| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area under transcendental or composite curve |
| Difficulty | Standard +0.3 This is a standard C3 question testing quotient rule differentiation, finding x-intercepts, and volumes of revolution. Part (i) is routine quotient rule application with chain rule. Part (ii) requires setting y=0 and substituting into the derivative. Part (iii) is a standard volume of revolution integral that simplifies nicely. All techniques are core C3 material with no novel insights required, making it slightly easier than average. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.07l Derivative of ln(x): and related functions1.07q Product and quotient rules: differentiation4.08d Volumes of revolution: about x and y axes |
\begin{enumerate}[label=(\roman*)]
\item Given that $y = \frac{4 \ln x - 3}{4 \ln x + 3}$, show that $\frac{dy}{dx} = \frac{24}{x(4 \ln x + 3)^2}$. [3]
\item Find the exact value of the gradient of the curve $y = \frac{4 \ln x - 3}{4 \ln x + 3}$ at the point where it crosses the $x$-axis. [4]
\item \includegraphics{figure_8iii}
The diagram shows part of the curve with equation
$$y = \frac{2}{x^2(4 \ln x + 3)}.$$
The region shaded in the diagram is bounded by the curve and the lines $x = 1$, $x = e$ and $y = 0$. Find the exact volume of the solid produced when this shaded region is rotated completely about the $x$-axis. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q8 [11]}}