If $f(x) = \left| \begin{array}{ccc} \cos x & 1 & 0 \\ 0 & 2 \cos x & 3 \\ 0 & 1 & 2 \cos x \end{array} \right|$,then $\lim_{x \rightarrow \pi} f(x)$ is equal to

For some $a, b$,let $f(x) = \left|\begin{array}{ccc} a+\frac{\sin x}{x} & 1 & b \\ a & 1+\frac{\sin x}{x} & b \\ a & 1 & b+\frac{\sin x}{x} \end{array}\right|, \quad x \neq 0$. If $\lim_{x \rightarrow 0} f(x) = \lambda + \mu a + \nu b$,then $(\lambda + \mu + \nu)^2$ is equal to:

If ${D_p} = \begin{vmatrix} p & 15 & 8 \\ p^2 & 35 & 9 \\ p^3 & 25 & 10 \end{vmatrix}$,then ${D_1} + {D_2} + {D_3} + {D_4} + {D_5} = $

Let for $i = 1, 2, 3$,$p_i(x)$ be a polynomial of degree $2$ in $x$,$p'_i(x)$ and $p''_i(x)$ be the first and second order derivatives of $p_i(x)$ respectively. Let $A(x) = \begin{bmatrix} p_1(x) & p'_1(x) & p''_1(x) \\ p_2(x) & p'_2(x) & p''_2(x) \\ p_3(x) & p'_3(x) & p''_3(x) \end{bmatrix}$ and $B(x) = [A(x)]^T A(x)$. Then the determinant of $B(x)$

If $f(x) = \left| \begin{array}{ccc} 2\cos^2 2x & \sin 2x & -\sin x \\ \sin 2x & 2\sin^2 x & \cos x \\ \sin x & -\cos x & 0 \end{array} \right|$,then the value of $\int_{0}^{\frac{\pi}{2}} f'(x) \,dx$ is equal to

The rank of the matrix $\begin{bmatrix} 4 & 2 & 1-x \\ 5 & k & 1 \\ 6 & 3 & 1+x \end{bmatrix}$ is $1$,then