| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area between curve and line |
| Difficulty | Moderate -0.3 Part (a) is a routine integration using substitution or recognition of a standard form, requiring only mechanical application of techniques. Part (b) involves setting up an area calculation with definite integration of 1/x and algebraic manipulation to reach the given answer, but follows a standard template for such problems. Both parts are slightly easier than average C3 questions due to their straightforward nature, though part (b) requires careful setup and simplification. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals1.08h Integration by substitution |
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $\int_1^2 \frac{2}{(4x - 1)^2} \, dx$. [4]
\item \includegraphics{figure_7b}
The diagram shows part of the curve $y = \frac{1}{x}$. The point $P$ has coordinates $(a, \frac{1}{a})$ and the point $Q$ has coordinates $(2a, \frac{1}{2a})$, where $a$ is a positive constant. The point $R$ is such that $PR$ is parallel to the $x$-axis and $QR$ is parallel to the $y$-axis. The region shaded in the diagram is bounded by the curve and by the lines $PR$ and $QR$. Show that the area of this shaded region is $\ln(\frac{4}{e})$. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q7 [10]}}