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Let w=e^2i(pi)/3 and a,b,c,x,y,z be non-zero complex number satisfying a+b+c=x a+bw+cw^2=y a+bw^2+cw=z Then, the value of |x^2| +|y^2| +|z^2| /|a^2| +|b^2| +|c^2| is (where w denote omega)

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Solution: We have ,

$\Rightarrow E=xarx+yary+zarz/left | a^2 ight |+left | b^2 ight |+left | c^2 ight |\ \Rightarrow N^r=(a+b+c)(ara+arb+arc)+(a+bomega +comega ^2)(ara+arbomega ^2+arcomega )+(a+bomega ^2+comega )(ara+arbomega +arcomega^2 ).$

Simpliying and using the result $1+omega +omega ^2=0$ ,we get

$\ E=3(left | a ight |^2+left | b ight |^2+left | c ight |^2)/left | a ight |^2+left | b ight |^2+left | c ight |^2=3$

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Deependra Verma

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