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Question

If i=1, then 4+5(12+32i)334+3(12+32i)365 is equal to

A
1i3
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B
1+i3
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C
i3
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D
i3
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Solution

The correct option is C i3
ω=12+i32 [cube root of unity]4+5(12+32i)334+3(12+32i)365=4+5ω334+3ω365=4+5.(ω3)111.ω+3.(ω3)121.ω2=4+5ω+3ω2 [ω3=1]=1+3+2ω+3ω+3ω2=1+2ω+3(1+ω+ω2)=1+2ω+3×0 [1+ω+ω2=0]=1+(1+3i)=3i

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