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]