| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Show dy/dx equals given expression |
| Difficulty | Moderate -0.3 This is a straightforward implicit differentiation question with standard techniques. Part (i) requires basic application of the chain rule, and part (ii) involves simple rearrangement and verification of the inverse derivative relationship. While it tests understanding of implicit differentiation, it's slightly easier than average as it's a direct application with no problem-solving insight required and the algebra is simple. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| \(y^2 = 4x + 7 \Rightarrow 2y\frac{dy}{dx} = 4 \Rightarrow \frac{dy}{dx} = \frac{2}{y}\) | M1, A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = \frac{1}{4}(y^2 - 7) \Rightarrow \frac{dx}{dy} = \frac{1}{4} \cdot 2y = \frac{y}{2} = \frac{1}{2}\) | B1, M1, A1 | 3 marks |
### (i)
$y^2 = 4x + 7 \Rightarrow 2y\frac{dy}{dx} = 4 \Rightarrow \frac{dy}{dx} = \frac{2}{y}$ | M1, A1 | 2 marks
### (ii)
$x = \frac{1}{4}(y^2 - 7) \Rightarrow \frac{dx}{dy} = \frac{1}{4} \cdot 2y = \frac{y}{2} = \frac{1}{2}$ | B1, M1, A1 | 3 marks
---You are given that $y^2 = 4x + 7$.
\begin{enumerate}[label=(\roman*)]
\item Use implicit differentiation to find $\frac{dy}{dx}$ in terms of $y$. [2]
\item Make $x$ the subject of the equation.
Find $\frac{dx}{dy}$ and hence show that in this case $\frac{dx}{dy} = \frac{1}{\frac{dx}{dy}}$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 Q1 [5]}}