in Figure 2.
  1. Find, in terms of \(m , g , d\) and \(a\), the magnitude of the force exerted on \(P\) by the inner surface of the hemisphere. The particle moves with constant speed \(v\).
  2. Find \(v\) in terms of \(g , a\) and \(d\).
Edexcel M3 2024 June Q3
9 marks Standard +0.8
  1. A particle \(P\) is moving along the \(x\)-axis.
At time \(t\) seconds, where \(t \geqslant 0\), the displacement of \(P\) from the origin \(O\) is \(x\) metres and \(P\) is moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. The acceleration of \(P\) is \(\frac { 3 \sqrt { x + 1 } } { 4 } \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction.
When \(t = 0 , x = 15\) and \(v = 8\)
  1. Show that \(v = ( x + 1 ) ^ { \frac { 3 } { 4 } }\)
  2. Find \(t\) in terms of \(v\).
Edexcel M3 2024 June Q4
10 marks Standard +0.3
  1. In a harbour, the water level rises and falls with the tides with simple harmonic motion.
On a particular day, the depths of water in the harbour at low and high tide are 4 m and 10 m respectively. Low tide occurs at 12:00 and high tide occurs at 18:20
  1. Find, in \(\mathrm { mh } ^ { - 1 }\), the speed at which the water level is rising on this particular day at 13:35 A ship can only safely enter the harbour when the depth of water is at least 8.5 m .
  2. Find the earliest time after 12:00 on this particular day at which it is safe for the ship to enter the harbour, giving your answer to the nearest minute.
Edexcel M3 2024 June Q5
12 marks Challenging +1.2
A uniform right solid circular cone \(C\) has radius \(r\) and height \(4 r\).
  1. Show, using algebraic integration, that the distance of the centre of mass of \(C\) from its vertex is \(3 r\).
    [0pt] [You may assume that the volume of \(C\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ] A uniform solid \(S\), shown below in Figure 3, is formed by removing from \(C\) a uniform solid right circular cylinder of height \(r\) and radius \(\frac { 1 } { 2 } r\), where the centre of one end of the cylinder coincides with the centre of the plane face of \(C\) and the axis of the cylinder coincides with the axis of \(C\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{176ae8f8-7de9-4993-825a-6067614526ae-12_661_1194_861_440} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure}
  2. Show that the distance of the centre of mass of \(S\) from the vertex of \(C\) is \(\frac { 75 } { 26 } r\) A rough plane is inclined at an angle \(\alpha\) to the horizontal.
    The solid \(S\) rests in equilibrium with its plane face in contact with the inclined plane.
    Given that \(S\) is on the point of toppling,
  3. find the exact value of \(\tan \alpha\)