| Date | None Specimen | Marks available | 2 | Reference code | SPNone.2.hl.TZ0.4 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 4 | Adapted from | N/A |
Question
The complex number \(z = - \sqrt 3 + {\text{i}}\) .
Find the modulus and argument of z , giving the argument in degrees.
[2]
a.
Find the cube root of z which lies in the first quadrant of the Argand diagram, giving your answer in Cartesian form.
[2]
b.
Find the smallest positive integer n for which \({z^n}\) is a positive real number.
[2]
c.
Markscheme
\(\bmod (z) = 2,{\text{ }}\arg (z) = 150^\circ \) A1A1
[2 marks]
a.
\({z^{\frac{1}{3}}} = {2^{\frac{1}{3}}}(\cos 50^\circ + {\text{i}}\sin 50^\circ )\) (M1)
\( = 0.810 + 0.965{\text{i}}\) A1
[2 marks]
b.
we require to find a multiple of 150 that is also a multiple of 360, so by any method, M1
n = 12 A1
Note: Only award 1 mark for part (c) if n = 12 is based on \(\arg (z) = - 30\) .
[2 marks]
c.
Examiners report
[N/A]
a.
[N/A]
b.
[N/A]
c.
Syllabus sections
Topic 1 - Core: Algebra » 1.5 » Complex numbers: the number \({\text{i}} = \sqrt { - 1} \) ; the terms real part, imaginary part, conjugate, modulus and argument.
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