If for $x \in \left( 0, \frac{1}{4} \right)$,the derivative of $\tan^{-1} \left( \frac{6x\sqrt{x}}{1 - 9x^3} \right)$ is $\sqrt{x} \cdot g(x)$,then $g(x)$ equals:

If $p \rightarrow (\sim p \vee q)$ is false,then the truth values of $p$ and $q$ are,respectively

$A$ body cools from $60\,^oC$ to $50\,^oC$ in $10\,minutes$. If the room temperature is $25\,^oC$ and assuming Newton's law of cooling to hold good,the temperature of the body at the end of the next $10\,minutes$ will be ......... $^oC$.

If the point $P(1,3)$ undergoes the following transformations successively:
$(i)$ Reflection with respect to the line $y=x$.
(ii) Translation through $3$ units along the positive direction of the $X$-axis.
(iii) Rotation through an angle of $\frac{\pi}{6}$ about the origin in the clockwise direction.
Then,the final position of the point $P$ is

The ratio of the mass percentages of $C$ and $H$,and $C$ and $O$ of a saturated acyclic organic compound $X$ are $4:1$ and $3:4$ respectively. Then,the moles of oxygen gas required for the complete combustion of two moles of organic compound $X$ is:

If $\omega$ is a complex cube root of unity,then $\sin \left\{\left(\omega^{10}+\omega^{23}\right) \pi-\frac{\pi}{4}\right\}$ is equal to