Let $f(x) = \sqrt 3 {{\rm{e}}^{2x}}\sin x + {{\rm{e}}^{2x}}\cos x$ , for $0 \le x \le \pi$ . Solve the equation $f(x) = 0$ .

${{\rm{e}}^{2x}}\left( {\sqrt 3 \sin x + \cos x} \right) = 0$     (A1)

${{\rm{e}}^{2x}} = 0$ not possible (seen anywhere)     (A1)

simplifying

e.g. $\sqrt 3 \sin x + \cos x = 0$ , $\sqrt 3 \sin x =  - \cos x$ , $\frac{{\sin x}}{{ - \cos x}} = \frac{1}{{\sqrt 3 }}$     A1 

EITHER

$\tan x = - \frac{1}{{\sqrt 3 }}$     A1

$x = \frac{{5\pi }}{6}$     A2     N4

OR

sketch of $30^\circ$ , $60^\circ$ , $90^\circ$ triangle with sides $1$, $2$, $\sqrt 3$     A1

work leading to $x = \frac{{5\pi }}{6}$     A1

verifying $\frac{{5\pi }}{6}$ satisfies equation     A1     N4

[6 marks]

Those who realized ${{\rm{e}}^{2x}}$ was a common factor usually earned the first four marks. Few could reason with the given information to solve the equation from there. There were many candidates who attempted some fruitless algebra that did not include factorization.