यदि `i = sqrt(-1)` हो, तो ` 4 + 5 (-(1)/(2) + (isqrt(3))/(2))^(334) + 3(-(1)/(2) + (isqrt(3))/(2))^(365)` बराबर है

Options

A. `1-isqrt(3)`
B. `-1+isqrt(3)`
C. `isqrt(3)`
D. `-isqrt(3)`

Answer

C

Explanation

यहाँ, `omega = - (1)/(2) + (sqrt(3))/(2)i`
`therefore 4+5 (-(1)/(2) + (isqrt(3))/(2))^(334) + 3(-(1)/(2) + (isqrt(3))/(2))^(365)`
` = 4 + 5omega^(334) + 3omega^(365)`
`= 4+ 5 (omega^(3))^(111) * omega + 3 (omega^(3))^(121) omega^(2)`
`=4 + 5omega + 3omega^(2) " " [because omega^(3) = 1]`
`= 1+ 3 + 2omega + 3omega + 3omega^(2)`
`= 1 + 2omega + 3(1 + omega + omega^(2))`
`=1 + 2omega + 3 xx 0 " " [because 1 + omega + omega^(2) = 0]`
`= 1 + (-1 + sqrt(3)i) =sqrt(3) i`