| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polar coordinates |
| Type | Sketch polar curve |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring conversion to polar form (standard technique), sketching a non-trivial polar curve (requires understanding of r = 3cos θ sin²θ behavior), a trigonometric substitution integral, and series manipulation culminating in a proof involving an infinite sum. While each part uses known techniques, the combination and the final series-to-logarithm proof require solid technical facility across multiple FP2 topics, placing it moderately above average difficulty. |
| Spec | 4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n4.08h Integration: inverse trig/hyperbolic substitutions4.09a Polar coordinates: convert to/from cartesian4.09b Sketch polar curves: r = f(theta) |
1
\begin{enumerate}[label=(\alph*)]
\item A curve has cartesian equation $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 3 x y ^ { 2 }$.
\begin{enumerate}[label=(\roman*)]
\item Show that the polar equation of the curve is $r = 3 \cos \theta \sin ^ { 2 } \theta$.
\item Hence sketch the curve.
\end{enumerate}\item Find the exact value of $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - 3 x ^ { 2 } } } \mathrm {~d} x$.
\item \begin{enumerate}[label=(\roman*)]
\item Write down the series for $\ln ( 1 + x )$ and the series for $\ln ( 1 - x )$, both as far as the term in $x ^ { 5 }$.
\item Hence find the first three non-zero terms in the series for $\ln \left( \frac { 1 + x } { 1 - x } \right)$.
\item Use the series in part (ii) to show that $\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } } = \ln 3$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2008 Q1 [18]}}