$A$ wall has two layers $A$ and $B$, each made of a different material. Both the layers have the same thickness. The thermal conductivity of the material of $A$ is twice that of $B$. Under thermal equilibrium, the temperature difference across the wall is $36^{\circ}C$. The temperature difference across the layer $A$ is ......... $^{\circ}C$

Two full turns of the circular scale of a screw gauge cover a distance of $1\,mm$ on the main scale. The total number of divisions on the circular scale is $50$. Further,it is found that the screw gauge has a zero error of $+0.03\,mm$. While measuring the diameter of a thin wire,a student notes the main scale reading of $3\,mm$ and the number of the circular scale division in line with the main scale is $35$. The diameter of the wire is .......... $mm$.

Let $a, b$,and $c$ be such that $\frac{1}{(1-x)(1-2x)(1-3x)} = \frac{a}{1-x} + \frac{b}{1-2x} + \frac{c}{1-3x}$. Then $\frac{a}{1} + \frac{b}{3} + \frac{c}{5}$ is equal to

Graphite is a soft solid lubricant and is extremely difficult to melt. What is the reason for this behavior of graphite?

An insulated cylindrical vessel is fitted with an insulated piston of negligible weight and negligible thickness at the midpoint of the vessel. The cylinder contains a gas at $0^{\circ} C$. When the gas is heated to $100^{\circ} C$,the piston moves through a length of $5 ~cm$. The total length of the cylindrical vessel in $cm$ is:

Among the following pairs of complexes,in which case is the $\Delta_0$ value higher for the first one than the second?