| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2019 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Maclaurin series for ln(trigonometric expressions) |
| Difficulty | Challenging +1.3 This is a Further Maths question requiring Maclaurin series via differentiation of a composite logarithmic-trigonometric function, followed by using the series to approximate π. Part (a) involves careful differentiation and evaluation at x=0 (moderate technical demand), while part (b) requires insight to set the series equal to zero and solve for x, then relate to π. The multi-step nature and need to connect series approximation to a geometric result elevates this above standard Further Maths exercises, but the techniques are well-practiced in FP2. |
| Spec | 4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
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\begin{enumerate}[label=(\alph*)]
\item Use differentiation to find the first two non-zero terms of the Maclaurin expansion of $\ln \left( \frac { 1 } { 2 } + \cos x \right)$.
\item By considering the root of the equation $\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0$ deduce that $\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }$.
\section*{END OF QUESTION PAPER}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2019 Q10 [7]}}