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

The Correct Option is 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, care all integers but not all simultaneously
equal.
$\Rightarrow $ If a = b then a$\ne$c and b $\ne$ c
Because difference of integers = integer
$\Rightarrow \, \, (b-c)^2 \ge \, 1$ 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 E (i), $z^2 \ge \frac{1}{2}(0+1+1)$
$\Rightarrow \, \, \, \, \, \, \, \, \, \, \, \, \, \, z^2 \ge 1$
Hence, minimum value of |z | is 1