Questions M3 (796 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
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]
Edexcel M3 2014 June Q3
11 marks Challenging +1.2
One end \(A\) of a light elastic string \(AB\), of modulus of elasticity \(mg\) and natural length \(a\), is fixed to a point on a rough plane inclined at an angle \(\theta\) to the horizontal. The other end \(B\) of the string is attached to a particle of mass \(m\) which is held at rest on the plane. The string \(AB\) lies along a line of greatest slope of the plane, with \(B\) lower than \(A\) and \(AB = a\). The coefficient of friction between the particle and the plane is \(\mu\), where \(\mu < \tan \theta\). The particle is released from rest.
  1. Show that when the particle comes to rest it has moved a distance \(2a(\sin \theta - \mu \cos \theta)\) down the plane. [6]
  2. Given that there is no further motion, show that \(\mu \geqslant \frac{1}{3} \tan \theta\). [5]
Edexcel M3 2014 June Q4
16 marks Challenging +1.2
\includegraphics{figure_2} A smooth sphere of radius \(a\) is fixed with a point \(A\) of its surface in contact with a fixed vertical wall. A particle is placed on the highest point of the sphere and is projected towards the wall and perpendicular to the wall with horizontal speed \(\sqrt{\frac{2ag}{5}}\), as shown in Figure 2. The particle leaves the surface of the sphere with speed \(V\).
  1. Show that \(V = \sqrt{\frac{4ag}{5}}\) [7]
The particle strikes the wall at the point \(X\).
  1. Find the distance \(AX\). [9]
Edexcel M3 2014 June Q5
13 marks Standard +0.8
\includegraphics{figure_3} A uniform solid right circular cylinder has height \(h\) and radius \(r\). The centre of one plane face is \(O\) and the centre of the other plane face is \(Y\). A cylindrical hole is made by removing a solid cylinder of radius \(\frac{1}{4}r\) and height \(\frac{1}{4}h\) from the end with centre \(O\). The axis of the cylinder removed is parallel to \(OY\) and meets the end with centre \(O\) at \(X\), where \(OX = \frac{1}{4}r\). One plane face of the cylinder removed coincides with the plane face through \(O\) of the original cylinder. The resulting solid \(S\) is shown in Figure 3.
  1. Show that the centre of mass of \(S\) is at a distance \(\frac{85h}{168}\) from the plane face containing \(O\). [7]
The solid \(S\) is freely suspended from \(O\). In equilibrium the line \(OY\) is inclined at an angle arctan(17) to the horizontal.
  1. Find \(r\) in terms of \(h\). [6]
Edexcel M3 2014 June Q6
14 marks Standard +0.8
A light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\), has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs freely at rest in equilibrium at the point \(E\). The distance of \(E\) below \(A\) is \((l + e)\).
  1. Find \(e\) in terms of \(l\). [2]
At time \(t = 0\), the particle is projected vertically downwards from \(E\) with speed \(\sqrt{gl}\).
  1. Prove that, while the string is taut, \(P\) moves with simple harmonic motion. [5]
  2. Find the amplitude of the simple harmonic motion. [3]
  3. Find the time at which the string first goes slack. [4]
AQA M3 2016 June Q1
4 marks Moderate -0.8
At a firing range, a man holds a gun and fires a bullet horizontally. The bullet is fired with a horizontal velocity of \(400 \text{ m s}^{-1}\). The mass of the gun is \(1.5\) kg and the mass of the bullet is \(30\) grams.
  1. Find the speed of recoil of the gun. [2 marks]
  2. Find the magnitude of the impulse exerted by the man on the gun in bringing the gun to rest after the bullet is fired. [2 marks]
AQA M3 2016 June Q2
6 marks Moderate -0.8
A lunar mapping satellite of mass \(m_1\) measured in kg is in an elliptic orbit around the moon, which has mass \(m_2\) measured in kg. The effective potential, \(E\), of the satellite is given by $$E = \frac{K^2}{2m_1r^2} - \frac{Gm_1m_2}{r}$$ where \(r\) measured in metres is the distance of the satellite from the moon, \(G\) Nm\(^2\)kg\(^{-2}\) is the universal gravitational constant, and \(K\) is the angular momentum of the satellite. By using dimensional analysis, find the dimensions of:
  1. \(E\), [3 marks]
  2. \(K\). [3 marks]
AQA M3 2016 June Q3
12 marks Standard +0.8
A ball is projected from a point \(O\) on horizontal ground with speed \(14 \text{ m s}^{-1}\) at an angle of elevation \(30°\) above the horizontal. The ball travels in a vertical plane through the point \(O\) and hits a point \(Q\) on a plane which is inclined at \(45°\) to the horizontal. The point \(O\) is \(6\) metres from \(P\), the foot of the inclined plane, as shown in the diagram. The points \(O\), \(P\) and \(Q\) lie in the same vertical plane. The line \(PQ\) is a line of greatest slope of the inclined plane. \includegraphics{figure_3}
  1. During its flight, the horizontal and upward vertical distances of the ball from \(O\) are \(x\) metres and \(y\) metres respectively. Show that \(x\) and \(y\) satisfy the equation $$y = x\frac{\sqrt{3}}{3} - \frac{x^2}{30}$$ Use \(\cos 30° = \frac{\sqrt{3}}{2}\) and \(\tan 30° = \frac{\sqrt{3}}{3}\). [5 marks]
  2. Find the distance \(PQ\). [7 marks]
AQA M3 2016 June Q4
14 marks Standard +0.3
A smooth uniform sphere \(A\), of mass \(m\), is moving with velocity \(8u\) in a straight line on a smooth horizontal table. A smooth uniform sphere \(B\), of mass \(4m\), has the same radius as \(A\) and is moving on the table with velocity \(u\). \includegraphics{figure_4} The sphere \(A\) collides directly with the sphere \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
    1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) immediately after the collision. [6 marks]
    2. The direction of motion of \(A\) is reversed by the collision. Show that \(e > a\), where \(a\) is a constant to be determined. [2 marks]
  2. Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac{2}{5}\). The sphere \(B\) collides with \(A\) again after rebounding from the wall. Show that \(e < b\), where \(b\) is a constant to be determined. [3 marks]
  3. Given that \(e = \frac{4}{7}\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) by the wall. [3 marks]
AQA M3 2016 June Q5
12 marks Challenging +1.8
A ball is projected from a point \(O\) above a smooth plane which is inclined at an angle of \(20°\) to the horizontal. The point \(O\) is at a perpendicular distance of \(1\) m from the inclined plane. The ball is projected with velocity \(22 \text{ m s}^{-1}\) at an angle of \(70°\) above the horizontal. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane for the first time at a point \(A\). \includegraphics{figure_5}
    1. Find the time taken by the ball to travel from \(O\) to \(A\). [4 marks]
    2. Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at \(A\). [4 marks]
  2. After striking \(A\), the ball rebounds and strikes the plane for a second time at a point further up than \(A\). The coefficient of restitution between the ball and the inclined plane is \(e\). Show that \(e < k\), where \(k\) is a constant to be determined. [4 marks]
AQA M3 2016 June Q6
14 marks Challenging +1.2
In this question use \(\cos 30° = \sin 60° = \frac{\sqrt{3}}{2}\). A smooth spherical ball, \(A\), is moving with speed \(u\) in a straight line on a smooth horizontal table when it hits an identical ball, \(B\), which is at rest on the table. Just before the collision, the direction of motion of \(A\) is parallel to a fixed smooth vertical wall. At the instant of collision, the line of centres of \(A\) and \(B\) makes an angle of \(60°\) with the wall, as shown in the diagram. \includegraphics{figure_6} The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Show that the speed of \(B\) immediately after the collision is \(\frac{1}{4}u(1 + e)\) and find, in terms of \(u\) and \(e\), the components of the velocity of \(A\), parallel and perpendicular to the line of centres, immediately after the collision. [7 marks]
  2. Subsequently, \(B\) collides with the wall. After colliding with the wall, the direction of motion of \(B\) is parallel to the direction of motion of \(A\) after its collision with \(B\). Show that the coefficient of restitution between \(B\) and the wall is \(\frac{1 + e}{7 - e}\). [7 marks]
AQA M3 2016 June Q7
13 marks Challenging +1.8
A quad-bike, a truck and a car are moving on a large, open, horizontal surface in a desert plain. Relative to the quad-bike, which is travelling due west at its maximum speed of \(10 \text{ m s}^{-1}\), the truck is moving on a bearing of \(340°\). Relative to the car, which is travelling due east at a speed of \(15 \text{ m s}^{-1}\), the truck is moving on a bearing of \(300°\).
  1. Show that the speed of the truck is approximately \(24.7 \text{ m s}^{-1}\) and that it is moving on a bearing of \(318°\), correct to the nearest degree. [8 marks]
  2. At the instant when the truck is at a distance of \(400\) metres from the quad-bike, the bearing of the truck from the quad-bike is \(060°\). The truck continues to move with the same velocity as in part (a). The quad-bike continues to move at a speed of \(10 \text{ m s}^{-1}\). Find the bearing, to the nearest degree, on which the quad-bike should travel in order to approach the truck as closely as possible. [5 marks]
Edexcel M3 Q1
7 marks Standard +0.3
A bird of mass 0.5 kg, flying around a vertical feeding post at a constant speed of 6 ms\(^{-1}\), banks its wings to move in a horizontal circle of radius 2 m. The aerodynamic lift \(L\) newtons is perpendicular to the bird's wings, as shown. \includegraphics{figure_1} Modelling the bird as a particle, find, to the nearest degree, the angle that its wings make with the vertical. [7 marks]