Question

If a,b,c are integers not all equal and $\omega$ is a cube root of unity $(\omega \mp 1)$, then the minimum value of $\mid a+b\omega +c{\omega }^2\mid$ is

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

Answer 1

The correct option is B 1

a, b, c not all equal integess

$|a+b\omega +c{\omega }^2|\to$ Always positive

$1+\omega +{\omega }^2=0$

if $b=c\ne a$ then

$|a+b(\omega +{\omega }^2)|$

a can not equal to b

so $a=b\pm 1$ for minimum diff in them

the minimum value of

$=|b\pm 1+6(\omega +{\omega }^2)|$

$=|\pm 1|$

$=1$

B is correct