The speed of light in media ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$ is $1.5\times {10}^8 \mathrm{m}/\mathrm{s}$ and $2.0\times {10}^8 \mathrm{m}/\mathrm{s}$ respectively. A ray of light enters from medium ${\mathrm{M}}_1$ to ${\mathrm{M}}_2$ at an incidence angle i. If the ray suffers total internal reflection, the value of i is

1. equal to ${\sin }^{-1}\left(\frac{2}{3}\right)$

2. equal to or less than ${\sin }^{-1}\left(\frac{3}{5}\right)$

3. equal to or greater than ${\sin }^{-1}\left(\frac{3}{4}\right)$

4. less than ${\sin }^{-1}\left(\frac{2}{3}\right)$

A plane-convex lens fits exactly into a plano-concave lens. Their plane surfaces are parallel to each other. If lenses are made of different materials of refractive indices \(\mu_1\) and \(\mu_2\) and \(R\) is the radius of curvature of the curved surface of the lenses, then the focal length of the combination is:

A person can see clearly objects only when they lie between 50 cm and 400 cm from his eyes. In order to increase the maximum distance of distinct vision to infinity, the type and power of the correcting lens, the person has to use will be-

1. convex, +2.25 diopter

2. concave, - 0.25 diopter

3. concave, - 0.2 diopter

4. convex, + 0.15 diopter

A ray of light is incident on the hypotenuse of a right-angled prism after travelling parallel

to the base inside the prism. If $\mu$ is the refractive index of the material of the prism, the

maximum value of the base angle for which light is totally reflected from the hypotenuse

is 

1. ${\sin }^{-1}\left(\frac{1}{\mu }\right)$                            

2. ${\tan }^{-1}\left(\frac{1}{\mu }\right)$

3. ${\sin }^{-1}\left(\frac{\mu -1}{\mu }\right)$                        

4. ${\cos }^{-1}\left(\frac{1}{\mu }\right)$

The focal length of the objective lens and the eye lens is 4 mm and 25 mm respectively in a compound microscope. The  distance between objective and eyepiece lens is 16 cm. Find its magnifying power for relaxed eye position-

1.  32.75                                 

2.  327.5 

3.  0.3275                               

4. None of the above

2. \(R\)
3. \(\frac{3}{2}R\)
4. \(R^2\)

A unit vector along incident ray of lights is $\hat{\mathrm{i}}$. The unit vector for the corresponding refracted ray of lights is $\hat{\mathrm{r}}$. $\hat{\mathrm{n}}$ is a unit vector normal to the boundary of the medium and directed towards the incident medium. If m be the refractive index of the medium, then Snell's law (2nd) of refraction is 

1. $\hat{\mathrm{i}}\times \hat{\mathrm{n}}=\mathrm{\mu }\left(\hat{\mathrm{n}}+\overrightarrow{\mathrm{r}}\right)$

2. $\hat{\mathrm{i}}.\hat{\mathrm{n}}=\mathrm{\mu }\left(\hat{\mathrm{n}}.\hat{\mathrm{r}}\right)$

3. $\hat{\mathrm{i}}\times \hat{\mathrm{n}}=\mathrm{\mu }\left(\hat{r}\times \hat{\mathrm{n}}\right)$

4. $\mathrm{\mu }\left(\hat{\mathrm{i}}\times \hat{\mathrm{n}}\right)=\hat{\mathrm{r}}\times \hat{\mathrm{n}}$

A bird is flying 3m above the surface of water. To a fish underwater, the height of the bird from the water surface appears to be $\left[\mathrm{Take} {\mathrm{\mu }}_{\mathrm{w}}=4/3\right]$

1.  2 m

2.  3.33 m

3.  4 m

4.  (4/3) m