Standard +0.3 This is a straightforward stationary point problem requiring the product rule and chain rule to differentiate cos x cos 2x, setting the derivative to zero, and solving a trigonometric equation. While it involves multiple techniques, it's a standard textbook exercise with no novel insight required, making it slightly easier than average.
3 A curve has equation \(y = \cos x \cos 2 x\). Find the \(x\)-coordinate of the stationary point on the curve in the interval \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.
Obtain correct derivative in any form, e.g. \(-\sin x \cos 2x - 2\cos x \sin 2x\)
A1
Use the correct double angle formulae to express derivative in \(\cos x\) and \(\sin x\), or \(\cos 2x\) and \(\sin x\)
M1
OR1: Use correct double angle formula to express \(y\) in terms of \(\cos x\) and attempt differentiation
M1
Use chain rule correctly
M1
Obtain correct derivative in any form, e.g. \(-6\cos^2 x \sin x + \sin x\)
A1
OR2: Use correct factor formula and attempt differentiation
M1
Obtain correct derivative in any form, e.g. \(-\frac{3}{2}\sin 3x - \frac{1}{2}\sin x\)
A1
Use correct trig formulae to express derivative in terms of \(\cos x\) and \(\sin x\), or \(\sin x\)
M1
Equate derivative to zero and obtain an equation in one trig function
M1
Obtain \(6\cos^2 x = 1\), \(6\sin^2 x = 5\), \(\tan^2 x = 5\) or \(3\cos 2x = -2\)
A1
Obtain answer \(x = 1.15\) (or \(65.9°\)) and no other in the given interval
A1
[6]
[Ignore answers outside the given interval.]
[SR: Solution attempts following the EITHER scheme for the first two marks can earn the second and third method marks as follows:]
Answer
Marks
Equate derivative to zero and obtain an equation in \(\tan 2x\) and \(\tan x\)
M1
Use correct double angle formula to obtain an equation in \(\tan x\)
M1
**EITHER:** Use correct product rule | M1 |
Obtain correct derivative in any form, e.g. $-\sin x \cos 2x - 2\cos x \sin 2x$ | A1 |
Use the correct double angle formulae to express derivative in $\cos x$ and $\sin x$, or $\cos 2x$ and $\sin x$ | M1 |
**OR1:** Use correct double angle formula to express $y$ in terms of $\cos x$ and attempt differentiation | M1 |
Use chain rule correctly | M1 |
Obtain correct derivative in any form, e.g. $-6\cos^2 x \sin x + \sin x$ | A1 |
**OR2:** Use correct factor formula and attempt differentiation | M1 |
Obtain correct derivative in any form, e.g. $-\frac{3}{2}\sin 3x - \frac{1}{2}\sin x$ | A1 |
Use correct trig formulae to express derivative in terms of $\cos x$ and $\sin x$, or $\sin x$ | M1 |
Equate derivative to zero and obtain an equation in one trig function | M1 |
Obtain $6\cos^2 x = 1$, $6\sin^2 x = 5$, $\tan^2 x = 5$ or $3\cos 2x = -2$ | A1 |
Obtain answer $x = 1.15$ (or $65.9°$) and no other in the given interval | A1 | [6]
[Ignore answers outside the given interval.]
[SR: Solution attempts following the EITHER scheme for the first two marks can earn the second and third method marks as follows:]
Equate derivative to zero and obtain an equation in $\tan 2x$ and $\tan x$ | M1 |
Use correct double angle formula to obtain an equation in $\tan x$ | M1 |
3 A curve has equation $y = \cos x \cos 2 x$. Find the $x$-coordinate of the stationary point on the curve in the interval $0 < x < \frac { 1 } { 2 } \pi$, giving your answer correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P3 2015 Q3 [6]}}