$100 \ g$ each of two solutions,one containing $20 \%$ by mass of $NaOH$ and the other $40 \%$ by mass of $NaOH$,are mixed. The density of the resulting solution is $1.25 \ g/mL$. Find the new $\% \ w/v$ of $NaOH$ in the final solution.

Two elements $A$ and $B$ form compounds having formula $AB_{2}$ and $AB_{4}$. When dissolved in $20 \ g$ of benzene $(C_{6}H_{6})$,$1 \ g$ of $AB_{2}$ lowers the freezing point by $2.3 \ K$ whereas $1.0 \ g$ of $AB_{4}$ lowers it by $1.3 \ K$. The molar depression constant for benzene is $5.1 \ K \ kg \ mol^{-1}$. Calculate atomic masses of $A$ and $B$.

$A$ saturated solution of $Ag_2SO_4$ shows a $0.003 \ K$ rise in boiling point. Calculate the $K_{sp}$ of $Ag_2SO_4$. Given: $K_b = 5 \ K \ kg \ mol^{-1}$ and $1 \ m = 1 \ M$.

Assertion : If one component of a solution obeys Raoult's law over a certain range of composition,the other component will not obey Henry's law in that range.
Reason : Raoult's law is a special case of Henry's law.

At $T(K)$, $2$ moles of liquid $A$ and $3$ moles of liquid $B$ are mixed. The vapour pressure of the ideal solution formed is $320 \ mm \ Hg$. At this stage, one mole of $A$ and one mole of $B$ are added to the solution. The vapour pressure is now measured as $328.6 \ mm \ Hg$. The vapour pressures of pure $A$ and pure $B$ (in $mm \ Hg$) are respectively:

$A$ solution of urea (molar mass $60 \, g \, mol^{-1}$) boils at $100.18 \, ^oC$ at atmospheric pressure. If $K_f$ and $K_b$ for water are $1.86$ and $0.512 \, K \, kg \, mol^{-1}$ respectively,the above solution will freeze at ........... $^oC$.