If $\log _{27}(\log _3 x) = \frac{1}{3}$, then the value of $x$ is

$x = \log \left( \frac{1}{y} + \sqrt{1 + \frac{1}{y^2}} \right) \Rightarrow y$ is equal to

If ${}^n P_r = 30240$ and ${}^n C_r = 252$,then the ordered pair $(n, r)$ is equal to

An element $P$ has atomic number $56$. What will be the formula of its halide?

Which of the following sets of quantum numbers is correct for an electron in a $4f$ orbital?

If $A$ is a square matrix such that $A(\operatorname{adj} A) = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix}$,then $\operatorname{det}(\operatorname{adj} A)$ is equal to