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OCR Further Mechanics AS Specimen Q2
7 marks Standard +0.3
\includegraphics{figure_2} A smooth wire is shaped into a circle of centre \(O\) and radius 0.8 m. The wire is fixed in a vertical plane. A small bead \(P\) of mass 0.03 kg is threaded on the wire and is projected along the wire from the highest point with a speed of \(4.2 \, \text{m s}^{-1}\). When \(OP\) makes an angle \(\theta\) with the upward vertical the speed of \(P\) is \(v \, \text{m s}^{-1}\) (see diagram).
  1. Show that \(v^2 = 33.32 - 15.68\cos\theta\). [4]
  2. Prove that the bead is never at rest. [1]
  3. Find the maximum value of \(v\). [2]
OCR Further Mechanics AS Specimen Q3
9 marks Standard +0.3
  1. Write down the dimension of density. [1]
The workings of an oil pump consist of a right, solid cylinder which is partially submerged in oil. The cylinder is free to oscillate along its central axis which is vertical. If the base area of the pump is \(0.4 \, \text{m}^2\) and the density of the oil is \(920 \, \text{kg m}^{-3}\) then the period of oscillation of the pump is 0.7 s. A student assumes that the period of oscillation of the pump is dependent only on the density of the oil, \(\rho\), the acceleration due to gravity, \(g\), and the surface area, \(A\), of the circular base of the pump. The student attempts to test this assumption by stating that the period of oscillation, \(T\), is given by \(T = C\rho^{\alpha} g^{\beta} A^{\gamma}\) where \(C\) is a dimensionless constant.
  1. Use dimensional analysis to find the values of \(\alpha\), \(\beta\) and \(\gamma\). [4]
  2. Hence give the value of \(C\) to 3 significant figures. [2]
  3. Comment, with justification, on the assumption made by the student that the formula for the period of oscillation of the pump was dependent on only \(\rho\), \(g\) and \(A\). [2]
OCR Further Mechanics AS Specimen Q4
10 marks Standard +0.8
A car of mass 1250 kg experiences a resistance to its motion of magnitude \(kv^2\) N, where \(k\) is a constant and \(v \, \text{m s}^{-1}\) is the car's speed. The car travels in a straight line along a horizontal road with its engine working at a constant rate of \(P\) W. At a point \(A\) on the road the car's speed is \(15 \, \text{m s}^{-1}\) and it has an acceleration of magnitude \(0.54 \, \text{m s}^{-2}\). At a point \(B\) on the road the car's speed is \(20 \, \text{m s}^{-1}\) and it has an acceleration of magnitude \(0.3 \, \text{m s}^{-2}\).
  1. Find the values of \(k\) and \(P\). [7]
The power is increased to 15 kW.
  1. Calculate the maximum steady speed of the car on a straight horizontal road. [3]
OCR Further Mechanics AS Specimen Q5
15 marks Standard +0.8
\includegraphics{figure_5} The masses of two spheres \(A\) and \(B\) are \(3m\) kg and \(m\) kg respectively. The spheres are moving towards each other with constant speeds \(2u \, \text{m s}^{-1}\) and \(u \, \text{m s}^{-1}\) respectively along the same straight line towards each other on a smooth horizontal surface (see diagram). The two spheres collide and the coefficient of restitution between the spheres is \(e\). After colliding, \(A\) and \(B\) both move in the same direction with speeds \(v \, \text{m s}^{-1}\) and \(w \, \text{m s}^{-1}\), respectively.
  1. Find an expression for \(v\) in terms of \(e\) and \(u\). [6]
  2. Write down unsimplified expressions in terms of \(e\) and \(u\) for
    1. the total kinetic energy of the spheres before the collision, [1]
    2. the total kinetic energy of the spheres after the collision. [2]
  3. Given that the total kinetic energy of the spheres after the collision is \(\lambda\) times the total kinetic energy before the collision, show that $$\lambda = \frac{27e^2 + 25}{52}.$$ [3]
  4. Comment on the cases when
    1. \(\lambda = 1\),
    2. \(\lambda = \frac{25}{52}\). [3]
OCR Further Mechanics AS Specimen Q6
13 marks Challenging +1.2
\includegraphics{figure_6} The fixed points \(A\), \(B\) and \(C\) are in a vertical line with \(A\) above \(B\) and \(B\) above \(C\). A particle \(P\) of mass 2.5 kg is joined to \(A\), to \(B\) and to a particle \(Q\) of mass 2 kg, by three light rods where the length of rod \(AP\) is 1.5 m and the length of rod \(PQ\) is 0.75 m. Particle \(P\) moves in a horizontal circle with centre \(B\). Particle \(Q\) moves in a horizontal circle with centre \(C\) at the same constant angular speed \(\omega\) as \(P\), in such a way that \(A\), \(B\), \(P\) and \(Q\) are coplanar. The rod \(AP\) makes an angle of \(60°\) with the downward vertical, rod \(PQ\) makes an angle of \(30°\) with the downward vertical and rod \(BP\) is horizontal (see diagram).
  1. Find the tension in the rod \(PQ\). [2]
  2. Find \(\omega\). [3]
  3. Find the speed of \(P\). [1]
  4. Find the tension in the rod \(AP\). [3]
  5. Hence find the magnitude of the force in rod \(BP\). Decide whether this rod is under tension or compression. [4]
OCR Further Pure Core 1 2021 November Q1
6 marks Moderate -0.8
  1. Sketch on a single Argand diagram the loci given by
    1. \(|z - 1 + 2\mathrm{i}| = 3\), [2]
    2. \(|z + 1| = |z - 2|\). [2]
  2. Indicate, by shading, the region of the Argand diagram for which \(|z - 1 + 2\mathrm{i}| \leqslant 3\) and \(|z + 1| \leqslant |z - 2|\). [2]
OCR Further Pure Core 1 2021 November Q2
8 marks Standard +0.3
You are given that \(\mathrm{f}(x) = \tan^{-1}(1 + x)\).
    1. Find the value of \(\mathrm{f}(0)\). [1]
    2. Determine the value of \(\mathrm{f}'(0)\). [2]
    3. Show that \(\mathrm{f}''(0) = -\frac{1}{2}\). [3]
  2. Hence find the Maclaurin series for \(\mathrm{f}(x)\) up to and including the term in \(x^2\). [2]
OCR Further Pure Core 1 2021 November Q3
8 marks Standard +0.8
A function \(\mathrm{f}(z)\) is defined on all complex numbers \(z\) by \(\mathrm{f}(z) = z^3 - 3z^2 + kz - 5\) where \(k\) is a real constant. The roots of the equation \(\mathrm{f}(z) = 0\) are \(\alpha\), \(\beta\) and \(\gamma\). You are given that \(\alpha^2 + \beta^2 + \gamma^2 = -5\).
  1. Explain why \(\mathrm{f}(z) = 0\) has only one real root. [3]
  2. Find the value of \(k\). [3]
  3. Find a cubic equation with integer coefficients that has roots \(\frac{1}{\alpha}\), \(\frac{1}{\beta}\) and \(\frac{1}{\gamma}\). [2]
OCR Further Pure Core 1 2021 November Q4
11 marks Standard +0.3
Points \(A\), \(B\) and \(C\) have coordinates \((4, 2, 0)\), \((1, 5, 3)\) and \((1, 4, -2)\) respectively. The line \(l\) passes through \(A\) and \(B\).
  1. Find a cartesian equation for \(l\). [3]
\(M\) is the point on \(l\) that is closest to \(C\).
  1. Find the coordinates of \(M\). [4]
  2. Find the exact area of the triangle \(ABC\). [4]
OCR Further Pure Core 1 2021 November Q5
4 marks Standard +0.8
Use de Moivre's theorem to find the constants \(A\), \(B\) and \(C\) in the identity \(\sin^3 \theta \equiv A \sin \theta + B \sin 3\theta + C \sin 5\theta\). [4]
OCR Further Pure Core 1 2021 November Q6
3 marks Standard +0.8
\(O\) is the origin of a coordinate system whose units are cm. The points \(A\), \(B\), \(C\) and \(D\) have coordinates \((1, 0)\), \((1, 4)\), \((6, 9)\) and \((0, 9)\) respectively. The arc \(BC\) is part of the curve with equation \(x^2 + (y - 10)^2 = 37\). The closed shape \(OABCD\) is formed, in turn, from the line segments \(OA\) and \(AB\), the arc \(BC\) and the line segments \(CD\) and \(DO\) (see diagram). A funnel can be modelled by rotating \(OABCD\) by \(2\pi\) radians about the \(y\)-axis. \includegraphics{figure_6} Find the volume of the funnel according to the model. [3]
OCR Further Pure Core 1 2021 November Q7
9 marks Standard +0.8
The diagram below shows the curve with polar equation \(r = \sin 3\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{3}\pi\). \includegraphics{figure_7}
  1. Find the values of \(\theta\) at the pole. [1]
  2. Find the polar coordinates of the point on the curve where \(r\) takes its maximum value. [2]
  3. In this question you must show detailed reasoning. Find the exact area enclosed by the curve. [4]
  4. Given that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\), find a cartesian equation for the curve. [2]
OCR Further Pure Core 1 2021 November Q8
8 marks Standard +0.3
You are given that \(\mathrm{f}(x) = 4 \sinh x + 3 \cosh x\).
  1. Show that the curve \(y = \mathrm{f}(x)\) has no turning points. [3]
  2. Determine the exact solution of the equation \(\mathrm{f}(x) = 5\). [5]
OCR Further Pure Core 1 2021 November Q9
5 marks Standard +0.3
You are given that the matrix \(\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\) represents a transformation T.
  1. You are given that the line with equation \(y = kx\) is invariant under T. Determine the value of \(k\). [4]
  2. Determine whether the line with equation \(y = kx\) in part (a) is a line of invariant points under T. [1]
OCR Further Pure Core 1 2021 November Q10
8 marks Challenging +1.2
Using an algebraic method, determine the least value of \(n\) for which \(\sum_{r=1}^{n} \frac{1}{(2r-1)(2r+1)} \geqslant 0.49\). [8]
OCR Further Pure Core 1 2021 November Q11
5 marks Standard +0.3
The displacement of a door from its equilibrium (closed) position is measured by the angle, \(\theta\) radians, which the door makes with its closed position. The door can swing either side of the equilibrium position so that \(\theta\) can take positive and negative values. The door is released from rest from an open position at time \(t = 0\). A proposed differential equation to model the motion of the door for \(t \geqslant 0\) is $$\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \lambda \frac{\mathrm{d}\theta}{\mathrm{d}t} + 3\theta = 0$$ where \(\lambda\) is a constant and \(\lambda \geqslant 0\).
    1. According to the model, for what value of \(\lambda\) will the motion of the door be simple harmonic? [1]
    2. Explain briefly why modelling the motion of the door as simple harmonic is unlikely to be realistic. [1]
  2. Find the range of values of \(\lambda\) for which the model predicts that the door will never pass through the equilibrium position. [2]
  3. Sketch a possible graph of \(\theta\) against \(t\) when \(\lambda\) lies outside the range found in part (b) but the motion is not simple harmonic. [1]
OCR Further Pure Core 2 2024 June Q1
5 marks Moderate -0.8
  1. Use the method of differences to show that \(\sum_{r=1}^{n}\left(\frac{1}{r} - \frac{1}{r+1}\right) = 1 - \frac{1}{n+1}\). [1]
  2. Hence determine the following sums.
    1. \(\sum_{r=1}^{90}\frac{1}{r} - \frac{1}{r+1}\) [1]
    2. \(\sum_{r=100}^{\infty}\frac{1}{r} - \frac{1}{r+1}\) [3]
OCR Further Pure Core 2 2024 June Q2
6 marks Moderate -0.8
In this question you must show detailed reasoning.
  1. Solve the equation \(x^2 - 6x + 58 = 0\). Give your solutions in the form \(a + bi\) where \(a\) and \(b\) are real numbers. [3]
  2. Determine, in exact form, \(\arg(-10 + (5\sqrt{12})i)^3\). [3]
OCR Further Pure Core 2 2024 June Q3
7 marks Standard +0.3
Matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A} = \begin{pmatrix} 4 & -3 \\ -2 & 2 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 3 & -5 \\ 0 & 1 \end{pmatrix}\).
  1. Find \(2\mathbf{A} - 4\mathbf{B}\). [2]
  2. Write down the matrix \(\mathbf{C}\) such that \(\mathbf{A}\mathbf{C} = 2\mathbf{A}\). [1]
  3. Find the value of \(\det \mathbf{A}\). [1]
  4. In this question you must show detailed reasoning. Use \(\mathbf{A}^{-1}\) to solve the equations \(4x - 3y = 7\) and \(-2x + 2y = 9\). [3]
OCR Further Pure Core 2 2024 June Q4
5 marks Challenging +1.2
In this question you must show detailed reasoning. The series \(S\) is defined as being the sum of the squares of all positive odd integers from \(1^2\) to \(779^2\). Determine the value of \(S\). [5]
OCR Further Pure Core 2 2024 June Q5
6 marks Challenging +1.2
Vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\), are given by \(\mathbf{a} = \mathbf{i} + (1-p)\mathbf{j} + (p+2)\mathbf{k}\), \(\mathbf{b} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}\) and \(\mathbf{c} = \mathbf{i} + 14\mathbf{j} + (p-3)\mathbf{k}\) where \(p\) is a constant. You are given that \(\mathbf{a} \times \mathbf{b}\) is perpendicular to \(\mathbf{c}\). Determine the possible values of \(p\). [6]
OCR Further Pure Core 2 2024 June Q6
11 marks Challenging +1.8
In polar coordinates, the equation of a curve, \(C\), is \(r = 6\sin(2\theta)\sinh\left(\frac{1}{3}\theta\right)\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). The pole of the polar coordinate system corresponds to the origin of the cartesian system and the initial line corresponds to the positive \(x\)-axis.
  1. Explain how you can tell that \(C\) comprises a single loop in the first quadrant, passing through the pole. [3]
The incomplete table below shows values of \(r\) for various values of \(\theta\).
\(\theta\)0\(\frac{1}{12}\pi\)\(\frac{1}{6}\pi\)\(\frac{1}{4}\pi\)\(\frac{1}{3}\pi\)\(\frac{5}{12}\pi\)\(\frac{1}{2}\pi\)
\(r\)00.2621.851
  1. Use the copy of the table and the polar coordinate system diagram given in the Printed Answer Booklet to complete the table and sketch \(C\). [3]
The point on \(C\) which is furthest away from the pole is denoted by \(A\) and the value of \(\theta\) at \(A\) is denoted by \(\phi\).
  1. Show that \(\phi\) satisfies the equation \(\phi = \frac{3}{4}\ln\left(\frac{6-\tan 2\phi}{6+\tan 2\phi}\right)\) [4]
  2. You are given that the relevant solution of the equation given in part (c) is \(\phi = 1.0207\) correct to 5 significant figures. Find the distance from \(A\) to the pole. Give your answer correct to 3 significant figures. [1]
OCR Further Pure Core 2 2024 June Q7
10 marks Challenging +1.8
  1. Express \(17\cosh x - 15\sinh x\) in the form \(e^{-x}(ae^{bx} + c)\) where \(a\), \(b\) and \(c\) are integers to be determined. [3]
A function is defined by \(f(x) = \frac{1}{\sqrt{17\cosh x - 15\sinh x}}\). The region bounded by the curve \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = \ln 3\) is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
  1. In this question you must show detailed reasoning. Use a suitable substitution, together with known results from the formula book, to show that the volume of \(S\) is given by \(k\pi\tan^{-1} q\) where \(k\) and \(q\) are rational numbers to be determined. [7]
OCR Further Pure Core 2 2024 June Q8
13 marks Standard +0.8
A children's play centre has two rooms, a room full of bouncy castles and a room full of ball pits. At any given instant, each child in the centre is playing either on the bouncy castles or in the ball pits. Each child can see one room from the other room and can decide to change freely between the two rooms. It is assumed that such changes happen instantaneously. The number of children playing on the bouncy castles at time \(t\) hours, is denoted by \(C\) and the corresponding number of children playing in the ball pits is \(P\). Because the number of children is large for most of the time, \(C\) and \(P\) are modelled as being continuous. When there is a different number of children in each room, some children will move from the room with more children to the room with fewer children. A researcher therefore decides to model \(C\) and \(P\) with the following coupled differential equations. $$\frac{dP}{dt} = \alpha(P-C) + \gamma t$$ $$\frac{dC}{dt} = \alpha(C-P)$$
  1. Explain why \(\alpha\) must be negative. [1]
After examining data, the researcher chooses \(\alpha = -2\) and \(\gamma = 32\).
  1. Show that \(P\) satisfies the second order differential equation \(\frac{d^2P}{dt^2} + 4\frac{dP}{dt} = 64t + 32\). [2]
    1. Find the complementary function for the differential equation from part (b). [1]
    2. Explain why a particular integral of the form \(P = at + b\) will not work in this situation. [1]
    3. Using a particular integral of the form \(P = at^2 + bt\), find the general solution of the differential equation from part (b). [3]
At a certain time there are 55 children playing in the ball pits and 24 children per hour are arriving at the ball pits.
  1. Use the model, starting from this time, to estimate the number of children in the ball pits 30 minutes later. [4]
  2. Explain why the model becomes unreliable as \(t\) gets very large. [1]
OCR Further Pure Core 2 2024 June Q9
12 marks Challenging +1.2
In this question, the argument of a complex number is defined as being in the range \([0, 2\pi)\). You are given that \(\omega_k\), where \(k = 0, 1, 2, ..., n-1\), are the \(n\) \(n\)th roots of unity for some integer \(n\), \(n \geqslant 3\), and that these are given in order of increasing argument (so that \(\omega_0 = 1\)).
  1. With the help of a diagram explain why \(\omega_k = (\omega_1)^k\) for \(k = 2, ..., n-1\). [3]
  2. Using the identity given in part (a), show that \(\sum_{k=0}^{n-1}\omega_k = 0\). [2]
  3. Show that if \(z\) is a complex number then \(z + z^* = 2\text{Re}(z)\). [1]
  4. Using the results from parts (b) and (c) show that \(\sum_{k=0}^{n-1}\text{Re}(\omega_k) = 0\). [1]
  5. With the help of a diagram explain why \(\text{Re}(\omega_k) = \text{Re}(\omega_{n-k})\) for \(k = 1, 2, ..., n-1\). [1]
You should now consider the case when \(n = 5\).
    1. Use parts (d) and (e) to deduce that \(\cos\frac{4\pi}{5} = a + b\cos\frac{2\pi}{5}\), for some rational constants \(a\) and \(b\). [2]
    2. Hence determine the exact value of \(\cos\frac{2\pi}{5}\). [2]