| Exam Board | OCR MEI |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find stationary points of parametric curve |
| Difficulty | Standard +0.8 This question requires finding dy/dx using the chain rule (dy/dθ ÷ dx/dθ), solving a trigonometric equation involving both sin θ and cos θ for stationary points, then substituting back to find exact coordinates. The algebra is non-trivial and requires careful manipulation of double angle formulas. Part (b) is straightforward observation. Overall, this is moderately challenging due to the multi-step parametric differentiation and trigonometric equation solving, but remains within standard A-level techniques. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{d\theta} = -2\sin\theta - 2\sin 2\theta\) | M1 | For differentiation of \(y\) wrt \(\theta\), may be as part of \(\frac{dy}{dx}\) |
| \(\sin\theta + 2\sin\theta\cos\theta = 0\) | M1 | For \(\frac{dy}{d\theta} = 0\) and use of sine or cosine double angle formula. If \(\frac{dx}{d\theta}\) is used incorrectly then give M0 |
| \(\sin\theta(1 + 2\cos\theta) = 0\) | *M1 | All on one side and factorised. If divided through by \(\sin\theta\) then \(1 = -2\cos\theta\) or \(1 + 2\cos\theta = 0\) will earn this mark |
| \(\sin\theta = 0\) or \(\cos\theta = -\frac{1}{2}\) | DM1 | Dependent on previous method mark. At least one trig value. Condone if divides through by sin |
| \(\theta = 0, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}\) | A1, A1 | A marks dependent on M4. At least 2 correct values; all four values correct with no extras |
| \((2, 3)\), \((2, -1)\) | A1 | At least 2 correct coordinates |
| \(\left(2 - \frac{\sqrt{3}}{2}, -1.5\right)\), \(\left(2 + \frac{\sqrt{3}}{2}, -1.5\right)\) | A1 | All four coordinates correct with no extras, must be exact form |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 2\) | B1 | — |
## Question 12(a):
$\frac{dy}{d\theta} = -2\sin\theta - 2\sin 2\theta$ | M1 | For differentiation of $y$ wrt $\theta$, may be as part of $\frac{dy}{dx}$
$\sin\theta + 2\sin\theta\cos\theta = 0$ | M1 | For $\frac{dy}{d\theta} = 0$ and use of sine or cosine double angle formula. If $\frac{dx}{d\theta}$ is used incorrectly then give M0
$\sin\theta(1 + 2\cos\theta) = 0$ | *M1 | All on one side and factorised. If divided through by $\sin\theta$ then $1 = -2\cos\theta$ or $1 + 2\cos\theta = 0$ will earn this mark
$\sin\theta = 0$ or $\cos\theta = -\frac{1}{2}$ | DM1 | Dependent on previous method mark. At least one trig value. Condone if divides through by sin
$\theta = 0, \frac{2\pi}{3}, \pi, \frac{4\pi}{3}$ | A1, A1 | A marks dependent on M4. At least 2 correct values; all four values correct with no extras
$(2, 3)$, $(2, -1)$ | A1 | At least 2 correct coordinates
$\left(2 - \frac{\sqrt{3}}{2}, -1.5\right)$, $\left(2 + \frac{\sqrt{3}}{2}, -1.5\right)$ | A1 | All four coordinates correct with no extras, must be exact form
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## Question 12(b):
$x = 2$ | B1 | —
---12 The diagram shows the curve with parametric equations\\
$x = \sin 2 \theta + 2 , y = 2 \cos \theta + \cos 2 \theta$, for $0 \leqslant \theta < 2 \pi$.\\
\includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-10_771_673_397_239}
\begin{enumerate}[label=(\alph*)]
\item In this question you must show detailed reasoning.
Determine the exact coordinates of all the stationary points on the curve.
\item Write down the equation of the line of symmetry of the curve.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 3 2024 Q12 [9]}}