IF ω is a cube root of unity but not equal to $1$ then minimum value of $∣ a + b w + c w^{2} ∣$ (where a ,b, c are integers but not all equal) is

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Solution

The correct option is C 1
Let y= $∣ a + b w + c w^{2} ∣$

for y to be minimum $y^{2}$ must be minimum.

$y^{2} =∣ a + b w + c w^{2} ∣^{2}$

$y^{2} = (a + b w + c w^{2}) (a + b w + c w^{2})$

$= \frac{1}{2} [(a - b)^{2} + (b - c)^{2} + (c - a)^{2}]$

Since a,b and c are not equal at a time so minimum value of $y^{2}$ occurs when any
two are same and third is differ by 1.

$\Rightarrow$ Minimum of y=1 (as $a, b, c$ are integers)

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IF ω is a cube root of unity but not equal to 1 then minimum value of ∣ a+bw+cw2∣ (where a ,b, c are integers but not all equal) is

IF ω is a cube root of unity but not equal to $1$ then minimum value of $∣ a + b w + c w^{2} ∣$ (where a ,b, c are integers but not all equal) is