1. (a) Express the three cube roots of \(5 + 10 \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 \leqslant \theta < 2 \pi\).
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   (b) The three cube roots of \(5 + 10 \mathrm { i }\) are plotted in an Argand diagram. The points are joined by straight lines to form a triangle. Find the area of this triangle, giving your answer correct to two significant figures.
   
2. The function \(f\) is defined by \(f ( x ) = \cosh \left( \frac { x } { 2 } \right)\).
   (a) State the Maclaurin series expansion for \(\cosh \left( \frac { x } { 2 } \right)\) up to and including the term in \(x ^ { 4 }\).
   

Another function \(g\) is defined by \(g ( x ) = x ^ { 2 } - 2\). The diagram below shows parts of the graphs of \(y = f ( x )\) and \(y = g ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{7316672a-ae33-4f5b-9c59-51ef43af8ff1-04_894_940_1471_552}
(b) The two graphs intersect at the point \(A\), as shown in the diagram. Use your answer from part (a) to find an approximation for the \(x\)-coordinate of \(A\), giving your answer correct to two decimal places.
(c) Using your answer to part (b), find an approximation for the area of the shaded region enclosed by the two graphs, the \(x\)-axis and the \(y\)-axis.
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