Let the three sides of a triangle be on the lines $4x - 7y + 10 = 0$,$x + y = 5$,and $7x + 4y = 15$. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines $x = 0$,$y = 0$,and $x + y = 1$ is

The number of radial nodes of $3s$ and $2p$ orbitals are respectively

The molar heat capacity of water at constant pressure is $75 \ J \ K^{-1} \ mol^{-1}$. When $1.0 \ kJ$ of heat is supplied to $100 \ g$ of water which is free to expand,the increase in temperature of water is ............. $K$.

Two identical vessels $A$ and $B$ with frictionless pistons contain the same ideal gas at the same temperature and the same volume $V$. The masses of gas in $A$ and $B$ are $m_A$ and $m_B$,respectively. The gases are allowed to expand isothermally to the same final volume $2V$. The changes in pressures of the gas in $A$ and $B$ are found to be $\Delta P$ and $1.5 \Delta P$,respectively. Then:

An object $2.4 \; m$ in front of a lens forms a sharp image on a film $12 \; cm$ behind the lens. $A$ glass plate $1 \; cm$ thick,of refractive index $1.50$ is interposed between the lens and the film with its plane faces parallel to the film. At what distance (from the lens) should the object be shifted to be in sharp focus on the film (in $; cm$)?

The number of rational terms in the expansion of $(3^{1/4} + 7^{1/6})^{144}$ is