| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find dy/dx at a point |
| Difficulty | Standard +0.3 This is a multi-part question covering standard C3 techniques: identifying asymptotes, finding stationary points, substitution in integration, and implicit differentiation. Part (iv) requires implicit differentiation of e^y = f(x) and evaluating at a point—straightforward application of the chain rule and quotient rule. All parts are routine textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.07n Stationary points: find maxima, minima using derivatives1.07s Parametric and implicit differentiation1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals1.08h Integration by substitution |
1 Fig. 7 shows the curve $y = _ { x - 1 }$. It has a minimum at the point P . The line $l$ is an asymptote to the curve.
\begin{figure}[h]
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\includegraphics[alt={},max width=\textwidth]{e0636807-d5bf-43c2-a484-68245e639cee-1_732_1049_467_547}
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\caption{Fig. 7}
\end{center}
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(i) Write down the equation of the asymptote $l$.\\
(ii) Find the coordinates of P .\\
(iii) Using the substitution $u = x - 1$, show that the area of the region enclosed by the $x$-axis, the curve and the lines $x = 2$ and $x = 3$ is given by
$$\int _ { 1 } ^ { 2 } \left( u + 2 + \frac { 4 } { u } \right) \mathrm { d } u$$
Evaluate this area exactly.\\
(iv) Another curve is defined by the equation $\mathrm { e } ^ { y } = \frac { x ^ { 2 } + 3 } { x - 1 }$. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$ by differentiating implicitly. Hence find the gradient of this curve at the point where $x = 2$.
\hfill \mbox{\textit{OCR MEI C3 Q1 [18]}}