The tourmaline crystal:

$A$ polariser is used to

An unpolarized light of certain intensity passes through a combination of two polarizers whose transmission axes are at $30^\circ$ and $90^\circ$, respectively, with respect to the horizontal axis. $A$ third polarizer with its transmission axis at $60^\circ$ with the horizontal axis is placed between the two existing polarizers. The ratio of the output intensities with and without the third polarizer is . . . . . . .

$A$ $20 \ cm$ length of a certain solution causes right-handed rotation of $38^{\circ}$. $A$ $30 \ cm$ length of another solution causes left-handed rotation of $24^{\circ}$. The optical rotation caused by a $30 \ cm$ length of a mixture of the above solutions in the volume ratio $1:2$ is

Consider the following statements $A$ and $B$. Identify the correct choice in the given answers.
$A$. The refractive index of the extra-ordinary ray depends on the angle of incidence in double refraction.
$B$. The vibrations of light waves acquire one-sidedness for both ordinary and extraordinary rays in double refraction.

The relation $I = I_0 \cos^2 \theta$ is known as (where $I_0$ is the intensity of incident light on the analyser,$I$ is the intensity of emergent light from the analyser,and $\theta$ is the angle between the plane of polarization and the axis of the analyser):