Question
Given the complex numbers \({z_1} = 1 + 3{\text{i}}\) and \({z_2} = – 1 – {\text{i}}\).
Write down the exact values of \(\left| {{z_1}} \right|\) and \(\arg ({z_2})\).[2]
Find the minimum value of \(\left| {{z_1} + \alpha{z_2}} \right|\), where \(\alpha \in \mathbb{R}\).[5]
Answer/Explanation
Markscheme
\(\left| {{z_1}} \right| = \sqrt {10} ;{\text{ }}\arg ({z_2}) = – \frac{{3\pi }}{4}{\text{ }}\left( {{\text{accept }}\frac{{5\pi }}{4}} \right)\) A1A1
[2 marks]
\(\left| {{z_1} + \alpha{z_2}} \right| = \sqrt {{{(1 – \alpha )}^2} + {{(3 – \alpha )}^2}} \) or the squared modulus (M1)(A1)
attempt to minimise \(2{\alpha ^2} – 8\alpha + 10\) or their quadratic or its half or its square root M1
obtain \(\alpha = 2\) at minimum (A1)
state \(\sqrt 2 \) as final answer A1
[5 marks]