Standard +0.3 This is a straightforward application of standard differentiation rules (chain rule for cos 2x, basic derivative for sin x) followed by solving a trigonometric equation using double angle formula. The restricted domain simplifies finding solutions. Slightly above average due to the algebraic manipulation required, but still a routine C4 question.
4 The equation of a curve is \(y = \cos 2 x + 2 \sin x\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of the stationary points on the curve for \(0 < x < \pi\).
\(\left(\frac{\pi}{2}, 1\right)\); \(\left(\frac{\pi}{6}, \frac{3}{2}\right)\) and \(\left(\frac{5\pi}{6}, \frac{3}{2}\right)\)
dep* A1; A1
\(-1\) (once) for degrees instead of radians. If B0 and/or B0 due to sign error, allow A1 for \(\left(\frac{\pi}{2},1\right)\). SC: If A0 but all 3 \(x\)-values correct, award SC A1. SC B2 for 3 correct answers without working
# Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use of $\sin 2x = +/-2\sin x\cos x$ or $+/-\cos\!\left(\frac{\pi}{2}-2x\right)$; or $\cos 2x = +/-2\cos^2 x +/- 1$ etc. | M1 | Seen anywhere in the solution |
| $\frac{dy}{dx} = -2\sin 2x$ (or $-4\sin x\cos x$); $+2\cos x$ | B1, B1 | |
| Set $\frac{dy}{dx} = 0$ | *M1 | |
| $\left(\frac{\pi}{2}, 1\right)$; $\left(\frac{\pi}{6}, \frac{3}{2}\right)$ and $\left(\frac{5\pi}{6}, \frac{3}{2}\right)$ | dep* A1; A1 | $-1$ (once) for degrees instead of radians. If B0 and/or B0 due to sign error, allow A1 for $\left(\frac{\pi}{2},1\right)$. SC: If A0 but all 3 $x$-values correct, award SC A1. SC B2 for 3 correct answers without working |
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4 The equation of a curve is $y = \cos 2 x + 2 \sin x$. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence find the coordinates of the stationary points on the curve for $0 < x < \pi$.
\hfill \mbox{\textit{OCR C4 2013 Q4 [6]}}