A plano-convex lens is made of material of refractive index n. When a small object is placed 30 cm away in front of the curved surface of the lens, an image of double the size of the object is produced. Due to reflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm away form the lens. Which of the following statement(s) is(are) true?

A transparent slab of thickness d has a refractive index n (z) that increases with z. Here, z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices n₁ and n₂ (> n₁), as shown in the figure. A ray of light is incident with angle $$\theta$$_i from medium 1 and emerges in medium 2 with refraction angle $$\theta$$_f with a lateral displacement $l$.


Which of the following statement(s) is (are) true?

Light guidance in an optical fibre can be understood by considering a structure comprising of thin solid glass cylinder of refractive index n₁ surrounded by a medium of lower refractive index n₂. The light guidance in the structure takes place due to successive total internal reflections at the interface of the media n₁ and n₂ as shown in the figure. All rays with the angle of incidence i less than a particular value i_m are confined in the medium of refractive index n₁. The numerical aperture (NA) of the structure is defined as sin i_m.

For two structures namely S₁ with n₁ = $${{\sqrt {45} } \over 4}$$ and n₂ = $${3 \over 2}$$, and S₂ with n₁ = $${8 \over 5}$$ and n₂ = $${7 \over 5}$$ and taking the refractive index of water to be $${4 \over 3}$$ and that to air to be 1, the correct options is/are :

Light guidance in an optical fibre can be understood by considering a structure comprising of thin solid glass cylinder of refractive index n₁ surrounded by a medium of lower refractive index n₂. The light guidance in the structure takes place due to successive total internal reflections at the interface of the media n₁ and n₂ as shown in the figure. All rays with the angle of incidence i less than a particular value i_m are confined in the medium of refractive index n₁. The numerical aperture (NA) of the structure is defined as sin i_m.

If two structures of same cross-sectional area, but different numeral apertures NA₁ and NA₂ (NA₂ < NA₁) are joined longitudinally, the numerical aperture of the combined structure is