Ray optics practice · Mixed difficulty

Option A ($-9.6 \mathrm{~cm}$) is correct because it is derived from using the magnification formula and the mirror formula.

Option D (If Statement-I is false but Statement-II is true) is correct because the twinkling of stars is due to atmospheric refraction, not reflection, and the velocity of light does change when it enters a different medium.

Option D ($\frac{3}{\sqrt{5}}$) is correct because it represents the apparent visible length of the road calculated using the given refractive index variation with height.

Option B (Air) is correct because the speed of light is maximum in a vacuum, and air is the closest to a vacuum among the given options.

Option A (${\mu _1} > {\mu _2}$) is correct because a higher refractive index in the core compared to the cladding minimizes light leakage due to total internal reflection.

Option B (2.32 cm) is correct because it is calculated using the lens formula twice, first for the eyepiece and then for the objective, considering the final image distance and the focal lengths.

Option C (Cylindrical lens) is correct because astigmatism is corrected by using a cylindrical lens, which has different curvatures in different planes.

Option C ($77^\circ$) is correct because it's the result of applying the critical angle formula with the given refractive indices.

The correct answer is B ($20^{\circ}$). The angle subtended by the moon's image is determined by the ratio of the focal lengths of the objective and eyepiece lenses.

Option C is correct. The sparking of a diamond is primarily due to total internal reflection, which traps light inside the diamond and causes it to reflect multiple times before exiting.

Option D (${n_1} > {n_2}$) is correct because total internal reflection requires the core to have a higher refractive index than the cladding.

Option B (Before retina) is correct because in a short-sighted person (myopic), the image is formed in front of the retina.

The correct answer is A ($\mathrm{0}\mathrm{.5}\,\,\mathrm{m}$) because it is calculated using the lens formula and the magnification formula for a real, inverted image.

Option B ($-5 \cos (t)$) is correct because acceleration is the second derivative of position with respect to time, and the second derivative of $5 \cos t$ is $-5 \cos t$.

Option A (0.3) is correct because it's derived from calculating the rate of change of magnification using the lens formula and the given object velocity.

Option D (Dispersion and total internal reflection) is correct because a rainbow is formed by the dispersion of sunlight into its constituent colors, followed by total internal reflection within water droplets.

Option B (Convex) is correct because a convex mirror always produces an erect and diminished image, which matches the given condition of a 6m object forming a 2m erect image.

Option A ($\upsilon = \frac{c}{{\sqrt {{ \in _r}{\mu _r}} }}$) is correct because the velocity of light in a medium is inversely proportional to the square root of the product of relative permittivity and permeability.

Option C is correct because the secondary mirror in a reflecting telescope redirects the light to bring the eyepiece to a more accessible location outside the main tube.

Option D (The image distance from the eye lens) is correct because the distance between the lens and the retina remains relatively constant, and the eye adjusts its focal length to focus objects at different distances onto the retina.

Option D ($40\\ cm$) is correct because the radius of curvature is twice the focal length for a spherical mirror.

Option B is correct because according to Rayleigh's scattering law, the intensity of scattered light is inversely proportional to the fourth power of the wavelength. Since wavelength is inversely proportional to frequency, the intensity is directly proportional to the fourth power of the frequency. Therefore, the ratio of frequencies is the fourth root of the intensity ratio, which is (256/81)^(1/4) = 4/3.

Option A (In primary rainbow, observer sees red colour on the top and violet on the bottom) is correct because of the order of refraction and reflection in water droplets.

Option C (Total internal reflection) is correct because optical fibers rely on total internal reflection to guide light along the fiber.

Option A ($\frac{1}{5}\mathrm{\\ m}\mathrm{\\ }/\mathrm{\\ }\mathrm{\\ s}$) is correct because it is derived from differentiating the mirror formula and substituting the given values.

Option A (50) is correct because the overall magnification of a compound microscope is the product of the magnifications of the objective and the eyepiece.

Option A is correct. The color of an object is determined by the frequencies it reflects, and since frequency doesn't change with the medium, a green object will still appear green.

Option C ($60^\circ$) is correct because when the ray is parallel to the base, the angle of refraction at the first face equals the angle of incidence at the second face, and for an equilateral prism, this angle is 30 degrees.

Option D is correct. Frequency remains unchanged during refraction because it's a property of the source, while speed, intensity, and wavelength change.

Option A is correct because violet light has the shortest wavelength and is therefore refracted the most by a converging lens, causing it to focus nearest to the lens.