The area (in square units) of the circle which touches the lines $4x + 3y = 15$ and $4x + 3y = 5$ is

The $pK_a$ of a weak acid $HA$ is $4.5$. The $pOH$ of an aqueous buffered solution of $HA$ in which $50\%$ of the $HA$ acid is ionized is

$A$ paramagnetic sample shows a net magnetization of $8 \ Am^{-1}$ when placed in an external magnetic field of $0.6 \ T$ at a temperature of $4 \ K$. When the same sample is placed in an external magnetic field of $0.2 \ T$ at a temperature of $16 \ K$,the magnetization will be

$\operatorname{sech}^{-1}(\sin \theta)$ is equal to

$A$ block of mass $8\, kg$ is at rest on a rough inclined plane as shown in the figure. The magnitude of the net force exerted by the surface on the block will be ......... $N$.