Consider the complex numbers $u = 2 + 3{\text{i}}$ and $v = 3 + 2{\text{i}}$.

(a)     Given that $\frac{1}{u} + \frac{{1}}{v} = \frac{{10}}{w}$, express w in the form $a + b{\text{i, }}a,{\text{ }}b \in \mathbb{R}$.

(b)     Find $w$* and express it in the form $r{e^{{\text{i}}\theta }}$.

(a)     METHOD 1

$\frac{1}{{2 + 3{\text{i}}}} + \frac{1}{{3 + 2{\text{i}}}} = \frac{{2 - 3{\text{i}}}}{{4 + 9}} + \frac{{3 - 2{\text{i}}}}{{9 + 4}}$     M1A1

$\frac{{10}}{w} = \frac{{5 - 5{\text{i}}}}{{13}}$     A1

$w = \frac{{130}}{{5 - 5{\text{i}}}}$

$= \frac{{130 \times 5 \times (1 + {\text{i}})}}{{50}}$

$w = 13 + 13{\text{i}}$     A1

[4 marks]

METHOD 2

$\frac{1}{{2 + 3{\text{i}}}} + \frac{1}{{3 + 2{\text{i}}}} = \frac{{3 + 2{\text{i}} + 2 + 3{\text{i}}}}{{(2 + 3{\text{i}})(3 + 2{\text{i}})}}$     M1A1

$\frac{{10}}{w} = \frac{{5 + 5{\text{i}}}}{{13{\text{i}}}}$     A1

$\frac{w}{{10}} = \frac{{13{\text{i}}}}{{5 + 5{\text{i}}}}$

$w = \frac{{130{\text{i}}}}{{(5 + 5{\text{i}})}} \times \frac{{(5 - 5{\text{i}})}}{{(5 - 5{\text{i}})}}$

$= \frac{{650 + 650{\text{i}}}}{{50}}$

$= 13 + 13{\text{i}}$     A1

[4 marks]

(b)     w* $= 13 - 13{\text{i}}$     A1

$z = \sqrt {338} {e^{ - \frac{\pi }{4}{\text{i}}}}{\text{ }}\left( { = 13\sqrt 2 {e^{ - \frac{\pi }{4}{\text{i}}}}} \right)$     A1A1

Note:     Accept $\theta  = \frac{{7\pi }}{4}$.

     Do not accept answers for $\theta$ given in degrees.

[3 marks]

Total [7 marks]