Questions — Edexcel M2 (623 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 M2 Specimen Q8
13 marks Standard +0.3
8. A particle \(A\) of mass \(m\) is moving with speed \(3 u\) on a smooth horizontal table when it collides directly with a particle \(B\) of mass \(2 m\) which is moving in the opposite direction with speed \(u\). The direction of motion of \(A\) is reversed by the collision. The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 3 } ( 1 + 4 e ) u\).
    (6)
  2. Show that \(e > \frac { 1 } { 8 }\).
    (3) Subsequently \(B\) hits a wall fixed at right angles to the line of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). After \(B\) rebounds from the wall, there is a further collision between \(A\) and \(B\).
  3. Show that \(e < \frac { 1 } { 4 }\).
    (4) END
Edexcel M2 Q2
Standard +0.3
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{173a2029-a0b8-437f-9339-5a1b6f30a8e3-004_378_652_294_630}
\end{figure} A thin uniform wire, of total length 20 cm , is bent to form a frame. The frame is in the shape of a trapezium \(A B C D\), where \(A B = A D = 4 \mathrm {~cm} , C D = 5 \mathrm {~cm}\), and \(A B\) is perpendicular to \(B C\) and \(A D\), as shown in Figure 1.
  1. Find the distance of the centre of mass of the frame from \(A B\). The frame has mass \(M\). A particle of mass \(k M\) is attached to the frame at \(C\). When the frame is freely suspended from the mid-point of \(B C\), the frame hangs in equilibrium with \(B C\) horizontal.
  2. Find the value of \(k\).
Edexcel M2 Q3
Standard +0.3
3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where \(c\) is a positive constant.When \(t = 1.5\) ,the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) .Find
  1. the value of \(c\) ,
  2. the acceleration of \(P\) when \(t = 1.5\) . \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } , \\ \text { where } c \text { is a positive constant.When } t = 1.5 \text { ,the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { .Find } \end{aligned}$$ (a)the value of \(c\) , 3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is D墐
    (b)the acceleration of \(P\) when \(t = 1.5\) .
Edexcel M2 Q6
Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{173a2029-a0b8-437f-9339-5a1b6f30a8e3-010_442_689_292_632}
\end{figure} A uniform pole \(A B\), of mass 30 kg and length 3 m , is smoothly hinged to a vertical wall at one end \(A\). The pole is held in equilibrium in a horizontal position by a light rod CD. One end \(C\) of the rod is fixed to the wall vertically below \(A\). The other end \(D\) is freely jointed to the pole so that \(\angle A C D = 30 ^ { \circ }\) and \(A D = 0.5 \mathrm {~m}\), as shown in Figure 2. Find
  1. the thrust in the rod \(C D\),
  2. the magnitude of the force exerted by the wall on the pole at \(A\). The rod \(C D\) is removed and replaced by a longer light rod \(C M\), where \(M\) is the mid-point of \(A B\). The rod is freely jointed to the pole at \(M\). The pole \(A B\) remains in equilibrium in a horizontal position.
  3. Show that the force exerted by the wall on the pole at \(A\) now acts horizontally.
Edexcel M2 2016 January Q1
8 marks Standard +0.3
  1. A car of mass 900 kg is travelling up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 25 }\). The car is travelling at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistance to motion from non-gravitational forces has a constant magnitude of 800 N . The car takes 10 seconds to travel from \(A\) to \(B\), where \(A\) and \(B\) are two points on the road.
    1. Find the work done by the engine of the car as the car travels from \(A\) to \(B\).
    When the car is at \(B\) and travelling at a speed of \(14 \mathrm {~ms} ^ { - 1 }\) the rate of working of the engine of the car is suddenly increased to \(P \mathrm {~kW}\), resulting in an initial acceleration of the car of \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The resistance to motion from non-gravitational forces still has a constant magnitude of 800 N .
  2. Find the value of \(P\).
Edexcel M2 2016 January Q2
10 marks Standard +0.3
2. A particle \(P\) of mass 0.7 kg is moving in a straight line on a smooth horizontal surface. The particle \(P\) collides with a particle \(Q\) of mass 1.2 kg which is at rest on the surface. Immediately before the collision the speed of \(P\) is \(6 \mathrm {~ms} ^ { - 1 }\). Immediately after the collision both particles are moving in the same direction. The coefficient of restitution between the particles is \(e\).
  1. Show that \(e < \frac { 7 } { 12 }\) Given that \(e = \frac { 1 } { 4 }\)
  2. find the magnitude of the impulse exerted on \(Q\) in the collision.
Edexcel M2 2016 January Q3
11 marks Standard +0.3
3.At time \(t\) seconds( \(t \geqslant 0\) )a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) ,where When \(t = 0\) the particle \(P\) is at the origin \(O\) .At time \(T\) seconds,\(P\) is at the point \(A\) and \(\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )\) ,where \(\lambda\) is a constant. Find
  1. the value of \(T\) ,
  2. the acceleration of \(P\) as it passes through the point \(A\) ,
  3. the distance \(O A\) . $$\mathbf { v } = \left( 6 t ^ { 2 } + 6 t \right) \mathbf { i } + \left( 3 t ^ { 2 } + 24 \right) \mathbf { j }$$ 的 When \(t = 0\) the particle \(P\) is at the origin \(O\) .At time \(T\) seconds,\(P\) is at the point \(A\) and \(\mathbf { v } = \lambda ( \mathbf { i } + \mathbf { j } )\) ,where \(\lambda\) is a constant. Find
    1. the value of \(T\) , \(\_\_\_\_\) "
Edexcel M2 2016 January Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6d100ff-dd4a-4591-a0a3-81761773045e-07_544_1264_251_338} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 4 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest at the point \(A\) on a rough fixed plane inclined at \(\alpha\) to the horizontal ground, where \(\sin \alpha = \frac { 3 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley and 2.5 m above the ground, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. The system is released from rest with the string taut. At the instant when \(Q\) hits the ground, \(P\) is at the point \(B\) on the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\).
  1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
  2. Find the total potential energy lost by the system as \(P\) moves from \(A\) to \(B\).
  3. Find, using the work-energy principle, the speed of \(P\) as it passes through \(B\).
Edexcel M2 2016 January Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6d100ff-dd4a-4591-a0a3-81761773045e-09_689_581_237_683} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform lamina \(A B C D E F\), shown in Figure 2, consists of two identical rectangles with sides of length \(a\) and \(3 a\). The mass of the lamina is \(M\). A particle of mass \(k M\) is attached to the lamina at \(E\). The lamina, with the attached particle, is freely suspended from \(A\) and hangs in equilibrium with \(A F\) at an angle \(\theta\) to the downward vertical. Given that \(\tan \theta = \frac { 4 } { 7 }\), find the value of \(k\).
(10)
Edexcel M2 2016 January Q6
11 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6d100ff-dd4a-4591-a0a3-81761773045e-11_757_1269_233_331} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\), of mass \(3 m\) and length \(2 a\), is freely hinged at \(A\) to a fixed point on horizontal ground. A particle of mass \(m\) is attached to the rod at the end \(B\). The system is held in equilibrium by a force \(\mathbf { F }\) acting at the point \(C\), where \(A C = b\). The rod makes an acute angle \(\theta\) with the ground, as shown in Figure 3. The line of action of \(\mathbf { F }\) is perpendicular to the rod and in the same vertical plane as the rod.
  1. Show that the magnitude of \(\mathbf { F }\) is \(\frac { 5 m g a } { b } \cos \theta\) The force exerted on the rod by the hinge at \(A\) is \(\mathbf { R }\), which acts upwards at an angle \(\phi\) above the horizontal, where \(\phi > \theta\).
  2. Find
    1. the component of \(\mathbf { R }\) parallel to the rod, in terms of \(m , g\) and \(\theta\),
    2. the component of \(\mathbf { R }\) perpendicular to the rod, in terms of \(a , b , m , g\) and \(\theta\).
  3. Hence, or otherwise, find the range of possible values of \(b\), giving your answer in terms of \(a\).
Edexcel M2 2016 January Q7
14 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6d100ff-dd4a-4591-a0a3-81761773045e-13_552_1296_255_317} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} At time \(t = 0\), a particle \(P\) of mass 0.7 kg is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) at an angle \(\theta ^ { \circ }\) to the horizontal. The particle moves freely under gravity. At time \(t = 2\) seconds, \(P\) passes through the point \(A\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving downwards at \(45 ^ { \circ }\) to the horizontal, as shown in Figure 4. Find
  1. the value of \(\theta\),
  2. the kinetic energy of \(P\) as it reaches the highest point of its path. For an interval of \(T\) seconds, the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is such that \(v \leqslant 6\)
  3. Find the value of \(T\).
Edexcel M2 2005 June Q1
7 marks Moderate -0.8
  1. A car of mass 1200 kg moves along a straight horizontal road. The resistance to motion of the car from non-gravitational forces is of constant magnitude 600 N . The car moves with constant speed and the engine of the car is working at a rate of 21 kW .
    1. Find the speed of the car.
    The car moves up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 14 }\).
    The car's engine continues to work at 21 kW , and the resistance to motion from nongravitational forces remains of magnitude 600 N .
  2. Find the constant speed at which the car can move up the hill.
Edexcel M2 2005 June Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{3847deb8-d86e-4254-828f-5d62f20c186f-03_378_652_294_630}
\end{figure} A thin uniform wire, of total length 20 cm , is bent to form a frame. The frame is in the shape of a trapezium \(A B C D\), where \(A B = A D = 4 \mathrm {~cm} , C D = 5 \mathrm {~cm}\), and \(A B\) is perpendicular to \(B C\) and \(A D\), as shown in Figure 1.
  1. Find the distance of the centre of mass of the frame from \(A B\). The frame has mass \(M\). A particle of mass \(k M\) is attached to the frame at \(C\). When the frame is freely suspended from the mid-point of \(B C\), the frame hangs in equilibrium with \(B C\) horizontal.
  2. Find the value of \(k\).
Edexcel M2 2005 June Q3
9 marks Moderate -0.3
3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } ,$$ where \(c\) is a positive constant.When \(t = 1.5\) ,the speed of \(P\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) .Find
  1. the value of \(c\) ,
  2. the acceleration of \(P\) when \(t = 1.5\) . \(\mathbf { r }\) metres relative to a fixed origin \(O\) ,and \(\mathbf { r }\) is given by $$\begin{aligned} \mathbf { r } = \left( 18 t - 4 t ^ { 3 } \right) \mathbf { i } + c t ^ { 2 } \mathbf { j } , \\ \text { where } c \text { is a positive constant.When } t = 1.5 \text { ,the speed of } P \text { is } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 } \text { .Find } \end{aligned}$$ (a)the value of \(c\) , 3.A particle \(P\) moves in a horizontal plane.At time \(t\) seconds,the position vector of \(P\) is D啨
    (b)the acceleration of \(P\) when \(t = 1.5\) .
Edexcel M2 2005 June Q4
10 marks Standard +0.3
4. A darts player throws darts at a dart board which hangs vertically. The motion of a dart is modelled as that of a particle moving freely under gravity. The darts move in a vertical plane which is perpendicular to the plane of the dart board. A dart is thrown horizontally with speed \(12.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It hits the board at a point which is 10 cm below the level from which it was thrown.
  1. Find the horizontal distance from the point where the dart was thrown to the dart board. The darts player moves his position. He now throws a dart from a point which is at a horizontal distance of 2.5 m from the board. He throws the dart at an angle of elevation \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 7 } { 24 }\). This dart hits the board at a point which is at the same level as the point from which it was thrown.
  2. Find the speed with which the dart is thrown.
Edexcel M2 2005 June Q5
14 marks Standard +0.3
5. Two small spheres \(A\) and \(B\) have mass \(3 m\) and \(2 m\) respectively. They are moving towards each other in opposite directions on a smooth horizontal plane, both with speed \(2 u\), when they collide directly. As a result of the collision, the direction of motion of \(B\) is reversed and its speed is unchanged.
  1. Find the coefficient of restitution between the spheres. Subsequently, \(B\) collides directly with another small sphere \(C\) of mass \(5 m\) which is at rest. The coefficient of restitution between \(B\) and \(C\) is \(\frac { 3 } { 5 }\).
  2. Show that, after \(B\) collides with \(C\), there will be no further collisions between the spheres.
Edexcel M2 2005 June Q6
12 marks Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{3847deb8-d86e-4254-828f-5d62f20c186f-09_442_689_292_632}
\end{figure} A uniform pole \(A B\), of mass 30 kg and length 3 m , is smoothly hinged to a vertical wall at one end \(A\). The pole is held in equilibrium in a horizontal position by a light rod CD. One end \(C\) of the rod is fixed to the wall vertically below \(A\). The other end \(D\) is freely jointed to the pole so that \(\angle A C D = 30 ^ { \circ }\) and \(A D = 0.5 \mathrm {~m}\), as shown in Figure 2. Find
  1. the thrust in the rod \(C D\),
  2. the magnitude of the force exerted by the wall on the pole at \(A\). The rod \(C D\) is removed and replaced by a longer light rod \(C M\), where \(M\) is the mid-point of \(A B\). The rod is freely jointed to the pole at \(M\). The pole \(A B\) remains in equilibrium in a horizontal position.
  3. Show that the force exerted by the wall on the pole at \(A\) now acts horizontally.
Edexcel M2 2005 June Q7
15 marks Standard +0.3
7. At a demolition site, bricks slide down a straight chute into a container. The chute is rough and is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The distance travelled down the chute by each brick is 8 m . A brick of mass 3 kg is released from rest at the top of the chute. When it reaches the bottom of the chute, its speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the potential energy lost by the brick in moving down the chute.
  2. By using the work-energy principle, or otherwise, find the constant frictional force acting on the brick as it moves down the chute.
  3. Hence find the coefficient of friction between the brick and the chute. Another brick of mass 3 kg slides down the chute. This brick is given an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the chute.
  4. Find the speed of this brick when it reaches the bottom of the chute.
Edexcel M2 2016 June Q1
13 marks Standard +0.3
  1. A particle \(P\) moves along a straight line. The speed of \(P\) at time \(t\) seconds ( \(t \geqslant 0\) ) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \left( p t ^ { 2 } + q t + r \right)\) and \(p , q\) and \(r\) are constants. When \(t = 2\) the speed of \(P\) has its minimum value. When \(t = 0 , v = 11\) and when \(t = 2 , v = 3\)
Find
  1. the acceleration of \(P\) when \(t = 3\)
  2. the distance travelled by \(P\) in the third second of the motion.
Edexcel M2 2016 June Q2
10 marks Standard +0.3
2. A car of mass 800 kg is moving on a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the car is moving up the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(3 P\) watts. When the car is moving down the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(P\) watts.
  1. Find
    1. the value of \(P\),
    2. the value of \(R\).
      (6) When the car is moving up the road at \(12.5 \mathrm {~ms} ^ { - 1 }\) the engine is switched off and the car comes to rest, without braking, in a distance \(d\) metres. The resistance to the motion of the car from non-gravitational forces is still modelled as a constant force of magnitude \(R\) newtons.
  2. Use the work-energy principle to find the value of \(d\).
Edexcel M2 2016 June Q3
6 marks Standard +0.3
3. A particle of mass 0.6 kg is moving with constant velocity ( \(c \mathbf { i } + 2 c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(c\) is a positive constant. The particle receives an impulse of magnitude \(2 \sqrt { 10 } \mathrm {~N} \mathrm {~s}\). Immediately after receiving the impulse the particle has velocity ( \(2 c \mathbf { i } - c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\). Find the value of \(c\).
(6)
Edexcel M2 2016 June Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{deb9e495-3bfb-4a46-9ee7-3eb421c33499-07_606_883_260_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(O B C\) is one quarter of a circular disc with centre \(O\) and radius 4 m . The points \(A\) and \(D\), on \(O B\) and \(O C\) respectively, are 3 m from \(O\). The uniform lamina \(A B C D\), shown shaded in Figure 1, is formed by removing the triangle \(O A D\) from \(O B C\). Given that the centre of mass of one quarter of a uniform circular disc of radius \(r\) is at a distance \(\frac { 4 \sqrt { 2 } } { 3 \pi } r\) from the centre of the disc,
  1. find the distance of the centre of mass of the lamina \(A B C D\) from \(A D\). The lamina is freely suspended from \(D\) and hangs in equilibrium.
  2. Find, to the nearest degree, the angle between \(D C\) and the downward vertical.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{deb9e495-3bfb-4a46-9ee7-3eb421c33499-09_915_1269_118_356} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure}
Edexcel M2 2016 June Q5
12 marks Standard +0.3
5. A non-uniform rod \(A B\), of mass 5 kg and length 4 m , rests with one end \(A\) on rough horizontal ground. The centre of mass of the rod is \(d\) metres from \(A\). The rod is held in limiting equilibrium at an angle \(\theta\) to the horizontal by a force \(\mathbf { P }\), which acts in a direction perpendicular to the rod at \(B\), as shown in Figure 2. The line of action of \(\mathbf { P }\) lies in the same vertical plane as the rod.
  1. Find, in terms of \(d , g\) and \(\theta\),
    1. the magnitude of the vertical component of the force exerted on the rod by the ground,
    2. the magnitude of the friction force acting on the rod at \(A\). Given that \(\tan \theta = \frac { 5 } { 12 }\) and that the coefficient of friction between the rod and the ground is \(\frac { 1 } { 2 }\),
  2. find the value of \(d\).
Edexcel M2 2016 June Q6
13 marks Standard +0.3
6. [In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is an upward vertical unit vector.] A particle \(P\) is projected from a fixed origin \(O\) with velocity ( \(3 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity and passes through the point \(A\) with position vector \(\lambda ( \mathbf { i } - \mathbf { j } ) \mathrm { m }\), where \(\lambda\) is a positive constant.
  1. Find the value of \(\lambda\).
  2. Find
    1. the speed of \(P\) at the instant when it passes through \(A\),
    2. the direction of motion of \(P\) at the instant when it passes through \(A\).
      HMAV SIHI NITIIIUM ION OC
      VILV SIHI NI JAHM ION OC
      VJ4V SIHI NI JIIYM ION OC
Edexcel M2 2016 June Q7
12 marks Standard +0.3
7. Two particles \(A\) and \(B\), of mass \(2 m\) and \(3 m\) respectively, are initially at rest on a smooth horizontal surface. Particle \(A\) is projected with speed \(3 u\) towards \(B\). Particle \(A\) collides directly with particle \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\)
  1. Find
    1. the speed of \(A\) immediately after the collision,
    2. the speed of \(B\) immediately after the collision. After the collision \(B\) hits a fixed smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(e\). The magnitude of the impulse received by \(B\) when it hits the wall is \(\frac { 27 } { 4 } m u\).
  2. Find the value of \(e\).
  3. Determine whether there is a further collision between \(A\) and \(B\) after \(B\) rebounds from the wall.