An open $U$-tube contains mercury. When $13.6 \,cm$ of water is poured into one of the arms of the tube,then the mercury rise in the other arm from its initial level is ....... $cm$

$A$ spherical marble of radius $1 \, cm$ is stuck in a circular hole of radius slightly smaller than its own radius (for calculation purpose,both can be taken same) at the bottom of a bucket filled with water up to a height of $10 \, cm$. If the mass of the marble is $20 \, g$,then the net force on the marble due to water is close to:

Is pressure a scalar or a vector quantity? Explain.

$A$ cubical block of wood, of length $10 \,cm$, floats at the interface between oil of density $800 \,kg/m^3$ and water. The lower surface of the block is $1.5 \,cm$ below the interface. If the depth of water is $10 \,cm$ below the interface and oil is up to $10 \,cm$ above the interface, then the difference in pressure at the lower and the upper face of the wooden block is:
(Assume density of water, $\rho_w = 1000 \,kg/m^3$ and acceleration due to gravity, $g = 10 \,m/s^2$) (in $\,Pa$)

Two substances of densities ${\rho _1}$ and ${\rho _2}$ are mixed in equal volumes and the relative density of the mixture is $4$. When they are mixed in equal masses,the relative density of the mixture is $3$. The values of ${\rho _1}$ and ${\rho _2}$ are

$A$ liquid mixture of volume $V$ has two liquids as its ingredients with densities $\alpha$ and $\beta$. If the density of the mixture is $\sigma$,then the mass of the first liquid in the mixture is ............