InterviewSolution
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InterviewSolution
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If w is a non real cube root of unity, then minimum value of `| a+ bw + c w^2|`is ( If a,b,c are not equal): (A) 0 (B) `(sqrt3)/2` (C) 1 (D) 2 |
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Answer» Let `z = a+b omega + c omega^2` We know, `1+ omega +omega^2 = 0` `:. omega^2 = -1-omega` `:. z = a+b omega+c (-1-omega)` ` z = a+ b omega-c - c omega` `z = (a-c)+(b-c) omega` Putting value of `omega`, `z = (a-c) +(b-c)(-1/2+sqrt3/2i)` `z = (a-b/2-c/2)+sqrt3/2(b-c)i` We know, `|z| = sqrt(x^2+y^2` So, here,`|z| = sqrt((a-b/2-c/2)^2+3/4(b-c)^2)` `|z| = sqrt(1/2[(a-b)^2+(b-c)^2+(c-a)^2])` To find minimum value of `|z|`, we will put, `a = b = k`and `c = k+1` Then,`|z| = sqrt(1/2(0+1^2+1^2)) = 1` So, minimum value of `|z|` will be `1`. |
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