Date | May 2021 | Marks available | 1 | Reference code | 21M.2.AHL.TZ1.8 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 1 |
Command term | Find | Question number | 8 | Adapted from | N/A |
Question
Consider the complex numbers z=2(cosπ5+i sinπ5) and w=8(cos2kπ5-i sin2kπ5), where k∈ℤ+.
Suppose that zw ∈ℤ.
Find the modulus of zw.
[1]
a.
Find the argument of zw in terms of k.
[2]
b.
Find the minimum value of k.
[3]
c.i.
For the value of k found in part (i), find the value of zw.
[1]
c.ii.
Markscheme
(|zw|=)16 A1
[1 mark]
a.
attempt to find arg(z)+arg(w) (M1)
arg(zw)=arg(z)+arg(w)
=π5-2kπ5(=(1-2k)π5) A1
[2 marks]
b.
zw ∈ℤ⇒arg(zw) is a multiple of π (M1)
⇒1-2k is a multiple of 5 (M1)
k=3 A1
[3 marks]
c.i.
zw =16(cos(-π)+i sin(-π))
-16 A1
[1 mark]
c.ii.
Examiners report
[N/A]
a.
[N/A]
b.
[N/A]
c.i.
[N/A]
c.ii.