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Edexcel F2 2018 June Q4
9 marks Challenging +1.2
4. A complex number \(z\) is represented by the point \(P\) in an Argand diagram. Given that $$| z + i | = 1$$
  1. sketch the locus of \(P\). The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac { 3 \mathrm { i } z - 2 } { z + \mathrm { i } } , \quad z \neq - \mathrm { i }$$
  2. Given that \(T\) maps \(| z + i | = 1\) to a circle \(C\) in the \(w\)-plane, find a cartesian equation of \(C\).
Edexcel F2 2018 June Q5
8 marks Standard +0.8
  1. (a) Express \(\frac { 4 r + 2 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions.
    (b) Hence, using the method of differences, prove that
$$\sum _ { r = 1 } ^ { n } \frac { 4 r + 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( a n + b ) } { 2 ( n + 1 ) ( n + 2 ) }$$ where \(a\) and \(b\) are constants to be found.
Edexcel F2 2018 June Q6
13 marks Challenging +1.2
  1. (a) Show that the transformation \(x = \mathrm { e } ^ { t }\) transforms the differential equation
$$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = x ^ { 2 } \quad x > 0$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 3 y = \mathrm { e } ^ { 2 t }$$ (b) Find the general solution of the differential equation (II), expressing \(y\) as a function of \(t\).
(c) Hence find the general solution of the differential equation (I).
Edexcel F2 2018 June Q7
11 marks Challenging +1.2
7.(a)Use de Moivre's theorem to show that $$\cos 7 \theta \equiv 64 \cos ^ { 7 } \theta - 112 \cos ^ { 5 } \theta + 56 \cos ^ { 3 } \theta - 7 \cos \theta$$ (b)Hence find the four distinct roots of the equation $$64 x ^ { 7 } - 112 x ^ { 5 } + 56 x ^ { 3 } - 7 x + 1 = 0$$ giving your answers to 3 decimal places where necessary.
Edexcel F2 2018 June Q8
11 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{27ac35ba-1969-4a37-a7c5-f4741c9c59a8-28_570_728_264_609} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curves with polar equations $$\begin{array} { l l } r = 2 \sin \theta & 0 \leqslant \theta \leqslant \pi \\ r = 1.5 - \sin \theta & 0 \leqslant \theta \leqslant 2 \pi \end{array}$$ The curves intersect at the points \(P\) and \(Q\).
  1. Find the polar coordinates of the point \(P\) and the polar coordinates of the point \(Q\). The region \(R\), shown shaded in Figure 1, is enclosed by the two curves.
  2. Find the exact area of \(R\), giving your answer in the form \(p \pi + q \sqrt { 3 }\), where \(p\) and \(q\) are rational numbers to be found.
Edexcel M1 2013 January Q1
7 marks Moderate -0.3
  1. Two particles \(P\) and \(Q\) have masses \(4 m\) and \(m\) respectively. The particles are moving towards each other on a smooth horizontal plane and collide directly. The speeds of \(P\) and \(Q\) immediately before the collision are \(2 u\) and \(5 u\) respectively. Immediately after the collision, the speed of \(P\) is \(\frac { 1 } { 2 } u\) and its direction of motion is reversed.
    1. Find the speed and direction of motion of \(Q\) after the collision.
    2. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.
    3. A steel girder \(A B\), of mass 200 kg and length 12 m , rests horizontally in equilibrium on two smooth supports at \(C\) and at \(D\), where \(A C = 2 \mathrm {~m}\) and \(D B = 2 \mathrm {~m}\). A man of mass 80 kg stands on the girder at the point \(P\), where \(A P = 4 \mathrm {~m}\), as shown in Figure 1.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-03_318_1061_420_440} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The man is modelled as a particle and the girder is modelled as a uniform rod.
  2. Find the magnitude of the reaction on the girder at the support at \(C\). The support at \(D\) is now moved to the point \(X\) on the girder, where \(X B = x\) metres. The man remains on the girder at \(P\), as shown in Figure 2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-03_324_1058_1292_443} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Given that the magnitudes of the reactions at the two supports are now equal and that the girder again rests horizontally in equilibrium, find
  3. the magnitude of the reaction at the support at \(X\),
  4. the value of \(x\).
Edexcel M1 2013 January Q3
8 marks Moderate -0.8
3. A particle \(P\) of mass 2 kg is attached to one end of a light string, the other end of which is attached to a fixed point \(O\). The particle is held in equilibrium, with \(O P\) at \(30 ^ { \circ }\) to the downward vertical, by a force of magnitude \(F\) newtons. The force acts in the same vertical plane as the string and acts at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-05_444_510_477_699} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Find
  1. the value of \(F\),
  2. the tension in the string.
Edexcel M1 2013 January Q4
9 marks Standard +0.3
4. A lifeboat slides down a straight ramp inclined at an angle of \(15 ^ { \circ }\) to the horizontal. The lifeboat has mass 800 kg and the length of the ramp is 50 m . The lifeboat is released from rest at the top of the ramp and is moving with a speed of \(12.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it reaches the end of the ramp. By modelling the lifeboat as a particle and the ramp as a rough inclined plane, find the coefficient of friction between the lifeboat and the ramp.
(9)
Edexcel M1 2013 January Q5
15 marks Standard +0.3
5.\(v \left( \mathrm {~m} \mathrm {~s} ^ { - 1 } \right) ^ { \wedge }\) Figure 4 The velocity-time graph in Figure 4 represents the journey of a train \(P\) travelling along a straight horizontal track between two stations which are 1.5 km apart.The train \(P\) leaves the first station,accelerating uniformly from rest for 300 m until it reaches a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) .The train then maintains this speed for \(T\) seconds before decelerating uniformly at \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) ,coming to rest at the next station.
  1. Find the acceleration of \(P\) during the first 300 m of its journey.
  2. Find the value of \(T\) . A second train \(Q\) completes the same journey in the same total time.The train leaves the first station,accelerating uniformly from rest until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then immediately decelerates uniformly until it comes to rest at the next station.
  3. Sketch on the diagram above,a velocity-time graph which represents the journey of train \(Q\) .
  4. Find the value of \(V\) .
Edexcel M1 2013 January Q6
11 marks Moderate -0.3
6. [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship sets sail at 9 am from a port \(P\) and moves with constant velocity. The position vector of \(P\) is \(( 4 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }\). At 9.30 am the ship is at the point with position vector \(( \mathbf { i } - 4 \mathbf { j } ) \mathrm { km }\).
  1. Find the speed of the ship in \(\mathrm { km } \mathrm { h } ^ { - 1 }\).
  2. Show that the position vector \(\mathbf { r } \mathrm { km }\) of the ship, \(t\) hours after 9 am , is given by \(\mathbf { r } = ( 4 - 6 t ) \mathbf { i } + ( 8 t - 8 ) \mathbf { j }\). At 10 am , a passenger on the ship observes that a lighthouse \(L\) is due west of the ship. At 10.30 am , the passenger observes that \(L\) is now south-west of the ship.
  3. Find the position vector of \(L\).
Edexcel M1 2013 January Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-12_337_1084_228_429} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows two particles \(A\) and \(B\), of mass \(2 m\) and \(4 m\) respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 4 }\). The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs vertically below \(P\). The system is released from rest with the string taut, with \(A\) at the point \(X\) and with \(B\) at a height \(h\) above the ground. For the motion until \(B\) hits the ground,
  1. give a reason why the magnitudes of the accelerations of the two particles are the same,
  2. write down an equation of motion for each particle,
  3. find the acceleration of each particle. Particle \(B\) does not rebound when it hits the ground and \(A\) continues moving up the plane towards \(P\). Given that \(A\) comes to rest at the point \(Y\), without reaching \(P\),
  4. find the distance \(X Y\) in terms of \(h\).
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
  4. 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
  3. 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.