Assertion (A): The stars which are not resolved in the image produced by the objective of a telescope can’t be further resolved by its eye piece.
Reason (R): The primary purpose of eyepiece of telescope is to provide the magnification of image produced by the objective.
The resolving power of a telescope is determined by the objective lens's diameter. The eyepiece's function is to magnify the image formed by the objective, not to enhance its resolution. Thus, details not resolved by the objective cannot be resolved by the eyepiece. Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
Assertion (A): If in YDSE, wavelength of light used is increased, angular width remain unchanged only linear width of fringes increases.
Reason (R): Only linear fringe width proportional to wavelength and angular fringe width does not depends on wavelength.
In YDSE, angular fringe width is \(\beta_\theta = \frac{\lambda}{\text{d}}\) and linear fringe width is \(\beta = \frac{\lambda \text{D}}{\text{d}}\). Both are directly proportional to the wavelength \(lambda\). Thus, Assertion (A) is false as angular width also changes. Reason (R) is false as angular width does depend on wavelength.
Assertion (A): As angle subtended by the diameter of objective lens at the focus of microscope increased, resolving limit also increases.
Reason (R): Resolving limit proportional to tangent of the angle subtended by the diameter of objective lens at the focus of microscope.
The resolving limit of a microscope is \(\text{RL} = \frac{\lambda}{\text{2n} \sin\theta}\). As the angle \(\theta\) increases, \(sin\theta\) increases, causing \(\text{RL}\) to decrease (better resolution). So, Assertion (A) is false. Reason (R) is also false as \(\text{RL}\) is inversely proportional to \(sin\theta\), not proportional to \(tan\theta\).
Assertion (A): When refractive index of medium is increased resolving power also increases.
Reason (R): In medium of higher refractive index wavelength is higher.
The resolving power of a microscope is \(\text{R.P.} = \frac{\text{2n} \sin\theta}{\lambda}\) (wavelength in vacuum). It is directly proportional to refractive index \(\text{n}\), so (A) is true. Wavelength in a medium is \(\lambda_\text{medium} = \frac{\lambda_\text{vacuum}}{\text{n}}\). Higher \(\text{n}\)
Assertion (A): The resolving power of a telescope is more if the diameter of the objective in more.
Reason (R): Objective lens of larger focal length collect more light.
The resolving power of a telescope is \(\text{R.P.} = \frac{\text{D}}{\text{1.22}\lambda}\) where \(\text{D}\) is the diameter. Thus, larger \(\text{D}\) means higher \(\text{R.P.}), so (A) is true. Light collection depends on aperture (diameter), not directly on focal length. So, Reason (R) is false.
Assertion (A): In single slit diffraction arrangement, instead of keeping the screen far away, often a converging lens is placed after the slit and a screen is placed at its focus.
Reason (R): Lens doesn’t introduce any extra path difference for a parallel beam.
Assertion (A) is true. Using a converging lens to focus the diffraction pattern at its focal plane is standard for Fraunhofer diffraction, simulating far-field conditions. Reason (R) is false. A lens works by introducing varying optical path lengths across its aperture to achieve focusing, thus creating path differences.
Assertion (A): The stars which are not resolved in the image produced by the objective of a telescope can’t be further resolved by its eye piece.
Reason (R): The primary purpose of eyepiece of telescope is to provide the magnification of image produced by the objective.
Assertion (A) is true. The resolving power is determined by the objective; the eyepiece only magnifies the existing image, it cannot resolve features not already resolved by the objective. Reason (R) is true; the eyepiece's primary role is magnification. Reason (R) correctly explains Assertion (A).
Assertion (A): Huygens’s principle can explain converging nature of convex lens.
Reason (R): Snell’s law can be derived from Huygens’s principle.
Assertion (A) is true. Huygens' principle, through its explanation of wavefront changes, can explain how a convex lens converges light.
Reason (R) is true. Snell's law of refraction, which governs how light behaves at interfaces, can be derived directly from Huygens' principle.
Reason (R) explains a fundamental principle (Snell's Law) that underpins the behavior of lenses, thus it correctly explains (A).
Assertion (A): In a YDSE, the two slits are at distance ‘a’ apart. Interference pattern is observed on a screen at a distance D from the slits. At a point on the screen which is directly opposite to the slit, a dark fringe is observed. Then the wavelength of wave is proportional to square of distance between slits.
Reason (R): The light ray coming from two slits do not interfere at the screen.
Assertion (A) is true. If a dark fringe occurs at \(y = a/2\) (point opposite one slit), the path difference is \(a^2/(2D)\). For a dark fringe, \(a^2/(2D) = (n + 1/2)\lambda\), implying \(lambda \propto a^2\).
Reason (R) is false. The core principle of YDSE is the interference of light waves from two coherent slits, which produces the observed pattern on the screen.
Assertion (A): When a monochromatic light beam is incident normally on a reflective surface, under some condition it is possible that all lights is transmitted without any reflection.
Reason (R): When light after passing through a polaroid is incident on a reflecting surface at angle of incidence equals to polarizing angle, then all light gets transmitted without any reflection.
Assertion (A) is false. Total transmission at normal incidence on a reflective surface is only possible if the refractive indices are identical, implying no actual reflection.
Reason (R) is false. At Brewster's angle, only the p-polarized component of light is completely transmitted. If the light passed by the polaroid is s-polarized, it would be reflected. Therefore, the statement 'all light gets transmitted' is not universally true for light passed by a polaroid without specifying its polarization.
Thus, both (A) and (R) are false.