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 M5 2006 June Q7
14 marks Challenging +1.8
Particles \(P\) and \(Q\) have mass \(3m\) and \(m\) respectively. Particle \(P\) is attached to one end of a light inextensible string and \(Q\) is attached to the other end. The string passes over a circular pulley which can freely rotate in a vertical plane about a fixed horizontal axis through its centre \(O\). The pulley is modelled as a uniform circular disc of mass \(2m\) and radius \(a\). The pulley is sufficiently rough to prevent the string slipping. The system is at rest with the string taut. A third particle \(R\) of mass \(m\) falls freely under gravity from rest for a distance \(a\) before striking and adhering to \(Q\). Immediately before \(R\) strikes \(Q\), particles \(P\) and \(Q\) are at rest with the string taut.
  1. Show that, immediately after \(R\) strikes \(Q\), the angular speed of the pulley is \(\frac{1}{3}\sqrt{\frac{g}{2a}}\). [5]
When \(R\) strikes \(Q\), there is an impulse in the string attached to \(Q\).
  1. Find the magnitude of this impulse. [3]
Given that \(P\) does not hit the pulley,
  1. find the distance that \(P\) moves upwards before first coming to instantaneous rest. [6]
Edexcel M5 2011 June Q1
4 marks Moderate -0.8
A particle moves from the point \(A\) with position vector \((3\mathbf{i} - \mathbf{j} + 3\mathbf{k})\) m to the point \(B\) with position vector \((\mathbf{i} - 2\mathbf{j} - 4\mathbf{k})\) m under the action of the force \((2\mathbf{i} - 3\mathbf{j} - \mathbf{k})\) N. Find the work done by the force. [4]
Edexcel M5 2011 June Q2
10 marks Challenging +1.8
A particle \(P\) moves in the \(x\)-\(y\) plane so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds satisfies the differential equation $$\frac{d^2\mathbf{r}}{dt^2} - 4\mathbf{r} = -3e^t\mathbf{j}$$ When \(t = 0\), the particle is at the origin and is moving with velocity \((2\mathbf{i} + \mathbf{j})\) ms\(^{-1}\). Find \(\mathbf{r}\) in terms of \(t\). [10]
Edexcel M5 2011 June Q4
12 marks Challenging +1.2
Two forces \(\mathbf{F}_1 = (3\mathbf{i} + \mathbf{k})\) N and \(\mathbf{F}_2 = (4\mathbf{i} + \mathbf{j} - \mathbf{k})\) N act on a rigid body. The force \(\mathbf{F}_1\) acts at the point with position vector \((2\mathbf{i} - \mathbf{j} + 3\mathbf{k})\) m and the force \(\mathbf{F}_2\) acts at the point with position vector \((-3\mathbf{i} + 2\mathbf{k})\) m. The two forces are equivalent to a single force \(\mathbf{R}\) acting at the point with position vector \((\mathbf{i} + 2\mathbf{j} + \mathbf{k})\) m together with a couple of moment \(\mathbf{G}\). Find,
  1. \(\mathbf{R}\), [2]
  2. \(\mathbf{G}\). [4]
A third force \(\mathbf{F}_3\) is now added to the system. The force \(\mathbf{F}_3\) acts at the point with position vector \((2\mathbf{i} - \mathbf{k})\) m and the three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are equivalent to a couple.
  1. Find the magnitude of the couple. [6]
Edexcel M5 2011 June Q6
7 marks Challenging +1.8
A uniform rod \(AB\) of mass \(4m\) is free to rotate in a vertical plane about a fixed smooth horizontal axis, \(L\), through \(A\). The rod is hanging vertically at rest when it is struck at its end \(B\) by a particle of mass \(m\). The particle is moving with speed \(u\), in a direction which is horizontal and perpendicular to \(L\), and after striking the rod it rebounds in the opposite direction with speed \(v\). The coefficient of restitution between the particle and the rod is \(1\). Show that \(u = 7v\). [7]
Edexcel M5 2011 June Q8
17 marks Challenging +1.3
A pendulum consists of a uniform rod \(PQ\), of mass \(3m\) and length \(2a\), which is rigidly fixed at its end \(Q\) to the centre of a uniform circular disc of mass \(m\) and radius \(a\). The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through the end \(P\) of the rod and is perpendicular to the rod.
  1. Show that the moment of inertia of the pendulum about \(L\) is \(\frac{33}{4}ma^2\). [5]
The pendulum is released from rest in the position where \(PQ\) makes an angle \(\alpha\) with the downward vertical. At time \(t\), \(PQ\) makes an angle \(\theta\) with the downward vertical.
  1. Show that the angular speed, \(\dot{\theta}\), of the pendulum satisfies $$\dot{\theta}^2 = \frac{40g(\cos\theta - \cos\alpha)}{33a}$$ [4]
  2. Hence, or otherwise, find the angular acceleration of the pendulum. [3]
Given that \(\alpha = \frac{\pi}{20}\) and that \(PQ\) has length \(\frac{8}{33}\) m,
  1. find, to 3 significant figures, an approximate value for the angular speed of the pendulum \(0.2\) s after it has been released from rest. [5]
Edexcel M5 2012 June Q1
9 marks Challenging +1.3
A particle \(P\) moves in a plane such that its position vector \(\mathbf{r}\) metres at time \(t\) seconds \((t > 0)\) satisfies the differential equation $$\frac{d\mathbf{r}}{dt} - \frac{2}{t}\mathbf{r} = 4\mathbf{i}$$ When \(t = 1\), the particle is at the point with position vector \((\mathbf{i} + \mathbf{j})\) m. Find \(\mathbf{r}\) in terms of \(t\). [9]
Edexcel M5 2012 June Q2
10 marks Challenging +1.2
A rocket, with initial mass 1500 kg, including 600 kg of fuel, is launched vertically upwards from rest. The rocket burns fuel at a rate of 15 kg s\(^{-1}\) and the burnt fuel is ejected vertically downwards with a speed of 1000 m s\(^{-1}\) relative to the rocket. At time \(t\) seconds after launch \((t \leqslant 40)\) the rocket has mass \(m\) kg and velocity \(v\) m s\(^{-1}\).
  1. Show that $$\frac{dv}{dt} + \frac{1000}{m}\frac{dm}{dt} = -9.8$$ [5]
  2. Find \(v\) at time \(t\), \(0 \leqslant t \leqslant 40\) [5]
Edexcel M5 2012 June Q3
12 marks Challenging +1.8
A uniform rod \(PQ\) of mass \(m\) and length \(3a\), is free to rotate about a fixed smooth horizontal axis \(L\), which passes through the end \(P\) of the rod and is perpendicular to the rod. The rod hangs at rest in equilibrium with \(Q\) vertically below \(P\). One end of a light inextensible string of length \(2a\) is attached to the rod at \(P\) and the other end is attached to a particle of mass \(3m\). The particle is held with the string taut, and horizontal and perpendicular to \(L\), and is then released. After colliding, the particle sticks to the rod forming a body \(B\).
  1. Show that the moment of inertia of \(B\) about \(L\) is \(15ma^2\). [2]
  2. Show that \(B\) first comes to instantaneous rest after it has turned through an angle \(\arccos\left(\frac{9}{25}\right)\). [10]
Edexcel M5 2012 June Q4
6 marks Challenging +1.8
A body consists of a uniform plane circular disc, of radius \(r\) and mass \(2m\), with a particle of mass \(3m\) attached to the circumference of the disc at the point \(P\). The line \(PQ\) is a diameter of the disc. The body is free to rotate in a vertical plane about a fixed smooth horizontal axis, \(L\), which is perpendicular to the plane of the disc and passes through \(Q\). The body is held with \(QP\) making an angle \(\beta\) with the downward vertical through \(Q\), where \(\sin \beta = 0.25\), and released from rest. Find the magnitude of the component, perpendicular to \(PQ\), of the force acting on the body at \(Q\) at the instant when it is released. [You may assume that the moment of inertia of the body about \(L\) is \(15mr^2\).] [6]
Edexcel M5 2012 June Q5
10 marks Standard +0.8
The points \(P\) and \(Q\) have position vectors \(4\mathbf{i} - 6\mathbf{j} - 12\mathbf{k}\) and \(2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k}\) respectively, relative to a fixed origin \(O\). Three forces, \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\), act along \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\) and \(\overrightarrow{QP}\) respectively, and have magnitudes 7 N, 3 N and \(3\sqrt{10}\) N respectively.
  1. Express \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) in vector form. [3]
  2. Show that the resultant of \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) is \((2\mathbf{i} - 10\mathbf{j} - 16\mathbf{k})\) N. [2]
  3. Find a vector equation of the line of action of this resultant, giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(\lambda\) is a parameter. [5]
Edexcel M5 2012 June Q7
16 marks Challenging +1.2
  1. A uniform lamina of mass \(m\) is in the shape of a triangle \(ABC\). The perpendicular distance of \(C\) from the line \(AB\) is \(h\). Prove, using integration, that the moment of inertia of the lamina about \(AB\) is \(\frac{1}{6}mh^2\). [7]
  2. Deduce the radius of gyration of a uniform square lamina of side \(2a\), about a diagonal. [3]
The points \(X\) and \(Y\) are the mid-points of the sides \(RQ\) and \(RS\) respectively of a square \(PQRS\) of side \(2a\). A uniform lamina of mass \(M\) is in the shape of \(PQXYS\).
  1. Show that the moment of inertia of this lamina about \(XY\) is \(\frac{79}{84}Ma^2\). [6]
Edexcel M5 2014 June Q1
8 marks Standard +0.8
A small bead is threaded on a smooth, straight horizontal wire which passes through the point \(A(-3, 1)\) and the point \(B(2, 5)\) in the \(x\)-\(y\) plane. The bead moves under the action of a horizontal force \(\mathbf{F}\) of magnitude \(8.5\) N whose line of action is parallel to the line with equation \(15x - 8y + 4 = 0\). The unit on both the \(x\) and \(y\) axes has length one metre. Find the work done by \(\mathbf{F}\) as it moves the bead from \(A\) to \(B\). [8]
Edexcel M5 2014 June Q2
9 marks Challenging +1.2
A particle \(P\) moves in a plane so that its position vector, \(\mathbf{r}\) metres at time \(t\) seconds, satisfies the differential equation $$\frac{d\mathbf{r}}{dt} + \mathbf{r} = t\mathbf{i} + e^{-t}\mathbf{j}$$ When \(t = 0\) the particle is at the point with position vector \((\mathbf{i} + \mathbf{j})\) m. Find \(\mathbf{r}\) in terms of \(t\). [9]
Edexcel M5 2014 June Q3
9 marks Standard +0.8
Three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) act on a rigid body at the points with position vectors \(\mathbf{r}_1\), \(\mathbf{r}_2\) and \(\mathbf{r}_3\) respectively. \(\mathbf{F}_1 = (2\mathbf{i} + 3\mathbf{j} - \mathbf{k})\) N and \(\mathbf{r}_1 = (\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) m, \(\mathbf{F}_2 = (\mathbf{i} - 4\mathbf{j} - 2\mathbf{k})\) N and \(\mathbf{r}_2 = (3\mathbf{i} - \mathbf{j} - \mathbf{k})\) m, \(\mathbf{F}_3 = (-3\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) N and \(\mathbf{r}_3 = (\mathbf{i} - 2\mathbf{j} + \mathbf{k})\) m. Show that the system is equivalent to a couple and find the magnitude of the vector moment of this couple. [9]
Edexcel M5 2014 June Q4
17 marks Challenging +1.8
A spacecraft is travelling in a straight line in deep space where all external forces can be assumed to be negligible. The spacecraft decelerates by ejecting fuel at a constant speed \(k\) relative to the spacecraft, in the direction of motion of the spacecraft. At time \(t\), the spacecraft has speed \(v\) and mass \(m\).
  1. Show, from first principles, that while the spacecraft is ejecting fuel, $$\frac{dv}{dm} - \frac{k}{m} = 0$$ [5]
At time \(t = 0\), the spacecraft has speed \(U\) and mass \(M\).
  1. Find the mass of the spacecraft when it comes to rest. [6]
Given that \(m = Me^{-\alpha t^2}\), where \(\alpha\) is a positive constant, and that the spacecraft comes to rest at time \(t = T\),
  1. find, in terms of \(U\) and \(T\) only, the distance travelled by the spacecraft in decelerating from speed \(U\) to rest. [6]
Edexcel M5 2014 June Q5
15 marks Challenging +1.2
A uniform rod \(AB\), of mass \(m\) and length \(2a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\). The axis \(L\) is perpendicular to the rod and passes through the point \(P\) of the rod, where \(AP = \frac{2}{3}a\).
  1. Find the moment of inertia of the rod about \(L\). [3]
The rod is held at rest with \(B\) vertically above \(P\) and is slightly displaced.
  1. Find the angular speed of the rod when \(PB\) makes an angle \(\theta\) with the upward vertical. [4]
  2. Find the magnitude of the angular acceleration of the rod when \(PB\) makes an angle \(\theta\) with the upward vertical. [3]
  3. Find, in terms of \(g\) and \(a\) only, the angular speed of the rod when the force acting on the rod at \(P\) is perpendicular to the rod. [5]
Edexcel M5 2014 June Q6
17 marks Challenging +1.8
  1. Prove, using integration, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(a\), about an axis through the centre of the disc and perpendicular to the plane of the disc is \(\frac{1}{2}ma^2\). [5]
[You may assume without proof that the moment of inertia of a uniform hoop of mass \(m\) and radius \(r\) about an axis through its centre and perpendicular to its plane is \(mr^2\).] \includegraphics{figure_1} A uniform plane shape \(S\) of mass \(M\) is formed by removing a uniform circular disc with centre \(O\) and radius \(a\) from a uniform circular disc with centre \(O\) and radius \(2a\), as shown in Figure 1. The shape \(S\) is free to rotate about a fixed smooth axis \(L\), which passes through \(O\) and lies in the plane of the shape.
  1. Show that the moment of inertia of \(S\) about \(L\) is \(\frac{5}{4}Ma^2\). [4]
The shape \(S\) is at rest in a horizontal plane and is free to rotate about the axis \(L\). A particle of mass \(M\) falls vertically and strikes \(S\) at the point \(A\), where \(OA = \frac{3}{2}a\) and \(OA\) is perpendicular to \(L\). The particle adheres to \(S\) at \(A\). Immediately before the particle strikes \(S\) the speed of the particle is \(u\).
  1. Find, in terms of \(M\) and \(u\), the loss in kinetic energy due to the impact. [8]
Edexcel M5 Specimen Q1
5 marks Standard +0.3
A bead of mass 0.125 kg is threaded on a smooth straight horizontal wire. The bead moves from rest at the point \(A\) with position vector \((2\mathbf{i} + \mathbf{j} - \mathbf{k})\) m relative to a fixed origin \(O\) to a point with position vector \((3\mathbf{i} - 4\mathbf{j} - \mathbf{k})\) m relative to \(O\) under the action of a force \(\mathbf{F} = (14\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\) N. Find
  1. the work done by \(\mathbf{F}\) as the bead moves from \(A\) to \(B\), [3]
  2. the speed of the bead at \(B\). [2]
Edexcel M5 Specimen Q2
7 marks Standard +0.8
  1. Prove, using integration, that the moment of inertia of a uniform rod, of mass \(m\) and length \(2a\), about an axis perpendicular to the rod through its centre is \(\frac{1}{3}ma^2\). [3]
A uniform wire of mass \(4m\) and length \(8a\) is bent into the shape of a square.
  1. Find the moment of inertia of the square about the axis through the centre of the square perpendicular to its plane. [4]
Edexcel M5 Specimen Q3
7 marks Challenging +1.2
Two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) and a couple \(\mathbf{G}\) act on a rigid body. The force \(\mathbf{F}_1 = (3\mathbf{i} + 4\mathbf{j})\) N acts through the point with position vector \(2\mathbf{i}\) m relative to a fixed origin \(O\). The force \(\mathbf{F}_2 = (2\mathbf{i} - \mathbf{j} + \mathbf{k})\) N acts through the point with position vector \((\mathbf{i} + \mathbf{j})\) m relative to \(O\). The forces and couple are equivalent to a single force \(\mathbf{F}\) acting through \(O\).
  1. Find \(\mathbf{F}\). [2]
  2. Find \(\mathbf{G}\). [5]
Edexcel M5 Specimen Q4
10 marks Challenging +1.8
A uniform circular disc, of mass \(2m\) and radius \(a\), is free to rotate in a vertical plane about a fixed, smooth horizontal axis through a point of its circumference. The axis is perpendicular to the plane of the disc. The disc hangs in equilibrium. A particle \(P\) of mass \(m\) is moving horizontally in the same plane as the disc with speed \(\sqrt{20ag}\). The particle strikes, and adheres to, the disc at one end of its horizontal diameter.
  1. Find the angular speed of the disc immediately after \(P\) strikes it. [7]
  2. Verify that the disc will turn through an angle of \(90°\) before first coming to instantaneous rest. [3]
Edexcel M5 Specimen Q5
10 marks Challenging +1.2
A uniform square lamina \(ABCD\) of side \(a\) and mass \(m\) is free to rotate in vertical plane about a horizontal axis through \(A\). The axis is perpendicular to the plane of the lamina. The lamina is released from rest when \(t = 0\) and \(AC\) makes a small angle with the downward vertical through \(A\).
  1. Show that the moment of inertia of the lamina about the axis is \(\frac{2}{3}ma^2\). [3]
  2. Show that the motion of the lamina is approximately simple harmonic. [5]
  3. Find the time \(t\) when \(AC\) is first vertical. [2]
Edexcel M5 Specimen Q6
11 marks Challenging +1.2
A uniform rod \(AB\) of mass \(m\) and length \(4a\) is free to rotate in a vertical plane about a horizontal axis through the point \(O\) of the rod, where \(OA = a\). The rod is slightly disturbed from rest when \(B\) is vertically above \(A\).
  1. Find the magnitude of the angular acceleration of the rod when it is horizontal. [4]
  2. Find the angular speed of the rod when it is horizontal. [2]
  3. Calculate the magnitude of the force acting on the rod at \(O\) when the rod is horizontal. [5]
Edexcel M5 Specimen Q7
12 marks Challenging +1.2
As a hailstone falls under gravity in still air, its mass increases. At time \(t\) the mass of the hailstone is \(m\). The hailstone is modelled as a uniform sphere of radius \(r\) such that $$\frac{dr}{dt} = kr,$$ where \(k\) is a positive constant.
  1. Show that \(\frac{dm}{dt} = 3km\). [2]
Assuming that there is no air resistance,
  1. show that the speed \(v\) of the hailstone at time \(t\) satisfies $$\frac{dv}{dt} = g - 3kv.$$ [4]
Given that the speed of the hailstone at time \(t = 0\) is \(u\),
  1. find an expression for \(v\) in terms of \(t\). [5]
  2. Hence show that the speed of the hailstone approaches the limiting value \(\frac{g}{3k}\). [1]