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If isqrt(-1), then 4+5(-(1)/(2)+(isqrt(3...

If `isqrt(-1)`, then `4+5(-(1)/(2)+(isqrt(3))/(2))^(334)+3(-(1)/(2)+(isqrt(3))/(2))^(365)` is

A

`1- isqrt(3)`

B

`-1+ I sqrt(3)`

C

`I sqrt(3)`

D

`-I sqrt(3)`

Text Solution

Verified by Experts

The correct Answer is:
C
We know that,
`w =-1/2+sqrt(3)/2i`
`therefore 4+5(-1/2+(isqrt(3))/(2))^334+3(-1/2+(isqrt(3))/2)^365`
`=4+5 omega^(334)+3 omega^(365)`
`=4+5.(omega^3)^111.omega+3.(omega^3)^121.omega^2`
`=4+5 omega + 3 omega^2" "[ because omega^3=1]`
`1+3+2 omega +3 omega+3 omega^2`
`=1+2 omega+3 (1+ omega+omega^2)=1+2 omega + 3xx 0`
`[ because 1+ omega+ omega^2=0]`
`=1+(-1 + sqrt(3i))=sqrt(3i)`
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