SPS SPS FM (SPS FM) 2021 March

1. Differentiate the following with respect to \(x\), simplifying your answers fully
   a) \(y = e ^ { 3 x } + \ln 2 x\)
   b) \(y = \left( 5 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } }\)
   c) \(y = \frac { 2 x } { \left( 5 - 3 x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } }\)
   d) \(y = e ^ { - \frac { 8 } { 3 } x } \ln \left( 1 + x ^ { 3 } \right)\)

1. Express \(2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta }\) in terms of \(\sec \theta\).
2. Hence solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta - \frac { 1 } { \cos \theta } = 4$$

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

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

4.
\includegraphics[max width=\textwidth, alt={}, center]{ef0c9a48-9d23-48ed-89b8-2e116114d7ed-07_527_718_191_651} The diagram shows the curves \(y = \mathrm { e } ^ { 3 x }\) and \(y = ( 2 x - 1 ) ^ { 4 }\). The shaded region is bounded by the two curves and the line \(x = \frac { 1 } { 2 }\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.

5. (a) 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. The temperature \(T ^ { \circ } \mathrm { C }\), 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 .
(b) Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs.
(c) Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 ^ { \circ } \mathrm { C }\).

6. $$\mathbf { M } = \left( \begin{array} { l l } 2 & 3
0 & 1 \end{array} \right) .$$ Prove by induction that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right)
0 & 1 \end{array} \right)\), for all positive integers \(n\).

7. This is the graph of \(y = \frac { 5 } { 4 x - 3 } - \frac { 3 } { 2 }\)
\includegraphics[max width=\textwidth, alt={}, center]{ef0c9a48-9d23-48ed-89b8-2e116114d7ed-10_513_547_207_817} 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 \(a , b , c \in Q\)

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

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