If $f(x) = \left| \begin{array}{ccc} \cos x & x & 1 \\ 2\sin x & x^2 & 2x \\ \tan x & x & 1 \end{array} \right|$,then find $\lim_{x \to 0} \frac{f'(x)}{x}$.

If $A(x) = \left| \begin{array}{ccc} 1 & 2 & 3 \\ x+1 & 2x+1 & 3x+1 \\ x^2+1 & 2x^2+1 & 3x^2+1 \end{array} \right|$, then $\int_0^1 A(x) \, dx$ equals

The rank of the following matrix $A$ is
$A = \begin{bmatrix} 1 & -2 & 3 & -4 \\ 2 & 9 & 4 & 5 \\ 4 & 5 & 10 & -3 \\ 1 & 11 & -1 & 9 \end{bmatrix}$

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 the points $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3)$ are collinear,then the rank of the matrix $\begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix}$ will always be less than

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 $3$.