| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Topic | Taylor series |
| Type | Use series to approximate integral |
| Difficulty | Standard +0.3 This is a structured Further Maths question with clear steps: finding derivatives of arcsin, writing its Maclaurin series (standard result), using it to approximate an integral, then computing the exact integral via integration by parts. While it's Further Maths content, each part follows routine procedures with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.08a Maclaurin series: find series for function4.08g Derivatives: inverse trig and hyperbolic functions |
5 Let $\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Determine $f ^ { \prime \prime } ( x )$.
\item Determine the first two non-zero terms of the Maclaurin expansion for $\mathrm { f } ( x )$.
\item By considering the first two non-zero terms of the Maclaurin expansion for $\mathrm { f } ( \mathrm { x } )$, find an approximation to $\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer correct to 6 decimal places.
\end{enumerate}\item By writing $\mathrm { f } ( x )$ as $\sin ^ { - 1 } ( x ) \times 1$, determine the value of $\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer in exact form.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q5 [10]}}