WBJEE · Maths · Complex Number
Let \(\omega(\neq 1)\) be a cubic root of unity. Then the minimum value of the set \(\left\{\left|a+b \omega+c \omega^2\right|^2 ; a, b, c\right.\) are distinct non-zero integers} equals
- A \(15\)
- B \(5\)
- C \(3\)
- D \(4\)
Answer & Solution
Correct Answer
(C) \(3\)
Step-by-step Solution
Detailed explanation
\(\left|a+b \omega+c \omega^2\right|^2=a^2+b^2+c^2-a b-b c-c a\) \(\begin{aligned} & =\frac{1}{2}\left[(a-b)^2+(b-c)^2+(c-a)^2\right] \\ & \therefore \operatorname{Min}=\frac{1+1+4}{2}=3 \quad(a, b, c \in\{1,2,3\}) \end{aligned}\)
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