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

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

Answer 1

Answer: (C)

$1$

Given,

$z=a+bw+c{w}^2$

$=a+bw-c-cw$

$=(a-c)+(b-c)w$

$=(a-c)+(b-c)(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)$ [$\because w$ is cube root of unity]

$\therefore z=a-c-\frac{b}{2}+\frac{c}{2}+(b-c)\frac{\sqrt{3}}{2}i$

$|z|=\sqrt{{(a-\frac{b}{2}-\frac{c}{2})}^2+\frac{3}{4}(b-c{)}^2}$

let $a=b=k,c=k+1$

$=\sqrt{\frac{1}{2}(0{)}^2+(1{)}^2+(1{)}^2}=1$