Cosider the cubic equation : `x^3-(1+costheta+sintheta)x^2+(costhetasintheta+costheta+sintheta)x-sinthetacostheta=0` whose roots are `x_1,x_2,x_3`. The value of `(x_1)^2+(x_2)^2+(x_3)^2` equals

Options

A. 1
B. 2
C. `2 cos theta`
D. `sin theta (sin theta+ cos theta)`

Answer

B

Explanation

`x^(3)-(1+cos theta + sin theta) x^(2) +(cos theta sin theta + cos theta + sin theta)x-sin theta cos theta=0`
Given cubic function is
`f(x)=(x-1)(x-cos theta) (x- sin theta)`
Therefore, roots are `1, sin theta`, and `cos theta`.
Hence, `x_(1)^(2)+x_(2)^(2)+x_(3)^(2)=1+sin^(2) theta+cos^(2) theta=2`