Matrices 1

I think this problem is a beautiful combination of Application of Derivatives(Rolle's Theorem) and Determinants.

As I mentioned above one can think of Rolle's seeing that $f(a) = 0$ . It is so because $C_{1}$ and $C_{2}$ are identical.

Also $f(b) = 0$ . It is so because $C_{1}$ and $C_{3}$ are identical.

Also the function $f(x)$ is a polynomial function of $\sin x, \cos x, \tan x$ hence, it will be continuous and differentiable in $\Big( 0,\dfrac{\pi}{2} \Big)$ and so in $[a,b]$ .

Now according to the statement of Rolle's theorem, if a function $f(x)$ if continuous in $[a,b]$ and diffrentiable in $(a,b)$ then there at least single root of $f'(x)$ .

Hence is the answer $1$ .