SPS SPS FM Pure (SPS FM Pure) 2026 November

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]

Prove by induction that, for all positive integers \(n\), $$\sum_{r=1}^{n}(2r-1)^2 = \frac{1}{3}n(4n^2-1)$$ [6]

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]

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]

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]

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]
2. 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]

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]

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]

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]

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]