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Question: If \(\omega\) is the cube root of unity, then \((3 + 5\omega + 3\omega^{2})^{2} + (3 + 3\omega + 5\...

If ω\omega is the cube root of unity, then

(3+5ω+3ω2)2+(3+3ω+5ω2)2(3 + 5\omega + 3\omega^{2})^{2} + (3 + 3\omega + 5\omega^{2})^{2}=

A

4

B

0

C

– 4

D

None of these

Answer

– 4

Explanation

Solution

(3+5ω+3ω2)2+(3+3ω+5ω2)2(3 + 5\omega + 3\omega^{2})^{2} + (3 + 3\omega + 5\omega^{2})^{2} =(3+3ω+3ω2+2ω)2+(3+3ω+3ω2+2ω2)2= (3 + 3\omega + 3\omega^{2} + 2\omega)^{2} + (3 + 3\omega + 3\omega^{2} + 2\omega^{2})^{2}

=(2ω)2+(2ω2)2=4ω2+4ω4=4(−1)=−4(∵1+ω+ω2=0,ω3=1)(2\omega)^{2} + (2\omega^{2})^{2} = 4\omega^{2} + 4\omega^{4} = 4( - 1) = - 4(\because 1 + \omega + \omega^{2} = 0,\omega^{3} = 1)