The equation of the line touching both the parabolas $y^{2}=4x$ and $x^{2}=-32y$ is

The dimensional formula of $\frac{1}{2} \mu_0 H^2$ (where $\mu_0 = \text{permeability of free space}$ and $H = \text{magnetic field intensity}$) is:

$A$ vibration magnetometer consists of two identical bar magnets placed one over the other such that they are perpendicular and bisect each other. The time period of oscillation in a horizontal magnetic field is $2^{5/4} \ s$. One of the magnets is removed and if the other magnet oscillates in the same field,then the time period in seconds is:

$A$ vector $\vec{a}$ has components $2p$ and $1$ with respect to a two-dimensional rectangular Cartesian system. This system is rotated through a certain angle $\theta$ about the origin in the counter-clockwise direction. If $\vec{a}$ has components $p+1$ and $1$ with respect to the new system,then:

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^2+m^2-n^2=0$ is

Two coherent light sources having intensity in the ratio $2x$ produce an interference pattern. The ratio $\frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}}$ will be