If $n(A) = 3$,$n(B) = 6$ and $A \subseteq B$,then the number of elements in $A \cup B$ is:

In a $\triangle ABC$,if $3a = b + c$,then $\cot \frac{B}{2} \cot \frac{C}{2}$ is equal to :

Which one of the following compounds gives aspirin on reacting with acetic anhydride in the presence of $H_2SO_4$?

Heating phthalic anhydride and concentrated sulfuric acid with which of the following yields phenolphthalein?

The maximum value of ${x^4}{e^{ - {x^2}}}$ is

An ideal gas is subjected to a cyclic process involving four thermodynamic states. The amounts of heat $(Q)$ and work $(W)$ involved in each of these states are:
$\begin{aligned} & Q_1=6000 \text{ J}, \quad Q_2=-5500 \text{ J}, \quad Q_3=-3000 \text{ J}, \quad Q_4=3500 \text{ J} \\ & W_1=2500 \text{ J}, \quad W_2=-1000 \text{ J}, \quad W_3=-1200 \text{ J}, \quad W_4=x \text{ J} \end{aligned}$
The ratio of the net work done by the gas to the total heat absorbed by the gas is $\eta$. The values of $x$ and $\eta$ respectively are: