Thermal expansion due to an increase in temperature...

$A$ piece of ice (heat capacity = $2100 \text{ J kg}^{-1} \text{ }^\circ\text{C}^{-1}$ and latent heat = $3.36 \times 10^5 \ J kg^{-1}$) of mass $m$ grams is at $-5^oC$ at atmospheric pressure. It is given $420 \ J$ of heat so that the ice starts melting. Finally, when the ice-water mixture is in equilibrium, it is found that $1 \ gm$ of ice has melted. Assuming there is no other heat exchange in the process, the value of $m$ is ...... $gm$.

Steam is passed into $22\, g$ of water at $20^{\circ}C$. The mass of water that will be present when the water acquires a temperature of $90^{\circ}C$ is ............ $g$ (Latent heat of steam is $540\, cal/g$) (in $, g$)

$Assertion :$ In the pressure-temperature $(P-T)$ phase diagram of water,the slope of the melting curve is found to be negative.
$Reason :$ Ice contracts on melting to water.

$A$ geyser heats water flowing at the rate of $3.0 \, L/min$ from $27 \, ^{\circ}C$ to $77 \, ^{\circ}C$. If the geyser operates on a gas burner,what is the rate of consumption (in $g/min$) of the fuel if its heat of combustion is $4.0 \times 10^{4} \, J/g$?

$A$ block of ice at $0^{\circ}C$ falls from a height of $1 \ km$ onto an insulating surface. If all its kinetic energy is converted into heat,what fraction of the ice will melt? (Take $g = 10 \ m/s^{2}$ and latent heat of fusion of ice $L = 3.36 \times 10^{5} \ J/kg$)