Ray Optics and Optical Instruments

Explanation: 

Statement I: Image formation needs regular reflection and/or refraction.
This statement is correct. Image formation relies on the interaction of light with surfaces. Here’s how:
Regular Reflection: This occurs when parallel rays of light strike a smooth surface and reflect back in a parallel manner. Mirrors are a prime example of regular reflection. This type of reflection allows for clear, focused images.
Refraction: This occurs when light passes from one medium to another (like from air to water or air to glass). The change in speed causes the light to bend, resulting in image formation. This is how lenses in cameras, telescopes, and our own eyes work.
Statement II: The variety in colour of objects we see around us is due to the constituent colours of the light incident on them.
This statement is also correct. The colors we perceive are based on the interaction between light and objects:
White Light: Sunlight or light from a bulb contains all the colors of the rainbow (the visible spectrum).
Selective Absorption: Objects absorb certain colors of light and reflect others. For instance, a red apple absorbs most colors except red, which it reflects back to our eyes.
Pigments: Many objects, especially those we paint or dye, contain pigments that absorb specific colors and reflect others.
Hence, option (3) is the correct answer.

Step: Find the time taken by the sunlight to pass through a glass slab.
The time taken by the sunlight to pass through a glass slab is given by; 
$t = \frac{\text{optical path length}}{\text{speed of light}}$
$\Rightarrow t = \frac{\mu \times L}{c}= \frac{1.5\times 5\times 10^{-3}}{3\times 10^{8}}= \left(\frac{5}{2}\right)\times 10^{-11}~\text{s}~~~[L = 5~\text{mm}]$
Hence, option (2) is the correct answer. 
 

Step: Find the correct variation of $\dfrac{1}{v}$ with $\dfrac{1}{u}$ for a concave mirror.
According to the mirror equation:
$\dfrac{1}{v}+\dfrac{1}{u}= \dfrac{1}{f}$
$\Rightarrow \dfrac{1}{v}= -\dfrac{1}{u}+\dfrac{1}{f}~~~...(1)~~~~[f = -ve,~\text{for concave mirror}]$
Equation $(1)$ represents a straight line having a negative slope and negative intercept. 
Therefore, the correct graph is shown below;
               
Hence, option (2) is the correct answer. 
 

Step: Find the relation between the angles $i$ and $i'$.
The angle of incidence $i$ and angle of emergence $i'$ on the prism is shown in the figure below;
          
For the minimum angle of deviation;
$i = i'$
$\Rightarrow r_1 = r_2$
Therefore, the correct relation between the angles is $i = i'$.
Hence, option (1) is the correct answer.
 

Step: Find the magnifying power of the telescope for viewing a distant object.
Given: $f_0 = 140~\text{cm}, f_e = 5~\text{cm}$
The magnifying power of a telescope for viewing a distant object is given by;
$m = \frac{f_0}{f_e}$
$\Rightarrow m = \frac{140}{5}= 28$
Therefore, the magnifying power of the telescope is $28$.
Hence, option (1) is the correct answer.
 

Step: Find the refractive index of the prism. 
Given: $A= 90^{\circ}$
Let the refractive index of the prism is $\mu$.
$A= r_1+ r_2$
$\Rightarrow r_2 = 90^{\circ} - r_1$
Apply Snell’s law at point $P$
$1\times \sin30^{\circ}= \mu \sin r_1\Rightarrow \mu \sin r_1 = \frac{1}{2}~~~...(1)$
Apply Snell’s law at point $Q$
$\mu\times \sin r_2= 1 \sin90^{\circ}\Rightarrow \mu \sin r_2= 1\Rightarrow \mu \sin{(90-r_1)}=1$
$\Rightarrow \mu \cos r_1= 1~~~...(2)$
Squaring and adding equations (1) and (2) we get; 
$\mu^2 = 1+\frac{1}{4}= \frac{5}{4}$
$\Rightarrow \mu = \frac{\sqrt{5}}{2}$
Therefore, the refractive index of the prism is $\frac{\sqrt{5}}{2}$.
Hence, option (1) is the correct answer.