| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation question requiring standard technique to find dy/dx, then solving dy/dx=0 for stationary points. The algebra is routine (differentiating x²+2y²=4x, then substituting back into the original equation), making it slightly easier than average but still requiring multiple connected steps. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2x + 4y\frac{dy}{dx} = 4\) | M1 | \(4y\frac{dy}{dx}\) seen; rearranging for \(y\) and differentiating explicitly is M0 |
| Correct equation | A1 | Ignore superfluous \(dy/dx = \ldots\) unless used subsequently |
| \(\frac{dy}{dx} = \frac{4-2x}{4y}\) | A1 | o.e., but mark final answer |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dy}{dx} = 0 \Rightarrow x = 2\) | B1dep | Dependent on correct derivative |
| \(4 + 2y^2 = 8 \Rightarrow y^2 = 2,\ y = \sqrt{2}\) or \(-\sqrt{2}\) | B1B1 | \(\sqrt{2},\ -\sqrt{2}\); can isw, penalise inexact answers of \(\pm 1.41\) or better once only; \(-1\) for extra solutions found from using \(y = 0\) |
| [3] |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2x + 4y\frac{dy}{dx} = 4$ | M1 | $4y\frac{dy}{dx}$ seen; rearranging for $y$ and differentiating explicitly is M0 |
| Correct equation | A1 | Ignore superfluous $dy/dx = \ldots$ unless used subsequently |
| $\frac{dy}{dx} = \frac{4-2x}{4y}$ | A1 | o.e., but mark final answer |
| **[3]** | | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 0 \Rightarrow x = 2$ | B1dep | Dependent on correct derivative |
| $4 + 2y^2 = 8 \Rightarrow y^2 = 2,\ y = \sqrt{2}$ or $-\sqrt{2}$ | B1B1 | $\sqrt{2},\ -\sqrt{2}$; can isw, penalise inexact answers of $\pm 1.41$ or better once only; $-1$ for extra solutions found from using $y = 0$ |
| **[3]** | | |
---2 A curve has equation $x ^ { 2 } + 2 y ^ { 2 } = 4 x$.\\
(i) By differentiating implicitly, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$.\\[0pt]
(ii) Hence find the exact coordinates of the stationary points of the curve. [You need not determine their nature.]
\hfill \mbox{\textit{OCR MEI C3 Q2 [6]}}