A transparent thin film of uniform thickness and refractive index n₁ = 1.4 is coated on the convex spherical surface of radius R at one end of a long solid glass cylinder of refractive index n₂ = 1.5, as shown in the figure. Rays of light parallel to the axis of the cylinder traversing through the film from air to glass get focused at distance f₁ from the film, while rays of light traversing from glass to air get focused at distance f₂ from the film. Then

Two ideal batteries of emf V₁ and V₂ and three resistances R₁, R₂ and R₃ are connected as shown in the figure. The current in resistance R₂ would be zero if

A uniform circular disc of mass 1.5 kg and radius 0.5 m is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude F = 0.5 N are applied simultaneously along the three sides of an equilateral triangle XYZ with its vertices on the perimeter of the disc (see figure). One second after applying the forces, the angular speed of the disc in rad s^-1 is

Two parallel wires in the plane of the paper are distance X₀ apart. A point charge is moving with speed u between the wires in the same plane at a distance X₁ from one of the wires. When the wires carry current of magnitude I in the same direction, the radius of curvature of the path of the point charge is R₁. In contrast, if the currents I in the two wires have directions opposite to each other, the radius of curvature of the path is R₂. If $${{{X_0}} \over {{X_1}}} = 3$$, and value of R₁/R₂ is