$x$ moles of $CO(NH_2)_2$ are present in $1200 \ g$ of water. If the freezing point of the solution is $-4.02 \ ^oC$,calculate the value of $x$. Given $k_f \ (H_2O) = 1.86 \ K \ kg \ mol^{-1}$.

If the freezing point of an aqueous urea solution is $271.14 \ K$ at $1 \ \text{atm}$ pressure (given $K_f$ of water = $1.86 \ K \ kg/mol$),then what is the mole fraction of urea in this solution? (Freezing point of pure water is $273 \ K$)

Identify the concentration of the solution from the following such that the values of $\Delta T_{f}$ and $K_{f}$ are the same.

What is the value of $K_{f}$ if $30 \ g$ urea (molar mass $60$) dissolved in $0.5 \ dm^{3}$ of water decreases the freezing point by $0.15^{\circ}C$?

$A$ solution of a nonvolatile solute is obtained by dissolving $1 \ g$ in $100 \ g$ of solvent,which decreases its freezing point by $0.3 \ K$. Calculate the cryoscopic constant of the solvent if the molar mass of the solute is $60 \ g \ mol^{-1}$.

Pure water freezes at $273 \ K$ and $1 \ bar$. The addition of $34.5 \ g$ of ethanol to $500 \ g$ of water changes the freezing point of the solution. Use the freezing point depression constant of water as $2 \ K \ kg \ mol^{-1}$. The figures shown below represent plots of vapour pressure $(V.P.)$ versus temperature $(T)$. [molecular weight of ethanol is $46 \ g \ mol^{-1}$]. Among the following,the option representing the change in the freezing point is: