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) = $

Let $f(x) = \begin{vmatrix} \cos x & x & 1 \\ 2 \sin x & x^2 & 2x \\ \tan x & x & 1 \end{vmatrix}$. Then $\lim_{x \to 0} \frac{f'(x)}{x} =$

The rank of $\left[\begin{array}{ccc}1 & -1 & 1 \\ 1 & 1 & -1 \\ -1 & 1 & 1\end{array}\right]$ is

If $y = \begin{vmatrix} f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c \end{vmatrix}$,prove that $\frac{dy}{dx} = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\ l & m & n \\ a & b & c \end{vmatrix}$.

The rank of the matrix $A = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 0 & 1 \\ 4 & 1 & 4 \end{bmatrix}$ is

Which of the following statements is false?
$1$. If $A$ is a skew-symmetric matrix of order $5 \times 5$,then the rank of $A$ is less than $5$.
$2$. If $P$ is a non-zero column matrix and $Q$ is a non-zero row matrix,then the rank of $PQ$ is $1$.
$3$. The rank of $\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 7 \end{bmatrix}$ is $2$.
$4$. If the lines $a_r x + b_r y + c_r = 0$ $(r = 1, 2, 3)$ are distinct and intersect at a point,then the rank of $\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$ is $3$.