If $f(x) = \left| \begin{array}{ccc} -\sin x & 2 \sin 2x & 4 \cos^2 x \\ \cos x & 4 \sin^2 x & 2 \sin 2x \\ 0 & -\cos x & \sin x \end{array} \right|$, then $f\left(\frac{5\pi}{4}\right) + f'\left(\frac{5\pi}{4}\right) = $

If there exists a $k^{\text{th}}$ order non-singular submatrix in a matrix $P$ of order $m \times n$,then the rank $(\rho)$ of $P$

If $f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x \end{vmatrix}$,then the ratio $f^{\prime \prime}(x) : f^{\prime}(x) =$

Let $A=\begin{bmatrix} -1 & -2 & -3 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{bmatrix}$, $B=\begin{bmatrix} 1 & -2 \\ -1 & 2 \end{bmatrix}$ and $C=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$. If $a, b$ and $c$ respectively denote the ranks of $A, B$ and $C$, then the correct order of these numbers is:

If $a_1, a_2, a_3, \dots, a_n$ form a geometric progression,find the value of the determinant: $\left| \begin{array}{ccc} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{array} \right|$.

The rank of the matrix $A=\begin{bmatrix} 1 & 1 & 1 & 3 \\ 2 & 2 & -1 & 3 \\ 1 & 1 & -1 & 1 \end{bmatrix}$ is