The number of real values of $x$ at which the function $f(x) = \left| \begin{array}{ccc} 1 & |x| & x^2 \\ 1 & |x-1| & (x-1)^2 \\ 1 & |x-2| & (x-2)^2 \end{array} \right|$ is not differentiable is

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:

The determinant $\left| \begin{array}{ccc} 4 + x^2 & -6 & -2 \\ -6 & 9 + x^2 & 3 \\ -2 & 3 & 1 + x^2 \end{array} \right|$ for $x \neq 0$ is not divisible by:

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:

The determinant $\left| \begin{array}{ccc} \cos(\theta + \phi) & -\sin(\theta + \phi) & \cos 2\phi \\ \sin \theta & \cos \theta & \sin \phi \\ -\cos \theta & \sin \theta & \cos \phi \end{array} \right|$ is :

If $A = \begin{vmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{vmatrix}$ and $B = \begin{vmatrix} x & 1 \\ 1 & x \end{vmatrix}$,then $\frac{dA}{dx}$ is equal to