$A$ cylindrical metallic rod in thermal contact with two heat reservoirs at its two ends conducts an amount of heat '$Q_1$' in time '$t$'. The metallic rod is melted and the material is formed into a rod of length four times the length of the original rod. The amount of heat conducted by the new rod when placed in thermal contact with the same two reservoirs in time '$t$' is '$Q_2$'. Then $\frac{Q_1}{Q_2}$ is:

$A$ rectangular ice box of total surface area of $1000 \,cm^2$ initially contains $1.5 \,kg$ of ice at $0^{\circ}C$. If the thickness of the walls of the box is $2 \,mm$ and the temperature outside the box is $42^{\circ}C$, then the mass of the ice remaining in the box after $160 \,minutes$ is (Thermal conductivity of the material of the box $= 10^{-2} \,W m^{-1} K^{-1}$ and latent heat of the fusion of ice $= 336 \times 10^3 \,J kg^{-1}$) (in $kg$)

Two rods with thermal conductivities $K$ and $3K$ and equal cross-sectional areas are joined as shown in the figure. Their lengths are $1 \ cm$ and $2 \ cm$,respectively. If the temperatures of the two ends of this composite rod are $0^{\circ}C$ and $100^{\circ}C$ (see figure),then the temperature $\phi$ of their interface is:

Two walls of thicknesses $d_1$ and $d_2$ and thermal conductivities $k_1$ and $k_2$ are in contact. In the steady state,if the temperatures at the outer surfaces are $T_1$ and $T_2$,the temperature at the common wall is

Two spheres made of different materials have radii in the ratio $2:1$ and wall thicknesses in the ratio $1:4$. They are completely filled with ice. If the time taken to melt the ice completely in the larger sphere is $25 \ min$ and in the smaller sphere is $16 \ min$,then the ratio of the thermal conductivities of the materials of the larger and smaller spheres is ..........

$A$ metal rod $AB$ of length $10x$ has its one end $A$ in ice at $0^{\circ}C$ and the other end $B$ in water at $100^{\circ}C$. If a point $P$ on the rod is maintained at $400^{\circ}C$,then it is found that equal amounts of water and ice evaporate and melt per unit time. The latent heat of evaporation of water is $540 \ cal/g$ and latent heat of melting of ice is $80 \ cal/g$. If the point $P$ is at a distance of $\lambda x$ from the ice end $A$,find the value of $\lambda$. (Neglect any heat loss to the surrounding.)