Challenging +1.8 This is a substantial Further Maths question requiring multiple advanced techniques: deriving a trigonometric identity using de Moivre's theorem, differentiating a composite function involving inverse trig functions, and evaluating an improper integral. However, each part follows standard FM procedures without requiring novel insights—part (a) is routine binomial expansion with complex numbers, part (b) uses the chain rule systematically with the result from (a), and part (c) recognizes the antiderivative from part (b). The multi-step nature and FM content place it above average difficulty but below the most challenging proof-based questions.
9 In this question you must show detailed reasoning.
Use de Moivre's theorem to determine constants \(A\), \(B\) and \(C\) such that
$$\sin ^ { 4 } \theta \equiv A \cos 4 \theta + B \cos 2 \theta + C .$$
The function f is defined by
\(\mathrm { f } ( x ) = \sin \left( 4 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) - 8 \sin \left( 2 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) + 12 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) , \quad x \in \mathbb { R } , 0 \leqslant x < 1\).
Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 5 \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }\).
\includegraphics[max width=\textwidth, alt={}, center]{478c66d2-16a0-41ef-9444-25cfcd47d11d-7_894_842_1000_260}
The diagram shows the curve with equation \(\mathrm { y } = \frac { 1 } { \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }\) for \(0 \leqslant x < 1\) and the asymptote \(x = 1\). The region \(R\) is the unbounded region between the curve, the \(x\)-axis, the line \(x = 0\) and the line \(x = 1\).
You are given that the area of \(R\) is finite.
9 In this question you must show detailed reasoning.
\begin{enumerate}[label=(\alph*)]
\item Use de Moivre's theorem to determine constants $A$, $B$ and $C$ such that
$$\sin ^ { 4 } \theta \equiv A \cos 4 \theta + B \cos 2 \theta + C .$$
The function f is defined by\\
$\mathrm { f } ( x ) = \sin \left( 4 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) - 8 \sin \left( 2 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) + 12 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) , \quad x \in \mathbb { R } , 0 \leqslant x < 1$.
\item Show that $\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 5 \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{478c66d2-16a0-41ef-9444-25cfcd47d11d-7_894_842_1000_260}
The diagram shows the curve with equation $\mathrm { y } = \frac { 1 } { \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }$ for $0 \leqslant x < 1$ and the asymptote $x = 1$. The region $R$ is the unbounded region between the curve, the $x$-axis, the line $x = 0$ and the line $x = 1$.
You are given that the area of $R$ is finite.
\item Determine the exact area of $R$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2023 Q9 [14]}}