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SPS SPS FM Pure 2021 May Q8
8 marks Hard +2.3
Let \(C = \sum_{r=0}^{20} \binom{20}{r} \cos(r\theta)\). Show that \(C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos(10\theta)\). [8]
SPS SPS FM Pure 2021 May Q9
12 marks Standard +0.8
During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by $$\frac{dy}{dt} = 0.3x - 0.2y \quad \text{and} \quad \frac{dz}{dt} = 0.2y + 0.1x$$ where \(x\), \(y\) and \(z\) are the amounts in kg of \(X\), \(Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substance \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.
  1. Show that \(x = Ae^{-0.4t}\), stating the value of \(A\). [3]
  2. Show that \(\frac{dx}{dt} + \frac{dy}{dt} + \frac{dz}{dt} = 0\). Comment on this result in the context of the industrial process. [4]
  3. Express \(y\) in terms of \(t\). [5]
SPS SPS SM Pure 2021 May Q1
7 marks Moderate -0.8
The function f is defined for all non-negative values of \(x\) by $$f(x) = 3 + \sqrt{x}.$$
  1. Evaluate \(f(169)\). [2]
  2. Find an expression for \(f^{-1}(x)\) in terms of \(x\). [2]
  3. On a single diagram sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\), indicating how the two graphs are related. [3]
SPS SPS SM Pure 2021 May Q2
4 marks Moderate -0.8
  1. Use the trapezium rule, with four strips each of width \(0.5\), to estimate the value of $$\int_0^2 e^{x^2} dx$$ giving your answer correct to 3 significant figures. [3]
  2. Explain how the trapezium rule could be used to obtain a more accurate estimate. [1]
SPS SPS SM Pure 2021 May Q3
6 marks Moderate -0.8
Vector \(\mathbf{v} = a\mathbf{i} + 0.6\mathbf{j}\), where \(a\) is a constant.
  1. Given that the direction of \(\mathbf{v}\) is \(45°\), state the value of \(a\). [1]
  2. Given instead that \(\mathbf{v}\) is parallel to \(8\mathbf{i} + 3\mathbf{j}\), find the value of \(a\). [2]
  3. Given instead that \(\mathbf{v}\) is a unit vector, find the possible values of \(a\). [3]
SPS SPS SM Pure 2021 May Q4
3 marks Standard +0.3
Prove that \(\sqrt{2}\cos(2\theta + 45°) \equiv \cos^2\theta - 2\sin\theta\cos\theta - \sin^2\theta\), where \(\theta\) is measured in degrees. [3]
SPS SPS SM Pure 2021 May Q5
8 marks Standard +0.3
  1. Show that \(\sqrt{\frac{1-x}{1+x}} \approx 1 - x + \frac{1}{2}x^2\), for \(|x| < 1\). [5]
  2. By taking \(x = \frac{2}{7}\), show that \(\sqrt{5} \approx \frac{111}{49}\). [3]
SPS SPS SM Pure 2021 May Q6
4 marks Challenging +1.2
Shona makes the following claim. "\(n\) is an even positive integer greater than \(2 \Rightarrow 2^n - 1\) is not prime" Prove that Shona's claim is true. [4]
SPS SPS SM Pure 2021 May Q7
11 marks Standard +0.3
A curve has parametric equations $$x = 2\sin t, \quad y = \cos 2t + 2\sin t$$ for \(-\frac{1}{2}\pi \leqslant t \leqslant \frac{1}{2}\pi\).
  1. Show that \(\frac{dy}{dx} = 1 - 2\sin t\) and hence find the coordinates of the stationary point. [5]
  2. Find the cartesian equation of the curve. [3]
  3. State the set of values that \(x\) can take and hence sketch the curve. [3]
SPS SPS SM Pure 2021 May Q8
12 marks Challenging +1.2
In this question you must show detailed reasoning. The \(n\)th term of a geometric progression is denoted by \(g_n\) and the \(n\)th term of an arithmetic progression is denoted by \(a_n\). It is given that \(g_1 = a_1 = 1 + \sqrt{5}\), \(g_2 = a_2\) and \(g_3 + a_3 = 0\). Given also that the geometric progression is convergent, show that its sum to infinity is \(4 + 2\sqrt{5}\). [12]
SPS SPS SM Pure 2021 May Q9
10 marks Challenging +1.8
In this question you must show detailed reasoning. \includegraphics{figure_9} The diagram shows the curve \(y = \frac{4\cos 2x}{3 - \sin 2x}\) for \(x > 0\), and the normal to the curve at the point \((\frac{1}{4}\pi, 0)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac{2}{3} + \frac{1}{128}\pi^2\). [10]
SPS SPS SM Pure 2021 May Q1
5 marks Standard +0.3
  1. For a small angle \(\theta\), where \(\theta\) is in radians, show that \(2\cos\theta + (1 - \tan\theta)^2 \approx 3 - 2\theta\). [3]
  2. Hence determine an approximate solution to \(2\cos\theta + (1 - \tan\theta)^2 = 28\sin\theta\). [2]
SPS SPS SM Pure 2021 May Q2
3 marks Easy -1.2
Solve the equation \(|2x - 1| = |x + 3|\). [3]
SPS SPS SM Pure 2021 May Q3
6 marks Standard +0.3
Solve the equation \(2^{4x-1} = 3^{5-2x}\), giving your answer in the form \(x = \frac{\log_{10} a}{\log_{10} b}\). [6]
SPS SPS SM Pure 2021 May Q4
3 marks Moderate -0.8
A sequence of transformations maps the curve \(y = e^x\) to the curve \(y = e^{2x+3}\). Give details of these transformations. [3]
SPS SPS SM Pure 2021 May Q5
8 marks Standard +0.3
A curve has equation \(x^3 - 3x^2y + y^2 + 1 = 0\).
  1. Show that \(\frac{dy}{dx} = \frac{6xy - 3x^2}{2y - 3x^2}\). [4]
  2. Find the equation of the normal to the curve at the point \((1, 2)\). [4]
SPS SPS SM Pure 2021 May Q6
7 marks Challenging +1.2
In this question you must show detailed reasoning. A circle touches the lines \(y = \frac{1}{2}x\) and \(y = 2x\) at \((6, 3)\) and \((3, 6)\) respectively. \includegraphics{figure_6} Find the equation of the circle. [7]
SPS SPS SM Pure 2021 May Q7
13 marks Challenging +1.2
It is given that there is exactly one value of \(x\), where \(0 < x < \pi\), that satisfies the equation $$3\tan 2x - 8\tan x = 4.$$
  1. Show that \(t = \sqrt[3]{\frac{1}{2} + \frac{1}{3}t - \frac{1}{3}t^2}\), where \(t = \tan x\). [3]
  2. Show by calculation that the value of \(t\) satisfying the equation in part (i) lies between 0.7 and 0.8. [2]
  3. Use an iterative process based on the equation in part (i) to find the value of \(t\) correct to 4 significant figures. Use a starting value of 0.75 and show the result of each iteration. [3]
  4. Solve the equation \(3\tan 4y - 8\tan 2y = 4\) for \(0 < y < \frac{1}{4}\pi\). [2]
SPS SPS SM Pure 2021 May Q8
9 marks Challenging +1.2
Find the general solution of the differential equation $$(2x^3 - 3x^2 - 11x + 6)\frac{dy}{dx} = y(20x - 35).$$ Give your answer in the form \(y = f(x)\). [9]
SPS SPS SM Pure 2021 May Q9
14 marks Challenging +1.3
  1. Show that the two non-stationary points of inflection on the curve \(y = \ln(1 + 4x^2)\) are at \(x = \pm\frac{1}{2}\). [6]
\includegraphics{figure_9} The diagram shows the curve \(y = \ln(1 + 4x^2)\). The shaded region is bounded by the curve and a line parallel to the \(x\)-axis which meets the curve where \(x = \frac{1}{2}\) and \(x = -\frac{1}{2}\).
  1. Show that the area of the shaded region is given by $$\int_0^{\ln 2} \sqrt{e^y - 1} \, dy.$$ [3]
  2. Show that the substitution \(e^y = \sec^2\theta\) transforms the integral in part (ii) to \(\int_0^{\frac{\pi}{4}} 2\tan^2\theta \, d\theta\). [2]
  3. Hence find the exact area of the shaded region. [3]
SPS SPS FM Mechanics 2021 January Q1
3 marks Moderate -0.5
A disc, of mass \(m\) and radius \(r\), rotates about an axis through its centre, perpendicular to the plane face of the disc. The angular speed of the disc is \(\omega\). A possible model for the kinetic energy \(E\) of the disc is $$E = km^ar^b\omega^c$$ where \(a\), \(b\) and \(c\) are constants and \(k\) is a dimensionless constant. Find the values of \(a\), \(b\) and \(c\). [3 marks]
SPS SPS FM Mechanics 2021 January Q2
11 marks Standard +0.8
The triangular region shown below is rotated through \(360°\) around the \(x\)-axis, to form a solid cone. \includegraphics{figure_1} The coordinates of the vertices of the triangle are \((0, 0)\), \((8, 0)\) and \((0, 4)\). All units are in centimetres.
  1. State an assumption that you should make about the cone in order to find the position of its centre of mass. [1 mark]
  2. Using integration, prove that the centre of mass of the cone is \(2\)cm from its plane face. [5 marks]
  3. The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding.
    1. Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree. [2 marks]
    2. Find the range of possible values for the coefficient of friction between the cone and the board. [3 marks]
SPS SPS FM Mechanics 2021 January Q3
8 marks Standard +0.8
\includegraphics{figure_2} Figure 1 represents the plan of part of a smooth horizontal floor, where \(W_1\) and \(W_2\) are two fixed parallel vertical walls. The walls are \(3\) metres apart. A particle lies at rest at a point \(O\) on the floor between the two walls, where the point \(O\) is \(d\) metres, \(0 < d \leq 3\), from \(W_1\). At time \(t = 0\), the particle is projected from \(O\) towards \(W_1\) with speed \(u\text{ms}^{-1}\) in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is \(\frac{2}{3}\). The particle returns to \(O\) at time \(t = T\) seconds, having bounced off each wall once.
  1. Show that \(T = \frac{45 - 5d}{4u}\). [6]
  2. The value of \(u\) is fixed, the particle still hits each wall once but the value of \(d\) can now vary. Find the least possible value of \(T\), giving your answer in terms of \(u\). You must give a reason for your answer. [2]
SPS SPS FM Mechanics 2021 January Q4
12 marks Standard +0.3
A car of mass \(600\)kg pulls a trailer of mass \(150\)kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude \(200\)N. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car is modelled as a force of magnitude \((200 + \lambda v)\)N, where \(\lambda\) is a constant. When the engine of the car is working at a constant rate of \(15\)kW, the car is moving at a constant speed of \(25\text{ms}^{-1}\).
  1. Show that \(\lambda = 8\). [4]
  2. Later on, the car is pulling the trailer up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = \frac{1}{15}\). The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude \(200\)N at all times. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude \((200 + 8v)\)N. The engine of the car is again working at a constant rate of \(15\)kW. When \(v = 10\), the towbar breaks. The trailer comes to instantaneous rest after moving a distance \(d\) metres up the road from the point where the towbar broke. Find the acceleration of the car immediately after the towbar breaks. [4]
  3. Use the work-energy principle to find the value of \(d\). [4]
SPS SPS FM Mechanics 2021 January Q5
6 marks Standard +0.3
\includegraphics{figure_3} A hemispherical shell of radius \(a\) is fixed with its rim uppermost and horizontal. A small bead, \(B\), is moving with constant angular speed, \(\omega\), in a horizontal circle on the smooth inner surface of the shell. The centre of the path of \(B\) is at a distance \(\frac{1}{4}a\) vertically below the level of the rim of the hemisphere, as shown in Figure 1. Find the magnitude of \(\omega\), giving your answer in terms of \(a\) and \(g\). [6]