If `Delta_(1) =|{:(x, sin theta, cos theta),(-sin theta, -x, 1),(cos theta, 1,x):}|` and `Delta_(2) =|{:(x, sin 2theta, cos 2theta),(-sin 2theta, -x, 1),(cos 2theta, 1,x):}|, x ne 0`, then for all `theta in (0, pi/2)`, Then, `Delta_(1) + Delta_(2) =-2x^(k)`. The value of k is ________.

Answer

3

Explanation

Here, `Delta_(1) =|{:(x, sin theta, cos theta),(-sin theta, -x,1),(cos theta,1,x):}|`
Expanding the determinant, we get
`Delta_(1)=-x^(3)` (which is independent of `theta`)
Similarly, `Delta_(2), Delta_(2)=|{:(x, sin 2theta, cos 2theta),(-sin 2theta, -x,1),(cos 2theta,1,x):}|=-x^(3)`