| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reduction Formulae |
| Type | Algebraic function with square root |
| Difficulty | Challenging +1.8 This is a Further Maths reduction formula question requiring multiple sophisticated techniques: recognizing a derivative relationship for part (a), deriving a reduction formula via integration by parts for part (b), and manipulating the formula to find a related integral in part (c). While the individual steps follow standard Further Maths methods, the question demands strong algebraic manipulation, insight into the derivative hint, and careful bookkeeping across multiple parts. It's significantly harder than typical A-level questions but represents expected Further Maths material rather than requiring exceptional creativity. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08i Integration by parts8.06a Reduction formulae: establish, use, and evaluate recursively |
7 Let $\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 2 } \frac { \mathrm { x } ^ { \mathrm { n } } } { \sqrt { \mathrm { x } ^ { 3 } + 1 } } \mathrm { dx }$ for integers $n > 0$.
\begin{enumerate}[label=(\alph*)]
\item By considering the derivative of $\sqrt { x ^ { 3 } + 1 }$ with respect to $x$, determine the exact value of $I _ { 2 }$.
\item Given that $n > 3$, show that $\left. ( 2 n - 1 ) \right| _ { n } = 3 \times 2 ^ { n - 1 } - \left. 2 ( n - 2 ) \right| _ { n - 3 }$.
\item Hence determine the exact value of $\int _ { 0 } ^ { 2 } x ^ { 5 } \sqrt { x ^ { 3 } + 1 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2024 Q7 [10]}}