Questions — Edexcel (10514 questions)

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Edexcel M3 2006 June Q6
13 marks Standard +0.3
A particle moving in a straight line starts from rest at the point \(O\) at time \(t = 0\). At time \(t\) seconds, the velocity \(v\) m s\(^{-1}\) of the particle is given by $$v = 3t(t - 4), \quad 0 \leq t \leq 5,$$ $$v = 75t^{-1}, \quad 5 \leq t \leq 10.$$
  1. Sketch a velocity-time graph for the particle for \(0 \leq t \leq 10\). [3]
  2. Find the set of values of \(t\) for which the acceleration of the particle is positive. [2]
  3. Show that the total distance travelled by the particle in the interval \(0 \leq t \leq 5\) is \(39\) m. [3]
  4. Find, to \(3\) significant figures, the value of \(t\) at which the particle returns to \(O\). [5]
Edexcel M3 2006 June Q7
13 marks Challenging +1.2
One end of a light inextensible string of length \(l\) is attached to a particle \(P\) of mass \(m\). The other end is attached to a fixed point \(A\). The particle is hanging freely at rest with the string vertical when it is projected horizontally with speed \(\sqrt{\frac{5gl}{2}}\).
  1. Find the speed of \(P\) when the string is horizontal. [4]
When the string is horizontal it comes into contact with a small smooth fixed peg which is at the point \(B\), where \(AB\) is horizontal, and \(AB < l\). Given that the particle then describes a complete semicircle with centre \(B\),
  1. Find the least possible value of the length \(AB\). [9]
Edexcel M3 2007 June Q1
9 marks Standard +0.3
The rudder on a ship is modelled as a uniform plane lamina having the same shape as the region \(R\) which is enclosed between the curve with equation \(y = 2x - x^2\) and the \(x\)-axis.
  1. Show that the area of \(R\) is \(\frac{4}{3}\). [4]
  2. Find the coordinates of the centre of mass of the lamina. [5]
Edexcel M3 2007 June Q2
10 marks Standard +0.3
An open container \(C\) is modelled as a thin uniform hollow cylinder of radius \(h\) and height \(h\) with a base but no lid. The centre of the base is \(O\).
  1. Show that the distance of the centre of mass of \(C\) from \(O\) is \(\frac{1}{4}h\). [5]
The container is filled with uniform liquid. Given that the mass of the container is \(M\) and the mass of the liquid is \(M\),
  1. find the distance of the centre of mass of the filled container from \(O\). [5]
Edexcel M3 2007 June Q3
9 marks Standard +0.8
A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the earth. The earth is modelled as a fixed sphere of radius \(R\). When \(S\) is at a distance \(x\) from the centre of the earth, the force exerted by the earth on \(S\) is directed towards the centre of the earth and has magnitude \(\frac{k}{x^2}\), where \(k\) is a constant.
  1. Show that \(k = mgR^2\). [2]
Given that \(S\) starts from rest when its distance from the centre of the earth is \(2R\), and that air resistance can be ignored,
  1. find the speed of \(S\) as it crashes into the surface of the earth. [7]
Edexcel M3 2007 June Q4
9 marks Standard +0.3
A light inextensible string of length \(l\) has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle moves with constant speed \(v\) in a horizontal circle with the string taut. The centre of the circle is vertically below \(A\) and the radius of the circle is \(r\). Show that $$gr^2 = v^2\sqrt{l^2 - r^2}.$$ [9]
Edexcel M3 2007 June Q5
11 marks Standard +0.3
A particle \(P\) moves on the \(x\)-axis with simple harmonic motion about the origin \(O\) as centre. When \(P\) is at a distance \(0.04\) m from \(O\), its speed is \(0.2\) m s\(^{-1}\) and the magnitude of its acceleration is \(1\) m s\(^{-2}\).
  1. Find the period of the motion. [3]
The amplitude of the motion is \(a\) metres. Find
  1. the value of \(a\), [3]
  2. the total time, within one complete oscillation, for which the distance \(OP\) is greater than \(\frac{3}{4}a\) metres. [5]
Edexcel M3 2007 June Q6
12 marks Standard +0.8
A particle \(P\) is free to move on the smooth inner surface of a fixed thin hollow sphere of internal radius \(a\) and centre \(O\). The particle passes through the lowest point of the spherical surface with speed \(U\). The particle loses contact with the surface when \(OP\) is inclined at an angle \(\alpha\) to the upward vertical.
  1. Show that \(U^2 = ag(2 + 3\cos \alpha)\). [7]
The particle has speed \(W\) as it passes through the level of \(O\). Given that \(\cos \alpha = \frac{1}{\sqrt{3}}\),
  1. show that \(W^2 = ag\sqrt{3}\). [5]
Edexcel M3 2007 June Q7
15 marks Challenging +1.2
\includegraphics{figure_1} A light elastic string, of natural length \(3l\) and modulus of elasticity \(\lambda\), has its ends attached to two points \(A\) and \(B\), where \(AB = 3l\) and \(AB\) is horizontal. A particle \(P\) of mass \(m\) is attached to the mid-point of the string. Given that \(P\) rests in equilibrium at a distance \(2l\) below \(AB\), as shown in Figure 1,
  1. show that \(\lambda = \frac{15mg}{16}\) [9]
The particle is pulled vertically downwards from its equilibrium position until the total length of the elastic string is \(7.8l\). The particle is released from rest.
  1. Show that \(P\) comes to instantaneous rest on the line \(AB\). [6]
Edexcel M3 2009 June Q1
9 marks Standard +0.3
A light elastic string has natural length \(8\) m and modulus of elasticity \(80\) N. The ends of the string are attached to fixed points \(P\) and \(Q\) which are on the same horizontal level and \(12\) m apart. A particle is attached to the mid-point of the string and hangs in equilibrium at a point \(4.5\) m below \(PQ\).
  1. Calculate the weight of the particle. [6]
  2. Calculate the elastic energy in the string when the particle is in this position. [3]
Edexcel M3 2009 June Q2
8 marks Standard +0.3
[The centre of mass of a uniform hollow cone of height \(h\) is \(\frac{1}{3}h\) above the base on the line from the centre of the base to the vertex.] \includegraphics{figure_1} A marker for the route of a charity walk consists of a uniform hollow cone fixed on to a uniform solid cylindrical ring, as shown in Figure 1. The hollow cone has base radius \(r\), height \(9h\) and mass \(m\). The solid cylindrical ring has outer radius \(r\), height \(2h\) and mass \(3m\). The marker stands with its base on a horizontal surface.
  1. Find, in terms of \(h\), the distance of the centre of mass of the marker from the horizontal surface. [5]
When the marker stands on a plane inclined at arctan \(\frac{1}{12}\) to the horizontal it is on the point of toppling over. The coefficient of friction between the marker and the plane is large enough to be certain that the marker will not slip.
  1. Find \(h\) in terms of \(r\). [3]
Edexcel M3 2009 June Q3
8 marks Standard +0.3
\includegraphics{figure_2} A particle \(P\) of mass \(m\) moves on the smooth inner surface of a hemispherical bowl of radius \(r\). The bowl is fixed with its rim horizontal as shown in Figure 2. The particle moves with constant angular speed \(\sqrt{\left(\frac{3g}{2r}\right)}\) in a horizontal circle at depth \(d\) below the centre of the bowl.
  1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the bowl on \(P\). [4]
  2. Find \(d\) in terms of \(r\). [4]
Edexcel M3 2009 June Q4
9 marks Standard +0.3
The finite region bounded by the \(x\)-axis, the curve \(y = \frac{1}{x}\), the line \(x = \frac{1}{4}\) and the line \(x = 1\), is rotated through one complete revolution about the \(x\)-axis to form a uniform solid of revolution.
  1. Show that the volume of the solid is \(21\pi\). [4]
  2. Find the coordinates of the centre of mass of the solid. [5]
Edexcel M3 2009 June Q5
11 marks Challenging +1.2
One end of a light inextensible string of length \(l\) is attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\), which is held at a point \(B\) with the string taut and \(AP\) making an angle arccos \(\frac{1}{4}\) with the downward vertical. The particle is released from rest. When \(AP\) makes an angle \(\theta\) with the downward vertical, the string is taut and the tension in the string is \(T\).
  1. Show that $$T = 3mg \cos \theta - \frac{mg}{2}.$$ [6]
\includegraphics{figure_3} At an instant when \(AP\) makes an angle of \(60°\) to the downward vertical, \(P\) is moving upwards, as shown in Figure 3. At this instant the string breaks. At the highest point reached in the subsequent motion, \(P\) is at a distance \(d\) below the horizontal through \(A\).
  1. Find \(d\) in terms of \(l\). [5]
Edexcel M3 2009 June Q6
14 marks Challenging +1.2
A cyclist and her bicycle have a combined mass of \(100\) kg. She is working at a constant rate of \(80\) W and is moving in a straight line on a horizontal road. The resistance to motion is proportional to the square of her speed. Her initial speed is \(4\) m s\(^{-1}\) and her maximum possible speed under these conditions is \(20\) m s\(^{-1}\). When she is at a distance \(x\) m from a fixed point \(O\) on the road, she is moving with speed \(v\) m s\(^{-1}\) away from \(O\).
  1. Show that $$v \frac{dv}{dx} = \frac{8000 - v^3}{10000v}.$$ [5]
  2. Find the distance she travels as her speed increases from \(4\) m s\(^{-1}\) to \(8\) m s\(^{-1}\). [5]
  3. Use the trapezium rule, with 2 intervals, to estimate how long it takes for her speed to increase from \(4\) m s\(^{-1}\) to \(8\) m s\(^{-1}\). [4]
Edexcel M3 2009 June Q7
16 marks Challenging +1.2
\includegraphics{figure_4} \(A\) and \(B\) are two points on a smooth horizontal floor, where \(AB = 5\) m. A particle \(P\) has mass \(0.5\) kg. One end of a light elastic spring, of natural length \(2\) m and modulus of elasticity \(16\) N, is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length \(1\) m and modulus of elasticity \(12\) N, are attached to \(P\) and \(B\), as shown in Figure 4.
  1. Find the extensions in the two springs when the particle is at rest in equilibrium. [5]
Initially \(P\) is at rest in equilibrium. It is then set in motion and starts to move towards \(B\). In the subsequent motion \(P\) does not reach \(A\) or \(B\).
  1. Show that \(P\) oscillates with simple harmonic motion about the equilibrium position. [4]
  2. Given that the initial speed of \(P\) is \(\sqrt{10}\) m s\(^{-1}\), find the proportion of time in each complete oscillation for which \(P\) stays within \(0.25\) m of the equilibrium position. [7]
Edexcel M3 2012 June Q1
9 marks Standard +0.3
A particle \(P\) is moving along the positive \(x\)-axis. At time \(t = 0\), \(P\) is at the origin \(O\). At time \(t\) seconds, \(P\) is \(x\) metres from \(O\) and has velocity \(v = 2e^{-t}\) m s\(^{-1}\) in the direction of \(x\) increasing.
  1. Find the acceleration of \(P\) in terms of \(x\). [3]
  2. Find \(x\) in terms of \(t\). [6]
Edexcel M3 2012 June Q2
8 marks Standard +0.3
A particle \(P\) moves in a straight line with simple harmonic motion about a fixed centre \(O\). The period of the motion is \(\frac{\pi}{2}\) seconds. At time \(t\) seconds the speed of \(P\) is \(v\) m s\(^{-1}\). When \(t = 0\), \(P\) is at \(O\) and \(v = 6\). Find
  1. the greatest distance of \(P\) from \(O\) during the motion, [3]
  2. the greatest magnitude of the acceleration of \(P\) during the motion, [2]
  3. the smallest positive value of \(t\) for which \(P\) is 1 m from \(O\). [3]
Edexcel M3 2012 June Q3
10 marks Standard +0.3
\includegraphics{figure_1} A particle \(Q\) of mass 5 kg is attached by two light inextensible strings to two fixed points \(A\) and \(B\) on a vertical pole. Each string has length 0.6 m and \(A\) is 0.4 m vertically above \(B\), as shown in Figure 1. Both strings are taut and \(Q\) is moving in a horizontal circle with constant angular speed 10 rad s\(^{-1}\). Find the tension in
  1. \(AQ\),
  2. \(BQ\). [10]
Edexcel M3 2012 June Q4
10 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows the cross-section \(AVBC\) of the solid \(S\) formed when a uniform right circular cone of base radius \(a\) and height \(a\), is removed from a uniform right circular cone of base radius \(a\) and height \(2a\). Both cones have the same axis \(VCO\), where \(O\) is the centre of the base of each cone.
  1. Show that the distance of the centre of mass of \(S\) from the vertex \(V\) is \(\frac{5}{4}a\). [5]
The mass of \(S\) is \(M\). A particle of mass \(kM\) is attached to \(S\) at \(B\). The system is suspended by a string attached to the vertex \(V\), and hangs freely in equilibrium. Given that \(VA\) is at an angle \(45°\) to the vertical through \(V\),
  1. find the value of \(k\). [5]
Edexcel M3 2012 June Q5
12 marks Standard +0.8
A fixed smooth sphere has centre \(O\) and radius \(a\). A particle \(P\) is placed on the surface of the sphere at the point \(A\), where \(OA\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is released from rest at \(A\). When \(OP\) makes an angle \(\theta\) to the upward vertical through \(O\), \(P\) is on the surface of the sphere and the speed of \(P\) is \(v\). Given that \(\cos \alpha = \frac{3}{5}\)
  1. show that $$v^2 = \frac{2ga}{5}(3 - 5\cos \theta)$$ [4]
  2. find the speed of \(P\) at the instant when it loses contact with the sphere. [8]
Edexcel M3 2012 June Q6
12 marks Standard +0.8
\includegraphics{figure_3} Figure 3 shows a uniform equilateral triangular lamina \(PRT\) with sides of length \(2a\).
  1. Using calculus, prove that the centre of mass of \(PRT\) is at a distance \(\frac{2\sqrt{3}}{3}a\) from \(R\). [6]
\includegraphics{figure_4} The circular sector \(PQU\), of radius \(a\) and centre \(P\), and the circular sector \(TUS\), of radius \(a\) and centre \(T\), are removed from \(PRT\) to form the uniform lamina \(QRSU\) shown in Figure 4.
  1. Show that the distance of the centre of mass of \(QRSU\) from \(U\) is \(\frac{2a}{3\sqrt{3} - \pi}\). [6]
Edexcel M3 2012 June Q7
14 marks Standard +0.8
A particle \(B\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.75 m and modulus of elasticity 24.5 N. The other end of the string is attached to a fixed point \(A\). The particle is hanging in equilibrium at the point \(E\), vertically below \(A\).
  1. Show that \(AE = 0.9\) m. [3]
The particle is held at \(A\) and released from rest. The particle first comes to instantaneous rest at the point \(C\).
  1. Find the distance \(AC\). [5]
  2. Show that while the string is taut, \(B\) is moving with simple harmonic motion with centre \(E\). [4]
  3. Calculate the maximum speed of \(B\). [2]
Edexcel M3 2014 June Q1
8 marks Standard +0.3
A particle \(P\) of mass \(0.25\) kg is moving along the positive \(x\)-axis under the action of a single force. At time \(t\) seconds \(P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) with speed \(v\) m s\(^{-1}\) where \(\frac{\mathrm{d}v}{\mathrm{d}x} = 3\). It is given that \(x = 2\) and \(v = 3\) when \(t = 0\)
  1. Find the magnitude of the force acting on \(P\) when \(x = 5\) [4]
  2. Find the value of \(t\) when \(x = 5\) [4]
Edexcel M3 2014 June Q2
13 marks Standard +0.8
\includegraphics{figure_1} A cone of semi-vertical angle \(60°\) is fixed with its axis vertical and vertex upwards. A particle 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 vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed \(\omega\), with the string making a constant angle \(60°\) with the horizontal, as shown in Figure 1.
  1. Find the tension in the string, in terms of \(m\), \(l\), \(\omega\) and \(g\). [7]
The particle remains on the surface of the cone.
  1. Show that the time for the particle to make one complete revolution is greater than $$2\pi\sqrt{\frac{l\sqrt{3}}{2g}}$$ [6]