| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area between curve and line |
| Difficulty | Standard +0.3 This is a structured multi-part question testing standard C3 techniques: solving sin equations, product rule differentiation, showing tangency by comparing gradients, and integration by parts. While part (iv) requires careful setup and integration by parts twice, each step follows predictable methods with clear guidance. The question is slightly above average due to the integration by parts component, but remains a typical exam question without requiring novel insight. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07q Product and quotient rules: differentiation1.08f Area between two curves: using integration1.08i Integration by parts |
Fig. 8 shows part of the curve $y = x \sin 3x$. It crosses the $x$-axis at P. The point on the curve with $x$-coordinate $\frac{1}{6}\pi$ is Q.
\includegraphics{figure_8}
\begin{enumerate}[label=(\roman*)]
\item Find the $x$-coordinate of P. [3]
\item Show that Q lies on the line $y = x$. [1]
\item Differentiate $x \sin 3x$. Hence prove that the line $y = x$ touches the curve at Q. [6]
\item Show that the area of the region bounded by the curve and the line $y = x$ is $\frac{1}{72}(\pi^2 - 8)$. [7]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 Q8 [17]}}