2016 H2 Mathematics Paper 2 Question 4

Complex Numbers

Answers

(a)

Out of syllabus
(bi)
Out of syllabus
(bii)
Smallest positive whole number n=7{n = 7}

Full solutions

(a)

Out of syllabus
(bi)
Out of syllabus
(bii)
arg⁡(w)=−π4\arg(w) = - \frac{\pi}{4}
arg⁡(w∗wn)=π2arg⁡(w∗)+arg⁡(wn)=π2+2kπ\begin{align*} \arg\left( w^* w^n \right) &= \frac{\pi}{2} \\ \arg\left( w^* \right) + \arg \left( w^n \right) &= \frac{\pi}{2} + 2k \pi \\ \end{align*}
where k∈Z{k \in \mathbb{Z}}
−(−π4)+n(−π4)=π2+2kπ−n(π4)=π4+2kπ−n=1+8kn=−1−8k\begin{gather*} -\left( - \frac{\pi}{4} \right) + n \left( - \frac{\pi}{4} \right) = \frac{\pi}{2} + 2k \pi \\ - n \left( \frac{\pi}{4} \right) = \frac{\pi}{4} + 2k \pi \\ -n = 1 + 8k \\ n = -1 - 8k \end{gather*}
For the smallest positive whole number value of n,{n, } k=−1{k=-1}
n=−1+8=7  ■\begin{align*} n &= -1 + 8 \\ &= 7 \; \blacksquare \end{align*}