| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find vertical tangent points |
| Difficulty | Standard +0.8 This question requires implicit differentiation (standard C3 technique) but then asks students to find where gradient is infinite (vertical tangent), requiring them to recognize this means the denominator equals zero and solve the resulting transcendental equation e^(2y) = 1/2, which involves logarithms and numerical work. The conceptual leap from 'infinite gradient' to 'denominator = 0' and solving the implicit equation elevates this above routine differentiation exercises. |
| Spec | 1.07s Parametric and implicit differentiation |
6 Fig. 6 shows the curve $\mathrm { e } ^ { 2 y } = x ^ { 2 } + y$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a66c2da5-46b4-4467-933e-179be04b03b1-3_739_1339_349_404}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{center}
\end{figure}
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x } { 2 \mathrm { e } ^ { 2 y } - 1 }$.\\
(ii) Hence find to 3 significant figures the coordinates of the point P , shown in Fig. 6, where the curve has infinite gradient.
Section B (36 marks)\\
\hfill \mbox{\textit{OCR MEI C3 2008 Q6 [8]}}