Questions — SPS SPS FM Pure (188 questions)

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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]