Challenging +1.8 This is a challenging Further Maths question requiring conversion between polar and Cartesian coordinates (algebraically non-trivial), identification of curve properties, and calculation of a volume of revolution using polar integration. The multi-step nature, the algebraic manipulation required in part (i)(a), and especially the volume calculation in part (ii) which requires setting up and evaluating a polar integral with careful attention to limits, place this well above average difficulty but not at the extreme end for Further Maths material.
(a) A curve has polar equation \(r = 2 - \sec \theta\). Show that the cartesian equation of the curve can be written in the form
$$y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }$$
The figure shows a sketch of part of the curve with equation \(y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{9d2db858-9c4d-4281-8e8d-9fb5cb11b8ca-4_681_695_667_685}
(b) Explain why the curve is symmetrical in the \(x\)-axis.
(c) The line \(x = a\) is an asymptote of the curve. State the value of \(a\).
The enclosed loop shown in the figure is rotated through \(180 ^ { \circ }\) about the \(x\)-axis.
Find the exact volume of the solid formed.
\section*{END OF QUESTION PAPER}
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\begin{enumerate}[label=(\roman*)]
\item (a) A curve has polar equation $r = 2 - \sec \theta$. Show that the cartesian equation of the curve can be written in the form
$$y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }$$
The figure shows a sketch of part of the curve with equation $y ^ { 2 } = \left( \frac { 2 x } { x + 1 } \right) ^ { 2 } - x ^ { 2 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{9d2db858-9c4d-4281-8e8d-9fb5cb11b8ca-4_681_695_667_685}\\
(b) Explain why the curve is symmetrical in the $x$-axis.\\
(c) The line $x = a$ is an asymptote of the curve. State the value of $a$.
\item The enclosed loop shown in the figure is rotated through $180 ^ { \circ }$ about the $x$-axis.
Find the exact volume of the solid formed.
\section*{END OF QUESTION PAPER}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2018 Q10 [14]}}