Questions — Edexcel (10514 questions)

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Edexcel M3 2015 January Q5
10 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3706a02d-95c6-4e7a-bf38-88b338d77892-09_270_919_267_557} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a uniform solid \(S\) formed by joining the plane faces of two solid right circular cones, of base radius \(r\), so that the centres of their bases coincide at \(O\). One cone, with vertex \(V\), has height \(4 r\) and the other cone has height \(k r\), where \(k > 4\)
  1. Find the distance of the centre of mass of \(S\) from \(O\).
    (4) The point \(A\) lies on the circumference of the common base of the cones. The solid is placed on a horizontal surface with VA in contact with the surface. Given that \(S\) rests in equilibrium,
  2. find the greatest possible value of \(k\). When \(S\) is suspended from \(A\) and hangs freely in equilibrium, \(O A\) makes an angle of \(12 ^ { \circ }\) with the downward vertical.
  3. Find the value of \(k\).
Edexcel M3 2015 January Q6
15 marks Standard +0.8
6. A smooth sphere, with centre \(O\) and radius \(a\), is fixed with its lowest point \(A\) on a horizontal floor. A particle \(P\) is placed on the surface of the sphere at the point \(B\), where \(B\) is vertically above \(A\). The particle is projected horizontally from \(B\) with speed \(\sqrt { \frac { a g } { 5 } }\) and moves along the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still in contact with the sphere, the speed of \(P\) is \(v\).
  1. Show that \(v ^ { 2 } = \frac { a g } { 5 } ( 11 - 10 \cos \theta )\). The particle leaves the surface of the sphere at the point \(C\).
    Find
  2. the speed of \(P\) at \(C\) in terms of \(a\) and \(g\),
  3. the size of the angle between the floor and the direction of motion of \(P\) at the instant immediately before \(P\) hits the floor.
Edexcel M3 2015 January Q7
16 marks Challenging +1.2
7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) on a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The string lies along a line of greatest slope of the plane. The particle rests in equilibrium at the point \(B\), where \(B\) is lower than \(A\) and \(A B = \frac { 6 } { 5 } a\).
  1. Show that \(\lambda = \frac { 5 } { 2 } m g\). The particle is now pulled down a line of greatest slope to the point \(C\), where \(B C = \frac { 1 } { 5 } a\), and released from rest.
  2. Show that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \frac { 2 a } { 5 g } }\)
  3. Find, in terms of \(g\), the greatest magnitude of the acceleration of \(P\) while the string is taut. The midpoint of \(B C\) is \(D\) and the string becomes slack for the first time at the point \(E\).
  4. Find, in terms of \(a\) and \(g\), the time taken by \(P\) to travel directly from \(D\) to \(E\).
Edexcel M3 2016 January Q1
6 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-02_503_524_121_712} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hemispherical bowl of internal radius \(2 r\) is fixed with its circular rim horizontal. A particle \(P\) is moving in a horizontal circle of radius \(r\) on the smooth inner surface of the bowl, as shown in Figure 1. Particle \(P\) is moving with constant angular speed \(\omega\). Show that \(\omega = \sqrt { \frac { g \sqrt { 3 } } { 3 r } }\)
Edexcel M3 2016 January Q2
8 marks Standard +0.3
2. A particle \(P\) is moving in a straight line. At time \(t\) seconds, the distance of \(P\) from a fixed point \(O\) on the line is \(x\) metres and the acceleration of \(P\) is \(( 6 - 2 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the direction of \(x\) increasing. When \(t = 0 , P\) is moving towards \(O\) with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  1. Find the velocity of \(P\) in terms of \(t\).
  2. Find the total distance travelled by \(P\) in the first 4 seconds.
Edexcel M3 2016 January Q3
9 marks Standard +0.8
3. A car of mass 800 kg is driven at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) round a bend in a race track. Around the bend, the track is banked at \(20 ^ { \circ }\) to the horizontal and the path followed by the car can be modelled as a horizontal circle of radius 20 m . The car is modelled as a particle. The coefficient of friction between the car tyres and the track is 0.5 Given that the tyres do not slip sideways on the track, find the maximum value of \(v\).
Edexcel M3 2016 January Q4
10 marks Standard +0.8
4. Fixed points \(A\) and \(B\) are on a horizontal ceiling, where \(A B = 4 a\). A light elastic string has natural length \(3 a\) and modulus of elasticity \(\lambda\). One end of the string is attached to \(A\) and the other end is attached to \(B\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The particle hangs freely in equilibrium at the point \(C\), where \(C\) is at a distance \(\frac { 3 } { 2 } a\) vertically below the ceiling.
  1. Show that \(\lambda = \frac { 5 m g } { 4 }\) (5) The point \(D\) is the midpoint of \(A B\). The particle is now raised vertically upwards to \(D\), and released from rest.
  2. Find the speed of \(P\) as it passes through \(C\).
    \includegraphics[max width=\textwidth, alt={}]{ffe0bc72-3136-48d9-9d5b-4a364d134070-05_542_51_2026_1982}VIIV SIHI NI JIIIM IONOOVI4V SIHI NI JIIYM ION OO
Edexcel M3 2016 January Q5
13 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-07_371_800_262_573} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) on a smooth plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The particle rests in equilibrium on the plane at the point \(B\) with the string lying along a line of greatest slope of the plane, as shown in Figure 2. Given that \(A B = \frac { 6 } { 5 } l\)
  1. show that \(\lambda = 3 \mathrm { mg }\) The particle is pulled down the line of greatest slope to the point \(C\), where \(B C = \frac { 1 } { 2 } l\), and released from rest.
  2. Show that, while the string remains taut, \(P\) moves with simple harmonic motion about centre \(B\).
  3. Find the greatest magnitude of the acceleration of \(P\) while the string remains taut. The point \(D\) is the midpoint of \(B C\). The time taken by \(P\) to move directly from \(D\) to the point where the string becomes slack for the first time is \(k \sqrt { \frac { l } { g } }\), where \(k\) is a constant.
  4. Find, to 2 significant figures, the value of \(k\).
Edexcel M3 2016 January Q6
14 marks Challenging +1.2
6.
  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 \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ]
    (5) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-09_351_597_598_678} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A uniform solid hemisphere of mass \(m\) and radius \(r\) is joined to a uniform solid right circular cone to form a solid \(S\). The cone has mass \(M\), base radius \(r\) and height \(4 r\). The vertex of the cone is \(O\). The plane face of the cone coincides with the plane face of the hemisphere, as shown in Figure 3.
  2. Find the distance of the centre of mass of \(S\) from \(O\). The point \(A\) lies on the circumference of the base of the cone. The solid is placed on a horizontal table with \(O A\) in contact with the table. The solid remains in equilibrium in this position.
  3. Show that \(M \geqslant \frac { 1 } { 10 } m\)
Edexcel M3 2016 January Q7
15 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-11_581_641_262_678} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can turn freely in a vertical plane about a horizontal axis through \(O\). The particle is projected with speed \(u\) from a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\), as shown in Figure 4. The particle moves in complete vertical circles. Given that \(\cos \alpha = \frac { 4 } { 5 }\)
  1. show that \(u > \sqrt { \frac { 2 g l } { 5 } }\) As the rod rotates, the least tension in the rod is \(T\) and the greatest tension is \(4 T\).
  2. Show that \(u = \sqrt { \frac { 17 } { 5 } g l }\)
    \includegraphics[max width=\textwidth, alt={}]{ffe0bc72-3136-48d9-9d5b-4a364d134070-12_2639_1830_121_121}
Edexcel M3 2017 January Q1
7 marks Standard +0.8
1. \begin{figure}[h]
[diagram]
\captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The shaded region \(R\) is bounded by the curve with equation \(y ^ { 2 } = 9 ( 4 - x )\), the positive \(x\)-axis and the positive \(y\)-axis, as shown in Figure 1. A uniform solid \(S\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.
Use algebraic integration to find the \(x\) coordinate of the centre of mass of \(S\).
Edexcel M3 2017 January Q2
7 marks Standard +0.3
2. A particle \(P\) of mass 0.6 kg is moving along the positive \(x\)-axis in the positive direction. The only force acting on \(P\) acts in the direction of \(x\) increasing and has magnitude \(\left( 3 t + \frac { 1 } { 2 } \right) \mathrm { N }\), 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 \(\frac { 10 } { 3 } \mathrm {~ms} ^ { - 1 }\).
  2. Find the distance \(O A\).
Edexcel M3 2017 January Q3
6 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{85d8fc7d-8863-419e-8eef-8751a6fb6315-04_647_684_260_635} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform right circular solid cylinder has radius \(4 a\) and height \(6 a\). A solid hemisphere of radius \(3 a\) is removed from the cylinder forming a solid \(S\). The upper plane face of the cylinder coincides with the plane face of the hemisphere. The centre of the upper plane face of the cylinder is \(O\) and this is also the centre of the plane face of the hemisphere, as shown in Figure 2. Find the distance from \(O\) to the centre of mass of \(S\).
(6)
Edexcel M3 2017 January Q4
12 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{85d8fc7d-8863-419e-8eef-8751a6fb6315-05_654_515_267_712} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\) and \(A B = 7 a\). A particle \(P\) of mass \(m\) is fixed to the string and moves with constant angular speed \(\omega\) in a horizontal circle of radius \(4 a\). The centre of the circle is \(C\), where \(C\) lies on \(A B\) and \(A C = 3 a\), as shown in Figure 3. Both parts of the string are taut.
  1. Show that the tension in \(A P\) is \(\frac { 5 } { 7 } m \left( 4 a \omega ^ { 2 } + g \right)\).
  2. Find the tension in \(B P\).
  3. Deduce that \(\omega \geqslant \sqrt { \frac { g } { k a } }\), stating the value of \(k\).
Edexcel M3 2017 January Q5
9 marks Standard +0.8
A particle \(P\) of mass \(4 m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity 3 mg . The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. The particle lies at rest at the point \(A\) on the table, where \(O A = \frac { 4 } { 3 } l\). The coefficient of friction between \(P\) and the table is \(\mu\).
  1. Show that \(\mu \geqslant \frac { 1 } { 4 }\) The particle is now moved along the table to the point \(B\), where \(O B = 2 l\), and released from rest. Given that \(\mu = \frac { 2 } { 5 }\)
  2. show that \(P\) comes to rest before the string becomes slack.
    (5)
Edexcel M3 2017 January Q6
17 marks Challenging +1.2
6. One end of a light elastic string, of natural length 5 l and modulus of elasticity 20 mg , is attached to a fixed point \(A\). A particle \(P\) of mass \(2 m\) is attached to the free end of the string and \(P\) hangs freely in equilibrium at the point \(B\).
  1. Find the distance \(A B\).
    (3) The particle is now pulled vertically downwards from \(B\) to the point \(C\) and released from rest. In the subsequent motion the string does not become slack.
  2. Show that \(P\) moves with simple harmonic motion with centre \(B\).
  3. Find the period of this motion. The greatest speed of \(P\) during this motion is \(\frac { 1 } { 5 } \sqrt { g l }\)
  4. Find the amplitude of this motion. The point \(D\) is the midpoint of \(B C\) and the point \(E\) is the highest point reached by \(P\).
  5. Find the time taken by \(P\) to move directly from \(D\) to \(E\).
Edexcel M3 2017 January Q7
17 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{85d8fc7d-8863-419e-8eef-8751a6fb6315-11_412_1054_260_447} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A hollow sphere has internal radius \(r\) and centre \(O\). A bowl with a plane circular rim is formed by removing part of the sphere. The bowl is fixed to a horizontal floor with the rim uppermost and horizontal. The point \(B\) is the lowest point of the inner surface of the bowl. The point \(A\), where angle \(A O B = 120 ^ { \circ }\), lies on the rim of the bowl, as shown in Figure 4. A particle \(P\) of mass \(m\) is projected from \(A\), with speed \(U\) at \(90 ^ { \circ }\) to \(O A\), and moves on the smooth inner surface of the bowl. The motion of \(P\) takes place in the vertical plane \(O A B\).
  1. Find, in terms of \(m , g , U\) and \(r\), the magnitude of the force exerted on \(P\) by the bowl at the instant when \(P\) passes through \(B\).
  2. Find, in terms of \(g , U\) and \(r\), the greatest height above the floor reached by \(P\). Given that \(U > \sqrt { 2 g r }\)
  3. show that, after leaving the surface of the bowl, \(P\) does not fall back into the bowl.
Edexcel M3 2018 January Q1
5 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-02_333_890_264_529} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform solid \(S\) consists of a right solid circular cone of base radius \(r\) and a right solid cylinder, also of radius \(r\). The cone has height \(4 h\) and the centre of the plane face of the cone is \(O\). The cylinder has height \(3 h\). The cone and cylinder are joined so that the plane face of the cone coincides with one of the plane faces of the cylinder, as shown in Figure 1. Find the distance from \(O\) to the centre of mass of \(S\).
Edexcel M3 2018 January Q2
5 marks Standard +0.3
  1. A particle of mass 0.9 kg is attached to one end of a light elastic string, of natural length 1.2 m and modulus of elasticity 29.4 N . The other end of the string is attached to a fixed point \(A\) on a ceiling.
The particle is held at \(A\) and then released from rest. The particle first comes to instantaneous rest at the point \(B\). Find the distance \(A B\).
(5)
Edexcel M3 2018 January Q3
10 marks Standard +0.3
A particle \(P\) of mass 0.4 kg moves along the \(x\)-axis in the positive direction. At time \(t = 0 , P\) passes through the origin \(O\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds \(P\) is \(x\) metres from \(O\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant force acting on \(P\) has magnitude \(\frac { 8 } { ( t + 4 ) ^ { 2 } } \mathrm {~N}\) and is directed towards \(O\).
  1. Show that \(v = \frac { 20 } { t + 4 } + 5\) When \(v = 6 , x = a + b \ln 5\), where \(a\) and \(b\) are integers.
  2. Using algebraic integration, find the value of \(a\) and the value of \(b\).
Edexcel M3 2018 January Q4
12 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-10_547_841_244_555} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A small smooth bead \(P\) is threaded on a light inextensible string of length \(8 a\). One end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The other end of the string is attached to the fixed point \(B\), where \(B\) is vertically above \(A\) and \(A B = 4 a\), as shown in Figure 2. The bead moves with constant angular speed, in a horizontal circle, centre \(A\), with \(A P\) horizontal. The bead remains in contact with the table and both parts of the string, \(A P\) and \(B P\), are taut. The time for \(P\) to complete one revolution is \(S\). Show that \(\quad S \geqslant \pi \sqrt { \frac { 6 a } { g } }\)
Edexcel M3 2018 January Q5
11 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-14_510_723_269_607} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the finite region \(R\) which is bounded by part of the curve with equation \(y = \sin x\), the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\). A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis. Using algebraic integration,
  1. show that the volume of \(S\) is \(\frac { \pi ^ { 2 } } { 4 }\)
  2. find, in terms of \(\pi\), the \(x\) coordinate of the centre of mass of \(S\).
Edexcel M3 2018 January Q6
15 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-18_483_730_242_609} \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 \(l\). The other end of the string is attached to a fixed point \(O\). The particle is held at the point \(A\), where \(O A = l\) and \(O A\) is horizontal. The particle is then projected vertically downwards from \(A\) with speed \(\sqrt { 2 g l }\), as shown in Figure 4 . 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 \(T = m g ( 3 \cos \theta + 2 )\) At the instant when the particle reaches the point \(B\), the string becomes slack.
  2. Find the speed of \(P\) at \(B\).
  3. Find the greatest height above \(O\) reached by \(P\) in the subsequent motion.
Edexcel M3 2018 January Q7
17 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-22_197_945_251_497} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The fixed points \(A\) and \(B\) are 4.2 m apart on a smooth horizontal floor. One end of a light elastic spring, of natural length 1.8 m and modulus of elasticity 20 N , is attached to a particle \(P\) and the other end is attached to \(A\). One end of another light elastic spring, of natural length 0.9 m and modulus of elasticity 15 N , 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 5.
  1. Show that \(A O = 2.7 \mathrm {~m}\). The particle \(P\) now receives an impulse acting in the direction \(O B\) and moves away from \(O\) towards \(B\). In the subsequent motion \(P\) does not reach \(B\).
  2. Show that \(P\) moves with simple harmonic motion about centre \(O\). The mass of \(P\) is 10 kg and the magnitude of the impulse is \(J \mathrm { Ns }\). Given that \(P\) first comes to instantaneous rest at the point \(C\) where \(A C = 2.9 \mathrm {~m}\),
    1. find the value of \(J\),
    2. find the time taken by \(P\) to travel a total distance of 0.5 m from when it first leaves \(O\).
Edexcel M3 2019 January Q1
6 marks Standard +0.8
  1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the displacement of \(P\) from the origin \(O\) is \(x\) metres and the acceleration of \(P\) is \(\left( \frac { 7 } { 2 } - 2 x \right) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the positive \(x\) direction. At time \(t = 0 , P\) passes through \(O\) moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) in the positive \(x\) direction. Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest.
    (6)