If `z_1, z_2 in C , z_1^2 in R , z_1(z_1^2-3z_2^2)=2` and `z_2(3z_1^2-z_2^2)=11`, then the value of `z_1^2+z_2^2` is

Options

A. 10
B. 12
C. 5
D. 8

Answer

C

Explanation

We have,
`z_(1)(z_(1)^(2)-3z_(2)^2) = 2 " "(1)`
`z_(2)(3z_(1)^(2)-z_(2)^(2)) = 11`
multiplying (2) by I and adding it to (1), we get
`z_(1)^(3) - 3z_(2)^(2)z_(1) + i(3z_(1)^(2)z_(2) - z_(2)^(3)) = 2 +11i`
`rArr (z_(1) +iz_(2))^(3) = 2 + 11i" "(3)`
Multiply (2) by i and subtracting it form (1), we get
`z_(1)^(3) - 3z_(2)^(2)z_(1) - i(3z_(1)^(2)z_(2)-z_(2)^(2)) = 2- 11i`
`rArr (z_(1) + iz_(2))^(3) = 2+ 11i`
Multiplying (3) and (4), we get
`(z_(1)^(2) + z_(2)^(2))^(3) = 2^(2) - 121i^(2) = 4 + 121 = 125`
`rArr z_(1)^(2) + _(2)^(2) = 5`