Optics

At the polarising angle, the reflected ray will be completely polarised and the transmitted ray will only be partially polarised. This occurs because the angle of incidence at the polarising angle produces a reflected ray that is fully polarised.

In the process of accommodation (adjusting focus) by the human eye, the eye lens changes its shape to focus on objects at different distances. However, the image distance remains constant for any object distance, as the image always forms on the retina.

In order for a person to see the entire image of a card in a plane mirror, the area of the mirror should be such that it can reflect the full image.
The size of the image will depend on the distance between the mirror and the card.
For the minimum area needed to see the entire card, the mirror only needs to cover one-fourth of the area of the card, assuming optimal positioning of the viewer and the card.
Thus, the correct answer is (d).

In single-slit diffraction, the angular position of the first dark fringe is given by: where is the wavelength of light and is the width of the slit.
For small angles, . The distance between the dark fringes on either side of the central bright fringe is given by: where is the distance from the slit to the screen, and is the distance between the dark fringes. Substituting the given values:
Thus, the correct answer is (d).

In the case of two parallel mirrors, multiple images of an object (in this case, the candle) will appear in both mirrors.
The distance between the images of the candle depends on the distance between the two mirrors, denoted as .
The candle is placed away from mirror 1, and the images will appear alternately on either side of the mirrors.
The positions of the images can be determined using the following formula for parallel mirrors: Hence, the positions of the nearest three images of the candle behind mirror 1 are: - (first image) - (second image) - (third image) Thus, the nearest three images are at a distance of 0.2d, 1.8d, and 2.2d behind mirror 1.

The power of a lens is given by the formula: where is the focal length of the lens. For a combination of lenses in contact, the total power is the sum of the individual powers: where is the power of the convex lens and is the power of the concave lens. The power of the convex lens with focal length 40 cm is: The power of the concave lens with focal length 25 cm is: Thus, the total power is: Therefore, the correct answer is (a).

For total internal reflection to occur at face AC, the angle of incidence at face AC must be greater than the critical angle .
The critical angle is given by: where and . Substituting the values: Thus, Now, for the light incident normally on face AB, it will refract inside the prism.
The angle of incidence at face AC will depend on the refractive index and geometry of the prism.
The light will experience total internal reflection if the angle at face AC is greater than .
The fraction of light that is reflected depends on the refractive indices and the angle of incidence.
Based on these parameters, the correct fraction of light that gets totally reflected at face AC is approximately 40%.
Thus, the correct answer is (a).

The lens formula is given by: where is the focal length, is the image distance, and is the object distance.
Given that the object distance is (negative because it is in front of the lens) and the focal length is , we can rearrange the lens formula to solve for :
Thus, the correct answer is (d).

In a flat mirror, the image formed is virtual, meaning it cannot be projected on a screen.
The virtual image appears behind the mirror, and it is laterally inverted.
The image formed is upright and of the same size as the object.
Thus, the correct answer is (c).

The magnification is given by: From the problem, we know that: Using the lens formula: where is the focal length, is the image distance, and is the object distance. Since , we can solve for and substitute into the lens equation to find the focal length.
After solving, we find .
Thus, the correct answer is (d).

In all mirrors, virtual images are formed when the object is placed at a distance closer than the focal point.
- Plane mirrors always form virtual images that are the same size as the object and appear behind the mirror. - Concave mirrors can form virtual images if the object is placed closer than the focal point. - Convex mirrors always form virtual images regardless of the object's position. Thus, virtual images can be formed by Plane mirrors and Convex mirrors.
Thus, the correct answer is (c).

For constructive interference, the path difference between the two waves should be an integer multiple of the wavelength, i.e., , where is an integer. This condition ensures that the two waves reinforce each other, leading to a larger amplitude at the points of constructive interference. When the path difference is an integer multiple of the wavelength, the crests of the waves align with the crests, and the troughs align with the troughs, resulting in constructive interference.

Hence, the correct answer is (d).

The focal length of a plano-convex lens is given by: For the combination of lenses, we can calculate the effective focal length by considering the refractive index of the lenses and the intervening medium (oil). Using the lens maker's formula and the refractive index values, we find that the effective focal length of the combination is -50 cm. Hence, the correct answer is (c).

The apparent position of the coin can be found using the formula for refraction: Where is the glass thickness, is the water thickness, and and are the refractive indices of glass and water respectively. By applying the refraction law, we find the apparent position of the coin to be 7.0 cm. Hence, the correct answer is (a).

Total internal reflection occurs when the angle of incidence exceeds the critical angle. The critical angle is given by:

where is the refractive index of the medium. The critical angle is the angle of incidence at which the refracted ray travels along the boundary of the two media, and beyond this angle, total internal reflection occurs, meaning no light escapes the medium. In this case, the refractive index is calculated as , which gives the condition for total internal reflection to occur.

Thus, the correct answer is (d).