Let $P=\left[\begin{array}{ccc}1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1\end{array}\right]$ and $I$ be the identity matrix of order $3$ . If $\left.Q=q_{i j}\right]$ is a matrix such that $P^{50}-Q=I$, then $\frac{q_{31}+q_{32}}{q_{21}}$ equals

If $a,b,c$ are distinct and rational numbers then $\left| {\begin{array}{*{20}{c}}
{\left( {{a^2} + {b^2} + {c^2}} \right)}&{ab + bc + ca}&{ab + bc + ca}\\
{ab + bc + ca}&{\left( {{a^2} + {b^2} + {c^2}} \right)}&{\left( {bc + ca + ab} \right)}\\
{ab + bc + ca}&{\left( {ab + bc + ca} \right)}&{\left( {{a^2} + {b^2} + {c^2}} \right)}
\end{array}} \right|$ is always 

If ${a^2} + {b^2} + {c^2} = - 2$ and $f(x) = \left| {\begin{array}{*{20}{c}}{1 + {a^2}x}&{(1 + {b^2})x}&{(1 + {c^2})x}\\{(1 + {a^2})x}&{1 + {b^2}x}&{(1 + {c^2})x}\\{(1 + {a^2})x}&{(1 + {b^2})x}&{1 + {c^2}x}\end{array}} \right|$ then $f(x)$ is a polynomial of degree

If $A$ and $B$ are $3 × 3$ matrices and $| A | \ne 0$, then which of the following are true?

The solutions of the equation $\left|\begin{array}{ccc}1+\sin ^{2} x & \sin ^{2} x & \sin ^{2} x \\ \cos ^{2} x & 1+\cos ^{2} x & \cos ^{2} x \\ 4 \sin 2 x & 4 \sin 2 x & 1+4 \sin 2 x\end{array}\right|=0,(0< x< \pi), \operatorname{are}$

If $D =\left| {\begin{array}{*{20}{c}}{{a^2}\, + \,\,1}&{ab}&{ac}\\{ba}&{{b^2}\, +\,\,1}&{bc}\\{ca}&{cb}&{{c^2}\, + \,\,1}\end{array}} \right|$ then $D =$