If $\hat{i}$ denotes a unit vector along the incident light ray,$\hat{r}$ a unit vector along the refracted ray into a medium of refractive index $\mu$,and $\hat{n}$ a unit vector normal to the boundary of the medium directed towards the incident medium,then the law of refraction is:

$A$ glass slab consists of thin uniform layers of progressively decreasing refractive indices $(RI)$ such that the $RI$ of any layer is $\mu - m \Delta \mu$. Here, $\mu$ and $\Delta \mu$ denote the $RI$ of the $0^{\text{th}}$ layer and the difference in $RI$ between any two consecutive layers, respectively. The integer $m = 0, 1, 2, 3, \ldots$ denotes the number of the successive layers. $A$ ray of light from the $0^{\text{th}}$ layer enters the $1^{\text{st}}$ layer at an angle of incidence of $30^{\circ}$. After undergoing the $m^{\text{th}}$ refraction, the ray emerges parallel to the interface. If $\mu = 1.5$ and $\Delta \mu = 0.015$, the value of $m$ is:

$A$ light ray is incident on a transparent sphere of refractive index $\mu = \sqrt{2}$ at an angle of incidence $i = 45^{\circ}$. What is the total deviation of a ray that enters the sphere,undergoes two internal reflections,and then refracts out into the air (in $^{\circ}$)?

Optical phenomena taking place for a mirror and a lens respectively are . . . . . . and . . . . . . .

Most materials have a refractive index,$n > 1$. So,when a light ray from air enters a naturally occurring material,then by Snell's law,$\frac{\sin \theta_1}{\sin \theta_2} = \frac{n_2}{n_1}$,it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism,the refractive index of the medium is given by the relation,$n = \left(\frac{c}{v}\right) = \pm \sqrt{\varepsilon_r \mu_r}$. Where $\varepsilon_r$ and $\mu_r$ are negative,one must choose the negative root of $n$. Such negative refractive index materials can now be artificially prepared and are called meta-materials. They exhibit significantly different optical behavior,without violating any physical laws. Since $n$ is negative,it results in a change in the direction of propagation of the refracted light. However,similar to normal materials,the frequency of light remains unchanged upon refraction even in meta-materials.
$1.$ Choose the correct statement.
$(A)$ The speed of light in the meta-material is $v = c|n|$.
$(B)$ The speed of light in the meta-material is $v = \frac{c}{|n|}$.
$(C)$ The speed of light in the meta-material is $v = c$.
$(D)$ The wavelength of the light in the meta-material $(\lambda_m)$ is given by $\lambda_m = \frac{\lambda_{\text{air}}}{|n|}$,where $\lambda_{\text{air}}$ is the wavelength of the light in air.
$2.$ For light incident from air on a meta-material,the appropriate ray diagram is:

Consider a light ray travelling in air incident into a medium of refractive index $\sqrt{2n}$. The incident angle is twice that of the refracting angle. Then,the angle of incidence will be