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SPS SPS FM Pure 2025 September Q3
5 marks Standard +0.3
A finite region is bounded by the curve with equation \(y = x + x^{\frac{3}{2}}\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) This region is rotated through \(2\pi\) radians about the \(x\)-axis. Show that the volume generated is \(\pi\left(a\sqrt{2} + b\right)\), where \(a\) and \(b\) are rational numbers to be determined. [5 marks]
SPS SPS FM Pure 2025 September Q4
13 marks Standard +0.8
The curve \(C\) has parametric equations $$x = 2\cos t, \quad y = \sqrt{3}\cos 2t, \quad 0 \leq t \leq \pi$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [2]
The point \(P\) lies on \(C\) where \(t = \frac{2\pi}{3}\) The line \(l\) is the normal to \(C\) at \(P\).
  1. Show that an equation for \(l\) is $$2x - 2\sqrt{3}y - 1 = 0$$ [5]
The line \(l\) intersects the curve \(C\) again at the point \(Q\).
  1. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers. [6]
SPS SPS FM Pure 2025 September Q5
6 marks Standard +0.3
  1. On the Argand diagram below, sketch the locus, \(L\), of points satisfying the equation $$\arg(z + i) = \frac{\pi}{6}$$ [2 marks] \includegraphics{figure_5}
  2. \(z_1\) is a point on \(L\) such that \(|z|\) is a minimum. Find the exact value of \(z_1\) in the form \(a + bi\) [4 marks]
SPS SPS FM Pure 2025 September Q8
7 marks Standard +0.8
A population of meerkats is being studied. The population is modelled by the differential equation $$\frac{dP}{dt} = \frac{1}{22}P(11 - 2P), \quad t \geq 0, \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, determine the time taken, in years, for this population of meerkats to double. [7]
SPS SPS FM Pure 2025 September Q9
18 marks Standard +0.3
A curve \(C\) has equation \(y = f(x)\) where $$f(x) = x + 2\ln(e - x)$$
    1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left(\frac{e}{2-e}\right)x + 2$$ [6 marks]
    2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer. [3 marks]
  1. The equation \(f(x) = 0\) has one positive root, \(\alpha\).
    1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer. [3 marks]
    2. Show that the roots of \(f(x) = 0\) satisfy the equation $$x = e - e^{\frac{x}{2}}$$ [2 marks]
    3. Use the recurrence relation $$x_{n+1} = e - e^{\frac{x_n}{2}}$$ with \(x_1 = 2\) to find the values of \(x_2\) and \(x_3\) giving your answers to three decimal places. [2 marks]
    4. Figure 1 below shows a sketch of the graphs of \(y = e - e^{\frac{x}{2}}\) and \(y = x\), and the position of \(x_1\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\) and \(x_3\) on the \(x\)-axis. [2 marks] \includegraphics{figure_9}
SPS SPS FM Pure 2026 November Q1
4 marks Moderate -0.3
The complex number \(z\) satisfies the equation \(z + 2iz^* + 1 - 4i = 0\). You are given that \(z = x + iy\), where \(x\) and \(y\) are real numbers. Determine the values of \(x\) and \(y\). [4]
SPS SPS FM Pure 2026 November Q2
6 marks Moderate -0.8
Prove by induction that, for all positive integers \(n\), $$\sum_{r=1}^{n}(2r-1)^2 = \frac{1}{3}n(4n^2-1)$$ [6]
SPS SPS FM Pure 2026 November Q3
9 marks Challenging +1.2
The figure below shows the curve with cartesian equation \((x^2 + y^2)^2 = xy\). \includegraphics{figure_3}
  1. Show that the polar equation of the curve is \(r^2 = a \sin b\theta\), where \(a\) and \(b\) are positive constants to be determined. [3]
  2. Determine the exact maximum value of \(r\). [2]
  3. Determine the area enclosed by one of the loops. [4]
SPS SPS FM Pure 2026 November Q4
9 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. The curves with equations $$y = \frac{3}{4}\sinh x \text{ and } y = \tanh x + \frac{1}{5}$$ intersect at just one point \(P\)
    1. Use algebra to show that the \(x\) coordinate of \(P\) satisfies the equation $$15e^{4x} - 48e^{3x} + 32e^x - 15 = 0$$ [3]
    2. Show that \(e^x = 3\) is a solution of this equation. [1]
    3. Hence state the exact coordinates of \(P\). [1]
  2. Show that $$\int_{-4}^{0} \frac{e^x}{x^2} dx = e^{-\frac{1}{4}}$$ [4]
SPS SPS FM Pure 2026 November Q5
5 marks Standard +0.8
Use the method of differences to prove that for \(n > 2\) $$\sum_{r=2}^{n} \frac{4}{r^2-1} = \frac{(pn+q)(n-1)}{n(n+1)}$$ where \(p\) and \(q\) are constants to be determined. [5]
SPS SPS FM Pure 2026 November Q6
10 marks Standard +0.3
  1. \(z_1 = a + bi\) and \(z_2 = c + di\) where \(a\), \(b\), \(c\) and \(d\) are real constants. Given that
    • \(b > d\)
    • \(z_1 + z_2\) is real
    • \(|z_1| = \sqrt{13}\)
    • \(|z_2| = 5\)
    • \(\text{Re}(z_2 - z_1) = 2\)
    show that \(a = 2\) and determine the value of each of \(b\), \(c\) and \(d\) [5]
    1. On the same Argand diagram
      showing the coordinates of any points of intersection with the axes. [2]
    2. Determine the range of possible values of \(|z - w|\) [3]
SPS SPS FM Pure 2026 November Q7
4 marks Standard +0.8
In this question you must show detailed reasoning. Evaluate \(\int_0^{\frac{1}{2}} \frac{2}{x^2 - x + 1} dx\). Give your answer in exact form. [4]
SPS SPS FM Pure 2026 November Q8
12 marks Challenging +1.3
In this question you must show detailed reasoning. The diagram shows the curve with equation \(y = \frac{x + 3}{\sqrt{x^2 + 9}}\). \includegraphics{figure_8} The region R, shown shaded in the diagram, is bounded by the curve, the \(x\)-axis, the \(y\)-axis, and the line \(x = 4\).
  1. Determine the area of R. Give your answer in the form \(p + \ln q\) where \(p\) and \(q\) are integers to be determined. [6]
The region R is rotated through \(2\pi\) radians about the \(x\)-axis.
  1. Determine the volume of the solid of revolution formed. Give your answer in the form \(\pi\left(a + b\ln\left(\frac{c}{d}\right)\right)\) where \(a\), \(b\), \(c\) and \(d\) are integers to be determined. [6]
SPS SPS FM Pure 2026 November Q9
8 marks Challenging +1.3
Given that $$y = \cos x \sinh x \quad x \in \mathbb{R}$$
  1. show that $$\frac{d^4y}{dx^4} = ky$$ where \(k\) is a constant to be determined. [5]
  2. Hence determine the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in simplest form. [3]
SPS SPS FM Pure 2026 November Q10
8 marks Challenging +1.2
The quartic equation $$2x^4 + Ax^3 - Ax^2 - 5x + 6 = 0$$ where \(A\) is a real constant, has roots \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\)
  1. Determine the value of $$\frac{3}{\alpha} + \frac{3}{\beta} + \frac{3}{\gamma} + \frac{3}{\delta}$$ [3]
Given that \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2 = -\frac{3}{4}\)
  1. determine the possible values of \(A\) [5]
SPS SPS FM Mechanics 2026 January Q1
8 marks Standard +0.3
A van of mass 600 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{16}\). The resistance to motion of the van from non-gravitational forces has constant magnitude \(R\) newtons. When the van is moving at a constant speed of 20 m s\(^{-1}\), the van's engine is working at a constant rate of 25 kW.
  1. Find the value of \(R\). [4]
The power developed by the van's engine is now increased to 30 kW. The resistance to motion from non-gravitational forces is unchanged. At the instant when the van is moving up the road at 20 m s\(^{-1}\), the acceleration of the van is \(a\) m s\(^{-2}\).
  1. Find the value of \(a\). [4]
SPS SPS FM Mechanics 2026 January Q2
12 marks Standard +0.3
\includegraphics{figure_2} The uniform L-shaped lamina \(OABCDE\), shown in Figure 2, is made from two identical rectangles. Each rectangle is 4 metres long and \(a\) metres wide. Giving each answer in terms of \(a\), find the distance of the centre of mass of the lamina from
  1. \(OE\). [4]
  2. \(OA\). [4]
The lamina is freely suspended from \(O\) and hangs in equilibrium with \(OE\) at an angle \(\theta\) to the downward vertical through \(O\), where \(\tan \theta = \frac{4}{3}\).
  1. Find the value of \(a\). [4]
SPS SPS FM Mechanics 2026 January Q3
8 marks Challenging +1.2
\includegraphics{figure_3} A light elastic string has natural length \(8a\) and modulus of elasticity \(5mg\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(AP = BP = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(AB\) it has speed \(\sqrt{80ag}\).
  1. Find \(L\) in terms of \(a\). [5]
  2. Find the initial acceleration of \(P\) in terms of \(g\). [3]
SPS SPS FM Mechanics 2026 January Q4
6 marks Standard +0.3
A hollow hemispherical bowl of radius \(a\) has a smooth inner surface and is fixed with its axis vertical. A particle \(P\) of mass \(m\) moves in horizontal circles on the inner surface of the bowl, at a height \(x\) above the lowest point of the bowl. The speed of \(P\) is \(\sqrt{\frac{g}{2}a}\). Find \(x\) in terms of \(a\). [6]
SPS SPS FM Mechanics 2026 January Q5
8 marks Challenging +1.2
\includegraphics{figure_5} \(AB\) and \(BC\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(ABC = 60°\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(AB\) at an angle \(\theta\) with \(AB\). It then strikes \(BC\) and rebounds at an angle \(\beta\) with \(BC\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is 40% of its kinetic energy before the first collision.
  1. Find the value of \(e\). [4]
  2. Find the size of angle \(\beta\). [4]
SPS SPS FM Mechanics 2026 January Q6
8 marks Challenging +1.2
\includegraphics{figure_6} A particle \(P\) of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m. The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass 0.04 kg is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius 0.8 m with angular speed \(\omega\) rad s\(^{-1}\). The particle \(Q\) moves in a horizontal circle of radius 1.4 m also with angular speed \(\omega\) rad s\(^{-1}\). The centres of the circles are vertically below \(O\), and \(O\), \(P\) and \(Q\) are always in the same vertical plane. The strings \(OP\) and \(PQ\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).
  1. Find the tension in the string \(OP\). [3]
  2. Find the value of \(\omega\). [3]
  3. Find the value of \(\beta\). [2]
SPS SPS FM Mechanics 2026 January Q7
9 marks Challenging +1.2
\includegraphics{figure_7} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac{1}{2}m\) respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere \(A\) is travelling with speed \(u\) and its direction of motion makes an angle \(\alpha\) with the line of centres. Sphere \(B\) is travelling with speed \(2u\) and its direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac{2}{3}\) and \(\alpha + \beta = 90°\).
  1. Find the component of the velocity of \(B\) parallel to the line of centres after the collision, giving your answer in terms of \(u\) and \(\alpha\). [4]
The direction of motion of \(B\) after the collision is parallel to the direction of motion of \(A\) before the collision.
  1. Find the value of \(\tan \alpha\). [5]
SPS SPS FM Mechanics 2026 January Q8
12 marks Challenging +1.2
\includegraphics{figure_1} A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre \(O\) and radius \(5a\). A uniform rod \(AB\), of length \(16a\) and weight \(W\), rests in equilibrium on the hemisphere with end \(A\) on the ground. The rod rests on the hemisphere at the point \(C\), where \(AC = 12a\) and angle \(CAO = \alpha\), as shown in Figure 1. Points \(A\), \(C\), \(B\) and \(O\) all lie in the same vertical plane.
  1. Explain why \(AO = 13a\) [1]
The normal reaction on the rod at \(C\) has magnitude \(kW\)
  1. Show that \(k = \frac{8}{13}\) [3]
The resultant force acting on the rod at \(A\) has magnitude \(R\) and acts upwards at \(\theta°\) to the horizontal.
  1. Find
    1. an expression for \(R\) in terms of \(W\)
    2. the value of \(\theta\)
    [8]
OCR FM1 AS 2021 June Q1
5 marks Moderate -0.3
A car of mass 1200 kg is driven on a long straight horizontal road. There is a constant force of 250 N resisting the motion of the car. The engine of the car is working at a constant power of 10 kW.
  1. The car can travel at constant speed \(v \text{ ms}^{-1}\) along the road. Find \(v\). [2]
  2. Find the acceleration of the car at an instant when its speed is \(30 \text{ ms}^{-1}\). [3]
OCR FM1 AS 2021 June Q2
6 marks Standard +0.8
A particle \(P\) of mass 5.6 kg is attached to one end of a light rod of length 2.1 m. The other end of the rod is freely hinged to a fixed point \(O\). The particle is initially at rest directly below \(O\). It is then projected horizontally with speed \(5 \text{ ms}^{-1}\). In the subsequent motion, the angle between the rod and the downward vertical at \(O\) is denoted by \(\theta\) radians, as shown in the diagram. \includegraphics{figure_2}
  1. Find the speed of \(P\) when \(\theta = \frac{1}{4}\pi\). [4]
  2. Find the value of \(\theta\) when \(P\) first comes to instantaneous rest. [2]