Question

The minimum value of $| a + b\omega + c \omega^2|$ , where $ a, b$ and $c$ are all not equal integers and $\omega \, (\ne 1)$ is a cube root of unity, is

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

Answer 1

Answer: (C)

Let $z=|a+b\omega+c\omega^2|$
$|a+b\omega+c\omega^2|^2=(a^2+b^2+c^2-ab-bc-ca)$
or $z^2=\frac{1}{2} \{(a-b)^2+(b-c)^2+(c-a)^2\} ...(i)$
Since, $a, b, c$ are all integers but not all simultaneously equal.
$\Rightarrow $ If $a = b$ then $a\ne c$ and $b \ne c$
Because difference of integers = integer
$\Rightarrow (b-c)^2 \ge \, 1$ {as minimum difference of two consecutive integers is ($\pm 1$)} also $(c - a)^2 \ge 1$
and we have taken $a = b \Rightarrow (a-b)^2=0$
From Eq. (i), $z^2 \ge \frac{1}{2}(0+1+1)$
$\Rightarrow z^2 \ge 1$
Hence, minimum value of$ |z |$ is $1$