Complex Numbers And Quadratic Equations question 306
Question: If $ \omega $ is a cube root of unity but not equal to 1 then minimum value of $ |a+b\omega +c{{\omega }^{2}}| $ (where a, b, c are integers but not all equal) is [IIT Screening 2005]
Options:
A) 0
B) $ \frac{\sqrt{3}}{2} $
C) 1
D) 2
Show Answer
Answer:
Correct Answer: C
Solution:
Let $ y=|a+b\omega +c{{\omega }^{2}}| $ for y to be minimum $ y^{2} $ must be minimum. $ y^{2}=|a+b\omega +c{{\omega }^{2}}{{|}^{2}} $ $ y^{2}=(a+b\omega +c{{\omega }^{2}})(a+b\omega +c{{\omega }^{2}}) $ = $ \frac{1}{2}[{{(a-b)}^{2}}+{{(b-c)}^{2}}+{{(c-a)}^{2}}] $ Since a, b and c are not equal at a time so minimum value of $ y^{2} $ occurs when any two are same and third is differ by 1. Þ Minimum of $ y=1 $ (as $ a,b,c $ are integers)
BETA
