Questions — OCR (4907 questions)

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OCR M3 2016 June Q1
6 marks Standard +0.3
\includegraphics{figure_1} A particle \(P\) of mass \(0.3\) kg is moving with speed \(0.4\) m s\(^{-1}\) in a straight line on a smooth horizontal surface when it is struck by a horizontal impulse. After the impulse acts \(P\) has speed \(0.6\) m s\(^{-1}\) and is moving in a direction making an angle \(30°\) with its original direction of motion (see diagram).
  1. Find the magnitude of the impulse and the angle its line of action makes with the original direction of motion of \(P\). [4]
Subsequently a second impulse acts on \(P\). After this second impulse acts, \(P\) again moves from left to right with speed \(0.4\) m s\(^{-1}\) in a direction parallel to its original direction of motion.
  1. State the magnitude of the second impulse, and show the direction of the second impulse on a diagram. [2]
OCR M3 2016 June Q2
8 marks Standard +0.3
A particle \(Q\) of mass \(0.2\) kg is projected horizontally with velocity \(4\) m s\(^{-1}\) from a fixed point \(A\) on a smooth horizontal surface. At time \(t\) s after projection \(Q\) is \(x\) m from \(A\) and is moving away from \(A\) with velocity \(v\) m s\(^{-1}\). There is a force of \(3\cos 2t\) N acting on \(Q\) in the positive \(x\)-direction.
  1. Find an expression for the velocity of \(Q\) at time \(t\). State the maximum and minimum values of the velocity of \(Q\) as \(t\) varies. [4]
  2. Find the average velocity of \(Q\) between times \(t = \pi\) and \(t = \frac{3}{2}\pi\). [4]
OCR M3 2016 June Q3
10 marks Challenging +1.2
\includegraphics{figure_3} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2m\) kg and \(3m\) kg respectively. The spheres are approaching each other on a horizontal surface when they collide. Before the collision \(A\) is moving with speed \(5\) m s\(^{-1}\) in a direction making an angle \(\alpha\) with the line of centres, where \(\cos \alpha = \frac{4}{5}\), and \(B\) is moving with speed \(3\frac{1}{4}\) m s\(^{-1}\) in a direction making an angle \(\beta\) with the line of centres, where \(\cos \beta = \frac{5}{13}\). A straight vertical wall is situated to the right of \(B\), perpendicular to the line of centres (see diagram). The coefficient of restitution between \(A\) and \(B\) is \(\frac{2}{5}\).
  1. Find the speed of \(A\) after the collision. Find also the component of the velocity of \(B\) along the line of centres after the collision. [7]
\(B\) subsequently hits the wall.
  1. Explain why \(A\) and \(B\) will have a second collision if the coefficient of restitution between \(B\) and the wall is sufficiently large. Find the set of values of the coefficient of restitution for which this second collision will occur. [3]
OCR M3 2016 June Q4
7 marks Challenging +1.8
\includegraphics{figure_4} \(A\) and \(C\) are two fixed points, \(1.5\) m apart, on a smooth horizontal plane. A light elastic string of natural length \(0.4\) m and modulus of elasticity \(20\) N has one end fixed to point \(A\) and the other end fixed to a particle \(B\). Another light elastic string of natural length \(0.6\) m and modulus of elasticity \(15\) N has one end fixed to point \(C\) and the other end fixed to the particle \(B\). The particle is released from rest when \(ABC\) forms a straight line and \(AB = 0.4\) m (see diagram). Find the greatest kinetic energy of particle \(B\) in the subsequent motion. [7]
OCR M3 2016 June Q5
11 marks Challenging +1.2
\includegraphics{figure_5} One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs at rest. \(P\) is then projected horizontally from this position with speed \(2\sqrt{ag}\). When the string makes an angle \(\theta\) with the upward vertical \(P\) has speed \(v\) (see diagram). The tension in the string is \(T\).
  1. Find an expression for \(T\) in terms of \(m\), \(g\) and \(\theta\), and hence find the height of \(P\) above its initial level when the string becomes slack. [6]
\(P\) is now projected horizontally from the same initial position with speed \(U\).
  1. Find the set of values of \(U\) for which the string does not remain taut in the subsequent motion. [5]
OCR M3 2016 June Q6
13 marks Standard +0.8
\includegraphics{figure_6} Two uniform rods \(AB\) and \(AC\) are freely jointed at \(A\). Rod \(AB\) is of length \(2l\) and weight \(W\); rod \(AC\) is of length \(4l\) and weight \(2W\). The rods rest in equilibrium in a vertical plane on two rough horizontal steps, so that \(AB\) makes an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac{3}{5}\), and \(AC\) makes an angle of \(\varphi\) with the horizontal, where \(\sin \varphi = \frac{1}{5}\) (see diagram). The force of the step acting on \(AB\) at \(B\) has vertical component \(R\) and horizontal component \(F\).
  1. By taking moments about \(A\) for the rod \(AB\), find an equation relating \(W\), \(R\) and \(F\). [3]
  2. Show that \(R = \frac{75}{68}W\), and find the vertical component of the force acting on \(AC\) at \(C\). [6]
  3. The coefficient of friction at \(B\) is equal to that at \(C\). Given that one of the rods is on the point of slipping, explain which rod this must be, and find the coefficient of friction. [4]
OCR M3 2016 June Q7
17 marks Challenging +1.2
A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of modulus of elasticity \(24mg\) N and natural length \(0.6\) m. The other end of the string is attached to a fixed point \(O\); the particle \(P\) rests in equilibrium at a point \(A\) with the string vertical.
  1. Show that the distance \(OA\) is \(0.625\) m. [2]
Another particle \(Q\), of mass \(3m\) kg, is released from rest from a point \(0.4\) m above \(P\) and falls onto \(P\). The two particles coalesce.
  1. Show that the combined particle initially moves with speed \(2.1\) m s\(^{-1}\). [3]
  2. Show that the combined particle initially performs simple harmonic motion, and find the centre of this motion and its amplitude. [5]
  3. Find the time that elapses between \(Q\) being released from rest and the combined particle first reaching the highest point of its subsequent motion. [7]
OCR M4 2006 June Q1
5 marks Standard +0.3
A straight rod \(AB\) of length \(a\) has variable density. At a distance \(x\) from \(A\) its mass per unit length is \(k(a + 2x)\), where \(k\) is a positive constant. Find the distance from \(A\) of the centre of mass of the rod. [5]
OCR M4 2006 June Q2
8 marks Standard +0.3
A flywheel takes the form of a uniform disc of mass 8 kg and radius 0.15 m. It rotates freely about an axis passing through its centre and perpendicular to the disc. A couple of constant moment is applied to the flywheel. The flywheel turns through an angle of 75 radians while its angular speed increases from 10 rad s\(^{-1}\) to 25 rad s\(^{-1}\).
  1. Find the moment of the couple about the axis. [5]
When the flywheel is rotating with angular speed 25 rad s\(^{-1}\), it locks together with a second flywheel which is mounted on the same axis and is at rest. Immediately afterwards, both flywheels rotate together with the same angular speed 9 rad s\(^{-1}\).
  1. Find the moment of inertia of the second flywheel about the axis. [3]
OCR M4 2006 June Q3
8 marks Standard +0.8
The region bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 2\) and the curve \(y = \frac{1}{x^2}\) for \(1 \leq x \leq 2\), is occupied by a uniform lamina of mass 24 kg. The unit of length is the metre. Find the moment of inertia of this lamina about the \(x\)-axis. [8]
OCR M4 2006 June Q4
10 marks Challenging +1.8
\includegraphics{figure_4} A uniform rod \(AB\), of mass \(m\) and length \(2a\), is freely hinged to a fixed point at \(A\). A particle of mass \(2m\) is attached to the rod at \(B\). A light elastic string, with natural length \(a\) and modulus of elasticity \(5mg\), passes through a fixed smooth ring \(R\). One end of the string is fixed to \(A\) and the other end is fixed to the mid-point \(C\) of \(AB\). The ring \(R\) is at the same horizontal level as \(A\), and is at a distance \(a\) from \(A\). The rod \(AB\) and the ring \(R\) are in a vertical plane, and \(RC\) is at an angle \(\theta\) above the horizontal, where \(0 < \theta < \frac{1}{2}\pi\), so that the acute angle between \(AB\) and the horizontal is \(2\theta\) (see diagram).
  1. By considering the energy of the system, find the value of \(\theta\) for which the system is in equilibrium. [7]
  2. Determine whether this position of equilibrium is stable or unstable. [3]
OCR M4 2006 June Q5
11 marks Challenging +1.2
A uniform rectangular lamina \(ABCD\) has mass 20 kg and sides of lengths \(AB = 0.6\) m and \(BC = 1.8\) m. It rotates in its own vertical plane about a fixed horizontal axis which is perpendicular to the lamina and passes through the mid-point of \(AB\).
  1. Show that the moment of inertia of the lamina about the axis is 22.2 kg m\(^2\). [3]
\includegraphics{figure_5} The lamina is released from rest with \(BC\) horizontal and below the level of the axis. Air resistance may be neglected, but a frictional couple opposes the motion. The couple has constant moment 44.1 N m about the axis. The angle through which the lamina has turned is denoted by \(\theta\) (see diagram).
  1. Show that the angular acceleration is zero when \(\cos \theta = 0.25\). [3]
  2. Hence find the maximum angular speed of the lamina. [5]
OCR M4 2006 June Q6
13 marks Challenging +1.2
\includegraphics{figure_6} A ship \(P\) is moving with constant velocity 7 m s\(^{-1}\) in the direction with bearing 110°. A second ship \(Q\) is moving with constant speed 10 m s\(^{-1}\) in a straight line. At one instant \(Q\) is at the point \(X\), and \(P\) is 7400 m from \(Q\) on a bearing of 050° (see diagram). In the subsequent motion, the shortest distance between \(P\) and \(Q\) is 1790 m.
  1. Show that one possible direction for the velocity of \(Q\) relative to \(P\) has bearing 036°, to the nearest degree, and find the bearing of the other possible direction of this relative velocity. [3]
Given that the velocity of \(Q\) relative to \(P\) has bearing 036°, find
  1. the bearing of the direction in which \(Q\) is moving, [4]
  2. the magnitude of the velocity of \(Q\) relative to \(P\), [2]
  3. the time taken for \(Q\) to travel from \(X\) to the position where the two ships are closest together, [3]
  4. the bearing of \(P\) from \(Q\) when the two ships are closest together. [1]
OCR M4 2006 June Q7
17 marks Challenging +1.2
\includegraphics{figure_7} A uniform rod \(AB\) has mass \(m\) and length \(6a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(C\) on the rod, where \(AC = a\). The angle between \(AB\) and the upward vertical is \(\theta\), and the force acting on the rod at \(C\) has components \(R\) parallel to \(AB\) and \(S\) perpendicular to \(AB\) (see diagram). The rod is released from rest in the position where \(\theta = \frac{1}{4}\pi\). Air resistance may be neglected.
  1. Find the angular acceleration of the rod in terms of \(a\), \(g\) and \(\theta\). [4]
  2. Show that the angular speed of the rod is \(\sqrt{\frac{2g(1 - 2\cos\theta)}{7a}}\). [3]
  3. Find \(R\) and \(S\) in terms of \(m\), \(g\) and \(\theta\). [6]
  4. When \(\cos\theta = \frac{1}{3}\), show that the force acting on the rod at \(C\) is vertical, and find its magnitude. [4]
OCR M4 2016 June Q1
4 marks Standard +0.3
A uniform square lamina, of mass 5 kg and side 0.2 m, is rotating about a fixed vertical axis that is perpendicular to the lamina and that passes through its centre. A couple of constant moment 0.06 N m is applied to the lamina. The lamina turns through an angle of 155 radians while its angular speed increases from 8 rad s\(^{-1}\) to \(\omega\) rad s\(^{-1}\). Find \(\omega\). [4]
OCR M4 2016 June Q2
9 marks Standard +0.3
\includegraphics{figure_2} Boat \(A\) is travelling with constant speed 7.9 m s\(^{-1}\) on a course with bearing 035°. Boat \(B\) is travelling with constant speed 10.5 m s\(^{-1}\) on a course with bearing 330°. At one instant, the boats are 1500 m apart with \(B\) on a bearing of 125° from \(A\) (see diagram).
  1. Find the magnitude and the bearing of the velocity of \(B\) relative to \(A\). [5]
  2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion. [2]
  3. Find the time taken from the instant when \(A\) and \(B\) are 1500 m apart to the instant when \(A\) and \(B\) are at the point of closest approach. [2]
OCR M4 2016 June Q3
13 marks Challenging +1.8
\includegraphics{figure_3} Two uniform rods \(AB\) and \(BC\), each of length \(a\) and mass \(m\), are rigidly joined together so that \(AB\) is perpendicular to \(BC\). The rod \(AB\) is freely hinged to a fixed point at \(A\). The rods can rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda mg\) is attached to \(B\). The other end of the string is attached to a fixed point \(D\) vertically above \(A\), where \(AD = a\). The string \(BD\) makes an angle \(\theta\) radians with the downward vertical (see diagram).
  1. Taking \(D\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = \frac{1}{2}mga(\sin 2\theta - 3\cos 2\theta) + \frac{1}{2}\lambda mga(2\cos \theta - 1)^2 - 2mga.$$ [5]
  2. Given that \(\theta = \frac{1}{3}\pi\) is a position of equilibrium, find the exact value of \(\lambda\). [4]
  3. Find \(\frac{d^2V}{d\theta^2}\) and hence determine whether the position of equilibrium at \(\theta = \frac{1}{3}\pi\) is stable or unstable. [4]
OCR M4 2016 June Q4
13 marks Standard +0.8
The region bounded by the curve \(y = 2e^{\frac{1}{2}x}\) for \(0 \leq x \leq 2\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), is occupied by a uniform lamina.
  1. Find the exact value of the \(y\)-coordinate of the centre of mass of the lamina. [6]
As shown in the diagram below, a uniform lamina occupies the closed region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = f(x)\) where $$f(x) = \begin{cases} 2e^{\frac{1}{2}x} & 0 \leq x \leq 2, \\ \frac{2}{3}(5-x)e & 2 \leq x \leq 5. \end{cases}$$ \includegraphics{figure_4}
  1. Find the exact value of the \(x\)-coordinate of the centre of mass of the lamina. [7]
OCR M4 2016 June Q5
18 marks Challenging +1.2
A uniform rod \(AB\) has mass \(2m\) and length \(4a\).
  1. Show by integration that the moment of inertia of the rod about an axis perpendicular to the rod through \(A\) is \(\frac{32}{3}ma^2\). [4]
The rod is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis through \(A\). A particle of mass \(m\) is moving horizontally in the plane in which the rod is free to rotate. The particle has speed \(v\), and strikes the rod at \(B\). In the subsequent motion the particle adheres to the rod and the combined rigid body \(Q\), consisting of the rod and the particle, starts to rotate.
  1. Find, in terms of \(v\) and \(a\), the initial angular speed of \(Q\). [4]
At time \(t\) seconds the angle between \(Q\) and the downward vertical is \(\theta\) radians.
  1. Show that \(\dot{\theta}^2 = k\frac{g}{a}(\cos \theta - 1) + \frac{9v^2}{400a^2}\), stating the value of the constant \(k\). [4]
  2. Find, in terms of \(a\) and \(g\), the set of values of \(v^2\) for which \(Q\) makes complete revolutions. [2]
When \(Q\) is horizontal, the force exerted by the axis on \(Q\) has vertically upwards component \(R\).
  1. Find \(R\) in terms of \(m\) and \(g\). [4]
OCR M4 2016 June Q6
15 marks Challenging +1.2
\includegraphics{figure_6} A compound pendulum consists of a uniform rod \(AB\) of length 1 m and mass 3 kg, a particle of mass 1 kg attached to the rod at \(A\) and a circular disc of radius \(\frac{1}{5}\) m, mass 6 kg and centre \(C\). The end \(B\) of the rod is rigidly attached to a point on the circumference of the disc in such a way that \(ABC\) is a straight line. The pendulum is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through the point \(P\) on the rod where \(AP = x\) m and \(x < \frac{1}{3}\) (see diagram).
  1. Show that the moment of inertia of the pendulum about the axis of rotation is \((10x^2 - 19x + 12)\) kg m\(^2\). [6]
The pendulum is making small oscillations about the equilibrium position, such that at time \(t\) seconds the angular displacement that the pendulum makes with the downward vertical is \(\theta\) radians.
  1. Find the angular acceleration of the pendulum, in terms of \(x\), \(g\) and \(\theta\). [4]
  2. Show that the motion is approximately simple harmonic, and show that the approximate period of oscillations, in seconds, is given by \(2\pi\sqrt{\frac{20x^2 - 38x + 24}{(19-20x)g}}\). [2]
  3. Hence find the value of \(x\) for which the approximate period of oscillations is least. [3]
OCR FP1 Q1
6 marks Moderate -0.5
Use the standard results for \(\sum_{r=1}^n r\) and \(\sum_{r=1}^n r^2\) to show that, for all positive integers \(n\), $$\sum_{r=1}^n (6r^2 + 2r + 1) = n(2n^2 + 4n + 3).$$ [6]
OCR FP1 Q2
6 marks Standard +0.3
The matrices \(\mathbf{A}\) and \(\mathbf{I}\) are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}\) and \(\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) respectively.
  1. Find \(\mathbf{A}^2\) and verify that \(\mathbf{A}^2 = 4\mathbf{A} - \mathbf{I}\). [4]
  2. Hence, or otherwise, show that \(\mathbf{A}^{-1} = 4\mathbf{I} - \mathbf{A}\). [2]
OCR FP1 Q3
7 marks Moderate -0.8
The complex numbers \(2 + 3i\) and \(4 - i\) are denoted by \(z\) and \(w\) respectively. Express each of the following in the form \(x + iy\), showing clearly how you obtain your answers.
  1. \(z + 5w\), [2]
  2. \(z*w\), where \(z*\) is the complex conjugate of \(z\), [3]
  3. \(\frac{1}{w}\). [2]
OCR FP1 Q4
6 marks Standard +0.3
Use an algebraic method to find the square roots of the complex number \(21 - 20i\). [6]
OCR FP1 Q5
7 marks Standard +0.3
  1. Show that $$\frac{r+1}{r+2} - \frac{r}{r+1} = \frac{1}{(r+1)(r+2)}.$$ [2]
  2. Hence find an expression, in terms of \(n\), for $$\frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots + \frac{1}{(n+1)(n+2)}.$$ [4]
  3. Hence write down the value of \(\sum_{r=1}^\infty \frac{1}{(r+1)(r+2)}\). [1]