SPS SPS FM Pure (SPS FM Pure) 2025 June

The complex number \(z\) satisfies the equation \(z + 2iz^* = 12 + 9i\). Find \(z\), giving your answer in the form \(x + iy\). [5]

1. Use binomial expansions to show that \(\sqrt{\frac{1 + 4x}{1 - x}} \approx 1 + \frac{5}{2}x - \frac{5}{8}x^2\) [6]

A student substitutes \(x = \frac{1}{2}\) into both sides of the approximation shown in part (a) in an attempt to find an approximation to \(\sqrt{6}\)

1. Give a reason why the student should not use \(x = \frac{1}{2}\) [1]
2. Substitute \(x = \frac{1}{11}\) into $$\sqrt{\frac{1 + 4x}{1 - x}} = 1 + \frac{5}{2}x - \frac{5}{8}x^2$$ to obtain an approximation to \(\sqrt{6}\). Give your answer as a fraction in its simplest form. [3]

Describe a sequence of transformations which maps the graph of $$y = |2x - 5|$$ onto the graph of $$y = |x|$$ [3 marks]

Given that $$y = \frac{3\sin \theta}{2\sin \theta + 2\cos \theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$ show that $$\frac{dy}{d\theta} = \frac{A}{1 + \sin 2\theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$ where \(A\) is a rational constant to be found. [5]

Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) satisfy the equation $$\mathbf{AB} = I + 2\mathbf{A}$$ where \(I\) is the identity matrix and \(\mathbf{B} = \begin{pmatrix} 3 & -2 \\ -4 & 8 \end{pmatrix}\) Find \(\mathbf{A}\). [3 marks]

1. Prove that $$1 - \cos 2\theta = \tan \theta \sin 2\theta, \quad \theta \neq \frac{(2n + 1)\pi}{2}, \quad n \in \mathbb{Z}$$ [3]
2. Hence solve, for \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), the equation $$(\sec^2 x - 5)(1 - \cos 2x) = 3\tan^2 x \sin 2x$$ Give any non-exact answer to 3 decimal places where appropriate. [6]

Fig. 10 shows the graph of \(x^3 + y^3 = xy\). \includegraphics{figure_10}

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [4]
2. P is the maximum point on the curve. The parabola \(y = kx^2\) intersects the curve at P. Find the value of the constant \(k\). [2]

1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$|z - 3| = 2$$ [1 mark] \includegraphics{figure_8}

1. There is a unique complex number \(w\) that satisfies both $$|w - 3| = 2 \quad \text{and} \quad \arg(w + 1) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\)
   1. [(b) (i)] Find the value of \(\alpha\). [2 marks]
   2. [(b) (ii)] Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). [4 marks]

\includegraphics{figure_9} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x, \quad x > 0\) The line \(l\) is the normal to \(C\) at the point \(P(e, e)\) The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis. Show that the exact area of \(R\) is \(Ae^2 + B\) where \(A\) and \(B\) are rational numbers to be found. [9]

Prove by induction that \(f(n) = 2^{4n} + 5^{2n} + 7^n\) is divisible by 3 for all positive integers \(n\). [5]

Fig. 15 shows the graph of \(f(x) = 2x + \frac{1}{x} + \ln x - 4\). \includegraphics{figure_11}

1. Show that the equation $$2x + \frac{1}{x} + \ln x - 4 = 0$$ has a root, \(\alpha\), such that \(0.1 < \alpha < 0.9\). [2]
2. Obtain the following Newton-Raphson iteration for the equation in part (i). $$x_{r+1} = x_r - \frac{2x_r^3 + x_r + x_r^2(\ln x_r - 4)}{2x_r^2 - 1 + x_r}$$ [3]
3. Explain why this iteration fails to find \(\alpha\) using each of the following starting values.
   1. \(x_0 = 0.4\) [2]
   2. \(x_0 = 0.5\) [2]
   3. \(x_0 = 0.6\) [2]

In this question you must show detailed reasoning. \includegraphics{figure_12} The curve \(C\) has parametric equations $$x = \frac{1}{\sqrt{2 + t}}, \quad y = \ln(1 + t), \quad 2 \leq t < \infty$$ The point \(P\) on curve \(C\) has \(x\)-coordinate \(\frac{1}{2}\).

1. Find the exact \(y\)-coordinate of \(P\). [1]

The tangent to \(C\) at \(P\) meets the \(y\)-axis at point \(Y\).

1. Determine the exact coordinates of \(Y\). [4]

The curve \(C\) and the line segment \(PY\) are rotated \(2\pi\) radians about the \(y\)-axis.

1. Determine the exact volume of the solid generated. Give your answer in the form \(\pi(\ln p + q)\), where \(p\) and \(q\) are rational numbers. [8]

[You are given that the volume of a cone with radius \(r\) and height \(h\) is \(\frac{1}{3}\pi r^2 h\)]

1. Using a suitable substitution, find $$\int \sqrt{1 - x^2} \, dx.$$ [4]
2. Show that the differential equation $$\frac{dy}{dx} = 2\sqrt{1 - x^2 - y^2 + x^2y^2},$$ given that \(y = 0\) when \(x = 0\), \(|x| < 1\) and \(|y| < 1\), has the solution $$y = x \cos\left(x\sqrt{1 - x^2}\right) + \sqrt{1 - x^2} \sin\left(x\sqrt{1 - x^2}\right).$$ [5]

The three dimensional non-zero vector \(\mathbf{u}\) has the following properties:

• The angle \(\theta\) between \(\mathbf{u}\) and the vector \(\begin{pmatrix} 1 \\ 5 \\ 9 \end{pmatrix}\) is acute.
• The (non-reflex) angle between \(\mathbf{u}\) and the vector \(\begin{pmatrix} 9 \\ 5 \\ 1 \end{pmatrix}\) is \(2\theta\).
• \(\mathbf{u}\) is perpendicular to the vector \(\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\).

Find the angle \(\theta\). [5]