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Physics Question on Ray optics and optical instruments

Monochromatic light is incident on a glass prism of angle AA. If the refractive index of the material of the prism is uu, a ray incident at an angle θ\theta, on the face ABAB would get transmitted through the face ACAC of the prism provided

A

θ<cos−1[μsin(A+sin−1(1μ))]\theta < cos^{-1}\Bigg[\mu sin\Bigg(A+sin^{-1}\Bigg(\frac{1}{\mu}\Bigg)\Bigg)\Bigg]

B

θ<sin−1[μ sin(A−sin−1(1μ))]\theta < sin^{-1}\Bigg[\mu\, sin\Bigg(A-sin^{-1}\Bigg(\frac{1}{\mu}\Bigg)\Bigg)\Bigg]

C

θ>cos−1[μ sin(A+sin−1(1μ))]\theta > cos^{-1}\Bigg[\mu\, sin\Bigg(A+sin^{-1}\Bigg(\frac{1}{\mu}\Bigg)\Bigg)\Bigg]

D

θ>sin−1[μ sin(A−sin−1(1μ))]\theta > sin^{-1}\Bigg[\mu\, sin\Bigg(A-sin^{-1}\Bigg(\frac{1}{\mu}\Bigg)\Bigg)\Bigg]

Answer

θ>sin−1[μ sin(A−sin−1(1μ))]\theta > sin^{-1}\Bigg[\mu\, sin\Bigg(A-sin^{-1}\Bigg(\frac{1}{\mu}\Bigg)\Bigg)\Bigg]

Explanation

Solution

$r_{2}