Two coherent narrow slits emitting light of wavelength $\lambda$ in the same phase are placed parallel to each other at a small separation of $3\lambda$. The light is collected on a screen $S$ which is placed at a distance $D$ from the slits. Find the smallest distance $x$ from the central point $O$ such that point $P$ is a maxima.

If $\frac{1}{x^4+x^2+1}=\frac{A x+B}{x^2+x+1}+\frac{C x+D}{x^2-x+1}$, then $C+D$ is equal to

Power radiated by a black body at temperature $T_1$ is $P$ and it radiates maximum energy at a wavelength $\lambda_1$. If the temperature of the black body is changed from $T_1$ to $T_2$, it radiates maximum energy at a wavelength $\frac{\lambda_1}{2}$. The power radiated at $T_2$ is (in $P$)

$\int_2^3 \frac{dx}{x^2-x}$ is equal to

The outer and inner radii of a hollow cylinder are $(4.23 \pm 0.01) \text{ cm}$ and $(3.89 \pm 0.01) \text{ cm}$ respectively. What is the thickness of the wall of the cylinder?

Condenser $A$ has a capacity of $15 \mu F$ when it is filled with a medium of dielectric constant $15$. Another condenser $B$ has a capacity of $1 \mu F$ with air between the plates. Both are charged separately by a battery of $100 V$. After charging,both are connected in parallel without the battery and the dielectric material being removed from condenser $A$. The common potential now is.......$V$