Standard +0.3 This is a straightforward stationary point problem requiring product rule differentiation of trigonometric functions, setting the derivative to zero, and solving a standard trigonometric equation. While it involves multiple trig identities and numerical solving, it follows a completely standard procedure with no novel insight required, making it slightly easier than average.
5 The curve with equation \(y = \sin x \cos 2 x\) has one stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinate of this point, giving your answer correct to 3 significant figures.
Equate derivative to zero and use double angle formulae
M1
Remove factor of \(\cos x\) and reduce to one single trig function
M1
Obtain \(6\sin^2 x = 1\), \(6\cos^2 x = 5\) or \(5\tan^2 x = 1\)
A1
Solve and obtain \(x = 0.421\)
A1
Total
[6]
[Alternative: Use double angle formula M1. Use chain rule to differentiate M1. Obtain correct derivative e.g. \(\cos\theta - 6\sin^2\theta\cos\theta\) A1, then as above.]
## Question 5:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use product rule | M1 | |
| Obtain correct derivative, e.g. $\cos x\cos 2x - 2\sin x\sin 2x$ | A1 | |
| Equate derivative to zero and use double angle formulae | M1 | |
| Remove factor of $\cos x$ and reduce to one single trig function | M1 | |
| Obtain $6\sin^2 x = 1$, $6\cos^2 x = 5$ or $5\tan^2 x = 1$ | A1 | |
| Solve and obtain $x = 0.421$ | A1 | |
| **Total** | **[6]** | [Alternative: Use double angle formula M1. Use chain rule to differentiate M1. Obtain correct derivative e.g. $\cos\theta - 6\sin^2\theta\cos\theta$ A1, then as above.] |
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5 The curve with equation $y = \sin x \cos 2 x$ has one stationary point in the interval $0 < x < \frac { 1 } { 2 } \pi$. Find the $x$-coordinate of this point, giving your answer correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P3 2016 Q5 [6]}}