Questions — Edexcel (10514 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 M4 2014 June Q6
Challenging +1.3
\includegraphics{figure_2} A railway truck of mass \(M\) approaches the end of a straight horizontal track and strikes a buffer. The buffer is parallel to the track, as shown in Figure 2. The buffer is modelled as a light horizontal spring \(PQ\), which is fixed at the end \(P\). The spring has a natural length \(a\) and modulus of elasticity \(Mn^2a\), where \(n\) is a positive constant. At time \(t = 0\), the spring has length \(a\) and the truck strikes the end \(Q\) with speed \(U\). A resistive force whose magnitude is \(Mkv\), where \(v\) is the speed of the truck at time \(t\), and \(k\) is a positive constant, also opposes the motion of the truck. At time \(t\), the truck is in contact with the buffer and the compression of the buffer is \(x\).
  1. Show that, while the truck is compressing the buffer $$\frac{\text{d}^2x}{\text{d}t^2} + k\frac{\text{d}x}{\text{d}t} + n^2x = 0$$ (4)
It is given that \(k = \frac{5n}{2}\)
  1. Find \(x\) in terms of \(U\), \(n\) and \(t\). (7)
  1. Find, in terms of \(U\) and \(n\), the greatest value of \(x\). (5)
Edexcel M4 2014 June Q1
8 marks Standard +0.8
A particle \(A\) has constant velocity \((3\mathbf{i} + \mathbf{j})\) m s\(^{-1}\) and a particle \(B\) has constant velocity \((\mathbf{i} - \mathbf{k})\) m s\(^{-1}\). At time \(t = 0\) seconds, the position vectors of the particles \(A\) and \(B\) with respect to a fixed origin \(O\) are \((-6\mathbf{i} + 4\mathbf{j} - 3\mathbf{k})\) m and \((-2\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\) m respectively.
  1. Show that, in the subsequent motion, the minimum distance between \(A\) and \(B\) is \(4\sqrt{2}\) m. [6]
  2. Find the position vector of \(A\) at the instant when the distance between \(A\) and \(B\) is a minimum. [2]
Edexcel M4 2014 June Q2
11 marks Standard +0.8
A car of mass 1000 kg is moving along a straight horizontal road. The engine of the car is working at a constant rate of 25 kW. When the speed of the car is \(v\) m s\(^{-1}\), the resistance to motion has magnitude \(10v\) newtons.
  1. Show that, at the instant when \(v = 20\), the acceleration of the car is 1.05 m s\(^{-2}\). [3]
  2. Find the distance travelled by the car as it accelerates from a speed of 10 m s\(^{-1}\) to a speed of 20 m s\(^{-1}\). [8]
Edexcel M4 2014 June Q3
8 marks Challenging +1.2
A small ball is moving on a smooth horizontal plane when it collides obliquely with a smooth plane vertical wall. The coefficient of restitution between the ball and the wall is \(\frac{1}{3}\). The speed of the ball immediately after the collision is half the speed of the ball immediately before the collision. Find the angle through which the path of the ball is deflected by the collision. [8]
Edexcel M4 2014 June Q4
8 marks Challenging +1.2
At noon two ships \(A\) and \(B\) are 20 km apart with \(A\) on a bearing of 230° from \(B\). Ship \(B\) is moving at 6 km h\(^{-1}\) on a bearing of 015°. The maximum speed of \(A\) is 12 km h\(^{-1}\). Ship \(A\) sets a course to intercept \(B\) as soon as possible.
  1. Find the course set by \(A\), giving your answer as a bearing to the nearest degree. [4]
  2. Find the time at which \(A\) intercepts \(B\). [4]
Edexcel M4 2014 June Q5
12 marks Challenging +1.8
\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(m\) and the mass of \(B\) is \(3m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before the collision, \(A\) is moving with speed \(3u\) at angle \(\alpha\) to the line of centres and \(B\) is moving with speed \(u\) at angle \(\beta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the two spheres is \(\frac{1}{5}\). It is given that \(\cos \alpha = \frac{1}{3}\) and \(\cos \beta = \frac{2}{3}\) and that \(\alpha\) and \(\beta\) are both acute angles.
  1. Find the magnitude of the impulse on \(A\) due to the collision in terms of \(m\) and \(u\). [8]
  2. Express the kinetic energy lost by \(A\) in the collision as a fraction of its initial kinetic energy. [4]
Edexcel M4 2014 June Q6
13 marks Challenging +1.8
A particle of mass \(m\) kg is attached to one end of a light elastic string of natural length \(a\) metres and modulus of elasticity \(5ma\) newtons. The other end of the string is attached to a fixed point \(O\) on a smooth horizontal plane. The particle is held at rest on the plane with the string stretched to a length \(2a\) metres and then released at time \(t = 0\). During the subsequent motion, when the particle is moving with speed \(v\) m s\(^{-1}\), the particle experiences a resistance of magnitude \(4mv\) newtons. At time \(t\) seconds after the particle is released, the length of the string is \((a + x)\) metres, where \(0 \leqslant x \leqslant a\).
  1. Show that, from \(t = 0\) until the string becomes slack, $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 4\frac{\mathrm{d}x}{\mathrm{d}t} + 5x = 0$$ [3]
  2. Hence express \(x\) in terms of \(a\) and \(t\). [6]
  3. Find the speed of the particle at the instant when the string first becomes slack, giving your answer in the form \(ka\), where \(k\) is a constant to be found correct to 2 significant figures. [4]
Edexcel M4 2014 June Q7
15 marks Challenging +1.2
\includegraphics{figure_2} A bead \(B\) of mass \(m\) is threaded on a smooth circular wire of radius \(r\), which is fixed in a vertical plane. The centre of the circle is \(O\), and the highest point of the circle is \(A\). A light elastic string of natural length \(r\) and modulus of elasticity \(kmg\) has one end attached to the bead and the other end attached to \(A\). The angle between the string and the downward vertical is \(\theta\), and the extension in the string is \(x\), as shown in Figure 2. Given that the string is taut,
  1. show that the potential energy of the system is $$2mgr[(k-1)\cos^2 \theta - k\cos \theta] + \text{constant}$$ [6]
Given also that \(k = 3\),
  1. find the positions of equilibrium and determine their stability. [9]
Edexcel M4 Specimen Q1
6 marks Moderate -0.3
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 \(4v\) N, where \(v\) m s\(^{-1}\) is its speed. At time \(t = 0\) s, \(P\) has a speed of 5 m s\(^{-1}\). Find \(v\) in terms of \(t\). [6]
Edexcel M4 Specimen Q2
6 marks Moderate -0.3
\includegraphics{figure_1} A girl swims in still water at 1 m s\(^{-1}\). She swims across a river which is 336 m wide and is flowing at 0.6 m 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, [3]
  2. the time that she takes to cross the river. [3]
Edexcel M4 Specimen Q3
10 marks Challenging +1.2
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 \(mkv^2\), 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. [9]
  2. State one physical factor which is ignored in this model. [1]
Edexcel M4 Specimen Q4
11 marks Challenging +1.2
\includegraphics{figure_2} 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\(\mathbf{i}\) + \(\mathbf{j}\)) m s\(^{-1}\), and sphere \(B\) has mass 5 kg and velocity (\(-\mathbf{i}\) + \(\mathbf{j}\)) m 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, [5]
  2. the coefficient of restitution between the spheres. [6]
Edexcel M4 Specimen Q5
12 marks Challenging +1.3
An elastic string spring of modulus \(2mg\) 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 \(2m\omega v\), 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 \(\frac{\mathrm{d}^2 x}{\mathrm{d} t^2} + 2\omega \frac{\mathrm{d} x}{\mathrm{d} t} + 2\omega^2 x = 0\). [7]
  2. Find the general solution of this differential equation. [4]
  3. Hence find the period of the damped harmonic motion. [1]
Edexcel M4 Specimen Q6
14 marks Standard +0.3
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 m 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 m 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. [6]
  2. Find the shortest distance between the car and the tractor. [8]
Edexcel M4 Specimen Q7
16 marks Challenging +1.8
\includegraphics{figure_3} A uniform rod \(AB\) has mass \(m\) and length \(2a\). 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 \(mg\), to the end \(B\) of the rod. The end \(B\) is always vertically below \(R\) and angle \(\angle RAB = \theta\), as shown in Fig. 3.
  1. Show that the potential energy of the system is $$mga(2\sin^2\theta - 3\sin\theta) + \text{constant}.$$ [6]
  2. Hence determine the value of \(\theta\), \(0 < \frac{\pi}{2}\), for which the system is in equilibrium. [5]
  3. Determine whether this position of equilibrium is stable or unstable. [5]
Edexcel M5 Q1
7 marks Standard +0.3
At time \(t = 0\), a particle \(P\) of mass \(3\) kg is at rest at the point \(A\) with position vector \((j - 3k)\) m. Two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) then act on the particle \(P\) and it passes through the point \(B\) with position vector \((8i - 3j + 5k)\) m. Given that \(\mathbf{F}_1 = (4i - 2j + 5k)\) N and \(\mathbf{F}_2 = (8i - 4j + 7k)\) N and that \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only two forces acting on \(P\), find the velocity of \(P\) as it passes through \(B\), giving your answer as a vector. [7]
Edexcel M5 Q2
11 marks Challenging +1.2
At time \(t\) seconds, the position vector of a particle \(P\) is \(\mathbf{r}\) metres, where \(\mathbf{r}\) satisfies the vector differential equation $$\frac{d^2\mathbf{r}}{dt^2} + 4\mathbf{r} = e^{2t} \mathbf{j}.$$ When \(t = 0\), \(P\) has position vector \((i + j)\) m and velocity \(2i\) m s\(^{-1}\). Find an expression for \(\mathbf{r}\) in terms of \(t\). [11]
Edexcel M5 Q3
9 marks Challenging +1.2
A spaceship is moving in a straight line in deep space and needs to increase its speed. This is done by ejecting fuel backwards from the spaceship at a constant speed \(c\) relative to the spaceship. When the speed of the spaceship is \(v\), its mass is \(m\).
  1. Show that, while the spaceship is ejecting fuel, $$\frac{dv}{dm} = -\frac{c}{m}.$$ [5]
The initial mass of the spaceship is \(m_0\) and at time \(t\) the mass of the spaceship is given by \(m = m_0(1 - kt)\), where \(k\) is a positive constant.
  1. Find the acceleration of the spaceship at time \(t\). [4]
Edexcel M5 Q4
13 marks Challenging +1.8
\includegraphics{figure_4} **Figure 1** A uniform lamina of mass \(M\) is in the shape of a right-angled triangle \(OAB\). The angle \(OAB\) is \(90°\), \(OA = a\) and \(AB = 2a\), as shown in Figure 1.
  1. Prove, using integration, that the moment of inertia of the lamina \(OAB\) about the edge \(OA\) is \(\frac{8}{3}Ma^2\). (You may assume without proof that the moment of inertia of a uniform rod of mass \(m\) and length \(2l\) about an axis through one end and perpendicular to the rod is \(\frac{4}{3}ml^2\).) [6]
The lamina \(OAB\) is free to rotate about a fixed smooth horizontal axis along the edge \(OA\) and hangs at rest with \(B\) vertically below \(A\). The lamina is then given a horizontal impulse of magnitude \(J\). The impulse is applied to the lamina at the point \(B\), in a direction which is perpendicular to the plane of the lamina. Given that the lamina first comes to instantaneous rest after rotating through an angle of \(120°\),
  1. find an expression for \(J\), in terms of \(M\), \(a\) and \(g\). [7]
Edexcel M5 Q5
16 marks Challenging +1.2
Two forces \(\mathbf{F}_1 = (2i + j)\) N and \(\mathbf{F}_2 = (-2j - k)\) N act on a rigid body. The force \(\mathbf{F}_1\) acts at the point with position vector \(\mathbf{r}_1 = (3i + j + k)\) m and the force \(\mathbf{F}_2\) acts at the point with position vector \(\mathbf{r}_2 = (i - 2j)\) m. A third force \(\mathbf{F}_3\) acts on the body such that \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are in equilibrium.
  1. Find the magnitude of \(\mathbf{F}_3\). [4]
  1. Find a vector equation of the line of action of \(\mathbf{F}_3\). [8]
The force \(\mathbf{F}_3\) is replaced by a fourth force \(\mathbf{F}_4\), acting through the origin \(O\), such that \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_4\) are equivalent to a couple.
  1. Find the magnitude of this couple. [4]
Edexcel M5 Q6
19 marks Challenging +1.8
A pendulum consists of a uniform rod \(AB\), of length \(4a\) and mass \(2m\), whose end \(A\) is rigidly attached to the centre \(O\) of a uniform square lamina \(PQRS\), of mass \(4m\) and side \(a\). The rod \(AB\) is perpendicular to the plane of the lamina. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(B\). The axis \(L\) is perpendicular to \(AB\) and parallel to the edge \(PQ\) of the square.
  1. Show that the moment of inertia of the pendulum about \(L\) is \(75ma^2\). [4]
The pendulum is released from rest when \(BA\) makes an angle \(\alpha\) with the downward vertical through \(B\), where \(\tan \alpha = \frac{3}{4}\). When \(BA\) makes an angle \(\theta\) with the downward vertical through \(B\), the magnitude of the component, in the direction \(AB\), of the force exerted by the axis \(L\) on the pendulum is \(X\).
  1. Find an expression for \(X\) in terms of \(m\), \(g\) and \(\theta\). [9]
Using the approximation \(\theta \approx \sin \theta\),
  1. find an estimate of the time for the pendulum to rotate through an angle \(\alpha\) from its initial rest position. [6]
Edexcel M5 Q1
7 marks Challenging +1.2
At time \(t = 0\), the position vector of a particle \(P\) is \(-3j\) m. At time \(t\) seconds, the position vector of \(P\) is \(\mathbf{r}\) metres and the velocity of \(P\) is \(\mathbf{v}\) m s\(^{-1}\). Given that $$\mathbf{v} - 2\mathbf{r} = 4e^t \mathbf{j},$$ find the time when \(P\) passes through the origin. [7]
Edexcel M5 Q2
13 marks Challenging +1.8
\includegraphics{figure_2} **Figure 1** A uniform circular disc has mass \(4m\), centre \(O\) and radius \(4a\). The line \(POQ\) is a diameter of the disc. A circular hole of radius \(2a\) is made in the disc with the centre of the hole at the point \(R\) on \(PQ\) where \(QR = 5a\), as shown in Figure 1. The resulting lamina is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(Q\) and is perpendicular to the plane of the lamina.
  1. Show that the moment of inertia of the lamina about \(L\) is \(69ma^2\). [7]
The lamina is hanging at rest with \(P\) vertically below \(Q\) when it is given an angular velocity \(\Omega\). Given that the lamina turns through an angle \(\frac{2\pi}{3}\) before it first comes to instantaneous rest,
  1. find \(\Omega\) in terms of \(g\) and \(a\). [6]
Edexcel M5 Q3
16 marks Challenging +1.3
A uniform lamina \(ABC\) of mass \(m\) is in the shape of an isosceles triangle with \(AB = AC = 5a\) and \(BC = 8a\).
  1. Show, using integration, that the moment of inertia of the lamina about an axis through \(A\), parallel to \(BC\), is \(\frac{9}{2}ma^2\). [6]
The foot of the perpendicular from \(A\) to \(BC\) is \(D\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through \(D\) and is perpendicular to the plane of the lamina. The lamina is released from rest when \(DA\) makes an angle \(\alpha\) with the downward vertical. It is given that the moment of inertia of the lamina about an axis through \(D\), perpendicular to \(BC\) and in the plane of the lamina, is \(\frac{8}{3}ma^2\).
  1. Find the angular acceleration of the lamina when \(DA\) makes an angle \(\theta\) with the downward vertical. [8]
Given that \(\alpha\) is small,
  1. find an approximate value for the period of oscillation of the lamina about the vertical. [2]
Edexcel M5 Q4
13 marks Standard +0.8
Two forces \(\mathbf{F}_1 = (i + 2j + 3k)\) N and \(\mathbf{F}_2 = (3i + j + 2k)\) N act on a rigid body. The force \(\mathbf{F}_1\) acts through the point with position vector \((2i + k)\) m and the force \(\mathbf{F}_2\) acts through the point with position vector \((j + 2k)\) m.
  1. If the two forces are equivalent to a single force \(\mathbf{R}\), find
    1. \(\mathbf{R}\), [2]
    2. a vector equation of the line of action of \(\mathbf{R}\), in the form \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}\). [6]
  1. If the two forces are equivalent to a single force acting through the point with position vector \((i + 2j + k)\) m together with a couple of moment \(\mathbf{G}\), find the magnitude of \(\mathbf{G}\). [5]