Questions — OCR MEI (4455 questions)

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OCR MEI Further Pure Core Specimen Q13
13 marks Challenging +1.2
Matrix M is given by \(\mathbf{M} = \begin{pmatrix} k & 1 & -5 \\ 2 & 3 & -3 \\ -1 & 2 & 2 \end{pmatrix}\), where \(k\) is a constant.
  1. Show that \(\det \mathbf{M} = 12(k - 3)\). [2]
  2. Find a solution of the following simultaneous equations for which \(x \neq z\). $$4x^2 + y^2 - 5z^2 = 6$$ $$2x^2 + 3y^2 - 3z^2 = 6$$ $$-x^2 + 2y^2 + 2z^2 = -6$$ [3]
    1. Verify that the point \((2, 0, 1)\) lies on each of the following three planes. $$3x + y - 5z = 1$$ $$2x + 3y - 3z = 1$$ $$-x + 2y + 2z = 0$$ [1]
    2. Describe how the three planes in part (iii) (A) are arranged in 3-D space. Give reasons for your answer. [4]
  4. Find the values of \(k\) for which the transformation represented by M has a volume scale factor of 6. [3]
OCR MEI Further Pure Core Specimen Q14
18 marks Challenging +1.2
  1. Starting with the result $$e^{i\theta} = \cos \theta + i \sin \theta,$$ show that
    1. \((\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta\) [2]
    2. \(\cos \theta = \frac{1}{2}(e^{i\theta} + e^{-i\theta})\). [2]
  2. Using the result in part (i) (A), obtain the values of the constants \(a\), \(b\), \(c\) and \(d\) in the identity $$\cos 6\theta = a \cos^6 \theta + b \cos^4 \theta + c \cos^2 \theta + d.$$ [6]
  3. Using the result in part (i) (B), obtain the values of the constants \(P\), \(Q\), \(R\) and \(S\) in the identity $$\cos^6 \theta = P \cos 6\theta + Q \cos 4\theta + R \cos 2\theta + S.$$ [5]
  4. Show that \(\cos \frac{\pi}{12} = \left(\frac{26 + 15\sqrt{3}}{64}\right)^{\frac{1}{4}}\). [3]
OCR MEI Further Pure Core Specimen Q15
8 marks Challenging +1.8
In this question you must show detailed reasoning. Show that $$\int_0^{\frac{\pi}{3}} \operatorname{arcsinh} 2x \, dx = \frac{2}{3} \ln 3 - \frac{1}{3}.$$ [8]
OCR MEI Further Pure Core Specimen Q16
18 marks Challenging +1.2
A small object is attached to a spring and performs oscillations in a vertical line. The displacement of the object at time \(t\) seconds is denoted by \(x\) cm. Preliminary observations suggest that the object performs simple harmonic motion (SHM) with a period of 2 seconds about the point at which \(x = 0\).
    1. Write down a differential equation to model this motion. [3]
    2. Give the general solution of the differential equation in part (i) (A). [1]
Subsequent observations indicate that the object's motion would be better modelled by the differential equation $$\frac{d^2x}{dt^2} + 2k \frac{dx}{dt} + (k^2 + 9)x = 0 \qquad (*)$$ where \(k\) is a positive constant.
    1. Obtain the general solution of (*). [3]
    2. State two ways in which the motion given by this model differs from that in part (i). [2]
The amplitude of the object's motion is observed to reduce with a scale factor of 0.98 from one oscillation to the next.
  1. Find the value of \(k\). [3]
At the start of the object's motion, \(x = 0\) and the velocity is 12 cm s\(^{-1}\) in the positive \(x\) direction.
  1. Find an equation for \(x\) as a function of \(t\). [4]
  2. Without doing any further calculations, explain why, according to this model, the greatest distance of the object from its starting point in the subsequent motion will be slightly less than 4 cm. [2]
OCR MEI Further Mechanics Major 2019 June Q1
5 marks Standard +0.3
Three forces represented by the vectors \(-4\mathbf{i} + \mathbf{j} + 2\mathbf{j}\) and \(k\mathbf{i} - 2\mathbf{j}\) act at the points with coordinates \((0, 0)\), \((3, 0)\) and \((0, 4)\) respectively.
  1. Given that the three forces form a couple, find the value of \(k\). [2]
  2. Find the magnitude and direction of the couple. [3]
OCR MEI Further Mechanics Major 2019 June Q2
4 marks Moderate -0.5
The Reynolds number, \(R\), is an important dimensionless quantity in fluid dynamics; it can be used to predict flow patterns when a fluid is in motion relative to a surface. The Reynolds number is defined as $$R = \frac{\rho ul}{\mu},$$ where \(\rho\) is the density of the fluid, \(u\) is the velocity of the fluid relative to the surface, \(l\) is the distance travelled by the fluid and \(\mu\) is the viscosity of the fluid. Find the dimensions of \(\mu\). [4]
OCR MEI Further Mechanics Major 2019 June Q3
5 marks Moderate -0.3
A ball of mass \(2\)kg is moving with velocity \((3\mathbf{i} - 2\mathbf{j})\)ms\(^{-1}\) when it is struck by a bat. The impulse on the ball is \((-8\mathbf{i} + 10\mathbf{j})\)Ns.
  1. Find the speed of the ball immediately after the impact. [4]
  2. State one modelling assumption you have used in answering part (a). [1]
OCR MEI Further Mechanics Major 2019 June Q4
6 marks Standard +0.8
\includegraphics{figure_4} Fig. 4 shows a uniform lamina ABCDE such that ABDE is a rectangle and BCD is an isosceles triangle. AB = 5a, AE = 4a and BC = CD. The point F is the midpoint of BD and FC = a.
  1. Find, in terms of \(a\), the exact distance of the centre of mass of the lamina from AE. [4]
The lamina is freely suspended from B and hangs in equilibrium.
  1. Find the angle between AB and the downward vertical. [2]
OCR MEI Further Mechanics Major 2019 June Q5
7 marks Standard +0.3
A particle P of mass 4 kilograms moves in such a way that its position vector at time \(t\) seconds is \(\mathbf{r}\) metres, where $$\mathbf{r} = 3t\mathbf{i} + 2e^{-3t}\mathbf{j}.$$
  1. Find the initial kinetic energy of P. [4]
  2. Find the time when the acceleration of P is 2 metres per second squared. [3]
OCR MEI Further Mechanics Major 2019 June Q6
7 marks Challenging +1.2
\includegraphics{figure_6} The rim of a smooth hemispherical bowl is a circle of centre O and radius \(a\). The bowl is fixed with its rim horizontal and uppermost. A particle P of mass \(m\) is released from rest at a point A on the rim as shown in Fig. 6. When P reaches the lowest point of the bowl it collides directly with a stationary particle Q of mass \(\frac{1}{2}m\). After the collision Q just reaches the rim of the bowl. Find the coefficient of restitution between P and Q. [7]
OCR MEI Further Mechanics Major 2019 June Q7
8 marks Challenging +1.8
In this question you must show detailed reasoning. \includegraphics{figure_7} Fig. 7 shows the curve with equation \(y = \frac{2}{3}\ln x\). The region R, shown shaded in Fig. 7, is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = \ln 2\). A uniform solid of revolution is formed by rotating the region R completely about the \(y\)-axis. Find the exact \(x\)-coordinate of the centre of mass of the solid. [8]
OCR MEI Further Mechanics Major 2019 June Q8
11 marks Standard +0.3
A car of mass 800kg travels up a line of greatest slope of a straight road inclined at \(5°\) to the horizontal. The power developed by the car is constant and equal to 25kW. The resistance to the motion of the car is constant and equal to 750N. The car passes through a point A on the road with speed \(7\)ms\(^{-1}\).
  1. Find
    [5]
The car later passes through a point B on the road where AB = 131m. The time taken to travel from A to B is 10.4s.
  1. Calculate the speed of the car at B. [6]
OCR MEI Further Mechanics Major 2019 June Q9
12 marks Challenging +1.2
\includegraphics{figure_9} A particle P of mass \(m\) is joined to a fixed point O by a light inextensible string of length \(l\). P is released from rest with the string taut and making an acute angle \(\alpha\) with the downward vertical, as shown in Fig. 9. At a time \(t\) after P is released the string makes an angle \(\theta\) with the downward vertical and the tension in the string is \(T\). Angles \(\alpha\) and \(\theta\) are measured in radians.
  1. Show that $$\left(\frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^2 = \frac{2g}{l}\cos\theta + k_1,$$ where \(k_1\) is a constant to be determined in terms of \(g\), \(l\) and \(\alpha\). [4]
  2. Show that $$T = 3mg\cos\theta + k_2,$$ where \(k_2\) is a constant to be determined in terms of \(m\), \(g\) and \(\alpha\). [3]
It is given that \(\alpha\) is small enough for \(\alpha^2\) to be negligible.
  1. Find, in terms of \(m\) and \(g\), the approximate tension in the string. [2]
  2. Show that the motion of P is approximately simple harmonic. [3]
OCR MEI Further Mechanics Major 2019 June Q10
8 marks Challenging +1.2
A particle P, of mass \(m\), moves on a rough horizontal table. P is attracted towards a fixed point O on the table by a force of magnitude \(\frac{kmg}{x^2}\), where \(x\) is the distance OP. The coefficient of friction between P and the table is \(\mu\). P is initially projected in a direction directly away from O. The velocity of P is first zero at a point A which is a distance \(a\) from O.
  1. Show that the velocity \(v\) of P, when P is moving away from O, satisfies the differential equation $$\frac{\mathrm{d}}{\mathrm{d}x}(v^2) + \frac{2kg}{x^2} + 2\mu g = 0.$$ [3]
  2. Verify that $$v^2 = 2gk\left(\frac{1}{x} - \frac{1}{a}\right) + 2\mu g(a-x).$$ [3]
  3. Find, in terms of \(k\) and \(a\), the range of values of \(\mu\) for which P remains at A. [2]
OCR MEI Further Mechanics Major 2019 June Q11
14 marks Standard +0.8
Two uniform smooth spheres A and B have equal radii and are moving on a smooth horizontal surface. The mass of A is 0.2kg and the mass of B is 0.6kg. The spheres collide obliquely. When the spheres collide the line joining their centres is parallel to \(\mathbf{i}\). Immediately before the collision the velocity of A is \(\mathbf{u}_A\)ms\(^{-1}\) and the velocity of B is \(\mathbf{u}_B\)ms\(^{-1}\). The coefficient of restitution between A and B is 0.5. Immediately after the collision the velocity of A is \((-4\mathbf{i} + 2\mathbf{j})\)ms\(^{-1}\) and the velocity of B is \((2\mathbf{i} + 3\mathbf{j})\)ms\(^{-1}\).
  1. Find \(\mathbf{u}_A\) and \(\mathbf{u}_B\). [7]
After the collision B collides with a smooth vertical wall which is parallel to \(\mathbf{j}\). The loss in kinetic energy of B caused by the collision with the wall is 1.152J.
  1. Find the coefficient of restitution between B and the wall. [3]
  2. Find the angle through which the direction of motion of B is deflected as a result of the collision with the wall. [4]
OCR MEI Further Mechanics Major 2019 June Q12
16 marks Challenging +1.2
\includegraphics{figure_12} The ends of a light inextensible string are fixed to two points A and B in the same vertical line, with A above B. The string passes through a small smooth ring of mass \(m\). The ring is fastened to the string at a point P. When the string is taut the angle APB is a right angle, the angle BAP is \(\theta\) and the perpendicular distance of P from AB is \(r\). The ring moves in a horizontal circle with constant angular velocity \(\omega\) and the string taut as shown in Fig. 12.
  1. By resolving horizontally and vertically, show that the tension in the part of the string BP is \(m(r\omega^2\cos\theta - g\sin\theta)\). [6]
  2. Find a similar expression, in terms of \(r\), \(\omega\), \(m\), \(g\) and \(\theta\), for the tension in the part of the string AP. [2]
It is given that AB = 5a and AP = 4a.
  1. Show that \(16a\omega^2 > 5g\). [3]
The ring is now free to move on the string but remains in the same position on the string as before. The string remains taut and the ring continues to move in a horizontal circle.
  1. Find the period of the motion of the ring, giving your answer in terms of \(a\), \(g\) and \(\pi\). [5]
OCR MEI Further Mechanics Major 2019 June Q13
17 marks Challenging +1.3
\includegraphics{figure_13} A step-ladder has two sides AB and AC, each of length \(4a\). Side AB has weight \(W\) and its centre of mass is at the half-way point; side AC is light. The step-ladder is smoothly hinged at A and the two parts of the step-ladder, AB and AC, are connected by a light taut rope DE, where D is on AB, E is on AC and AD = AE = \(a\). A man of weight \(4W\) stands at a point F on AB, where BF = \(x\). The system is in equilibrium with B and C on a smooth horizontal floor and the sides AB and AC are each at an angle \(\theta\) to the vertical, as shown in Fig. 13.
  1. By taking moments about A for side AB of the step-ladder and then for side AC of the step-ladder show that the tension in the rope is $$W\left(1 + \frac{2x}{a}\right)\tan\theta.$$ [7]
The rope is elastic with natural length \(\frac{1}{2}a\) and modulus of elasticity \(W\).
  1. Show that the condition for equilibrium is that $$x = \frac{1}{2}a(8\cos\theta - \cot\theta - 1).$$ [5]
In this question you must show detailed reasoning.
  1. Hence determine, in terms of \(a\), the maximum value of \(x\) for which equilibrium is possible. [5]
END OF QUESTION PAPER
OCR MEI Further Mechanics Major 2022 June Q1
5 marks Moderate -0.8
\includegraphics{figure_1} Three forces of magnitudes 4 N, 7 N and \(P\) N act at a point in the directions shown in the diagram. The forces are in equilibrium.
  1. Draw a closed figure to represent the three forces. [1]
  2. Hence, or otherwise, find the following.
    1. The value of \(\theta\). [2]
    2. The value of \(P\). [2]
OCR MEI Further Mechanics Major 2022 June Q2
4 marks Standard +0.3
\includegraphics{figure_2} A particle is projected with speed \(v\) from a point O on horizontal ground. The angle of projection is \(\theta\) above the horizontal. The particle passes, in succession, through two points A and B, each at a height \(h\) above the ground and a distance \(d\) apart, as shown in the diagram. You are given that \(d^2 = \frac{v^\alpha \sin^2 2\theta}{g^\beta} - \frac{8h^2 \cos^2 \theta}{g}\). Use dimensional analysis to find \(\alpha\) and \(\beta\). [4]
OCR MEI Further Mechanics Major 2022 June Q3
6 marks Standard +0.3
A particle, of mass 2 kg, is placed at a point A on a rough horizontal surface. There is a straight vertical wall on the surface and the point on the wall nearest to A is B. The distance AB is 5 m. The particle is projected with speed 4.2 m s\(^{-1}\) along the surface from A towards B. The particle hits the wall directly and rebounds. The coefficient of friction between the particle and the surface is 0.1.
  1. Determine the speed of the particle immediately before impact with the wall. [4]
The magnitude of the impulse that the wall exerts on the particle is 9.8 N s.
  1. Find the speed of the particle immediately after impact with the wall. [2]
OCR MEI Further Mechanics Major 2022 June Q4
7 marks Standard +0.3
\includegraphics{figure_4} The diagram shows a particle P, of mass 0.1 kg, which is attached by a light inextensible string of length 0.5 m to a fixed point O. P moves with constant angular speed 5 rad s\(^{-1}\) in a horizontal circle with centre vertically below O. The string is inclined at an angle \(\theta\) to the vertical.
  1. Determine the tension in the string. [3]
  2. Find the value of \(\theta\). [2]
  3. Find the kinetic energy of P. [2]
OCR MEI Further Mechanics Major 2022 June Q5
7 marks Standard +0.3
At time \(t\) seconds, where \(t \geq 0\), a particle P of mass 2 kg is moving on a smooth horizontal surface. The particle moves under the action of a constant horizontal force of \((-2\mathbf{i} + 6\mathbf{j})\) N and a variable horizontal force of \((2\cos 2t \mathbf{i} + 4\sin t \mathbf{j})\) N. The acceleration of P at time \(t\) seconds is \(\mathbf{a}\) m s\(^{-2}\).
  1. Find \(\mathbf{a}\) in terms of \(t\). [2]
The particle P is at rest when \(t = 0\).
  1. Determine the speed of P at the instant when \(t = 2\). [5]
OCR MEI Further Mechanics Major 2022 June Q6
7 marks Standard +0.3
In this question the box should be modelled as a particle. A box of mass \(m\) kg is placed on a rough slope which makes an angle of \(\alpha\) with the horizontal.
  1. Show that the box is on the point of slipping if \(\mu = \tan \alpha\), where \(\mu\) is the coefficient of friction between the box and the slope. [2]
A box of mass 5 kg is pulled up a rough slope which makes an angle of 15° with the horizontal. The box is subject to a constant frictional force of magnitude 3 N. The speed of the box increases from 2 m s\(^{-1}\) at a point A on the slope to 5 m s\(^{-1}\) at a point B on the slope with B higher up the slope than A. The distance AB is 10 m. \includegraphics{figure_6} The pulling force has constant magnitude \(P\) N and acts at a constant angle of 25° above the slope, as shown in the diagram.
  1. Use the work–energy principle to determine the value of \(P\). [5]
OCR MEI Further Mechanics Major 2022 June Q7
12 marks Standard +0.3
Two small uniform smooth spheres A and B, of masses 2 kg and 3 kg respectively, are moving in opposite directions along the same straight line towards each other on a smooth horizontal surface. Sphere A has speed 2 m s\(^{-1}\) and B has speed 1 m s\(^{-1}\) before they collide. The coefficient of restitution between A and B is \(e\).
  1. Show that the velocity of B after the collision, in the original direction of motion of A, is \(\frac{1}{5}(1 + 6e)\) m s\(^{-1}\) and find a similar expression for the velocity of A after the collision. [5]
  2. The following three parts are independent of each other, and each considers a different scenario regarding the collision between A and B.
    1. In the collision between A and B the spheres coalesce to form a combined body C. State the speed of C after the collision. [1]
    2. In the collision between A and B the direction of motion of A is reversed. Find the range of possible values of \(e\). [2]
    3. The total loss in kinetic energy due to the collision is 3 J. Determine the value of \(e\). [4]
OCR MEI Further Mechanics Major 2022 June Q8
13 marks Standard +0.8
A particle P is projected from a fixed point O with initial velocity \(u\mathbf{i} + ku\mathbf{j}\), where \(k\) is a positive constant. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the horizontal and vertically upward directions respectively. P moves with constant gravitational acceleration of magnitude \(g\). At time \(t \geq 0\), particle P has position vector \(\mathbf{r}\) relative to O.
  1. Starting from an expression for \(\ddot{\mathbf{r}}\), use integration to derive the formula $$\mathbf{r} = ut\mathbf{i} + \left(kut - \frac{1}{2}gt^2\right)\mathbf{j}.$$ [4]
The position vector \(\mathbf{r}\) of P at time \(t \geq 0\) can be expressed as \(\mathbf{r} = x\mathbf{i} + y\mathbf{j}\), where the axes Ox and Oy are horizontally and vertically upwards through O respectively. The axis Ox lies on horizontal ground.
  1. Show that the path of P has cartesian equation $$gy^2 - 2ku^2x + 2u^2y = 0.$$ [3]
  2. Hence find, in terms of \(g\), \(k\) and \(u\), the maximum height of P above the ground during its motion. [3]
The maximum height P reaches above the ground is equal to the distance OA, where A is the point where P first hits the ground.
  1. Determine the value of \(k\). [3]