$A$ small quantity of water of mass $m$ at temperature $\theta^{\circ} C$ is mixed with a large mass $M$ of ice which is at its melting point. If $s$ is the specific heat capacity of water and $L$ is the latent heat of fusion of ice,then the mass of ice melted is:

$50\, g$ of ice at $0\,^{\circ}C$ is dropped into a calorimeter containing $100\, g$ of water at $30\,^{\circ}C$. If the thermal capacity of the calorimeter is zero,then the amount of ice left in the mixture at equilibrium is ........ $g$.

The temperatures of equal masses of three different liquids $A, B$ and $C$ are $15^{\circ} C, 24^{\circ} C$ and $30^{\circ} C$ respectively. The resultant temperature when liquids $A$ and $B$ are mixed is $20^{\circ} C$ and when liquids $B$ and $C$ are mixed is $26^{\circ} C$. Then the ratio of specific heat capacities of the liquids $A, B$ and $C$ is

The temperatures of equal masses of three different liquids $x$,$y$,and $z$ are $10^{\circ}C$,$20^{\circ}C$,and $30^{\circ}C$ respectively. The temperature of the mixture when $x$ is mixed with $y$ is $16^{\circ}C$,and the temperature when $y$ is mixed with $z$ is $26^{\circ}C$. The temperature of the mixture when $x$ and $z$ are mixed will be ...... $^{\circ}C$.

$1 \,kg$ of ice at $-20^{\circ} C$ is mixed with $2 \,kg$ of water at $90^{\circ} C$. Assuming that there is no loss of energy to the environment,the final temperature of the mixture is ............ $^{\circ} C$. (Assume,latent heat of ice $= 334.4 \,kJ/kg$,specific heat of water and ice are $4.18 \,kJ \,kg^{-1} K^{-1}$ and $2.09 \,kJ \,kg^{-1} K^{-1}$,respectively.)

$A$ vessel contains $110 \, g$ of water. The heat capacity of the vessel is equivalent to $10 \, g$ of water. The initial temperature of water in the vessel is $10^{\circ}C$. If $220 \, g$ of hot water at $70^{\circ}C$ is poured into the vessel,the final temperature,neglecting radiation loss,will be........ $^{\circ}C$