A complex number z is given by \(z = \frac{{a + {\text{i}}}}{{a - {\text{i}}}},{\text{ }}a \in \mathbb{R}\).

(a)     Determine the set of values of a such that

          (i)     z is real;

          (ii)     z is purely imaginary.

(b)     Show that \(\left| z \right|\) is constant for all values of a.

(a)     \(\frac{{a + {\text{i}}}}{{a - {\text{i}}}} \times \frac{{a + {\text{i}}}}{{a + {\text{i}}}}\)     M1

\( = \frac{{{a^2} - 1 + 2a{\text{i}}}}{{{a^2} + 1}}{\text{ }}\left( { = \frac{{{a^2} - 1}}{{{a^2} + 1}} + \frac{{2a}}{{{a^2} + 1}}{\text{i}}} \right)\)     A1

          (i)     z is real when \(a = 0\)     A1

          (ii)     z is purely imaginary when \(a =  \pm 1\)     A1

Note:     Award M1A0A1A0 for \(\frac{{{a^2} - 1 + 2a{\text{i}}}}{{{a^2} - 1}}{\text{ }}\left( { = 1 + \frac{{2a}}{{{a^2} - 1}}{\text{i}}} \right)\) leading to \(a = 0\) in (i).

[4 marks]

(b)     METHOD 1

attempting to find either \(\left| z \right|\) or \({\left| z \right|^2}\) by expanding and simplifying

eg \({\left| z \right|^2} = \frac{{{{\left( {{a^2} - 1} \right)}^2} + 4{a^2}}}{{{{\left( {{a^2} + 1} \right)}^2}}} = \frac{{{a^4} + 2{a^2} + 1}}{{{{\left( {{a^2} + 1} \right)}^2}}}\)     M1

\( = \frac{{{{\left( {{a^2} + 1} \right)}^2}}}{{{{\left( {{a^2} + 1} \right)}^2}}}\)

\({\left| z \right|^2} = 1 \Rightarrow \left| z \right| = 1\)     A1

METHOD 2

\(\left| z \right| = \frac{{\left| {a + {\text{i}}} \right|}}{{\left| {a - {\text{i}}} \right|}}\)     M1

\(\left| z \right| = \frac{{\sqrt {{a^2} + 1} }}{{\sqrt {{a^2} + 1} }} \Rightarrow \left| z \right| = 1\)     A1

[2 marks]

Total [6 marks]

Part (a) was reasonably well done. When multiplying and dividing by the conjugate of \(a - {\text{i}}\), some candidates incorrectly determined their denominator as \({a^2} - 1\).

In part (b), a significant number of candidates were able to correctly expand and simplify \(\left| z \right|\) although many candidates appeared to not understand the definition of \(\left| z \right|\).