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OCR Further Pure Core 2 2018 September Q9
15 marks Challenging +1.2
The quantity of grass on an island at time \(t\) years is \(x\), in appropriate units. At time \(t = 0\) some rabbits are introduced to the island. The population of rabbits on the island at time \(t\) years is \(y\), in units of \(100\)s of rabbits. An ecologist who is studying the island suggests that the following pair of simultaneous first order differential equations can be used to model the population of rabbits and quantity of grass for \(t \geq 0\). $$\frac{dx}{dt} = 3x - 2y,$$ $$\frac{dy}{dt} = y + 5x$$
    1. Show that \(\frac{d^2x}{dt^2} = a\frac{dx}{dt} + bx\) where \(a\) and \(b\) are constants which should be found. [2]
    2. Find the general solution for \(x\) in real form. [3]
  2. Find the corresponding general solution for \(y\). [3]
At time \(t = 0\) the quantity of grass on the island was \(4\) units. The number of rabbits introduced at this time was \(500\).
  1. Find the particular solutions for \(x\) and \(y\). [5]
  2. The ecologist finds that the model predicts that there will be no grass at time \(T\), when there are still rabbits on the island. Find the value of \(T\). [1]
  3. State one way in which the model is not appropriate for modelling the quantity of grass and the population of rabbits for \(0 \leq t \leq T\). [1]
OCR Further Mechanics 2018 September Q1
5 marks Moderate -0.3
A car of mass 850 kg is being driven uphill along a straight road inclined at \(7°\) to the horizontal. The resistance to motion is modelled as a constant force of magnitude 140 N. At a certain instant the car's speed is \(12 \text{ms}^{-1}\) and its acceleration is \(0.4 \text{ms}^{-2}\).
  1. Calculate the power of the car's engine at this instant. [3]
  2. Find the constant speed at which the car could travel up the hill with the engine generating this power. [2]
OCR Further Mechanics 2018 September Q2
6 marks Standard +0.3
A particle of mass 0.8 kg is moving in a straight line on a smooth horizontal surface with constant speed \(12 \text{ms}^{-1}\) when it is struck by a horizontal impulse. Immediately after the impulse acts, the particle is moving with speed \(9 \text{ms}^{-1}\) at an angle of 50° to its original direction of motion (see diagram). \includegraphics{figure_2} Find
  1. the magnitude of the impulse, [3]
  2. the angle that the impulse makes with the original direction of motion of the particle. [3]
OCR Further Mechanics 2018 September Q3
6 marks Standard +0.3
Assume that the earth moves round the sun in a circle of radius \(1.50 \times 10^8\) km at constant speed, with one complete orbit taking 365 days. Given that the mass of the earth is \(5.97 \times 10^{24}\) kg,
  1. calculate the magnitude of the force exerted by the sun on the earth, giving your answer in newtons, [5]
  2. state the direction in which this force acts. [1]
OCR Further Mechanics 2018 September Q4
13 marks Standard +0.8
\(A\) and \(B\) are two points a distance of 5 m apart on a horizontal ceiling. A particle \(P\) of mass \(m\) kg is attached to \(A\) and \(B\) by light elastic strings. The particle hangs in equilibrium at a distance of 4 m from \(A\) and 3 m from \(B\) so that angle \(APB = 90°\) (see diagram). \includegraphics{figure_4} The string joining \(P\) to \(A\) has natural length 2 m and modulus of elasticity \(\lambda_A\) N. The string joining \(P\) to \(B\) also has natural length 2 m but has modulus of elasticity \(\lambda_B\) N.
    1. Show that \(\lambda_B = \frac{3}{4}\lambda_A\). [4]
    2. Find an expression for \(\lambda_A\) in terms of \(m\) and \(g\). [3]
  2. Find, in terms of \(m\) and \(g\), the total elastic potential energy stored in the strings. [2]
The string joining \(P\) to \(A\) is detached from \(A\) and a second particle, \(Q\), of mass \(0.3m\) kg is attached to the free end of the string. \(Q\) is then gently lowered into a position where the system hangs vertically in equilibrium.
  1. Find the distance of \(Q\) below \(B\) in this equilibrium position. [4]
OCR Further Mechanics 2018 September Q5
10 marks Standard +0.3
One end of a non-uniform rod is freely hinged to a fixed point so that the rod can rotate about the point. When the rod rotates with angular velocity \(\omega\) it can be shown that the kinetic energy \(E\) of the rod is given by \(E = \frac{1}{2}I\omega^2\), where \(I\) is a quantity called the moment of inertia of the rod.
  1. Deduce the dimensions of \(I\). [3]
  2. Given that the rod has mass \(m\) and length \(r\), suggest an expression for \(I\), explaining any additional symbols that you use. [3]
A student notices that the formula \(E = \frac{1}{2}I\omega^2\) looks similar to the formula \(E = \frac{1}{2}mv^2\) for the kinetic energy of a particle, with angular velocity for the rod corresponding to velocity for the particle, and moment of inertia corresponding to mass. Assuming a similar correspondence between angular acceleration (i.e. \(\frac{d\omega}{dt}\)) and acceleration, the student thinks that an equation for angular motion of the rod corresponding to Newton's second law for the particle should be \(F = I\alpha\), where \(F\) is the force applied to the rod and \(\alpha\) is the resulting angular acceleration.
  1. Use dimensional analysis to show that the student's suggestion is incorrect. [2]
  2. State the dimensions of a quantity \(x\) for which the equation \(Fx = I\alpha\) would be dimensionally consistent. [1]
  3. Explain why the fact that the equation in part (iv) is dimensionally consistent does not necessarily mean that it is correct. [1]
OCR Further Mechanics 2018 September Q6
10 marks Standard +0.8
A particle \(P\) of mass \(m\) moves along the positive \(x\)-axis. When its displacement from the origin \(O\) is \(x\) its velocity is \(v\), where \(v \geqslant 0\). It is subject to two forces: a constant force \(T\) in the positive \(x\) direction, and a resistive force which is proportional to \(v^2\).
  1. Show that \(v^2 = \frac{1}{k}\left(T - Ae^{-\frac{2kx}{m}}\right)\) where \(A\) and \(k\) are constants. [5]
\(P\) starts from rest at \(O\).
  1. Find an expression for the work done against the resistance to motion as \(P\) moves from \(O\) to the point where \(x = 1\). [4]
  2. Find an expression for the limiting value of the velocity of \(P\) as \(x\) increases. [1]
OCR Further Mechanics 2018 September Q7
9 marks Standard +0.8
A uniform solid hemisphere has radius 0.4 m. A uniform solid cone, made of the same material, has base radius 0.4 m and height 1.2 m. A solid, \(S\), is formed by joining the hemisphere and the cone so that their circular faces coincide. \(O\) is the centre of the joint circular face and \(V\) is the vertex of the cone. \(G\) is the centre of mass of \(S\).
  1. Explain briefly why \(G\) lies on the line through \(O\) and \(V\). [1]
  2. Show that the distance of \(G\) from \(O\) is 0.12 m. (The volumes of a hemisphere and cone are \(\frac{2}{3}\pi r^3\) and \(\frac{1}{3}\pi r^2 h\) respectively.) [5]
\includegraphics{figure_7} \(S\) is suspended from two light vertical strings, one attached to \(V\) and the other attached to a point on the circumference of the joint circular face, and hangs in equilibrium with \(OV\) horizontal (see diagram).
  1. The weight of \(S\) is \(W\). Find the magnitudes of the tensions in the strings in terms of \(W\). [3]
OCR Further Mechanics 2018 September Q8
16 marks Challenging +1.8
A point \(O\) is situated a distance \(h\) above a smooth horizontal plane, and a particle \(A\) of mass \(m\) is attached to \(O\) by a light inextensible string of length \(h\). A particle \(B\) of mass \(2m\) is at rest on the plane, directly below \(O\), and is attached to a point \(C\) on the plane, where \(BC = l\), by a light inextensible string of length \(l\). \(A\) is released from rest with the string \(OA\) taut and making an acute angle \(\theta\) with the downward vertical (see diagram). \includegraphics{figure_8} \(A\) moves in a vertical plane perpendicular to \(CB\) and collides directly with \(B\). As a result of this collision, \(A\) is brought to rest and \(B\) moves on the plane in a horizontal circle with centre \(C\). After \(B\) has made one complete revolution the particles collide again.
  1. Show that, on the next occasion that \(A\) comes to rest, the string \(OA\) makes an angle \(\phi\) with the downward vertical through \(O\), where \(\cos \phi = \frac{3 + \cos \theta}{4}\). [9]
\(A\) and \(B\) collide again when \(AO\) is next vertical.
  1. Find the percentage of the original energy of the system that remains immediately after this collision. [5]
  2. Explain why the total momentum of the particles immediately before the first collision is the same as the total momentum of the particles immediately after the second collision. [1]
  3. Explain why the total momentum of the particles immediately before the first collision is different from the total momentum of the particles immediately after the third collision. [1]
OCR Further Additional Pure 2018 September Q1
5 marks Standard +0.8
  1. Write the number \(100011_n\), where \(n \geq 2\), as a polynomial in \(n\). [1]
  2. Show that \(n^2 + n + 1\) is a factor of this expression. [2]
  3. Hence show that \(100011_n\) is composite in any number base \(n \geq 2\). [2]
OCR Further Additional Pure 2018 September Q2
10 marks Challenging +1.8
In this question, you must show detailed reasoning. A curve is defined parametrically by \(x = t^3 - 3t + 1\), \(y = 3t^2 - 1\), for \(0 \leq t \leq 5\). Find, in exact form,
  1. the length of the curve, [6]
  2. the area of the surface generated when the curve is rotated completely about the \(x\)-axis. [4]
OCR Further Additional Pure 2018 September Q3
11 marks Standard +0.8
The function \(w = f(x, y, z)\) is given by \(f(x, y, z) = x^2yz + 2xy^2z + 3xyz^2 - 24xyz\), for \(x, y, z \neq 0\).
    1. Find
    2. Hence find the values of \(a\), \(b\), \(c\) and \(d\) for which \(w\) has a stationary value when \(d = f(a, b, c)\). [5]
  2. You are given that this stationary value is a local minimum of \(w\). Find values of \(x\), \(y\) and \(z\) which show that it is not a global minimum of \(w\). [2]
OCR Further Additional Pure 2018 September Q4
12 marks Challenging +1.2
The points \(A\), \(B\), \(C\) and \(P\) have coordinates \((a, 0, 0)\), \((0, b, 0)\), \((0, 0, c)\) and \((a, b, c)\) respectively, where \(a\), \(b\) and \(c\) are positive constants. The plane \(\Pi\) contains \(A\), \(B\) and \(C\).
    1. Use the scalar triple product to determine
    2. Hence show that the distance from \(P\) to \(\Pi\) is twice the distance from \(O\) to \(\Pi\). [2]
    1. Determine a vector which is normal to \(\Pi\). [2]
    2. Hence determine, in terms of \(a\), \(b\) and \(c\) only, the distance from \(P\) to \(\Pi\). [3]
OCR Further Additional Pure 2018 September Q5
11 marks Hard +2.3
  1. You are given that \(N = \binom{p-1}{r}\), where \(p\) is a prime number and \(r\) is an integer such that \(1 \leq r \leq p - 1\). By considering the number \(N \times r!\), prove that \(N \equiv (-1)^r \pmod{p}\). [5]
  2. You are given that \(M = \binom{2p}{p}\), where \(p\) is an odd prime number. Prove that \(M \equiv 2 \pmod{p}\). [6]
OCR Further Additional Pure 2018 September Q6
12 marks Hard +2.3
A class of students is set the task of finding a group of functions, under composition of functions, of order 6. Student P suggests that this can be achieved by finding a function \(f\) for which \(f^6(x) = x\) and using this as a generator for the group.
  1. Explain why the suggestion by Student P might not work. [2]
Student Q observes that their class has already found a group of order 6 in a previous task; a group consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), under the operation of matrix multiplication.
  1. Explain why such a group is only possible if \(\det(\mathbf{M}) = 1\) or \(-1\). [2]
  2. Write down values of \(a\), \(b\), \(c\) and \(d\) that would give a suitable matrix \(\mathbf{M}\) for which \(\mathbf{M}^6 = \mathbf{I}\) and \(\det(\mathbf{M}) = 1\). [1]
Student Q believes that it is possible to construct a rational function \(f\) in the form \(f(x) = \frac{ax + b}{cx + d}\) so that the group of functions is isomorphic to the matrix group which is generated by the matrix \(\mathbf{M}\) of part (iii).
    1. Write down and simplify the function \(f\) that, according to Student Q, corresponds to \(\mathbf{M}\). [1]
    2. By calculating \(\mathbf{M}^2\), show that Student Q's suggestion does not work. [2]
    3. Find a different function \(f\) that will satisfy the requirements of the task. [4]
OCR Further Additional Pure 2018 September Q7
14 marks Challenging +1.8
The members of the family of the sequences \(\{u_n\}\) satisfy the recurrence relation $$u_{n+1} = 10u_n - u_{n-1} \text{ for } n \geq 1. \quad (*)$$
  1. Determine the general solution of (*). [3]
  2. The sequences \(\{a_n\}\) and \(\{b_n\}\) are members of this family of sequences, corresponding to the initial terms \(a_0 = 1\), \(a_1 = 5\) and \(b_0 = 0\), \(b_1 = 2\) respectively.
    1. Find the next two terms of each sequence. [1]
    2. Prove that, for all non-negative integers \(n\), \((a_n)^2 - 6(b_n)^2 = 1\). [8]
    3. Determine \(\lim_{n \to \infty} \frac{a_n}{b_n}\). [2]
OCR Further Pure Core 2 2018 December Q1
6 marks Easy -1.2
  1. The Argand diagram below shows the two points which represent two complex numbers, \(z_1\) and \(z_2\). \includegraphics{figure_1} On the copy of the diagram in the Printed Answer Booklet
    • draw an appropriate shape to illustrate the geometrical effect of adding \(z_1\) and \(z_2\),
    • indicate with a cross (\(\times\)) the location of the point representing the complex number \(z_1 + z_2\).
    [2]
  2. You are given that \(\arg z_3 = \frac{1}{4}\pi\) and \(\arg z_4 = \frac{3}{8}\pi\). In each part, sketch and label the points representing the numbers \(z_3\), \(z_4\) and \(z_3z_4\) on the diagram in the Printed Answer Booklet. You should join each point to the origin with a straight line.
    1. \(|z_3| = 1.5\) and \(|z_4| = 1.2\) [2]
    2. \(|z_3| = 0.7\) and \(|z_4| = 0.5\) [2]
OCR Further Pure Core 2 2018 December Q2
8 marks Moderate -0.8
In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the \(x\)-axis. A is the matrix which represents S.
  1. Write down A. [1]
  2. By considering the transformation represented by \(\mathbf{A}^{-1}\), determine the matrix \(\mathbf{A}^{-1}\). [2]
Matrix B is given by \(\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\). T is the transformation represented by B.
  1. Describe T. [1]
  2. Determine the matrix which represents the transformation S followed by T. [2]
  3. Demonstrate, by direct calculation, that \((\mathbf{BA})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}\). [2]
OCR Further Pure Core 2 2018 December Q3
6 marks Standard +0.8
In this question you must show detailed reasoning. Solve the equation \(2\cosh^2 x + 5\sinh x - 5 = 0\) giving each answer in the form \(\ln(p + q\sqrt{r})\) where \(p\) and \(q\) are rational numbers, and \(r\) is an integer, whose values are to be determined. [6]
OCR Further Pure Core 2 2018 December Q4
6 marks Standard +0.3
You are given that the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{2a-a^2}{3} & 0 \\ 0 & 0 & 1 \end{pmatrix}\), where \(a\) is a positive constant, represents the transformation R which is a reflection in 3-D.
  1. State the plane of reflection of R. [1]
  2. Determine the value of \(a\). [3]
  3. With reference to R explain why \(\mathbf{A}^2 = \mathbf{I}\), the \(3\times 3\) identity matrix. [2]
OCR Further Pure Core 2 2018 December Q5
7 marks Standard +0.8
  1. Find the shortest distance between the point \((-6, 4)\) and the line \(y = -0.75x + 7\). [2]
Two lines, \(l_1\) and \(l_2\), are given by $$l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -4 \end{pmatrix} \text{ and } l_2: \mathbf{r} = \begin{pmatrix} 11 \\ -1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}.$$
  1. Find the shortest distance between \(l_1\) and \(l_2\). [3]
  2. Hence determine the geometrical arrangement of \(l_1\) and \(l_2\). [2]
OCR Further Pure Core 2 2018 December Q6
9 marks Standard +0.3
Three matrices, A, B and C, are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}\) where \(a\) is a constant.
  1. Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
  2. Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]
For a certain value of \(a\), \(\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}\)
  1. Find
    [3]
OCR Further Pure Core 2 2018 December Q7
7 marks Challenging +1.8
C is the locus of numbers, \(z\), for which \(\text{Im}\left(\frac{z + 7i}{z - 24}\right) = \frac{1}{4}\). By writing \(z = x + iy\) give a complete description of the shape of C on an Argand diagram. [7]
OCR Further Pure Core 2 2018 December Q8
7 marks Challenging +1.8
\includegraphics{figure_8} The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the \(x\)-axis is rotated by \(2\pi\) radians around the \(x\)-axis to form a solid of revolution, S. Determine the exact volume of S. [7]
OCR Further Pure Core 2 2018 December Q9
5 marks Standard +0.8
  1. By using Euler's formula show that \(\cosh(\text{iz}) = \cos z\). [3]
  2. Hence, find, in logarithmic form, a root of the equation \(\cos z = 2\). [You may assume that \(\cos z = 2\) has complex roots.] [2]