Questions — Edexcel M3 (510 questions)

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Edexcel M3 2013 June Q1
7 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{daa795f0-2c5e-4617-a295-fbe74c22be4a-02_679_568_210_680} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hollow right circular cone, of base radius \(a\) and height \(h\), is fixed with its axis vertical and vertex downwards, as shown in Figure 1. A particle moves with constant speed \(v\) in a horizontal circle of radius \(\frac { 1 } { 3 } a\) on the smooth inner surface of the cone. Show that \(v = \sqrt { } \left( \frac { 1 } { 3 } h g \right)\).
Edexcel M3 2013 June Q2
7 marks Standard +0.8
2. A particle of mass 4 kg is moving along the horizontal \(x\)-axis under the action of a single force which acts in the positive \(x\)-direction. At time \(t\) seconds the force has magnitude \(\left( 1 + 3 t ^ { \frac { 1 } { 2 } } \right) \mathrm { N }\).
When \(t = 0\) the particle has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Find the work done by the force in the interval \(0 \leqslant t \leqslant 4\)
Edexcel M3 2013 June Q3
10 marks Standard +0.3
3. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 2 m and modulus of elasticity 20 N . The other end of the spring is attached to a fixed point \(A\). The particle \(P\) is held at rest at the point \(B\), which is 1 m vertically below \(A\), and then released.
  1. Find the acceleration of \(P\) immediately after it is released from rest. The particle comes to instantaneous rest for the first time at the point \(C\).
  2. Find the distance \(B C\).
Edexcel M3 2013 June Q4
10 marks Standard +0.8
  1. A particle \(P\) is moving along the positive \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 4 } { ( x + 2 ) }\). When \(t = 0 , P\) is at \(O\). Find
    1. the distance of \(P\) from \(O\) when \(t = 2\)
    2. the magnitude and direction of the acceleration of \(P\) when \(t = 2\)
Edexcel M3 2013 June Q5
12 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{daa795f0-2c5e-4617-a295-fbe74c22be4a-08_504_1429_212_264} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Part of a hollow spherical shell, centre \(O\) and radius \(r\), forms a bowl with a plane circular rim. The bowl is fixed to a horizontal surface at \(A\) with the rim uppermost and horizontal. The point \(A\) is the lowest point of the bowl. The point \(B\), where \(\angle A O B = \alpha\) and \(\tan \alpha = \frac { 3 } { 4 }\), is on the rim of the bowl, as shown in Figure 2. A small smooth marble \(M\) is placed inside the bowl at \(A\), and given an initial horizontal speed \(\sqrt { } ( g r )\). The motion of \(M\) takes place in the vertical plane \(O A B\).
  1. Show that the speed of \(M\) as it reaches \(B\) is \(\sqrt { } \left( \frac { 3 } { 5 } g r \right)\). After leaving the surface of the bowl at \(B , M\) moves freely under gravity and first strikes the horizontal surface at the point \(C\). Given that \(r = 0.4 \mathrm {~m}\),
  2. find the distance \(A C\).
Edexcel M3 2013 June Q6
15 marks Challenging +1.2
6. (a) A uniform lamina is in the shape of a quadrant of a circle of radius \(a\). Show, by integration, that the centre of mass of the lamina is at a distance of \(\frac { 4 a } { 3 \pi }\) from each of its straight edges. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{daa795f0-2c5e-4617-a295-fbe74c22be4a-10_809_802_484_571} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A second uniform lamina \(A B C D E F A\) is shown shaded in Figure 3. The straight sides \(A C\) and \(A E\) are perpendicular and \(A C = A E = 2 a\). In the figure, the midpoint of \(A C\) is \(B\), the midpoint of \(A E\) is \(F\), and \(A B D F\) and \(D G E F\) are squares of side \(a\). \(B C D\) is a quadrant of a circle with centre \(B\). \(D G E\) is a quadrant of a circle with centre \(G\).
(b) Find the distance of the centre of mass of the lamina from the side \(A E\). The lamina is smoothly hinged to a horizontal axis which passes through \(E\) and is perpendicular to the plane of the lamina. The lamina has weight \(W\) newtons. The lamina is held in equilibrium in a vertical plane, with \(A\) vertically above \(E\), by a horizontal force of magnitude \(X\) newtons applied at \(C\).
(c) Find \(X\) in terms of \(W\).
Edexcel M3 2013 June Q7
14 marks Challenging +1.2
  1. Two points \(A\) and \(B\) are 4 m apart on a smooth horizontal surface. A light elastic string, of natural length 0.8 m and modulus of elasticity 15 N , has one end attached to the point A. A light elastic string, of natural length 0.8 m and modulus of elasticity 10 N , has one end attached to the point \(B\). A particle \(P\) of mass 0.2 kg is attached to the free end of each string. The particle rests in equilibrium on the surface at the point \(C\) on the straight line between \(A\) and \(B\).
    1. Show that the length of \(A C\) is 1.76 m .
    The particle \(P\) is now held at the point \(D\) on the line \(A B\) such that \(A D = 2.16 \mathrm {~m}\). The particle is then released from rest and in the subsequent motion both strings remain taut.
  2. Show that \(P\) moves with simple harmonic motion.
  3. Find the speed of \(P\) as it passes through the point \(C\).
  4. Find the time from the instant when \(P\) is released from \(D\) until the instant when \(P\) is first moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Edexcel M3 2013 June Q1
6 marks Moderate -0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-02_515_976_285_484} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A rough disc is rotating in a horizontal plane with constant angular speed 20 revolutions per minute about a fixed vertical axis through its centre \(O\). A particle \(P\) rests on the disc at a distance 0.4 m from \(O\), as shown in Figure 1. The coefficient of friction between \(P\) and the disc is \(\mu\). The particle \(P\) is on the point of slipping. Find the value of \(\mu\).
Edexcel M3 2013 June Q2
9 marks Standard +0.3
2. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis in the positive \(x\)-direction. The only force on \(P\) is a force of magnitude \(\left( 2 t + \frac { 1 } { 2 } \right) \mathrm { N }\) acting in the direction of \(x\) increasing, where \(t\) seconds is the time after \(P\) leaves the origin \(O\). When \(t = 0\), \(P\) is at rest at \(O\).
  1. Find an expression, in terms of \(t\), for the velocity of \(P\) at time \(t\) seconds. The particle passes through the point \(A\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the distance \(O A\).
Edexcel M3 2013 June Q3
8 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-04_707_1006_258_427} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Two particles \(P\) and \(Q\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string of length 6l. The string passes through a small smooth fixed ring at the point \(A\). The particle \(Q\) is hanging freely at a distance \(l\) vertically below \(A\). The particle \(P\) is moving in a horizontal circle with constant angular speed \(\omega\). The centre \(O\) of the circle is vertically below \(A\). The particle \(Q\) does not move and \(A P\) makes a constant angle \(\theta\) with the downward vertical, as shown in Figure 2. Show that
  1. \(\theta = 60 ^ { \circ }\)
  2. \(\omega = \sqrt { } \left( \frac { 2 g } { 5 l } \right)\)
Edexcel M3 2013 June Q4
9 marks Standard +0.8
  1. A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.2 m . 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 rest at a point \(B\) on the plane, where \(O B = 1.5 \mathrm {~m}\). When \(P\) is at \(B\), the tension in the string is 20 N . The particle is released from rest.
    1. Find the speed of \(P\) when \(O P = 1.2 \mathrm {~m}\).
    The particle comes to rest at the point \(C\).
  2. Find the distance \(B C\).
Edexcel M3 2013 June Q5
13 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-08_622_1186_251_443} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = ( x + 1 ) ^ { 2 }\), the \(x\)-axis, the \(y\)-axis and the line with equation \(x = 2\), as shown in Figure 3. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\).
  1. Use algebraic integration to find the \(x\) coordinate of the centre of mass of \(S\).
    (8) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-08_558_492_1263_703} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} A uniform solid hemisphere is fixed to \(S\) to form a solid \(T\). The hemisphere has the same radius as the smaller plane face of \(S\) and its plane face coincides with the smaller plane face of \(S\), as shown in Figure 4. The mass per unit volume of the hemisphere is 10 times the mass per unit volume of \(S\). The centre of the circular plane face of \(T\) is \(A\). All lengths are measured in centimetres.
  2. Find the distance of the centre of mass of \(T\) from \(A\).
Edexcel M3 2013 June Q6
14 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-10_191_972_276_484} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The points \(A\) and \(B\) are 3.75 m apart on a smooth horizontal floor. A particle \(P\) has mass 0.8 kg . One end of a light elastic spring, of natural length 1.5 m and modulus of elasticity 24 N , is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length 0.75 m and modulus of elasticity 18 N , are attached to \(P\) and \(B\). The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 5.
  1. Show that \(A O = 2.4 \mathrm {~m}\). The point \(C\) lies on the straight line \(A O B\) between \(O\) and \(B\). The particle \(P\) is held at \(C\) and released from rest.
  2. Show that \(P\) moves with simple harmonic motion. The maximum speed of \(P\) is \(\sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the time taken by \(P\) to travel 0.3 m from \(C\).
Edexcel M3 2013 June Q7
16 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-12_499_833_262_664} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} A particle \(P\) of mass \(5 m\) 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 6. The particle is projected vertically downwards with speed \(\sqrt { } \left( \frac { 9 a g } { 5 } \right)\). 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 \(T\).
  1. Show that \(T = 3 m g ( 5 \cos \theta + 3 )\). At the instant when the particle reaches the point \(B\) the string becomes slack.
  2. Find the speed of \(P\) at \(B\). At time \(t = 0 , P\) is at \(B\). At time \(t\), before the string becomes taut once more, the coordinates of \(P\) are \(( x , y )\) referred to horizontal and vertical axes with origin \(O\). The \(x\)-axis is directed along \(O A\) produced and the \(y\)-axis is vertically upward.
  3. Find
    1. \(x\) in terms of \(t , a\) and \(g\),
    2. \(y\) in terms of \(t , a\) and \(g\).
Edexcel M3 2014 June Q1
9 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e500e20b-9060-4c69-af13-fb97b9a86dfd-02_389_524_221_712} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hemispherical bowl of internal radius \(4 r\) 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 \(\sqrt { \frac { 3 g } { 8 r } }\) The point \(C\) lies on \(O A\), as shown in Figure 1.
Find, in terms of \(r\), the distance \(O C\).
Edexcel M3 2014 June Q2
9 marks Standard +0.3
2. 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 magnitude \(\frac { k } { x ^ { 2 } }\), where \(k\) is a constant. At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a fixed sphere of radius \(R\).
  1. Show that \(k = m g R ^ { 2 }\). When \(P\) is at a height \(\frac { R } { 4 }\) above the surface of the Earth, the speed of \(P\) is \(\sqrt { \frac { g R } { 2 } }\) Given that air resistance can be ignored,
  2. find, in terms of \(R\), the greatest distance from the centre of the Earth reached by \(P\).
Edexcel M3 2014 June Q3
7 marks Challenging +1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e500e20b-9060-4c69-af13-fb97b9a86dfd-05_639_422_223_769} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a container in the shape of a uniform right circular conical shell of height 6r. The radius of the open circular face is \(r\). The container is suspended by two vertical strings attached to two points at opposite ends of a diameter of the open circular face. It hangs with the open circular face uppermost and axis vertical. Molten wax is poured into the container. The wax solidifies and adheres to the container, forming a uniform solid right circular cone. The depth of the wax in the container is \(2 r\). The container together with the wax forms a solid \(S\). The mass of the container when empty is \(m\) and the mass of the wax in the container is \(3 m\).
  1. Find the distance of the centre of mass of the solid \(S\) from the vertex of the container. One of the strings is now removed and the solid \(S\) hangs freely in equilibrium suspended by the remaining vertical string.
  2. Find the size of the angle between the axis of the container and the downward vertical.
Edexcel M3 2014 June Q4
11 marks Standard +0.8
4. \begin{figure}[h]
[diagram]
\captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(3 m g\), is fixed to a point \(A\) on a fixed plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\) A small ball of mass \(2 m\) is attached to the free end of the string. The ball is held at a point \(C\) on the plane, where \(C\) is below \(A\) and \(A C = l\) as shown in Figure 3. The string is parallel to a line of greatest slope of the plane. The ball is released from rest. In an initial model the plane is assumed to be smooth.
  1. Find the distance that the ball moves before first coming to instantaneous rest. In a refined model the plane is assumed to be rough. The coefficient of friction between the ball and the plane is \(\mu\). The ball first comes to instantaneous rest after moving a distance \(\frac { 2 } { 5 } l\).
  2. Find the value of \(\mu\).
Edexcel M3 2014 June Q5
11 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e500e20b-9060-4c69-af13-fb97b9a86dfd-09_529_713_223_612} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows the region \(R\) bounded by part of the curve with equation \(y = \cos x\), the \(x\)-axis and the \(y\)-axis. A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.
  1. Show that the volume of \(S\) is \(\frac { \pi ^ { 2 } } { 4 }\)
  2. Find, using algebraic integration, the \(x\) coordinate of the centre of mass of \(S\).
Edexcel M3 2014 June Q6
13 marks Standard +0.3
6. A particle \(P\) 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. The particle is hanging freely at rest, with the string vertical, when it is projected horizontally with speed \(U\). The particle moves in a complete vertical circle.
  1. Show that \(U \geqslant \sqrt { 5 a g }\) As \(P\) moves in the circle the least tension in the string is \(T\) and the greatest tension is \(k T\). Given that \(U = 3 \sqrt { a g }\)
  2. find the value of \(k\).
Edexcel M3 2014 June Q7
15 marks Standard +0.3
7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(l\). The other end of the spring is attached to a fixed point \(A\). The particle is hanging freely in equilibrium at the point \(B\), where \(A B = 1.5 l\)
  1. Show that the modulus of elasticity of the spring is \(2 m g\). The particle is pulled vertically downwards from \(B\) to the point \(C\), where \(A C = 1.8 \mathrm { l }\), and released from rest.
  2. Show that \(P\) moves in simple harmonic motion with centre \(B\).
  3. Find the greatest magnitude of the acceleration of \(P\). The midpoint of \(B C\) is \(D\). The point \(E\) lies vertically below \(A\) and \(A E = 1.2 l\)
  4. Find the time taken by \(P\) to move directly from \(D\) to \(E\).
Edexcel M3 2015 June Q1
7 marks Standard +0.3
  1. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 1.2 m and modulus of elasticity \(\lambda\) newtons. The other end of the spring is attached to a fixed point \(A\) on a ceiling. The particle is hanging freely in equilibrium at a distance 1.5 m vertically below \(A\).
    1. Find the value of \(\lambda\).
    The particle is now raised to the point \(B\), where \(B\) is vertically below \(A\) and \(A B = 0.8 \mathrm {~m}\). The spring remains straight. The particle is released from rest and first comes to instantaneous rest at the point \(C\).
  2. Find the distance \(A C\).
Edexcel M3 2015 June Q2
10 marks Standard +0.8
2. The finite region bounded by the \(x\)-axis, the curve with equation \(y = 2 \mathrm { e } ^ { x }\), the \(y\)-axis and the line \(x = 1\) is rotated through one complete revolution about the \(x\)-axis to form a uniform solid. Use algebraic integration to
  1. show that the volume of the solid is \(2 \pi \left( \mathrm { e } ^ { 2 } - 1 \right)\),
  2. find, in terms of e, the \(x\) coordinate of the centre of mass of the solid.
Edexcel M3 2015 June Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00388805-5d60-4327-a10e-c0df74a0cb75-05_776_791_223_573} \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 \(4 l\). The ends of the string are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). Both strings are taut and \(A P\) makes an angle of \(30 ^ { \circ }\) with \(A B\), as shown in Figure 1. The ball is moving in a horizontal circle with constant angular speed \(\omega\).
  1. Find, in terms of \(m , g , l\) and \(\omega\),
    1. the tension in \(A P\),
    2. the tension in \(B P\).
  2. Show that \(\omega ^ { 2 } \geqslant \frac { g \sqrt { 3 } } { 3 l }\).
Edexcel M3 2015 June Q4
12 marks Standard +0.3
  1. A vehicle of mass 900 kg moves along a straight horizontal road. At time \(t\) seconds the resultant force acting on the vehicle has magnitude \(\frac { 63000 } { k t ^ { 2 } } \mathrm {~N}\), where \(k\) is a positive constant. The force acts in the direction of motion of the vehicle. At time \(t\) seconds, \(t \geqslant 1\), the speed of the vehicle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the vehicle is a distance \(x\) metres from a fixed point \(O\) on the road. When \(t = 1\) the vehicle is at rest at \(O\) and when \(t = 4\) the speed of the vehicle is \(10.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that \(v = 14 - \frac { 14 } { t }\)
    2. Hence deduce that the speed of the vehicle never reaches \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    3. Use the trapezium rule, with 4 equal intervals, to estimate the value of \(x\) when \(v = 7\)