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The minimum value of |alpha bomega+comeg...

The minimum value of `|alpha bomega+comega^2|` , where a, b and c are all not equal integer and `omega( ne 1)` is a cube root of unity , is

A

`sqrt(3)`

B

`1/2`

C

1

D

0

Text Solution

Verified by Experts

The correct Answer is:
C
Let `z|alpha+bomega +comega^2|`
`rArr z^2 = |alpha + b omega+ comega^2|=(a^2 + b^2 +c^2-ab-bc -ca)or z^2=1/2{(a-b)^2+(b-c)^2+(c-b)^2}`
since a,b,c are all intergers but not all simulataneously equal
`rArr ` If a=b then `a ne c and be ne c `
Because difference of intergerse = interger
` rArr (b-c)^2 le 1 ` and we have taken `a= b rArr (a -b)^2 =0`
From Eq.(i) `z^2 ge 1/2 (0+1+1)`
`rArr z^2 ge 1`
Hence minimum value of |Z| is 1
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