| Date | May 2017 | Marks available | 2 | Reference code | 17M.1.hl.TZ1.2 |
| Level | HL only | Paper | 1 | Time zone | TZ1 |
| Command term | Find | Question number | 2 | Adapted from | N/A |
Question
Consider the complex numbers \({z_1} = 1 + \sqrt 3 {\text{i, }}{z_2} = 1 + {\text{i}}\) and \(w = \frac{{{z_1}}}{{{z_2}}}\).
By expressing \({z_1}\) and \({z_2}\) in modulus-argument form write down the modulus of \(w\);
By expressing \({z_1}\) and \({z_2}\) in modulus-argument form write down the argument of \(w\).
Find the smallest positive integer value of \(n\), such that \({w^n}\) is a real number.
Markscheme
\({z_1} = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)\) and \({z_2} = \sqrt 2 {\text{cis}}\left( {\frac{\pi }{4}} \right)\) A1A1
Note: Award A1A0 for correct moduli and arguments found, but not written in mod-arg form.
\(\left| w \right| = \sqrt 2 \) A1
[3 marks]
\({z_1} = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)\) and \({z_2} = \sqrt 2 {\text{cis}}\left( {\frac{\pi }{4}} \right)\) A1A1
Note: Award A1A0 for correct moduli and arguments found, but not written in mod-arg form.
\(\arg w = \frac{\pi }{{12}}\) A1
Notes: Allow FT from incorrect answers for \({z_1}\) and \({z_2}\) in modulus-argument form.
[1 mark]
EITHER
\(\sin \left( {\frac{{\pi n}}{{12}}} \right) = 0\) (M1)
OR
\(\arg ({w^n}) = \pi \) (M1)
\(\frac{{n\pi }}{{12}} = \pi \)
THEN
\(\therefore n = 12\) A1
[2 marks]
