| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Simpson's rule application |
| Difficulty | Standard +0.3 This is a slightly easier than average C3 question. Part (i) is standard product rule differentiation with straightforward algebra to find stationary points. Part (ii) applies Simpson's rule mechanically with given strip count. Part (iii) requires recognizing symmetry to deduce area B from area A, but this is a simple conceptual step. The question tests routine techniques without requiring novel insight or complex problem-solving. |
| Spec | 1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.09f Trapezium rule: numerical integration |
\includegraphics{figure_8}
The diagram shows the curve with equation $y = x^8 e^{-x^2}$. The curve has maximum points at $P$ and $Q$. The shaded region $A$ is bounded by the curve, the line $y = 0$ and the line through $Q$ parallel to the $y$-axis. The shaded region $B$ is bounded by the curve, the line $y = 0$ and the line $PQ$.
\begin{enumerate}[label=(\roman*)]
\item Show by differentiation that the $x$-coordinate of $Q$ is 2. [5]
\item Use Simpson's rule with 4 strips to find an approximation to the area of region $A$. Give your answer correct to 3 decimal places. [4]
\item Deduce an approximation to the area of region $B$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q8 [11]}}