Explanation

(c)
Let $x = |a + b\omega + c\omega^{2}|$
$\Rightarrow \ \ \ x^{2} = (a^{2} + b^{2} + c^{2} - ab - bc - ca)$
$\Rightarrow \ \ \ x^{2} = \frac{1}{2}\{(a - b)^{2} + (b - c)^{2} + (c - a)^{2}\}$ …(i)
Since $a,\ b,\ c$ are all integers but not all simultaneously equal
$\Rightarrow \ \ \$ If $a = b$, then $a \neq c$ and $b \neq c$
As, difference of integers=integer
$\Rightarrow \ \ \ (b - c)^{2} \geq 1$
[as minimum difference of two consecutive integers is ($\pm 1$)]
Also, $(c - a)^{2} \geq 1$
$\therefore$ From Eq. (i),
$x^{2} = \frac{1}{2}\lbrack(a - b)^{2} + (b - c)^{2} + (c - a)^{2}\rbrack$
$\geq \frac{1}{2}\lbrack 0 + 1 + 1\rbrack$
$\Rightarrow \ \ \ x^{2} \geq 1$
Hence, minimum value of $x$ is 1