$A$ liquid of mass $m$ and specific heat $c$ is heated to a temperature $2T$. Another liquid of mass $m/2$ and specific heat $2c$ is heated to a temperature $T$. If these two liquids are mixed, the resulting temperature of the mixture is:

$2 \ kg$ of ice at $-20^{\circ} C$ is mixed with $5 \ kg$ of water at $20^{\circ} C$. The final mass of water formed is: (in $kg$)

An unknown metal of mass $192 \, g$ heated to a temperature of $100 \, ^oC$ was immersed into a brass calorimeter of mass $128 \, g$ containing $240 \, g$ of water at a temperature of $8.4 \, ^oC$. Calculate the specific heat of the unknown metal if the water temperature stabilizes at $21.5 \, ^oC$. (Specific heat of brass is $394 \, J \, kg^{-1} \, K^{-1}$, specific heat of water is $4186 \, J \, kg^{-1} \, K^{-1}$) ......... $J \, kg^{-1} \, K^{-1}$

At a power station,heat is removed from the heat exchanger by cooling water at $6.7 \times 10^9 \ J$ per minute. The cooling water enters at $6.0 \ ^oC$ and leaves at $14.0 \ ^oC$. [Take the specific heat capacity of water $4200 \ J/kg \ ^oC$]. Which of the following is the rate of water flow?

$A$ massless spring $(k = 800\, N/m)$,attached with a mass $(500\, g)$ is completely immersed in $1\, kg$ of water. The spring is stretched by $2\, cm$ and released so that it starts vibrating. What would be the order of magnitude of the change in the temperature of water when the vibrations stop completely? (Assume that the water container and spring receive negligible heat and specific heat of mass $= 400\, J/kg\, K$,specific heat of water $= 4184\, J/kg\, K$)

$2\, kg$ of ice at $-20^{\circ}C$ is mixed with $5\, kg$ of water at $20^{\circ}C$ in an insulating vessel having a negligible heat capacity. Calculate the final mass of water remaining in the container. It is given that the specific heats of water and ice are $1\, kcal/kg/^{\circ}C$ and $0.5\, kcal/kg/^{\circ}C$ respectively,while the latent heat of fusion of ice is $80\, kcal/kg$.