| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - logarithmic functions |
| Difficulty | Standard +0.3 This is a multi-part question covering standard C3 techniques: finding x-intercepts (routine logarithm manipulation), finding stationary points (differentiate and solve dy/dx=0), verifying perpendicular tangents (product of gradients = -1), and integration by parts with a given result to apply. Each part is straightforward application of learned methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation1.08e Area between curve and x-axis: using definite integrals1.08i Integration by parts |
1 Fig. 7 shows the curve
$$y = 2 x - x \ln x , \text { where } x > 0 .$$
The curve crosses the $x$-axis at A , and has a turning point at B . The point C on the curve has $x$-coordinate 1 . Lines CD and BE are drawn parallel to the $y$-axis.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{74cc215f-bd55-489d-aa4b-0f67c2c8de52-1_529_1259_657_602}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}
(i) Find the $x$-coordinate of A , giving your answer in terms of e .\\
(ii) Find the exact coordinates of B .\\
(iii) Show that the tangents at A and C are perpendicular to each other.\\
(iv) Using integration by parts, show that
$$\int x \ln x \mathrm {~d} x = \frac { 1 } { 2 } x ^ { 2 } \ln x - \frac { 1 } { 4 } x ^ { 2 } + c$$
Hence find the exact area of the region enclosed by the curve, the $x$-axis and the lines CD and BE .
\hfill \mbox{\textit{OCR MEI C3 Q1 [18]}}