Consider a homogeneous system of three linear equations in three unknowns represented by $AX=O$. If $X=\left[\begin{array}{c}l \\ m \\ 0\end{array}\right]$, where $l \neq 0, m \neq 0, l, m \in \mathbb{R}$, represents an infinite number of solutions of this system, then the rank of $A$ is:

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 $A = \begin{bmatrix} 1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6 \end{bmatrix}$ and the rank of $A$ is $2$,then the value of $x$ is equal to

$f(x) = \left| \begin{array}{ccc} x^3 & x^2 & 3x^2 \\ 1 & -6 & 4 \\ p & p^2 & p^3 \end{array} \right|$,where $p$ is a constant,then $\frac{d^3f(x)}{dx^3}$ is

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

If $y(x) = \left| \begin{array}{ccc} \sin x & \cos x & \sin x + \cos x + 1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{array} \right|$,$x \in R$,then $\frac{d^2 y}{d x^2} + y$ is equal to