What is the product obtained from the dehydration of glycerol?

$CH_3-CH_2-CH=O$ $\xrightarrow[(2) H^{+}]{(1) \text{Tollen's reagent}} X$ $\xrightarrow[(2) \text{dry distillation}]{(1) Ca(OH)_2} Y$
$Y$ is:

$A$ liquid does not wet the solid surface if the angle of contact is

$A$ and $B$ are two radioactive substances whose half-lives are $1$ and $2$ years respectively. Initially, $10 \, g$ of $A$ and $1 \, g$ of $B$ are taken. The time (approximate) after which they will have the same quantity remaining is ........... $years$.

$A$ system is provided with $50 \ J$ of heat and the work done on the system is $10 \ J$. What is the change in internal energy of the system in joules?

$\lim _{x \rightarrow 0} \frac{\tan ^3 x-\sin ^3 x}{x^5}$ is equal to