| Exam Board | OCR MEI |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Classify nature of stationary points |
| Difficulty | Standard +0.3 This is a straightforward stationary point question requiring standard differentiation of polynomial and trigonometric terms, then sign-checking the derivative at given bounds and using the second derivative test. The algebraic manipulation is routine and the question provides helpful bounds to verify rather than requiring independent solving. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.09a Sign change methods: locate roots |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = \frac{4}{\pi} - \frac{8x}{\pi^2} - \cos x\) | M1 | 3.1a |
| \(x = 2.6\), \(\frac{dy}{dx} = 0.0226...\) | M1 | 3.1a |
| \(x = 2.7\), \(\frac{dy}{dx} = -0.0112...\) | A1 | 1.1 |
| Gradient zero for a value between 2.6 and 2.7 | E1 | 2.2a |
| Gradient positive, zero, negative so max | B1 | 2.4 |
## Question 10:
$\frac{dy}{dx} = \frac{4}{\pi} - \frac{8x}{\pi^2} - \cos x$ | M1 | 3.1a | Attempt to differentiate. Both an $x$ term and a $\cos$ term needed, condone other errors. If $\pi^2$ is treated as a variable and incorrectly differentiated e.g. to $2\pi$ then M0. Similarly with $\pi$.
$x = 2.6$, $\frac{dy}{dx} = 0.0226...$ | M1 | 3.1a | At least one substitution into their expression
$x = 2.7$, $\frac{dy}{dx} = -0.0112...$ | A1 | 1.1 | Both correct (at least 2 d.p., rounded or truncated)
Gradient zero for a value between 2.6 and 2.7 | E1 | 2.2a | Can be implied by 'sign change' or sketch. Dependent on M2 but can be earned following **M2A0**
Gradient positive, zero, negative so max | B1 | 2.4 | Allow $2^{nd}$ derivative used at turning point found BC ($x = 2.67$, $2^{nd}$ deriv $= -0.356$) or 2 relevant values e.g. $-0.295$ and $-0.383$ seen and stated as $< 0$ therefore max
**[5 marks]**
---10 In this question you must show detailed reasoning.
Fig. C2.2 indicates that the curve $\mathrm { y } = \frac { 4 \mathrm { x } ( \pi - \mathrm { x } ) } { \pi ^ { 2 } } - \sin \mathrm { x }$ has a stationary point near $x = 3$.
\begin{itemize}
\item Verify that the $x$-coordinate of this stationary point is between 2.6 and 2.7.
\item Show that this stationary point is a maximum turning point.
\end{itemize}
\hfill \mbox{\textit{OCR MEI Paper 3 2022 Q10 [5]}}