SPS SPS FM (SPS FM) 2021 April

1. i) 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) \(y = e ^ { - \frac { 8 } { 3 } x } \ln \left( 1 + x ^ { 3 } \right)\)
   ii) Integrate with respect to \(x\)

\begin{displayquote}

1. \(\frac { 7 } { ( 2 x - 5 ) ^ { 5 } } - \frac { 3 } { 2 x - 5 }\)
2. \(\frac { 4 x ^ { 2 } + 5 x - 3 } { 2 x - 5 }\) \end{displayquote}

1. 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]{11e1fe10-7d93-4a49-9151-83b40391a329-07_520_714_196_653} 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.

6. This is the graph of \(y = \frac { 5 } { 4 x - 3 } - \frac { 3 } { 2 }\)
\includegraphics[max width=\textwidth, alt={}, center]{11e1fe10-7d93-4a49-9151-83b40391a329-09_753_917_301_630} 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 \(a , b , c \in Q\)

7. $$\mathbf { M } = \left( \begin{array} { l l } 2 & 3
0 & 1 \end{array} \right) .$$ 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 \(n\).

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 \(\mathrm { f } ( x )\). Spare Paper