Let the numbers $2, b, c$ be in an $A.P.$ and $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & b & c \\ 4 & b^2 & c^2 \end{bmatrix}$. If $\det(A) \in [2, 16]$,then $c$ lies in the interval:

Read the following mathematical statements carefully:
$I$. There can exist two triangles such that the sides of one triangle are all less than $1 \text{ cm}$ while the sides of the other triangle are all bigger than $10 \text{ m}$,but the area of the first triangle is larger than the area of the second triangle.
$II$. If $x, y, z$ are all different real numbers,then $\frac{1}{(x - y)^2} + \frac{1}{(y - z)^2} + \frac{1}{(z - x)^2} = \left( \frac{1}{x - y} + \frac{1}{y - z} + \frac{1}{z - x} \right)^2$.
$III$. $\log_3 x \cdot \log_4 x \cdot \log_5 x = (\log_3 x \cdot \log_4 x) + (\log_4 x \cdot \log_5 x) + (\log_5 x \cdot \log_3 x)$ is true for exactly one real value of $x$.
$IV$. $A$ matrix has $12$ elements. The number of possible orders it can have is $6$. Now indicate the correct alternative.

If $A = \begin{bmatrix} 1 + a^2 + a^4 & 1 + ab + a^2b^2 & 1 + ac + a^2c^2 \\ 1 + ab + a^2b^2 & 1 + b^2 + b^4 & 1 + bc + b^2c^2 \\ 1 + ac + a^2c^2 & 1 + bc + b^2c^2 & 1 + c^2 + c^4 \end{bmatrix}$ and $\det(A) = \det(4I)$,where $I$ is a $3 \times 3$ identity matrix,then $(a - b)^3 + (b - c)^3 + (c - a)^3$ can be equal to -

If $ A = \frac{1}{\pi} \begin{bmatrix} \sin^{-1}(\pi x) & \tan^{-1}(\frac{x}{\pi}) \\ \tan^{-1}(\frac{x}{\pi}) & \cot^{-1}(\pi x) \end{bmatrix} $ and $ B = \begin{bmatrix} -\cos^{-1}(\pi x) & \tan^{-1}(\frac{x}{\pi}) \\ \sin^{-1}(\frac{x}{\pi}) & -\tan^{-1}(\pi x) \end{bmatrix} $,then $ A - B $ is

Which of the following statements is correct about two square matrices $A$ and $B$ of the same order $n$?

The least positive integer $n$ such that $\left(\begin{array}{cc}\cos \frac{\pi}{4} & \sin \frac{\pi}{4} \\ -\sin \frac{\pi}{4} & \cos \frac{\pi}{4}\end{array}\right)^{n}$ is an identity matrix of order $2$ is