Questions — Edexcel M4 (159 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 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 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 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
Edexcel M4 2018 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0f141c7-ecd0-4f62-bfad-76c81c2d6396-02_538_881_278_534} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\) has mass \(m\) and length 4a. The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string, of natural length \(a\) and modulus \(\frac { 1 } { 4 } m g\), is attached to the end \(B\) of the rod. The other end of the string is attached to a small light smooth ring \(R\). The ring can move freely on a smooth horizontal wire which is fixed at a height \(a\) above \(A\), and in a vertical plane through \(A\). The angle between the rod and the horizontal is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\), as shown in Figure 1. Given that the elastic string is vertical,
  1. show that the potential energy of the system is $$2 m g a \left( \sin ^ { 2 } \theta - \sin \theta \right) + \text { constant }$$
  2. Show that when \(\theta = \frac { \pi } { 6 }\) the rod is in stable equilibrium.
Edexcel M4 2018 June Q2
2. A small ball \(B\), moving on a smooth horizontal plane, collides with a fixed smooth vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(\alpha\). The coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 4 }\). The kinetic energy of \(B\) immediately after the collision is \(60 \%\) of its kinetic energy immediately before the collision. Find, in degrees, the size of angle \(\alpha\).
Edexcel M4 2018 June Q3
3. When a man walks due West at a constant speed of \(4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the wind appears to be blowing from due South. When he runs due North at a constant speed of \(8 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the speed of the wind appears to be \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
The velocity of the wind relative to the Earth is constant with magnitude \(w \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
Find the two possible values of \(w\).
Edexcel M4 2018 June Q4
4. A particle \(P\) of mass 0.5 kg moves in a horizontal straight line. At time \(t\) seconds \(( t \geqslant 0 )\), the displacement of \(P\) from a fixed point \(O\) of the line is \(x\) metres, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(P\) is moving in the direction of \(x\) increasing. A force of magnitude \(k x\) newtons acts on \(P\) in the direction \(P O\). The motion of \(P\) is also subject to a resistance of magnitude \(\lambda v\) newtons. Given that $$x = ( 1.5 + 10 t ) \mathrm { e } ^ { - 4 t }$$ find
  1. the value of \(k\) and the value of \(\lambda\),
  2. the distance from \(P\) to \(O\) when \(P\) is instantaneously at rest.
Edexcel M4 2018 June Q5
5. A horizontal square field, \(P Q R S\), has sides of length 75 m . Ali is at corner \(P\) of the field and Beth is at corner \(Q\) of the field. Ali starts to walk in a straight line along the diagonal of the field from \(P\) to \(R\) at a constant speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Beth sees Ali start to walk, waits 10 seconds, and then walks from \(Q\) to intercept Ali. Beth walks in a straight line at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the time from the instant Beth leaves \(Q\) until the instant that she intercepts Ali,
  2. the direction Beth should take.
Edexcel M4 2018 June Q6
6. A particle of mass \(m\) is projected vertically upwards in a resisting medium. As the particle moves upwards, the speed \(v\) of the particle is given by $$v ^ { 2 } = k g \left( 5 \mathrm { e } ^ { - \frac { x } { 2 k } } - 4 \right)$$ where \(x\) is the distance of the particle above the point of projection and \(k\) is a positive constant.
  1. Show that the magnitude of the resistance to the motion of the particle is \(\frac { m v ^ { 2 } } { 4 k }\).
    (4)
  2. Find, in terms of \(k\), the greatest height reached by the particle above the point of projection.
  3. Show that the time taken by the particle to reach its greatest height above the point of projection is \(\sqrt { \frac { 4 k } { g } } \arctan \left( \frac { 1 } { 2 } \right)\)
Edexcel M4 2018 June Q7
7. Two smooth uniform spheres \(A\) and \(B\), of mass 2 kg and 3 kg respectively, and of equal radius, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Show that, at the instant when \(A\) and \(B\) collide, their line of centres is parallel to \(- \mathbf { i } + \mathbf { j }\).
  2. Find the velocity of \(B\) immediately after the collision.
  3. Find the coefficient of restitution between \(A\) and \(B\).
Edexcel M4 Q1
  1. A smooth sphere \(S\) is moving on a smooth horizontal plane with speed \(u\) when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of \(S\) makes an angle of \(30 ^ { \circ }\) with the wall. The coefficient of restitution between \(S\) and the wall is \(\frac { 1 } { 3 }\).
Find the speed of \(S\) immediately after the collision.
Edexcel M4 Q2
2. A car of mass 1000 kg , moving along a straight horizontal road, is driven by an engine which produces a constant power of 12 kW . The only resistance to the motion of the car is air resistance of magnitude \(10 v ^ { 2 } \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car. Find the distance travelled by the car as its speed increases from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(8 marks)
Edexcel M4 Q3
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d57ea92a-4d6a-46bf-a6aa-bbd5083e8726-3_469_1163_1217_443}
\end{figure} A smooth uniform sphere \(A\), moving on a smooth horizontal table, collides with a second identical sphere \(B\) which is at rest on the table. When the spheres collide the line joining their centres makes an angle of \(30 ^ { \circ }\) with the direction of motion of \(A\), as shown in Fig. 1. The coefficient of restitution between the spheres is \(e\). The direction of motion of \(A\) is deflected through an angle \(\theta\) by the collision. Show that \(\tan \theta = \frac { ( 1 + e ) \sqrt { 3 } } { 5 - 3 e }\).
(10 marks)
Edexcel M4 Q4
4. A body falls vertically from rest and is subject to air resistance of a magnitude which is proportional to its speed. Given that its terminal speed is \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time it takes for the body to attain a speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(10 marks)
Edexcel M4 Q5
5. A particle \(P\) of mass \(m\) is fixed to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m a n ^ { 2 }\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at a point which is a distance \(2 a\) vertically below \(O\). The air resistance is modelled as having magnitude \(2 m n v\), where \(v\) is the speed of \(P\).
  1. Show that, when the extension of the string is \(x\), $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 n \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 n ^ { 2 } x = g$$
  2. Find \(x\) in terms of \(t\).
Edexcel M4 Q6
6. Two particles \(P\) and \(Q\) have constant velocity vectors \(\mathbf { v } _ { P }\) and \(\mathbf { v } _ { Q }\) respectively. The magnitude of the velocity of \(P\) relative to \(Q\) is equal to the speed of \(P\). If the direction of motion of one of the particles is reversed, the magnitude of the velocity of \(P\) relative to \(Q\) is doubled. Find
  1. the ratio of the speeds of \(P\) and \(Q\),
  2. the cosine of the angle between the directions of motion of \(P\) and \(Q\).
Edexcel M4 Q7
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d57ea92a-4d6a-46bf-a6aa-bbd5083e8726-5_955_855_349_573}
\end{figure} A smooth wire \(A B\), in the shape of a circle of radius \(r\), is fixed in a vertical plane with \(A B\) vertical. A small smooth ring \(R\) of mass \(m\) is threaded on the wire and is connected by a light inextensible string to a particle \(P\) of mass \(m\). The length of the string is greater than the diameter of the circle. The string passes over a small smooth pulley which is fixed at the highest point \(A\) of the wire and angle \(R \hat { A } P = \theta\), as shown in Fig. 2.
  1. Show that the potential energy of the system is given by $$2 m g r \left( \cos \theta - \cos ^ { 2 } \theta \right) + \text { constant. }$$
  2. Hence determine the values of \(\theta , \theta \geq 0\), for which the system is in equilibrium. (6 marks)
  3. Determine the stability of each position of equilibrium. END
Edexcel M4 Specimen Q1
  1. A particle \(P\) of mass 2 kg moves in a straight line along a smooth horizontal plane. The only horizontal force acting on \(P\) is a resistance of magnitude \(4 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is its speed. At time \(t = 0 \mathrm {~s} , P\) has a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find \(v\) in terms of \(t\).
    (6)
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4737e682-1e1d-4c1a-91c2-7d051cb43aac-2_470_979_657_591}
\end{figure} A girl swims in still water at \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She swims across a river which is 336 m wide and is flowing at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She sets off from a point \(A\) on one bank and lands at a point \(B\), which is directly opposite \(A\), on the other bank as shown in Fig. 1. Find
  1. the direction, relative to the earth, in which she swims,
  2. the time that she takes to cross the river.
Edexcel M4 Specimen Q3
3. A ball of mass \(m\) is thrown vertically upwards from the ground. When its speed is \(v\) the magnitude of the air resistance is modelled as being \(m k v ^ { 2 }\), where \(k\) is a positive constant. The ball is projected with speed \(\sqrt { \frac { g } { k } }\). By modelling the ball as a particle,
  1. find the greatest height reached by the ball.
  2. State one physical factor which is ignored in this model.
Edexcel M4 Specimen Q4
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{4737e682-1e1d-4c1a-91c2-7d051cb43aac-3_417_986_303_534}
\end{figure} Two smooth uniform spheres \(A\) and \(B\), of equal radius, are moving on a smooth horizontal plane. Sphere \(A\) has mass 3 kg and velocity \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and sphere \(B\) has mass 5 kg and velocity \(( - \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When the spheres collide the line joining their centres is parallel to \(\mathbf { i }\), as shown in Fig. 2. Given that the direction of \(A\) is deflected through a right angle by the collision, find
  1. the velocity of \(A\) after the collision,
  2. the coefficient of restitution between the spheres.
Edexcel M4 Specimen Q5
5. An elastic string spring of modulus \(2 m g\) and natural length \(l\) is fixed at one end. To the other end is attached a mass \(m\) which is allowed to hang in equilibrium. The mass is then pulled vertically downwards through a distance \(l\) and released from rest. The air resistance is modelled as having magnitude \(2 m \omega v\), where \(v\) is the speed of the particle and \(\omega = \sqrt { \frac { g } { l } }\). The particle is at distance \(x\) from its equilibrium position at time \(t\).
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 \omega ^ { 2 } x = 0\).
  2. Find the general solution of this differential equation.
  3. Hence find the period of the damped harmonic motion.
Edexcel M4 Specimen Q6
6. Two horizontal roads cross at right angles. One is directed from south to north, and the other from east to west. A tractor travels north on the first road at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at noon is 200 m south of the junction. A car heads west on the second road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at noon is 960 m east of the junction.
  1. Find the magnitude and direction of the velocity of the car relative to the tractor.
  2. Find the shortest distance between the car and the tractor.
Edexcel M4 Specimen Q7
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{4737e682-1e1d-4c1a-91c2-7d051cb43aac-4_558_1180_845_440}
\end{figure} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The end \(A\) is smoothly hinged at a fixed point on a fixed straight horizontal wire. A smooth light ring \(R\) is threaded on the wire. The ring \(R\) is attached by a light elastic string, of natural length \(a\) and modulus of elasticity \(m g\), to the end \(B\) of the rod. The end \(B\) is always vertically below \(R\) and angle \(\angle R A B = \theta\), as shown in Fig. 3.
  1. Show that the potential energy of the system is $$m g a \left( 2 \sin ^ { 2 } \theta - 3 \sin \theta \right) + \text { constant }$$ (6)
  2. Hence determine the value of \(\theta , \theta < \frac { \pi } { 2 }\), for which the system is in equilibrium.
  3. Determine whether this position of equilibrium is stable or unstable. END
Edexcel M4 2005 January Q5
  1. Show that, when \(\angle B F O = \theta\), the potential energy of the system is $$\frac { 1 } { 10 } m g a ( 8 \cos \theta - 5 ) ^ { 2 } - 2 m g a \cos ^ { 2 } \theta + \text { constant } .$$
  2. Hence find the values of \(\theta\) for which the system is in equilibrium.
  3. Determine the nature of the equilibrium at each of these positions.
Edexcel M4 Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-03_457_638_233_598} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A fixed smooth plane is inclined to the horizontal at an angle of \(45 ^ { \circ }\). A particle \(P\) is moving horizontally and strikes the plane. Immediately before the impact, \(P\) is moving in a vertical plane containing a line of greatest slope of the inclined plane. Immediately after the impact, \(P\) is moving in a direction which makes an angle of \(30 ^ { \circ }\) with the inclined plane, as shown in Figure 1. Find the fraction of the kinetic energy of \(P\) which is lost in the impact.
Edexcel M4 Q2
2. At time \(t = 0\), a particle \(P\) of mass \(m\) is projected vertically upwards with speed \(\sqrt { \frac { g } { k } }\), where \(k\) is a constant. At time \(t\) the speed of \(P\) is \(v\). The particle \(P\) moves against air resistance whose magnitude is modelled as being \(m k v ^ { 2 }\) when the speed of \(P\) is \(v\). Find, in terms of \(k\), the distance travelled by \(P\) until its speed first becomes half of its initial speed.
Edexcel M4 Q3
  1. At noon a motorboat \(P\) is 2 km north-west of another motorboat \(Q\). The motorboat \(P\) is moving due south at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motorboat \(Q\) is pursuing motorboat \(P\) at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and sets a course in order to get as close to motorboat \(P\) as possible.
    1. Find the course set by \(Q\), giving your answer as a bearing to the nearest degree.
    2. Find the shortest distance between \(P\) and \(Q\).
    3. Find the distance travelled by \(Q\) from its position at noon to the point of closest approach.
Edexcel M4 Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-08_479_807_246_571} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light inextensible string of length \(2 a\) has one end attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\). A second light inextensible string of length \(L\), where \(L > \frac { 12 a } { 5 }\), has one of its ends attached to \(P\) and passes over a small smooth peg fixed at a point \(B\). The line \(A B\) is horizontal and \(A B = 2 a\). The other end of the second string is attached to a particle of mass \(\frac { 7 } { 20 } m\), which hangs vertically below \(B\), as shown in Figure 2.
  1. Show that the potential energy of the system, when the angle \(P A B = 2 \theta\), is $$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
  2. Show that there is only one value of \(\cos \theta\) for which the system is in equilibrium and find this value.
  3. Determine the stability of the position of equilibrium.