Question

Let $\omega$ be a cube root of unity not equal to 1. Then the maximum possible value of $|a+b\omega +c{\omega }^2|$ where a, b, c $ϵ$ {+1,-1} is

• A. $0$
• B. $2$
• C. $\sqrt{3}$
• D. $1+\sqrt{3}$

Answer 1

Answer: (B)

$2$

${\omega }^3=1$

$\therefore {\omega }^3-1=0$ or $(\omega -1)({\omega }^2+\omega +1)=0$

Since $\omega \ne 1,\omega =\frac{-1+i\sqrt{3}}{2},{\omega }^2=\frac{-1-i\sqrt{3}}{2}$

$|a+b\omega +c{\omega }^2|=|a-\frac{(b+c)}{2}+i\left(\frac{\sqrt{3}}{2}\right)(b-c\left)\right|$

Square of the modulus is given by ${\left(\frac{2a-b-c}{2}\right)}^2+{\left(\frac{\sqrt{3}(b-c)}{2}\right)}^2$

This can be maximised by substituting $a=1,b=1,c=-1$ or $a=-1,b=1,c=1$

Both the substitutions yield the same result, of 4.

$\Rightarrow$ maximum modulus $=2$