| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - logarithmic functions |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard C3 techniques: verifying a point lies on a curve (substitution), finding stationary points by setting dy/dx = 0, using the second derivative test, and integration by parts for ∫ln x dx. All steps are routine textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07n 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. 8 shows the curve $y = 3 \ln x + x - x ^ { 2 }$.\\
The curve crosses the $x$-axis at P and Q , and has a turning point at R . The $x$-coordinate of Q is approximately 2.05 .
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{65ac8807-cd93-450f-adb5-dc6864f8470c-1_720_834_578_681}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{center}
\end{figure}
(i) Verify that the coordinates of P are $( 1,0 )$.\\
(ii) Find the coordinates of R , giving the $y$-coordinate correct to 3 significant figures.
Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$, and use this to verify that R is a maximum point.\\
(iii) Find $\int \ln x \mathrm {~d} x$.
Hence calculate the area of the region enclosed by the curve and the $x$-axis between P and Q , giving your answer to 2 significant figures.
\hfill \mbox{\textit{OCR MEI C3 Q1 [17]}}