| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2017 |
| Session | Specimen |
| Marks | 9 |
| Topic | Reduction Formulae |
| Type | Trigonometric power reduction |
| Difficulty | Challenging +1.2 This is a standard reduction formula question from Further Maths requiring integration by parts to derive the formula (routine technique) and then application with substitution. While it requires multiple steps and careful algebraic manipulation, the techniques are well-practiced in Further Maths syllabi. The cos²θ substitution in part (ii) adds modest complexity but follows a predictable pattern. Slightly above average difficulty due to the multi-step nature and being Further Maths content, but not requiring novel insight. |
| Spec | 1.08i Integration by parts8.06a Reduction formulae: establish, use, and evaluate recursively |
\textbf{In this question you must show detailed reasoning.}
It is given that $I_n = \int_0^\pi \sin^n \theta \, d\theta$ for $n \geq 0$.
\begin{enumerate}[label=(\roman*)]
\item Prove that $I_n = \frac{n-1}{n} I_{n-2}$ for $n \geq 2$. [5]
\item Evaluate $I_1$ and use the reduction formula to determine the exact value of $\int_0^\pi \cos^2 \theta \sin^5 \theta \, d\theta$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2017 Q5 [9]}}