Standard +0.3 This is a straightforward FP2 question testing standard techniques: Maclaurin series manipulation (routine subtraction of known series), finding range of validity, and converting polar to cartesian coordinates. All parts follow textbook methods with no novel insight required. The polar curve conversion uses a given hint (r + y = a) which significantly reduces difficulty. Slightly above average difficulty only because it's Further Maths content, but these are standard exercises within FP2.
Use the Maclaurin series for \(\ln(1 + x)\) and \(\ln(1 - x)\) to obtain the first three non-zero terms in the Maclaurin series for \(\ln\left(\frac{1 + x}{1 - x}\right)\). State the range of validity of this series. [4]
Find the value of \(x\) for which \(\frac{1 + x}{1 - x} = 3\). Hence find an approximation to \(\ln 3\), giving your answer to three decimal places. [4]
A curve has polar equation \(r = \frac{a}{1 + \sin \theta}\) for \(0 \leq \theta \leq \pi\), where \(a\) is a positive constant. The points on the curve have cartesian coordinates \(x\) and \(y\).
By plotting suitable points, or otherwise, sketch the curve. [3]
Show that, for this curve, \(r + y = a\) and hence find the cartesian equation of the curve. [5]
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Use the Maclaurin series for $\ln(1 + x)$ and $\ln(1 - x)$ to obtain the first three non-zero terms in the Maclaurin series for $\ln\left(\frac{1 + x}{1 - x}\right)$. State the range of validity of this series. [4]
\item Find the value of $x$ for which $\frac{1 + x}{1 - x} = 3$. Hence find an approximation to $\ln 3$, giving your answer to three decimal places. [4]
\end{enumerate}
\item A curve has polar equation $r = \frac{a}{1 + \sin \theta}$ for $0 \leq \theta \leq \pi$, where $a$ is a positive constant. The points on the curve have cartesian coordinates $x$ and $y$.
\begin{enumerate}[label=(\roman*)]
\item By plotting suitable points, or otherwise, sketch the curve. [3]
\item Show that, for this curve, $r + y = a$ and hence find the cartesian equation of the curve. [5]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2009 Q1 [16]}}