Questions — OCR MEI (4301 questions)

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OCR MEI M4 2015 June Q1
12 marks Challenging +1.8
1 A rocket is launched vertically upwards from rest. The initial mass of the rocket, including fuel and payload, is \(m _ { 0 }\) and the propulsion system ejects mass at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. The only other force acting on the rocket is its weight. The acceleration due to gravity is constant throughout the motion. At time \(t\) after launch the speed of the rocket is \(v\).
  1. Show that while mass is being ejected from the rocket \(v = u \ln \left( \frac { m _ { 0 } } { m _ { 0 } - k t } \right) - g t\). The rocket initially has 2400 kg of fuel which is ejected at a constant rate of \(100 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\) with constant speed \(3000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket.
  2. Given that the rocket must reach a speed of \(7910 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before releasing its payload, find the maximum possible value of \(m _ { 0 }\).
OCR MEI M4 2015 June Q2
12 marks Challenging +1.8
2 Fig. 2 shows a system in a vertical plane. A uniform rod AB of length \(2 a\) and mass \(m\) is freely hinged at A . The angle that AB makes with the horizontal is \(\theta\), where \(- \frac { 2 } { 3 } \pi < \theta < \frac { 2 } { 3 } \pi\). Attached at B is a light spring BC of natural length \(a\) and stiffness \(\frac { m g } { a }\). The other end of the spring is attached to a small light smooth ring C which can slide freely along a vertical rail. The rail is at a distance of \(a\) from A and the spring is always horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8ea28e6f-528c-4e3c-9562-6c964043747e-2_737_703_1356_680} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Find the potential energy, \(V\), of the system and hence show that \(\frac { \mathrm { d } V } { \mathrm {~d} \theta } = m g a \cos \theta ( 1 - 4 \sin \theta )\).
  2. Hence find the positions of equilibrium of the system and investigate their stability.
OCR MEI M4 2015 June Q3
24 marks Challenging +1.8
3 A particle of mass 4 kg moves along the \(x\)-axis. At time \(t\) seconds the particle is \(x \mathrm {~m}\) from the origin O and has velocity \(v \mathrm {~ms} ^ { - 1 }\). A driving force of magnitude \(20 t \mathrm { t } ^ { - t } \mathrm {~N}\) is applied in the positive \(x\) direction. Initially \(v = 2\) and the particle is at O .
  1. Find, in either order, the impulse of the force over the first 3 seconds and the velocity of the particle after 3 seconds. From time \(t = 3\) a resistive force of magnitude \(\frac { 1 } { 2 } t \mathrm {~N}\) and the driving force are applied until the particle comes to rest.
  2. Show that, after the resistive force is applied, the only time at which the resultant force on the particle is zero is when \(t = \ln 40\). Hence find the maximum velocity of the particle during the motion.
  3. Given that the time \(T\) seconds at which the particle comes to rest is given by the equation \(T = \sqrt { 121 - 80 \mathrm { e } ^ { - T } ( 1 + T ) }\), without solving the equation deduce that \(T \approx 11\).
  4. Use a numerical method to find \(T\) correct to 4 decimal places.
OCR MEI M4 2015 June Q4
24 marks Challenging +1.8
4 A solid cylinder of radius \(a \mathrm {~m}\) and length \(3 a \mathrm {~m}\) has density \(\rho \mathrm { kg } \mathrm { m } ^ { - 3 }\) given by \(\rho = k \left( 2 + \frac { x } { a } \right)\) where \(x \mathrm {~m}\) is the distance from one end and \(k\) is a positive constant. The mass of the cylinder is \(M \mathrm {~kg}\) where \(M = \frac { 21 } { 2 } \pi a ^ { 3 } k\). Let A and B denote the circular faces of the cylinder where \(x = 0\) and \(x = 3 a\), respectively.
  1. Show by integration that the moment of inertia, \(I _ { \mathrm { A } } \mathrm { kg } \mathrm { m } ^ { 2 }\), of the cylinder about a diameter of the face A is given by \(I _ { \mathrm { A } } = \frac { 109 } { 28 } M a ^ { 2 }\).
  2. Show that the centre of mass of the cylinder is \(\frac { 12 } { 7 } a \mathrm {~m}\) from A .
  3. Using the parallel axes theorem, or otherwise, show that the moment of inertia, \(I _ { \mathrm { B } } \mathrm { kg } \mathrm { m } ^ { 2 }\), of the cylinder about a diameter of the face B is given by \(I _ { \mathrm { B } } = \frac { 73 } { 28 } M a ^ { 2 }\). You are now given that \(M = 4\) and \(a = 0.7\). The cylinder is at rest and can rotate freely about a horizontal axis which is a diameter of the face B as shown in Fig. 4. It is struck at the bottom of the curved surface by a small object of mass 0.2 kg which is travelling horizontally at speed \(20 \mathrm {~ms} ^ { - 1 }\) in the vertical plane which is both perpendicular to the axis of rotation and contains the axis of symmetry of the cylinder. The object sticks to the cylinder at the point of impact. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8ea28e6f-528c-4e3c-9562-6c964043747e-4_606_435_1087_817} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure}
  4. Find the initial angular speed of the combined object after the collision. \section*{END OF QUESTION PAPER}
OCR MEI M4 2016 June Q1
12 marks Challenging +1.2
1 A car of mass \(m\) moves horizontally in a straight line. At time \(t\) the car is a distance \(x\) from a point O and is moving away from O with speed \(v\). There is a force of magnitude \(k v ^ { 2 }\), where \(k\) is a constant, resisting the motion of the car. The car's engine has a constant power \(P\). The terminal speed of the car is \(U\).
  1. Show that $$m v ^ { 2 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = P \left( 1 - \frac { v ^ { 3 } } { U ^ { 3 } } \right)$$
  2. Show that the distance moved while the car accelerates from a speed of \(\frac { 1 } { 4 } U\) to a speed of \(\frac { 1 } { 2 } U\) is $$\frac { m U ^ { 3 } } { 3 P } \ln A$$ stating the exact value of the constant \(A\). Once the car attains a speed of \(\frac { 1 } { 2 } U\), no further power is supplied by the car's engine.
  3. Find, in terms of \(m , P\) and \(U\), the time taken for the speed of the car to reduce from \(\frac { 1 } { 2 } U\) to \(\frac { 1 } { 4 } U\).
OCR MEI M4 2016 June Q2
12 marks Challenging +1.8
2 A thin rigid rod PQ has length \(2 a\). Its mass per unit length, \(\rho\), is given by \(\rho = k \left( 1 + \frac { x } { 2 a } \right)\) where \(x\) is the distance from P and \(k\) is a positive constant. The mass of the rod is \(M\) and the moment of inertia of the rod about an axis through P perpendicular to PQ is \(I\).
  1. Show that \(I = \frac { 14 } { 9 } M a ^ { 2 }\). The rod is initially at rest with Q vertically below P . It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through P . The rod is struck a horizontal blow perpendicular to the fixed axis at the point where \(x = \frac { 3 } { 2 } a\). The magnitude of the impulse of this blow is \(J\).
  2. Find, in terms of \(a , J\) and \(M\), the initial angular speed of the rod.
  3. Find, in terms of \(a , g\) and \(M\), the set of values of \(J\) for which the rod makes complete revolutions.
OCR MEI M4 2016 June Q3
24 marks Challenging +1.8
3 Fig. 3 shows a uniform rigid rod AB of length \(2 a\) and mass \(2 m\). The rod is freely hinged at A so that it can rotate in a vertical plane. One end of a light inextensible string of length \(l\) is attached to B . The string passes over a small smooth fixed pulley at C , where C is vertically above A and \(\mathrm { AC } = 6 a\). A particle of mass \(\lambda m\), where \(\lambda\) is a positive constant, is attached to the other end of the string and hangs freely, vertically below C . The rod makes an angle \(\theta\) with the upward vertical, where \(0 \leqslant \theta \leqslant \pi\). You may assume that the particle does not interfere with the rod AB or the section of the string BC . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3fdb2cff-0f74-4c88-b25a-759bfab1675a-3_878_615_667_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Find the potential energy, \(V\), of the system relative to a situation in which the rod AB is horizontal, and hence show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 m g a \sin \theta \left( \frac { 3 \lambda } { \sqrt { 10 - 6 \cos \theta } } - 1 \right) .$$
  2. Show that \(\theta = 0\) and \(\theta = \pi\) are the only values of \(\theta\) for which the system is in equilibrium whatever the value of \(\lambda\).
  3. Show that, if there is a third value of \(\theta\) for which the system is in equilibrium, then \(\frac { 2 } { 3 } < \lambda < \frac { 4 } { 3 }\).
  4. Given that there are three positions of equilibrium, establish whether each of these positions is stable or unstable. It is given that, for small values of \(\theta\), $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } \approx 2 m g a \left[ \left( \frac { 3 } { 2 } \lambda - 1 \right) \theta - \left( \frac { 13 } { 16 } \lambda - \frac { 1 } { 6 } \right) \theta ^ { 3 } \right] .$$
  5. Investigate the stability of the equilibrium position given by \(\theta = 0\) in the case when \(\lambda = \frac { 2 } { 3 }\).
OCR MEI M4 2016 June Q4
24 marks Challenging +1.8
4 A raindrop falls from rest through a stationary cloud. The raindrop has mass \(m\) and speed \(v\) when it has fallen a distance \(x\). You may assume that resistances to motion are negligible.
  1. Derive the equation of motion $$m v \frac { \mathrm {~d} v } { \mathrm {~d} x } + v ^ { 2 } \frac { \mathrm {~d} m } { \mathrm {~d} x } = m g .$$ Initially the mass of the raindrop is \(m _ { 0 }\). Two different models for the mass of the raindrop are suggested.
    In the first model \(m = m _ { 0 } \mathrm { e } ^ { k _ { 1 } x }\), where \(k _ { 1 }\) is a positive constant.
  2. Show that $$v ^ { 2 } = \frac { g } { k _ { 1 } } \left( 1 - \mathrm { e } ^ { - 2 k _ { 1 } x } \right) ,$$ and hence state, in terms of \(g\) and \(k _ { 1 }\), the terminal velocity of the raindrop according to this first model. In the second model \(m = m _ { 0 } \left( 1 + k _ { 2 } x \right)\), where \(k _ { 2 }\) is a positive constant.
  3. By considering the expression obtained from differentiating \(v ^ { 2 } \left( 1 + k _ { 2 } x \right) ^ { 2 }\) with respect to \(x\), show that, for the second model, the equation of motion in part (i) may be written as $$\frac { d } { d x } \left[ v ^ { 2 } \left( 1 + k _ { 2 } x \right) ^ { 2 } \right] = 2 g \left( 1 + k _ { 2 } x \right) ^ { 2 } .$$ Hence find an expression for \(v ^ { 2 }\) in terms of \(g , k _ { 2 }\) and \(x\).
  4. Suppose that the models give the same value for the speed of the raindrop at the instant when it has doubled its initial mass. Find the exact value of \(\frac { k _ { 1 } } { k _ { 2 } }\), giving your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\)
    are integers. are integers. \section*{END OF QUESTION PAPER}
OCR MEI Further Pure Core AS 2018 June Q1
4 marks Easy -1.2
1 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are defined as follows: $$\mathbf { A } = \left( \begin{array} { l } 1
OCR MEI Further Pure Core AS 2018 June Q3
5 marks Moderate -0.8
3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 0 & 3
1 & - 1 & 3 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 3 \end{array} \right)$$ Calculate all possible products formed from two of these three matrices. 2 Find, to the nearest degree, the angle between the vectors \(\left( \begin{array} { r } 1 \\ 0 \\ - 2 \end{array} \right)\) and \(\left( \begin{array} { r } - 2 \\ 3 \\ - 3 \end{array} \right)\). 3 Find real numbers \(a\) and \(b\) such that \(( a - 3 i ) ( 5 - i ) = b - 17 i\).
OCR MEI Further Pure Core AS 2018 June Q4
5 marks Moderate -0.8
4 Find a cubic equation with real coefficients, two of whose roots are \(2 - \mathrm { i }\) and 3.
OCR MEI Further Pure Core AS 2018 June Q5
7 marks Moderate -0.3
5 A transformation of the \(x - y\) plane is represented by the matrix \(\left( \begin{array} { r r } \cos \theta & 2 \sin \theta \\ 2 \sin \theta & - \cos \theta \end{array} \right)\), where \(\theta\) is a positive acute angle.
  1. Write down the image of the point \(( 2,3 )\) under this transformation.
  2. You are given that this image is the point ( \(a , 0\) ). Find the value of \(a\).
OCR MEI Further Pure Core AS 2018 June Q6
4 marks Moderate -0.5
6 Find the invariant line of the transformation of the \(x - y\) plane represented by the matrix \(\left( \begin{array} { r r } 2 & 0 \\ 4 & - 1 \end{array} \right)\).
OCR MEI Further Pure Core AS 2018 June Q7
9 marks Standard +0.3
7
  1. Express \(\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 }\) as a single fraction.
  2. Find how many terms of the series $$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 3 \times 5 } + \frac { 2 } { 5 \times 7 } + \ldots + \frac { 2 } { ( 2 r - 1 ) ( 2 r + 1 ) } + \ldots$$ are needed for the sum to exceed 0.999999.
OCR MEI Further Pure Core AS 2018 June Q8
6 marks Standard +0.3
8 Prove by induction that \(\left( \begin{array} { l l } 1 & 1 \\ 0 & 2 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & 2 ^ { n } - 1 \\ 0 & 2 ^ { n } \end{array} \right)\) for all positive integers \(n\).
OCR MEI Further Pure Core AS 2018 June Q9
9 marks Moderate -0.3
9 Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{ z : | z | \leqslant 4 \sqrt { 2 } \} \cap \left\{ z : \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi \right\} .$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9ef04b56-c6e5-46ea-a485-fe872932e9d8-3_549_520_397_751} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find, in modulus-argument form, the complex number represented by the point P .
  2. Find, in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q .
  3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
    • \(3 + 5 \mathrm { i }\)
    • \(5.5 ( \cos 0.8 + \mathrm { i } \sin 0.8 )\)
      lie within this region.
OCR MEI Further Pure Core AS 2018 June Q10
8 marks Standard +0.3
10 Three planes have equations $$\begin{aligned} - x + 2 y + z & = 0 \\ 2 x - y - z & = 0 \\ x + y & = a \end{aligned}$$ where \(a\) is a constant.
  1. Investigate the arrangement of the planes:
    • when \(a = 0\);
    • when \(a \neq 0\).
    • Chris claims that the position vectors \(- \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , 2 \mathbf { i } - \mathbf { j } - \mathbf { k }\) and \(\mathbf { i } + \mathbf { j }\) lie in a plane. Determine whether or not Chris is correct.
OCR MEI Further Pure Core AS 2019 June Q1
3 marks Standard +0.3
1 In this question you must show detailed reasoning.
Find \(\sum _ { r = 1 } ^ { 100 } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\), giving your answer correct to 4 decimal places.
OCR MEI Further Pure Core AS 2019 June Q2
3 marks Standard +0.3
2 The roots of the equation \(3 x ^ { 2 } - x + 2 = 0\) are \(\alpha\) and \(\beta\).
Find a quadratic equation with integer coefficients whose roots are \(2 \alpha - 3\) and \(2 \beta - 3\).
OCR MEI Further Pure Core AS 2019 June Q3
6 marks Moderate -0.3
3 In this question you must show detailed reasoning.
\(\mathbf { A }\) and \(\mathbf { B }\) are matrices such that \(\mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 2 & 1 \\ - 1 & 1 \end{array} \right)\).
  1. Find \(\mathbf { A B }\).
  2. Given that \(\mathbf { A } = \left( \begin{array} { l l } \frac { 1 } { 3 } & 1 \\ 0 & 1 \end{array} \right)\), find \(\mathbf { B }\).
OCR MEI Further Pure Core AS 2019 June Q4
8 marks Standard +0.3
4
  1. Find \(\mathbf { M } ^ { - 1 }\), where \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 1 & 1 & 2 \\ - 2 & 1 & 2 \end{array} \right)\).
  2. Hence find, in terms of the constant \(k\), the point of intersection of the planes $$\begin{aligned} x + 2 y + 3 z & = 19 \\ - x + y + 2 z & = 4 \\ - 2 x + y + 2 z & = k \end{aligned}$$
  3. In this question you must show detailed reasoning. Find the acute angle between the planes \(x + 2 y + 3 z = 19\) and \(- x + y + 2 z = 4\).
OCR MEI Further Pure Core AS 2019 June Q5
6 marks Moderate -0.3
5 Prove by induction that, for all positive integers \(n , \sum _ { r = 1 } ^ { n } \frac { 1 } { 3 ^ { r } } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 ^ { n } } \right)\).
OCR MEI Further Pure Core AS 2019 June Q6
11 marks Standard +0.8
6 A linear transformation \(T\) of the \(x - y\) plane has an associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { c c } \lambda & k \\ 1 & \lambda - k \end{array} \right)\), and \(\lambda\)
and \(k\) are real constants. and \(k\) are real constants.
  1. You are given that \(\operatorname { det } \mathbf { M } > 0\) for all values of \(\lambda\).
    1. Find the range of possible values of \(k\).
    2. What is the significance of the condition \(\operatorname { det } \mathbf { M } > 0\) for the transformation T? For the remainder of this question, take \(k = - 2\).
  2. Determine whether there are any lines through the origin that are invariant lines for the transformation T.
  3. The transformation T is applied to a triangle with area 3 units \({ } ^ { 2 }\). The area of the resulting image triangle is 15 units \({ } ^ { 2 }\).
    Find the possible values of \(\lambda\).
OCR MEI Further Pure Core AS 2019 June Q7
12 marks Standard +0.8
7
  1. Sketch on a single Argand diagram
    1. the set of points for which \(| z - 1 - 3 i | = 3\),
    2. the set of points for which \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).
  2. Find, in exact form, the two values of \(z\) for which \(| z - 1 - 3 i | = 3\) and \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).
OCR MEI Further Pure Core AS 2022 June Q1
6 marks Moderate -0.3
1
    1. Write the following simultaneous equations as a matrix equation. $$\begin{aligned} x + y + 2 z & = 7 \\ 2 x - 4 y - 3 z & = - 5 \\ - 5 x + 3 y + 5 z & = 13 \end{aligned}$$
    2. Hence solve the equations.
  2. Determine the set of values of the constant \(k\) for which the matrix equation $$\left( \begin{array} { c c } k + 1 & 1 \\ 2 & k \end{array} \right) \binom { x } { y } = \binom { 23 } { - 17 }$$ has a unique solution.