$\frac{1}{e^{3x}}(e^x + e^{5x}) = a_0 + a_1x + a_2x^2 + \ldots$
$\Rightarrow 2a_1 + 2^3a_3 + 2^5a_5 + \ldots$ is equal to

Consider the following reaction,
$C_2H_5Cl + AgCN \xrightarrow{EtOH / H_2O} \underline{X} \text{ (major)}$
Which one of the following statements is true for $\underline{X}$?
$(I)$ It gives propionic acid on hydrolysis
$(II)$ It has an ester functional group
$(III)$ It has a nitrogen linked to ethyl carbon
$(IV)$ It has a cyanide group

The product of the lengths of the perpendiculars drawn from the point $(-1, 5)$ to the pair of lines $2x^2 - xy - 3y^2 + 6x + y + 4 = 0$ is

Doping silicon with boron produces a

Chemical $A$ is used for water softening to remove temporary hardness. $A$ reacts with sodium carbonate to produce caustic soda. When $CO_2$ is passed through a solution of $A$,it turns milky. What is the chemical formula of $A$?

$A$ micelle formed during the cleansing action by soap is