Express $z$ in the form $a + {\text{i}}b$, where $a,\,b \in \mathbb{Q}$.

Find the exact value of the modulus of $z$.

Find the argument of $z$, giving your answer to 4 decimal places.

$z = \frac{{\left( {2 + 7{\text{i}}} \right)}}{{\left( {6 + 2{\text{i}}} \right)}} \times \frac{{\left( {6 - 2{\text{i}}} \right)}}{{\left( {6 - 2{\text{i}}} \right)}}$     (M1)

$= \frac{{26 + 38{\text{i}}}}{{40}} = \left( {\frac{{13 + 19{\text{i}}}}{{20}} = 0.65 + 0.95{\text{i}}} \right)$     A1

[2 marks]

attempt to use $\left| z \right| = \sqrt {{a^2} + {b^2}}$    (M1)

$\left| z \right| = \sqrt {\frac{{53}}{{40}}} \left( { = \frac{{\sqrt {530} }}{{20}}} \right)$ or equivalent      A1

Note: A1 is only awarded for the correct exact value.

[2 marks]

EITHER

arg $z$ = arg(2 + 7i) − arg(6 + 2i)      (M1)

OR

arg $z$ = arctan$\left( {\frac{{19}}{{13}}} \right)$         (M1)

THEN

arg $z$ = 0.9707 (radians) (= 55.6197 degrees)     A1

Note: Only award the last A1 if 4 decimal places are given.

[2 marks]