Two cubes $A$ and $B$ of same dimensions are made of different materials with densities $\rho_A$ and $\rho_{\mathrm{B}}$, respectively. Cube A floats in water with a fraction $\eta$ of its volume immersed. When cube $B$ is placed on top of cube $A$, it is found that cube $A$ is just entirely immersed, while cube $B$ is entirely above the surface of the water. What is the ratio $\rho_{\mathrm{B}} / \rho_{\mathrm{A}}$ ?

An object is placed in front of a convex lens. A real inverted image of double its size is formed. When the object is moved closer to the lens by a distance $d$, the image shifts away from the lens by a distance $8 d$ from its previous position. What is the magnitude of the magnification in the final setup?

The frequency of the whistle of a train moving with a constant speed is observed by a stationary detector on the platform to be $v_1$. The frequency of the same whistle is detected to be $v_2$ inside another train moving on a parallel track, at a speed $v$ towards the frst train. If the speed of sound in air is taken to be $v_s$ ound, what is the ratio $v / v_s$ ound?

A current $I$ flows through a cylindrical cable of length $L$ and uniform cross-sectional area $A$. The power dissipated due to the current is $P_1$. The cable is cut into two equal halves. If the cross-sectional area and the current flowing through the two halves remain unchanged and the power dissipated in each half is $P_2$, which of these options is correct?