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Let z = ((sqrt(3))/(2) + (i)/(2))^(5)+((...

Let `z = ((sqrt(3))/(2) + (i)/(2))^(5)+((sqrt(3))/(2)-(i)/(2))^(5)`. If R(z) and I(z), respectively, denote the real and imaginary parts of z, then

A

`R(z) gt 0 and I(z) gt 0`

B

`I(z)=0`

C

`R(z) lt 0 and I(z) gt 0`

D

`R(z)-3`

Text Solution

Verified by Experts

The correct Answer is:
B
Given `z= (sqrt(3)/2+ i/ 2)^5+(sqrt(3)/2-i/2)^5`
`because ` Euler's form of
`(sqrt(3)/2+i/2=( cos""pi/6+ I sin pi/6 ) = e^i(pi//6)`
`sqrt(3)/2- i/2= cos ((-pi )/(6)+isin ""pi/6)=e^(i(pi//6))`
So, `z= (e^(i pi//6))^5+(e^(-ipi//6))^5=e^(i(5pi)/6)+e^(-i(5pi)/6)`
`=(cos""(5pi)/6+ i sin ""(5pi)/6)+(cos ""(5pi)/6- isin "" (5pi)/6)`
`= 2 cos""(5 pi)/6 " "[ because e^(i theta) = cos theta + i sin theta]`
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