| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Find stationary points coordinates |
| Difficulty | Moderate -0.3 This is a straightforward C2 differentiation question requiring standard techniques: differentiating polynomials and reciprocals, finding stationary points by setting dy/dx=0, identifying decreasing intervals, and finding tangent equations. All steps are routine textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| [two terms differentiated] | B1, B1 | B1 each term |
| their \(\dfrac{dy}{dx} = 0\) | M1 | s.o.i. |
| correct step | DM1 | s.o.i. |
| \(x = \frac{1}{2}\) c.a.o. | A1 |
## Question 5:
| Answer | Mark | Guidance |
|--------|------|----------|
| [two terms differentiated] | B1, B1 | B1 each term |
| their $\dfrac{dy}{dx} = 0$ | M1 | s.o.i. |
| correct step | DM1 | s.o.i. |
| $x = \frac{1}{2}$ c.a.o. | A1 | |
**[5 marks]**
---5 Differentiate $4 x ^ { 2 } + \frac { 1 } { x }$ and hence find the $x$-coordinate of the stationary point of the curve $y = 4 x ^ { 2 } + \frac { 1 } { x }$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{bba82ee6-90b2-4f03-9bb9-0371ff711a09-3_639_1027_302_542}
\captionsetup{labelformat=empty}
\caption{Fig. 11}
\end{center}
\end{figure}
The equation of the curve shown in Fig. 11 is $y = x ^ { 3 } - 6 x + 2$.\\
(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(ii) Find, in exact form, the range of values of $x$ for which $x ^ { 3 } - 6 x + 2$ is a decreasing function.\\
(iii) Find the equation of the tangent to the curve at the point $( - 1,7 )$.
Find also the coordinates of the point where this tangent crosses the curve again.
\hfill \mbox{\textit{OCR MEI C2 Q5 [5]}}