The molar heat of formation of $NH_4NO_{3(s)}$ is $-367.54 \ kJ$ and those of $N_2O_{(g)}$ and $H_2O_{(l)}$ are $+81.46 \ kJ$ and $-285.78 \ kJ$ respectively at $25 \ ^oC$ and $1.0 \ atm$ pressure. Calculate $\Delta U$ at $25 \ ^oC$ for the reaction:
$NH_4NO_{3(s)} \to N_2O_{(g)} + 2H_2O_{(l)}$ (in $kJ$)

The height of the cone of maximum volume inscribed in a sphere of radius $R$ is

The major product of the following reaction is

Three rods $AB$,$BC$,and $BD$ made of the same material and having the same cross-section have been joined as shown in the figure. The ends $A$,$C$,and $D$ are held at temperatures of $20^{\circ}C$,$80^{\circ}C$,and $80^{\circ}C$ respectively. If each rod is of the same length,then the temperature at the junction $B$ of the three rods is: (in $^{\circ}C$)

Removal of shoot tips is a very useful technique to boost the production of tea-leaves. This is because

$\cos ^{-1}\left(\frac{-1}{2}\right)-2 \sin ^{-1}\left(\frac{1}{2}\right)+3 \cos ^{-1}\left(\frac{-1}{\sqrt{2}}\right)-4 \tan ^{-1}(-1)$ equals: