If $1\; g$ of steam at $100^{\circ}C$ is mixed with $1\; g$ of ice at $0^{\circ}C$,then the resultant temperature of the mixture is ........ $^{\circ}C$.

$A$ $20 \,g$ bullet whose specific heat is $5000 \,J/(kg \cdot ^{\circ}C)$ and moving at $2000 \,m/s$ plunges into a $1.0 \,kg$ block of wax whose specific heat is $3000 \,J/(kg \cdot ^{\circ}C)$. Both the bullet and the wax are at $25^{\circ}C$. Assuming that $(i)$ the bullet comes to rest in the wax and $(ii)$ all its kinetic energy is converted into heat,the final temperature of the wax (in $^{\circ}C$) is close to:

$A$ water cooler of storage capacity $120$ litres can cool water at a constant rate of $P$ watts. In a closed circulation system (as shown schematically in the figure),the water from the cooler is used to cool an external device that generates constantly $3 \text{ kW}$ of heat (thermal load). The temperature of water fed into the device cannot exceed $30^{\circ}\text{C}$ and the entire stored $120$ litres of water is initially cooled to $10^{\circ}\text{C}$. The entire system is thermally insulated. The minimum value of $P$ (in watts) for which the device can be operated for $3$ hours is (Specific heat of water is $4.2 \text{ kJ kg}^{-1} \text{K}^{-1}$ and the density of water is $1000 \text{ kg m}^{-3}$)

$Assertion :$ 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.

Fill in the blanks:
$1.$ The temperature of freezing point ...... and temperature of boiling point ...... are taken in a Fahrenheit thermometer.
$2.$ The zeroth law of thermodynamics defines ...... and the first law of thermodynamics defines ......
$3.$ At ...... temperature,water and water vapour have equal density.
$4.$ The ...... has the maximum value of specific heat.

Consider two thermometers $T_1$ and $T_2$ of equal length,which can be used to measure temperature over the range $\theta_1$ to $\theta_2$. $T_1$ contains mercury as the thermometric liquid,while $T_2$ contains bromine. The volumes of the two liquids are the same at the temperature $\theta_1$. The volumetric coefficients of expansion of mercury and bromine are $18 \times 10^{-5} \, K^{-1}$ and $108 \times 10^{-5} \, K^{-1}$,respectively. The increase in length of each liquid is the same for the same increase in temperature. If the diameters of the capillary tubes of the two thermometers are $d_1$ and $d_2$,respectively,then the ratio $d_1: d_2$ is closest to: