If two distinct chords drawn from the point $(p, q)$ to the circle $x^{2} + y^{2} = px + qy$ (where $pq \neq 0$) are bisected by the $x$-axis,then:

The product of the lengths of perpendiculars drawn from any point on the hyperbola $x^2 - 2y^2 - 2 = 0$ to its asymptotes is:

In an industrial process,$10\, kg$ of water per hour is to be heated from $20^\circ C$ to $80^\circ C$. To do this,steam at $200^\circ C$ is passed from a boiler into a copper coil immersed in water. The steam condenses in the coil and is returned to the boiler as water at $90^\circ C$. How many $kg$ of steam is required per hour? (Specific heat of steam $= 0.5\, cal/g^\circ C$,Latent heat of vaporisation $= 540\, cal/g$,Specific heat of water $= 1\, cal/g^\circ C$)

$\frac{d}{d x}\left[a \tan ^{-1} x+b \log \left(\frac{x-1}{x+1}\right)\right]=\frac{1}{x^4-1}$
$\Rightarrow a-2 b$ is equal to

$A$ dipole having dipole moment $p$ is placed in front of a solid uncharged conducting sphere as shown in the diagram. The net potential at point $A$ lying on the surface of the sphere is:

$n$ moles of an ideal gas at temperature $T$ (in kelvin) occupy $V \ L$ of volume,exerting a pressure of $P$ atmospheres. What is the concentration (in $mol / L$)?