IBDP Maths analysis and approaches Topic: SL 2.7 :Solving quadratic equations using the quadratic formula HL Paper 1

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]

a.

Find the minimum value of \(\left| {{z_1} + \alpha{z_2}} \right|\), where \(\alpha \in \mathbb{R}\).[5]

b.
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]

a.

\(\left| {{z_1} + \alpha{z_2}} \right| = \sqrt {{{(1 – \alpha )}^2} + {{(3 – \alpha )}^2}} \) 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]

b.
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