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 $f(x) = \begin{vmatrix} 1 + \sin x + \sin 2x + \sin 3x & \frac{3 + \sin 2x}{2} & \frac{-2 + \sin 3x}{3} \\ 3 + 4 \sin x & \frac{3}{2} & \frac{4}{3} \sin x \\ 1 + \sin x & \frac{1}{2} \sin x & \frac{1}{3} \end{vmatrix}$,then $\int_0^{\pi / 2} (f(x) + f^{\prime}(x)) dx =$

Suppose $\left| \begin{array}{cc} f'(x) & f(x) \\ f''(x) & f'(x) \end{array} \right| = 0$ where $f(x)$ is a continuously differentiable function with $f'(x) \ne 0$ and satisfies $f(0) = 1$ and $f'(0) = 2$. Then the number of solution$(s)$ of the equation $f(x) = x^2$ is equal to:

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 system of equations $a_1 x + b_1 y + c_1 z = 0$,$a_2 x + b_2 y + c_2 z = 0$,and $a_3 x + b_3 y + c_3 z = 0$ has only the trivial solution,then the rank of the matrix $A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$ is:

If $f(x) = \left| \begin{array}{ccc} \cos(x+a+b) & \sin(x+a+b) & 10 \\ \cos(x+b+c) & \sin(x+b+c) & 10 \\ \cos(x+c+a) & \sin(x+c+a) & 10 \end{array} \right|$,then find the value of $f(2019)^{f(2020)} - f(2020)^{f(2019)}$.