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. 3
B. 4
C. 5
D. 6

Answer

C

Explanation

Now if `1=sin theta`, we get `theta=pi//2`
If `1= cos theta`, then `theta=0, 2pi` and if `sin theta=cos theta`, we get `tan theta=1`.
therefore,
`theta=pi/4, (5pi)/4`
Therefore, the number of values of `theta` in `[0, 2pi]` is 5.