A parallel plate capacitor with air medium between the plates has a capacitance of $$10 \mu \mathrm{F}$$. The area of capacitor is divided into two equal halves and filled with two media (as shown in figure) having dielectric constant $$K_1=2$$ and $$\mathrm{K}_2=4$$. The capacitance of the system will be

Refractive index of a glass convex lens is 1.5. The radius of curvature of each of the two surfaces of the lens is $$20 \mathrm{~cm}$$. The ratio of the power of the lens when immersed in a liquid of refractive index 1.25 to that when placed in air is

Earth is assumed to be a sphere of radius R. If '$$\mathrm{g}_\phi$$' is value of effective acceleration due to gravity at latitude $$30^{\circ}$$ and '$$g$$' is the value at equator, then the value of $$\left|g-g_\phi\right|$$ is ($$\omega$$ is angular velocity of rotation of earth, $$\cos 30^{\circ}=\frac{\sqrt{3}}{2}$$ )

Four identical uniform solid spheres each of same mass '$$M$$' and radius '$$R$$' are placed touching each other as shown in figure, with centres A, B, C, D. $$\mathrm{I}_{\mathrm{A}}, \mathrm{I}_{\mathrm{B}}, \mathrm{I}_{\mathrm{C}}$$ and $$\mathrm{I}_{\mathrm{D}}$$ are the moment of inertia of these spheres respectively about an axis passing through centre and perpendicular to the plane. The difference in $$\mathrm{I}_{\mathrm{A}}$$, and $$\mathrm{I}_{\mathrm{B}}$$ is