An object is placed $20 \ cm$ in front of a $4 \ cm$ thick plane mirror. The image of the object is finally formed at $45 \ cm$ from the object itself. The refractive index of the material of the unpolished side of the mirror is (considering near normal incidence):

$A$ printed page is kept pressed by a transparent cube of edge $t$. The refractive index of the cube varies as $\mu(z) = 1 + \frac{z}{t}$, where $z$ is the vertical distance from the bottom of the cube. If viewed from the top, then the printed letters appear to be shifted by an amount

$A$ glass tumbler having an inner depth of $17.5 \, cm$ is kept on a table. $A$ student starts pouring water $(\mu = 4/3)$ into it while looking at the surface of water from above. When he feels that the tumbler is half-filled,he stops pouring water. Up to what height,the tumbler is actually filled? (In $cm$)

$A$ ray of light is incident at an angle of incidence $60^{\circ}$ on a glass slab of refractive index $\sqrt{3}$. After refraction,the light ray emerges out from the other parallel face,and the lateral shift between the incident ray and the emergent ray is $4 \sqrt{3} \, cm$. The thickness of the glass slab is . . . $cm$.

What is the lateral shift of a ray refracted through a parallel-sided glass slab of thickness $h$ in terms of the angle of incidence $i$ and angle of refraction $r$,if the glass slab is placed in air medium?

$A$ man standing in a swimming pool looks at a stone lying at the bottom. The depth of the swimming pool is $h$. At what distance from the surface of water is the image of the stone formed? (Line of vision is normal; Refractive index of water is $n$)