Questions — OCR MEI (4455 questions)

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OCR MEI Further Mechanics Major 2022 June Q9
11 marks Challenging +1.8
[In this question you may use the facts that for a uniform solid right circular cone of height \(h\) and base radius \(r\) the volume is \(\frac{1}{3}\pi r^2 h\) and the centre of mass is \(\frac{1}{4}h\) above the base on the line from the centre of the base to the vertex.] \includegraphics{figure_9} The diagram shows the shaded region S bounded by the curve \(y = e^{\frac{1}{4}x}\) for \(0 \leq x \leq 2\), the \(x\)-axis, the \(y\)-axis, and the line \(y = \frac{1}{4}e(6-x)\). The line \(y = \frac{1}{4}e(6-x)\) meets the curve \(y = e^{\frac{1}{4}x}\) at the point A with coordinates \((2, e)\). The region S is rotated through \(2\pi\) radians about the \(x\)-axis to form a uniform solid of revolution T.
  1. Show that the \(x\)-coordinate of the centre of mass of T is \(\frac{3(5e^2 + 1)}{7e^2 - 3}\). [8]
Solid T is freely suspended from A and hangs in equilibrium.
  1. Determine the angle between AO, where O is the origin, and the vertical. [3]
OCR MEI Further Mechanics Major 2022 June Q10
10 marks Standard +0.8
\includegraphics{figure_10} A small toy car runs along a track in a vertical plane. The track consists of three sections: a curved section AB, a horizontal section BC which rests on the floor, and a circular section that starts at C with centre O and radius \(r\) m. The section BC is tangential to the curved section at B and tangential to the circular section at C, as shown in the diagram. The car, of mass \(m\) kg, is placed on the track at A, at a height \(h\) m above the floor, and released from rest. The car runs along the track from A to C and enters the circular section at C. It can be assumed that the track does not obstruct the car moving on to the circular section at C. The track is modelled as being smooth, and the car is modelled as a particle P.
  1. Show that, while P remains in contact with the circular section of the track, the magnitude of the normal contact force between P and the circular section is $$mg\left(3\cos\theta - 2 + \frac{2h}{r}\right)\text{N},$$ where \(\theta\) is the angle between OC and OP. [7]
  2. Hence determine, in terms of \(r\), the least possible value of \(h\) so that P can complete a vertical circle. [2]
  3. Apart from not modelling the car as a particle, state one refinement that would make the model more realistic. [1]
OCR MEI Further Mechanics Major 2022 June Q11
8 marks Challenging +1.2
\includegraphics{figure_11} The diagram shows two small identical uniform smooth spheres, A and B, just before A collides with B. Sphere B is at rest on a horizontal table with its centre vertically above the edge of the table. Sphere A is projected vertically upwards so that, just before it collides with B, the speed of A is \(U\) m s\(^{-1}\) and it is in contact with the vertical side of the table. The point of contact of A with the vertical side of the table and the centres of the spheres are in the same vertical plane.
  1. Show that on impact the line of centres makes an angle of 30° with the vertical. [1]
The coefficient of restitution between A and B is \(\frac{1}{2}\). After the impact B moves freely under gravity.
  1. Determine, in terms of \(U\) and \(g\), the time taken for B to first return to the table. [7]
OCR MEI Further Mechanics Major 2022 June Q12
13 marks Challenging +1.8
\includegraphics{figure_12} The diagram shows a uniform square lamina ABCD, of weight \(W\) and side-length \(a\). The lamina is in equilibrium in a vertical plane that also contains the point O. The vertex A rests on a smooth plane inclined at an angle of 30° to the horizontal. The vertex B rests on a smooth plane inclined at an angle of 60° to the horizontal. OA is a line of greatest slope of the plane inclined at 30° to the horizontal and OB is a line of greatest slope of the plane inclined at 60° to the horizontal. The side AB is inclined at an angle \(\theta\) to the horizontal and the lamina is kept in equilibrium in this position by a clockwise couple of magnitude \(\frac{1}{8}aW\).
  1. By resolving horizontally and vertically, determine, in terms of \(W\), the magnitude of the normal contact force between the plane and the lamina at B. [6]
  2. By taking moments about A, show that \(\theta\) satisfies the equation $$2(\sqrt{3} + 2)\sin\theta - 2\cos\theta = 1.$$ [5]
  3. Verify that \(\theta = 22.4°\), correct to 1 decimal place. [2]
OCR MEI Further Mechanics Major 2022 June Q13
17 marks Challenging +1.3
In this question take \(g = 10\). A particle P of mass 0.15 kg is attached to one end of a light elastic string of modulus of elasticity 13.5 N and natural length 0.45 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 0.5 m. [3]
At time \(t = 0\), P is projected vertically downwards from A with a speed of 1.25 m s\(^{-1}\). Throughout the subsequent motion, P experiences a variable resistance \(R\) newtons which is of magnitude 0.6 times its speed (in m s\(^{-1}\)).
  1. Given that the downward displacement of P from A at time \(t\) seconds is \(x\) metres, show that, while the string remains taut, \(x\) satisfies the differential equation $$\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 200x = 0.$$ [3]
  2. Verify that \(x = \frac{5}{56}e^{-2t}\sin(14t)\). [6]
  3. Determine whether the string becomes slack during the motion. [5]
OCR MEI Further Mechanics Major 2023 June Q1
4 marks Moderate -0.8
A car of mass 800 kg moves in a straight line along a horizontal road. There is a constant resistance to the motion of the car of magnitude 600 N. When the car is travelling at a speed of \(15 \text{ m s}^{-1}\) the power developed by the car is 27 kW. Determine the acceleration of the car when it is travelling at \(15 \text{ m s}^{-1}\). [4]
OCR MEI Further Mechanics Major 2023 June Q2
4 marks Standard +0.3
\includegraphics{figure_2} Two small uniform smooth spheres A and B have masses 0.5 kg and 2 kg respectively. The two spheres are travelling in the same direction in the same straight line on a smooth horizontal surface. Sphere A is moving towards B with speed \(6 \text{ m s}^{-1}\) and B is moving away from A with speed \(2 \text{ m s}^{-1}\) (see diagram). Spheres A and B collide. After this collision A moves with speed \(0.2 \text{ m s}^{-1}\). Determine the possible speeds with which B moves after the collision. [4]
OCR MEI Further Mechanics Major 2023 June Q3
5 marks Standard +0.3
\includegraphics{figure_3} The diagram shows a particle P, of mass 0.2 kg, which is attached by a light inextensible string of length 0.75 m to a fixed point O. Particle P moves with constant angular speed \(\omega \text{ rad s}^{-1}\) in a horizontal circle with centre vertically below O. The string is inclined at an angle \(\theta\) to the vertical. The greatest tension that the string can withstand without breaking is 15 N.
  1. Find the greatest possible value of \(\theta\), giving your answer to the nearest degree. [2]
  2. Determine the greatest possible value of \(\omega\). [3]
OCR MEI Further Mechanics Major 2023 June Q4
6 marks Standard +0.8
\includegraphics{figure_4} A rigid lamina of negligible mass is in the form of a rhombus ABCD, where AC = 6 m and BD = 8 m. Forces of magnitude 2 N, 4 N, 3 N and 5 N act along its sides AB, BC, CD and DA, respectively, as shown in the diagram. A further force F N, acting at A, and a couple of magnitude G N m are also applied to the lamina so that it is in equilibrium.
  1. Determine the magnitude and direction of F. [4]
  2. Determine the value of G. [2]
OCR MEI Further Mechanics Major 2023 June Q5
7 marks Standard +0.3
A particle P of mass \(m\) kg is projected with speed \(u \text{ m s}^{-1}\) along a rough horizontal surface. During the motion of P, a constant frictional force of magnitude \(F\) N acts on P. When the velocity of P is \(v \text{ m s}^{-1}\), it experiences a force of magnitude \(kv\) N due to air resistance, where \(k\) is a constant.
  1. Determine the dimensions of \(k\). [3]
At time \(T\) s after projection P comes to rest. A formula approximating the value of \(T\) is $$T = \frac{mu}{F} - \frac{kmu^2}{2F^2} + \frac{1}{3}k^2m^{\alpha}u^{\beta}F^{\gamma}.$$
  1. Use dimensional analysis to find \(\alpha\), \(\beta\) and \(\gamma\). [4]
OCR MEI Further Mechanics Major 2023 June Q6
9 marks Standard +0.3
At time \(t\) seconds, where \(t \geq 0\), a particle P has position vector \(\mathbf{r}\) metres, where $$\mathbf{r} = (2t^2 - 12t + 6)\mathbf{i} + (t^3 + 3t^2 - 8t)\mathbf{j}.$$ The velocity of P at time \(t\) seconds is \(v \text{ m s}^{-1}\).
  1. Find \(v\) in terms of \(t\). [1]
  2. Determine the speed of P at the instant when it is moving parallel to the vector \(\mathbf{i} - 4\mathbf{j}\). [5]
  3. Determine the value of \(t\) when the magnitude of the acceleration of P is \(20.2 \text{ m s}^{-2}\). [3]
OCR MEI Further Mechanics Major 2023 June Q7
9 marks Standard +0.3
One end of a rope is attached to a block A of mass 2 kg. The other end of the rope is attached to a second block B of mass 4 kg. Block A is held at rest on a fixed rough ramp inclined at \(30°\) to the horizontal. The rope is taut and passes over a small smooth pulley P which is fixed at the top of the ramp. The part of the rope from A to P is parallel to a line of greatest slope of the ramp. Block B hangs vertically below P, at a distance \(d\) m above the ground, as shown in the diagram. \includegraphics{figure_7} Block A is more than \(d\) m from P. The blocks are released from rest and A moves up the ramp. The coefficient of friction between A and the ramp is \(\frac{1}{2\sqrt{3}}\). The blocks are modelled as particles, the rope is modelled as light and inextensible, and air resistance can be ignored.
  1. Determine, in terms of \(g\) and \(d\), the work done against friction as A moves \(d\) m up the ramp. [3]
  2. Given that the speed of B immediately before it hits the ground is \(1.75 \text{ m s}^{-1}\), use the work–energy principle to determine the value of \(d\). [5]
  3. Suggest one improvement, apart from including air resistance, that could be made to the model to make it more realistic. [1]
OCR MEI Further Mechanics Major 2023 June Q8
8 marks Challenging +1.8
\includegraphics{figure_8} The diagram shows the shaded region R bounded by the curve \(y = \sqrt{3x + 4}\), the \(x\)-axis, the \(y\)-axis, and the straight line that passes through the points \((k, 0)\) and \((4, 4)\), where \(0 < k < 4\). Region R is occupied by a uniform lamina.
  1. Determine, in terms of \(k\), an expression for the \(y\)-coordinate of the centre of mass of the lamina. Give your answer in the form \(\frac{a + bk}{c + dk}\), where \(a\), \(b\), \(c\) and \(d\) are integers to be determined. [6]
  2. Show that the \(y\)-coordinate of the centre of mass of the lamina cannot be \(\frac{3}{2}\). [2]
OCR MEI Further Mechanics Major 2023 June Q9
12 marks Challenging +1.3
In this question take \(g = 10\). A small ball P is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation of \((\alpha + \theta)\) from a point O at the bottom of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{5}{12}\) and \(\tan \theta = \frac{3}{4}\). The ball subsequently hits the plane at a point A, where OA is a line of greatest slope of the plane, as shown in the diagram. \includegraphics{figure_9}
  1. Determine the following, in either order.
    [9]
After P hits the plane at A it continues to move away from O. Immediately after hitting the plane at A the direction of motion of P makes an angle \(\beta\) with the horizontal.
  1. Determine the maximum possible value of \(\beta\), giving your answer to the nearest degree. [3]
OCR MEI Further Mechanics Major 2023 June Q10
16 marks Challenging +1.8
\includegraphics{figure_10} A hollow sphere has centre O and internal radius \(r\). A bowl is formed by removing part of the sphere. The bowl is fixed to a horizontal floor, with its circular rim horizontal and the centre of the rim vertically above O. The point A lies on the rim of the bowl such that AO makes an angle of \(30°\) with the horizontal (see diagram). A particle P of mass \(m\) is projected from A, with speed \(u\), where \(u > \sqrt{\frac{gr}{2}}\), in a direction perpendicular to AO and moves on the smooth inner surface of the bowl. The motion of P takes place in the vertical plane containing O and A. The particle P passes through a point B on the inner surface, where OB makes an acute angle \(\theta\) with the vertical.
  1. Determine, in terms of \(m\), \(g\), \(u\), \(r\) and \(\theta\), the magnitude of the force exerted on P by the bowl when P is at B. [7]
The difference between the magnitudes of the force exerted on P by the bowl when P is at points A and B is \(4mg\).
  1. Determine, in terms of \(r\), the vertical distance of B above the floor. [4]
It is given that when P leaves the inner surface of the bowl it does not fall back into the bowl.
  1. Show that \(u^2 > 2gr\). [5]
OCR MEI Further Mechanics Major 2023 June Q11
12 marks Challenging +1.8
\includegraphics{figure_11} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a right-angled triangle ABC, with AB perpendicular to AC, which lies in a vertical plane. The length of AB is 3 cm, and the length of AC is 12 cm. The prism is resting in equilibrium on a horizontal surface and against a vertical wall. The side AC of the prism makes an angle \(\theta\) with the horizontal. A horizontal force of magnitude \(P\) N is now applied to the prism at B. This force acts towards the wall in the vertical plane which passes through the centre of mass G of the prism and is perpendicular to the wall. The weight of the prism is 15 N and the coefficients of friction between the prism and the surface, and between the prism and the wall, are each \(\frac{1}{2}\).
  1. Show that the least value of \(P\) needed to move the prism is given by $$P = \frac{40 \cos \theta + 95 \sin \theta}{16 \sin \theta - 13 \cos \theta}.$$ [8]
  2. Determine the range in which the value of \(\theta\) must lie. [4]
OCR MEI Further Mechanics Major 2023 June Q12
13 marks Challenging +1.2
Two small uniform smooth spheres A and B are of equal radius and have masses \(m\) and \(\lambda m\) respectively. The spheres are on a smooth horizontal surface. Sphere A is moving on the surface with velocity \(u_1 \mathbf{i} + u_2 \mathbf{j}\) towards B, which is at rest. The spheres collide obliquely. When the spheres collide, the line joining their centres is parallel to \(\mathbf{i}\). The coefficient of restitution between A and B is \(e\).
    1. Explain why, when the spheres collide, the impulse of A on B is in the direction of \(\mathbf{i}\). [1]
    2. Determine this impulse in terms of \(\lambda\), \(m\), \(e\) and \(u_1\). [6]
The loss in kinetic energy due to the collision between A and B is \(\frac{1}{8}mu_1^2\).
  1. Determine the range of possible values of \(\lambda\). [6]
OCR MEI Further Mechanics Major 2023 June Q13
15 marks Challenging +1.2
A particle P of mass \(m\) is fixed to one end of a light spring of natural length \(a\) and modulus of elasticity \(man^2\), where \(n > 0\). The other end of the spring is attached to the ceiling of a lift. The lift is at rest and P is hanging vertically in equilibrium.
  1. Find, in terms of \(g\) and \(n\), the extension in the spring. [3]
At time \(t = 0\) the lift begins to accelerate upwards from rest. At time \(t\), the upward displacement of the lift from its initial position is \(y\) and the extension of the spring is \(x\).
  1. Express, in terms of \(g\), \(n\), \(x\) and \(y\), the upward displacement of P from its initial position at time \(t\). [2]
  2. Given that \(\ddot{y} = kt\), where \(k\) is a positive constant, express the upward acceleration of P in terms of \(\ddot{x}\), \(k\) and \(t\). [1]
  3. Show that \(x\) satisfies the differential equation $$\ddot{x} + n^2 x = kt + g.$$ [3]
  4. Verify that \(x = \frac{1}{n^2}(knt + gn - k \sin(nt))\). [4]
  5. By considering \(\ddot{x}\) comment on the motion of P relative to the ceiling of the lift for all times after the lift begins to move. [2]
OCR MEI Further Mechanics Major 2024 June Q1
4 marks Moderate -0.3
A car A of mass 1200 kg is about to tow another car B of mass 800 kg in a straight line along a horizontal road by means of a tow-rope attached between A and B. The tow-rope is modelled as being light and inextensible. Just before the tow-rope tightens, A is travelling at a speed of \(1.5 \text{ m s}^{-1}\) and B is at rest. Just after the tow-rope tightens, both cars have a speed of \(v \text{ m s}^{-1}\).
  1. Find the value of \(v\). [2]
  2. Calculate the magnitude of the impulse on A when the tow-rope tightens. [2]
OCR MEI Further Mechanics Major 2024 June Q2
9 marks Moderate -0.3
One end of a light spring is attached to a fixed point. A mass of 2 kg is attached to the other end of the spring. The spring hangs vertically in equilibrium. The extension of the spring is 0.05 m.
  1. Find the stiffness of the spring. [2]
  2. Find the energy stored in the spring. [2]
  3. Find the dimensions of stiffness of a spring. [1]
A particle P of mass \(m\) is performing complete oscillations with amplitude \(a\) on the end of a light spring with stiffness \(k\). The spring hangs vertically and the maximum speed \(v\) of P is given by the formula $$v = Cm^{\alpha}a^{\beta}k^{\gamma},$$ where C is a dimensionless constant.
  1. Use dimensional analysis to determine \(\alpha\), \(\beta\), and \(\gamma\). [4]
OCR MEI Further Mechanics Major 2024 June Q3
5 marks Challenging +1.2
\includegraphics{figure_3} A circular hole with centre C and radius \(r\) m, where \(r < 0.5\), is cut in a uniform circular disc with centre O and radius 0.5 m. The hole touches the rim of the disc at A (see diagram). The centre of mass, G, of the remainder of the disc is on the rim of the hole. Determine the value of \(r\). [5]
OCR MEI Further Mechanics Major 2024 June Q4
8 marks Standard +0.8
\includegraphics{figure_4} A uniform rod AB has mass 3 kg and length 4 m. The end A of the rod is in contact with rough horizontal ground. The rod rests in equilibrium on a smooth horizontal peg 1.5 m above the ground, such that the rod is inclined at an angle of \(25°\) to the ground (see diagram). The rod is in a vertical plane perpendicular to the peg.
  1. Determine the magnitude of the normal contact force between the peg and the rod. [3]
  2. Determine the range of possible values of the coefficient of friction between the rod and the ground. [5]
OCR MEI Further Mechanics Major 2024 June Q5
7 marks Standard +0.3
A car of mass 850 kg is travelling along a straight horizontal road. The power developed by the car is constant and is equal to 18 kW. There is a constant resistance to motion of magnitude 600 N.
  1. Find the greatest steady speed at which the car can travel. [2]
Later in the journey, while travelling at a speed of \(15 \text{ m s}^{-1}\), the car comes to the bottom of a straight hill which is inclined at an angle of \(\sin^{-1}\left(\frac{1}{40}\right)\) to the horizontal. The power developed by the car remains constant at 18 kW. The magnitude of the resistance force is no longer constant but changes such that the total work done against the resistance force in ascending the hill is 103 000 J. The car takes 10 seconds to ascend the hill and at the top of the hill the car is travelling at \(18 \text{ m s}^{-1}\).
  1. Determine the distance the car travels from the bottom to the top of the hill. [5]
OCR MEI Further Mechanics Major 2024 June Q6
6 marks Challenging +1.2
In this question you must show detailed reasoning. In this question, positions are given relative to a fixed origin, O. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the \(x\)- and \(y\)-directions respectively in a horizontal plane. Distances are measured in centimetres and the time, \(t\), is measured in seconds, where \(0 \leq t \leq 5\). A small radio-controlled toy car C moves on a horizontal surface which contains O. The acceleration of C is given by \(2\mathbf{i} + t\mathbf{j} \text{ cm s}^{-2}\). When \(t = 4\), the displacement of C from O is \(16\mathbf{i} + \frac{32}{3}\mathbf{j}\) cm, and the velocity of C is \(8\mathbf{i} \text{ cm s}^{-1}\). Determine a cartesian equation for the path of C for \(0 < t < 5\). You are not required to simplify your answer. [6]
OCR MEI Further Mechanics Major 2024 June Q7
9 marks Standard +0.8
The region bounded by the curve \(y = x^3 - 3x^2 + 4\), the positive \(x\)-axis and the positive \(y\)-axis is occupied by a uniform lamina L. The vertices of L are O, A and B, where O is the origin, A is a point on the positive \(x\)-axis and B is a point on the positive \(y\)-axis (see diagram). \includegraphics{figure_7}
  1. Determine the coordinates of the centre of mass of L. [5]
The lamina L is the cross-section through the centre of mass of a uniform solid prism M. The prism M is placed on an inclined plane, which makes an angle of \(30°\) with the horizontal, so that OA lies along a line of greatest slope of the plane with O lower down the plane than A. It is given that M does not slip on the plane.
  1. Determine whether M will topple in this case. Give a reason to support your answer. [2]
The prism M is now placed on the same inclined plane so that OB lies along a line of greatest slope of the plane with O lower down the plane than B. It is given that M still does not slip on the plane.
  1. Determine whether M will topple in this case. Give a reason to support your answer. [2]