A lead bullet penetrates into a solid object and melts. Assuming that $40\%$ of its kinetic energy is used to heat it, the initial speed of bullet is
(Given, initial temperature of the bullet $=127°\mathrm{C}$,
Melting point of the bullet $=327°\mathrm{C}$,
Latent heat of fusion of lead $=2.5\times {10}^4 \mathrm{J}{ \mathrm{kg}}^{-1}$,
Specific heat capacity of lead $=125 \mathrm{J} \mathrm{kg} {\mathrm{K}}^{-1}$)

As per the given figure, two plates $A$ and $B$ of thermal conductivity $K$ and $2K$ are joined together to form a compound plate. The thickness of plates are $4.0 \mathrm{cm}$ and $2.5 \mathrm{cm}$ respectively and the area of cross-section is $120{ \mathrm{cm}}^2$ for each plate. The equivalent thermal conductivity of the compound plate is $\left(1+\frac{5}{\alpha }\right)K$, then the value of $\alpha$ will be _____ .

A unit scale is to be prepared whose length does not change with temperature and remains $20 \mathrm{cm}$, using a bimetallic strip made of brass and iron each of different length. The length of both components would change in such a way that difference between their lengths remains constant. If length of brass is $40 \mathrm{cm}$ and length of iron will be _____ $\mathrm{cm}$.

$\left({\alpha }_{\mathrm{iron} }=1.2\times {10}^{-5}{ \mathrm{K}}^{-1}\right.$ and $\left.{\alpha }_{\mathrm{brass} }=1.8\times {10}^{-5}{ \mathrm{K}}^{-1}\right)$.

A block of ice of mass $120 \mathrm{g}$ at temperature $0°\mathrm{C}$ is put in $300 \mathrm{g}$ of water at $25°\mathrm{C}$. The $x \mathrm{g}$ of ice melts as the temperature of the water reaches $0°\mathrm{C}$. The value of $x$ is

[Use: Specific heat capacity of water $=4200 \mathrm{J} {\mathrm{kg}}^{-1}{ \mathrm{K}}^{-1}$, Latent heat of ice $=3.5\times {10}^5 \mathrm{J} {\mathrm{kg}}^{-1}$]

If $K_1$ and $K_2$ are the thermal conductivities $L_1$ and $L_2$ are the lengths and $A_1$ and $A_2$ are the cross sectional areas of steel and copper rods respectively such that $\frac{K_2}{K_1}=9,\frac{A_1}{A_2}=2,\frac{L_1}{L_2}=2$. Then, for the arrangement as shown in the figure. The value of temperature $T$ of the steel - copper junction in the steady state will be