Consider the cubic equation `x^(3)-(1+ cos theta+sin theta)x^(2)+(cos theta sin theta +cos theta + sin theta) x- sin theta cos theta=0` whose roots are `x_(1), x_(2)` and `x_(3)`.
The greatest possible difference between two of the roots if `theta in [0, 2pi]` is

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`