Complex Numbers And Quadratic Equations question 258
Question: If $ \omega (\ne 1) $ is a cube root of unity and $ {{(1+\omega )}^{7}}=A+B\omega $ , then $ A $ and $ B $ are respectively, the numbers [IIT 1995]
Options:
A) 0, 1
B) 1, 0
C) 1, 1
D) $ -1,\ 1 $
Show Answer
Answer:
Correct Answer: C
Solution:
$ {{(1+\omega )}^{7}}=A+B\omega \Rightarrow {{(-{{\omega }^{2}})}^{7}}=A+B{{\omega }^{2}}$
Þ $ =\frac{\cos 4\theta +i\sin 4\theta }{\cos 4\theta -i\sin 4\theta } $
Þ $ {{\omega }^{2}}.{{\omega }^{12}}=-A-B\omega $
Þ $ A+B\omega +{{\omega }^{2}}=0 $
Þ $ A=1,B=1 $ $ (\because 1+\omega +{{\omega }^{2}}=0) $
BETA
