Standard +0.8 Part (a) is a standard Further Maths polar area integral requiring the double angle formula and integration of cos²(2θ). Part (b) involves routine Maclaurin series construction but the final approximation requires recognizing odd/even function properties and careful algebraic manipulation. The combination of topics and multi-step reasoning elevates this above average difficulty, though each component uses standard techniques.
Fig. 1 shows the curve with polar equation \(r = a ( 1 - \cos 2 \theta )\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant.
\begin{figure}[h]
\end{figure}
Find the area of the region enclosed by the curve.
Given that \(\mathrm { f } ( x ) = \arctan ( \sqrt { 3 } + x )\), find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
Hence find the Maclaurin series for \(\arctan ( \sqrt { 3 } + x )\), as far as the term in \(x ^ { 2 }\).
Hence show that, if \(h\) is small, \(\int _ { - h } ^ { h } x \arctan ( \sqrt { 3 } + x ) \mathrm { d } x \approx \frac { 1 } { 6 } h ^ { 3 }\).
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\begin{enumerate}[label=(\alph*)]
\item Fig. 1 shows the curve with polar equation $r = a ( 1 - \cos 2 \theta )$ for $0 \leqslant \theta \leqslant \pi$, where $a$ is a positive constant.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{43b4c7ed-3556-4d87-8aef-0111fe747982-2_529_620_577_799}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
Find the area of the region enclosed by the curve.
\item \begin{enumerate}[label=(\roman*)]
\item Given that $\mathrm { f } ( x ) = \arctan ( \sqrt { 3 } + x )$, find $\mathrm { f } ^ { \prime } ( x )$ and $\mathrm { f } ^ { \prime \prime } ( x )$.
\item Hence find the Maclaurin series for $\arctan ( \sqrt { 3 } + x )$, as far as the term in $x ^ { 2 }$.
\item Hence show that, if $h$ is small, $\int _ { - h } ^ { h } x \arctan ( \sqrt { 3 } + x ) \mathrm { d } x \approx \frac { 1 } { 6 } h ^ { 3 }$.
\end{enumerate}\end{enumerate}
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