SPS SPS FM (SPS FM) 2021 April

1. Differentiate the following with respect to \(x\), simplifying your answers fully
   1. \(y = e^{3x} + \ln 2x\) [1]
   2. \(y = (5 + x^2)^{\frac{3}{2}}\) [1]
   3. \(y = \frac{2x}{(5-3x^2)^{\frac{1}{2}}}\) [2]
   4. \(y = e^{-\frac{3}{x}} \ln(1 + x^3)\) [2]
2. Integrate with respect to \(x\)
   1. \(\frac{7}{(2x-5)^8} - \frac{3}{2x-5}\) [2]
   2. \(\frac{4x^2+5x-3}{2x-5}\) [3]

solve, for \(0° < \theta < 360°\), the equation $$2 \tan^2 \theta - \frac{1}{\cos \theta} = 4.$$ [4]

$$\text{f}(x) = x^2 - 2x - 3, \quad x \in \mathbb{R}, x \geq 1.$$

1. Write down the domain and range of \(\text{f}^{-1}\) [2]
2. Sketch the graph of \(\text{f}^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]
3. Find the gradient of \(f^{-1}(x)\) when \(f^{-1}(x) = -\frac{5}{3}\) [3]

1. Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos (\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\) Give the values of \(R\) and \(\alpha\) to 3 significant figures. [3]

The temperature \(T\) °C, of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac{\pi t}{12} \right) + 5\sin \left( \frac{\pi t}{12} \right), \quad 0 \leq t < 24,$$ where \(t\) hours is the number of hours after 1200.

1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
2. Calculate, to the nearest half hour, the times when the temperature is predicted to be 12 °C. [6]

This is the graph of \(y = \frac{5}{4x-3} - \frac{3}{2}\) \includegraphics{figure_6} Find the area between the graph, the \(x\) axis, and the lines \(x = 1\) and \(x = 7\) in the form \(a \ln b + c\) where \(a,b,c \in Q\) [6]

$$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$ Prove by induction that \(\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}\), for all positive integers \(n\). [6]

The function f is defined, for any complex number \(z\), by $$\text{f}(z) = \frac{iz - 1}{iz + 1}.$$ Suppose throughout that \(x\) is a real number.

1. Show that $$\text{Re f}(x) = \frac{x^2 - 1}{x^2 + 1} \quad \text{and} \quad \text{Im f}(x) = \frac{2x}{x^2 + 1}.$$ [2]
2. Show that \(\text{f}(x)\text{f}(x)^* = 1\), where \(\text{f}(x)^*\) is the complex conjugate of \(\text{f}(x)\). [2]