Questions — Edexcel (10514 questions)

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Edexcel M3 2015 June Q4
12 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-07_408_509_246_705} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of mass \(3 m\) 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 \(O A\) is horizontal and \(O A = a\). The particle is projected vertically downwards from \(A\) with speed \(u\), as shown in Figure 2. The particle moves in complete vertical circles.
  1. Show that \(u ^ { 2 } \geqslant 3 a g\). Given that the greatest tension in the string is three times the least tension in the string, (b) show that \(u ^ { 2 } = 6 a g\).
Edexcel M3 2015 June Q5
17 marks Challenging +1.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-09_205_941_262_513} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two fixed points \(A\) and \(B\) are 5 m apart on a smooth horizontal floor. 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 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 15 N , has one end attached to \(P\) and the other end attached to \(B\). Initially \(P\) rests in equilibrium at the point \(O\), as shown in Figure 3.
  1. Show that \(A O = 3 \mathrm {~m}\). The particle is now pulled towards \(A\) and released from rest at the point \(C\), where \(A C B\) is a straight line and \(O C = 1 \mathrm {~m}\).
  2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion.
  3. Find the speed of \(P\) at the instant when the string \(P B\) becomes slack. The particle first comes to instantaneous rest at the point \(D\).
  4. Find the distance \(D B\).
Edexcel M3 2015 June Q6
17 marks Challenging +1.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-11_442_727_237_603} \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,
  1. show that the volume of \(S\) is \(\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),
  2. 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]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-11_478_472_1407_762} \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.
  3. 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.
  4. Find the size of the angle between \(V A\) and the vertical.
Edexcel M3 2017 June Q1
6 marks Standard +0.3
  1. The region enclosed by the curve with equation \(y = \frac { 1 } { 2 } \sqrt { x }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 4\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\). Use algebraic integration to find the exact value of the \(x\) coordinate of the centre of mass of \(S\).
    (6)
"都 D \(\_\_\_\_\) 1
Edexcel M3 2017 June Q2
9 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a67e3644-13fa-4196-a2ef-ea1e26f5726c-04_264_438_269_753} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform solid right circular cone \(R\), with vertex \(V\), has base radius \(4 r\) and height \(4 h\). A right circular cone \(S\), also with vertex \(V\) and the same axis of symmetry as \(R\), has base radius \(3 r\) and height \(3 h\). The cone \(S\) is cut away from the cone \(R\) leaving a solid \(T\). The centre of the larger plane face of \(T\) is \(O\). Figure 1 shows the solid \(T\).
  1. Find the distance from \(O\) to the centre of mass of \(T\). The point \(A\) lies on the circumference of the smaller plane face of \(T\). The solid is freely suspended from \(A\) and hangs in equilibrium. Given that \(h = r\)
  2. find the size of the angle between \(O A\) and the downward vertical.
Edexcel M3 2017 June Q3
9 marks Challenging +1.2
3. A particle \(P\) of mass 0.5 kg moves in a straight line with simple harmonic motion, completing 4 oscillations per second. The particle comes to instantaneous rest at the fixed points \(A\) and \(B\), where \(A B = 0.5 \mathrm {~m}\).
  1. Find the maximum magnitude of the acceleration of \(P\). When \(P\) is moving at its maximum speed it receives an impulse. The direction of this impulse is opposite to the direction in which \(P\) is moving when it receives the impulse. The impulse causes \(P\) to reverse its direction of motion but \(P\) continues to move with simple harmonic motion. The centre and period of this new simple harmonic motion are the same as the centre and period of the original simple harmonic motion. The amplitude is now half the original amplitude.
  2. Find the magnitude of the impulse.
    \section*{II} " ; O L
Edexcel M3 2017 June Q4
11 marks Standard +0.8
4. A light elastic string has natural length 0.4 m and modulus of elasticity 49 N . A particle \(P\) of mass 0.3 kg is attached to one end of the string. The other end of the string is attached to a fixed point \(A\) on a ceiling. The particle is released from rest at \(A\) and falls vertically. The particle first comes to instantaneous rest at the point \(B\).
  1. Find the distance \(A B\). The particle is now held at the point 0.6 m vertically below \(A\) and released from rest.
  2. Find the speed of \(P\) immediately before it hits the ceiling.
Edexcel M3 2017 June Q5
12 marks Standard +0.8
5. A particle \(P\) of mass 0.4 kg moves on the positive \(x\)-axis under the action of a single force. The force is directed towards the origin \(O\) and has magnitude \(\frac { k } { x ^ { 2 } }\) newtons, where \(O P = x\) metres and \(k\) is a constant. Initially \(P\) is moving away from \(O\). At \(x = 2\) the speed of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(x = 5\) the speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the value of \(k\). The particle first comes to instantaneous rest at the point \(A\).
  2. Find the value of \(x\) at \(A\).
Edexcel M3 2017 June Q6
13 marks Standard +0.3
6. The path followed by a motorcycle round a circular race track is modelled as a horizontal circle of radius 50 m . The track is banked at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The motorcycle travels round the track at constant speed. The motorcycle is modelled as a particle and air resistance can be ignored. In an initial model it is assumed that there is no sideways friction between the motorcycle tyres and the track.
  1. Find the speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), of the motorcycle. In a refined model it is assumed that there is sideways friction. The coefficient of friction between the motorcycle tyres and the track is \(\frac { 1 } { 4 }\). It is still assumed that air resistance can be ignored and that the motorcycle is modelled as a particle. The motorcycle's path is unchanged. Using this model,
  2. find the maximum speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), at which the motorcycle can travel without slipping sideways.
Edexcel M3 2017 June Q7
15 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a67e3644-13fa-4196-a2ef-ea1e26f5726c-20_442_967_283_486} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light inextensible string of length \(a\) has one end attached to a fixed point \(O\) on a horizontal plane. A particle \(P\) is attached to the other end of the string. The particle is held at the point \(A\), where \(A\) is vertically above \(O\) and \(O A = a\). The particle is then projected horizontally with speed \(\sqrt { 10 a g }\), as shown in Figure 2. The particle strikes the plane at the point \(B\). After rebounding from the plane, \(P\) passes through \(A\). The coefficient of restitution between the plane and \(P\) is \(e\).
  1. Show that \(e \geqslant \frac { 1 } { 2 }\) The point \(C\) is above the horizontal plane such that \(O C = a\) and angle \(C O B = 120 ^ { \circ }\) As the particle reaches \(C\), the string breaks. The particle now moves freely under gravity and strikes the plane at the point \(D\).
    Given that \(e = \frac { \sqrt { 3 } } { 2 }\)
  2. find the size of the angle between the horizontal and the direction of motion of \(P\) at \(D\).
Edexcel M3 2018 June Q1
4 marks Moderate -0.8
  1. A light elastic string of modulus of elasticity 29.4 N has one end attached to a fixed point \(A\). A particle \(P\) of mass 1.5 kg is attached to the other end of the string and \(P\) hangs freely in equilibrium 0.5 m vertically below \(A\). Find the natural length of the string.
Edexcel M3 2018 June Q2
11 marks Standard +0.3
2. A particle \(P\) is moving in a straight line with simple harmonic motion about the fixed point \(O\) as centre. When \(P\) is a distance 0.02 m from \(O\), the speed of \(P\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the magnitude of the acceleration of \(P\) is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  1. Find the period of the motion. The amplitude of the motion is \(a\) metres. Find
  2. the value of \(a\),
  3. the total length of time during each complete oscillation for which \(P\) is within \(\frac { 1 } { 2 } a\) metres of \(O\). metres of \(O\).
Edexcel M3 2018 June Q3
12 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-08_583_549_210_760} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A light inextensible string of length \(7 l\) has one end attached to a fixed point \(A\) and the other end attached to a fixed point \(B\), where \(A\) is vertically above \(B\) and \(A B = 5\) l. A particle of mass \(m\) is attached to the string at the point \(C\) where \(A C = 4 l\), as shown in Figure 1. The particle moves in a horizontal circle with constant angular speed \(\omega\). Both parts of the string are taut.
  1. Find, in terms of \(m , g , l\) and \(\omega\),
    1. the tension in \(A C\),
    2. the tension in \(B C\). The time taken by the particle to complete one revolution is \(R\).
      Given that \(R \leqslant k \pi \sqrt { \frac { l } { 5 g } }\)
  2. find the least possible value of \(k\).
Edexcel M3 2018 June Q4
8 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-12_469_844_269_552} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a light elastic string, of modulus of elasticity \(\lambda\) newtons and natural length 0.6 m . One end of the string is attached to a fixed point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass 0.5 kg . The string lies along a line of greatest slope of the plane. The particle is held at rest on the plane at the point \(B\), where \(B\) is lower than \(A\) and \(A B = 1.2 \mathrm {~m}\). The particle then receives an impulse of magnitude 1.5 N s in the direction parallel to the string, causing \(P\) to move up the plane towards \(A\). The coefficient of friction between \(P\) and the plane is 0.7 . Given that \(P\) comes to rest at the instant when the string becomes slack, find the value of \(\lambda\).
Edexcel M3 2018 June Q5
13 marks Standard +0.8
  1. A particle \(P\) of mass 0.8 kg moves along the \(x\)-axis in the positive \(x\) direction under the action of a resultant force. This force acts in the direction of \(x\) increasing. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O , P\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the force has magnitude \(\frac { 4 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}\).
When \(t = 0 , P\) is at rest at \(O\).
  1. Show that \(v ^ { 2 } = 5 \left( \frac { ( x + 1 ) ^ { 2 } - 1 } { ( x + 1 ) ^ { 2 } } \right)\) When \(t = 2 , P\) is at the point \(A\). When \(t = 4 , P\) is at the point \(B\).
  2. Using algebraic integration, find the distance \(A B\).
Edexcel M3 2018 June Q6
13 marks Challenging +1.2
6. A uniform solid right circular cone has base radius \(r\) and height \(h\).
  1. Use algebraic integration to show that the distance of the centre of mass of the cone from its vertex is \(\frac { 3 } { 4 } h\).
    [0pt] [You may assume that the volume of a cone of base radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) ] \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-20_394_716_632_621} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A solid \(S\) is formed by joining a uniform right circular solid cone of mass \(5 m\) to a uniform solid hemisphere, of radius \(r\) and mass \(k m\) where \(k < 20\). The cone has base radius \(r\) and height \(6 r\). The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the cone is \(O\) and the point \(A\) is on the circular edge of this plane face, as shown in Figure 3.
  2. Find the distance from \(O\) to the centre of mass of \(S\). The solid is suspended from \(A\) and hangs freely in equilibrium. The angle between the axis of the cone and the horizontal is \(30 ^ { \circ }\).
  3. Find, to the nearest whole number, the value of \(k\).
Edexcel M3 2018 June Q7
14 marks Standard +0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-24_575_821_214_566} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A smooth solid sphere, with centre \(O\) and radius \(r\), is fixed with its lowest point on a horizontal plane. A particle is placed on the surface of the sphere at the highest point of the sphere. The particle is then projected horizontally with speed \(u\) and starts to move on the surface of the sphere. The particle leaves the surface of the sphere at the point \(A\) where \(O A\) makes an angle \(\alpha , \alpha > 0\), with the upward vertical, as shown in Figure 4.
  1. Show that \(\cos \alpha = \frac { 1 } { 3 g r } \left( u ^ { 2 } + 2 g r \right)\)
  2. Show that \(u < \sqrt { g r }\) After leaving the surface of the sphere, the particle strikes the plane with speed \(3 \sqrt { \frac { g r } { 2 } }\)
  3. Find the value of \(\cos \alpha\).
Edexcel M3 2020 June Q1
6 marks Moderate -0.3
1.
VILV SIHI NI JIIIM IONOOVIIN SIHI NI JIIIM IONOOVARV SIHI NI JIIIM ION OC
\includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-03_62_37_2659_1914}
Edexcel M3 2020 June Q2
8 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ace84823-db30-463e-b24b-f0cd7df73746-04_542_831_301_552} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A smooth bead of weight 12 N is threaded onto a light elastic string of natural length 3 m . The points \(A\) and \(B\) are on a horizontal ceiling, with \(A B = 3 \mathrm {~m}\). One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). The bead hangs freely in equilibrium, 2 m below the ceiling, as shown in Figure 2.
  1. Find the tension in the string.
  2. Show that the modulus of elasticity of the string is 11.25 N . The bead is now pulled down to a point vertically below its equilibrium position and released from rest.
  3. Find the elastic energy stored in the string at the instant when the bead is moving at its maximum speed.
Edexcel M3 2020 June Q3
7 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ace84823-db30-463e-b24b-f0cd7df73746-06_615_1134_290_409} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 12 N . The other end of the spring is attached to a fixed point \(A\) on a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. Initially \(P\) is held at rest on the plane at the point \(B\), where \(B\) is below \(A\), with \(A B = 0.3 \mathrm {~m}\) and \(A B\) lies along a line of greatest slope of the plane. The point \(C\) lies on the plane with \(A C = 1 \mathrm {~m}\), as shown in Figure 3. The coefficient of friction between \(P\) and the plane is 0.3 After being released \(P\) passes through the point \(C\). Find the speed of \(P\) at the instant it passes through \(C\).
Edexcel M3 2020 June Q4
10 marks Challenging +1.2
4.
  1. Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius \(a\) is a distance \(\frac { 3 } { 8 } a\) 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]{ace84823-db30-463e-b24b-f0cd7df73746-08_444_764_539_591} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} A uniform solid hemisphere has mass \(m\) and radius \(a\). A particle of mass \(k m\) is attached to a point \(A\) on the circumference of the plane face of the hemisphere to form the loaded solid \(S\). The centre of the plane face of the hemisphere is the point \(O\), as shown in Figure 4. The loaded solid \(S\) is placed on a horizontal plane. The curved surface of \(S\) is in contact with the plane and \(S\) rests in equilibrium with \(O A\) making an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \sqrt { 3 }\)
  2. Find the exact value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-09_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel M3 2020 June Q5
12 marks Standard +0.8
5. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis in the positive \(x\) direction. At time \(t\) seconds, \(t \geqslant 1 , P\) is \(x\) metres from the origin \(O\) and is moving with speed \(v \mathrm {~ms} ^ { - 1 }\). The resultant force acting on \(P\) has magnitude \(\frac { 2 } { x ^ { 3 } } \mathrm {~N}\) and is directed towards \(O\). When \(t = 1 , x = 1\) and \(v = 3\) Show that
  1. \(v ^ { 2 } = \frac { 4 } { x ^ { 2 } } + 5\)
  2. \(t = \frac { a + \sqrt { b x ^ { 2 } + c } } { d }\), where \(a , b , c\) and \(d\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-13_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel M3 2020 June Q6
15 marks Challenging +1.2
6. A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 3 } { 4 } \mathrm { mg }\). A particle \(P\) of mass \(m\) is attached to one end of the string. The other end of the string is attached to a fixed point \(A\). Particle \(P\) hangs freely in equilibrium at the point \(O\), vertically below \(A\).
  1. Find the distance \(O A\). The particle \(P\) is now pulled vertically down to a point \(B\), where \(A B = 3 a\), and released from rest.
  2. Show that, throughout the subsequent motion, \(P\) performs only simple harmonic motion, justifying your answer. The point \(C\) is vertically below \(A\), where \(A C = 2 a\).
    Find, in terms of \(a\) and \(g\),
  3. the speed of \(P\) at the instant that it passes through \(C\),
  4. the time taken for \(P\) to move directly from \(B\) to \(C\). \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-17_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel M3 2020 June Q7
17 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ace84823-db30-463e-b24b-f0cd7df73746-20_808_542_264_703} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A particle of mass \(m\) is attached to one end of a light inextensible string of length 8a. The other end of the string is fixed to the point \(O\) on the smooth horizontal surface of a desk. The point \(E\) is on the edge of the desk, where \(O E = 5 a\) and \(O E\) is perpendicular to the edge of the desk. The particle is held at the point \(A\), vertically above \(O\), with the string taut. The particle is projected horizontally from \(A\) with speed \(\sqrt { 8 a g }\) in the direction \(O E\), as shown in Figure 5. When the particle is above the level of \(O E\) the particle is moving in a vertical circle with radius \(8 a\). Given that, when the string makes an angle \(\theta\) with the upward vertical through \(O\), the tension in the string is \(T\),
  1. show that \(T = 3 m g ( 1 - \cos \theta )\) At the instant when the string is horizontal, the particle passes through the point \(B\).
  2. Find the instantaneous change in the tension in the string as the particle passes through \(B\). The particle hits the vertical side \(E F\) of the desk and rebounds. As a result of the impact, the particle loses one third of the kinetic energy it had immediately before the impact. In the subsequent motion the string becomes slack when it makes an angle \(\alpha\) with the upward vertical through \(O\).
  3. Show that \(\cos \alpha = \frac { 7 } { 12 }\) DO NOT WRITEIN THIS AREA
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
    \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-23_2255_50_314_34}
    Leave
    blank
    Q7
Edexcel M3 2021 June Q1
6 marks Challenging +1.2
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b99b3eb0-9bca-42e3-bea9-3b0454a872db-02_622_730_251_694} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hollow toy is formed by joining a uniform right circular conical shell \(C\), with radius \(4 a\) and height \(3 a\), to a uniform hemispherical shell \(H\), with radius \(4 a\). The circular edge of \(C\) coincides with the circular edge of \(H\), as shown in Figure 1. The mass per unit area of \(C\) is \(\lambda\) and the mass per unit area of \(H\) is \(k \lambda\) where \(k\) is a constant.
Given that the centre of mass of the toy is a distance \(4 a\) from the vertex of the cone, find the value of \(k\).