What is $R_f$?

Benzoic acid is an organic compound. Its crude sample can be purified by crystallisation from hot water. What characteristic differences in the properties of benzoic acid and the impurity make this process of purification suitable?

Glycerol boils at $290\,^{\circ}C$ with slight decomposition. Impure glycerine can be purified by

Which of the following statements are correct?
$A$. Glycerol is purified by vacuum distillation because it decomposes at its normal boiling point.
$B$. Aniline is purified by steam distillation as it is immiscible in water.
$C$. Ethanol can be separated from an ethanol-water mixture by azeotropic distillation because it forms an azeotrope.
$D$. An organic compound is pure if the mixed $M.P.$ remains the same.
Choose the most appropriate answer from the options given below:

By the crystallisation of an impure compound,if the mother liquor becomes coloured,what shall be done to remove the colour?

When the $TLC$ solvent reaches a height of $10 \, cm$,compound $(A)$ reaches $8 \, cm$ and compound $(B)$ reaches $6 \, cm$. Calculate the $R_f$ values of $(A)$ and $(B)$.