Given three points $P, Q, R$ with $P(5, 3)$ and $R$ lies on the $x-$ axis. If the equation of $RQ$ is $x - 2y = 2$ and $PQ$ is parallel to the $x-$ axis,then the centroid of $\Delta PQR$ lies on the line

In the ground state,an element has $13$ electrons in the $M$ shell. The element is

$A$ plane passes through the point $(3, 5, 7)$. If the direction ratios of its normal are equal to the intercepts made by the plane $x + 3y + 2z = 9$ with the coordinate axes,then the equation of that plane is

For the electrochemical cell, $Mg ( s )\left| Mg ^{2+}( aq , 1 M ) \| Cu ^{2+}( aq , 1 M )\right| Cu ( s )$ the standard emf of the cell is $2.70 V$ at $300 K$. When the concentration of $Mg ^{2+}$ is changed to $x M$, the cell potential changes to $2.67 V$ at $300 K$. The value of $x$ is.

(given, $\frac{ F }{ R }=11500 K V ^{-1}$, where $F$ is the Faraday constant and $R$ is the gas constant, The value of $\ln (10)=2.30$ )

The value of $\frac{\cot 54^\circ}{\tan 36^\circ} + \frac{\tan 20^\circ}{\cot 70^\circ}$ is

$A$ cylindrical vessel open at the top is $20 \ cm$ high and $10 \ cm$ in diameter. $A$ circular hole whose cross-sectional area is $1 \ cm^2$ is cut at the centre of the bottom of the vessel. Water flows from a tube above it into the vessel at the rate of $100 \ cm^3 s^{-1}$. The height of water in the vessel under steady state is ....... $cm$ (Take $g = 1000 \ cm s^{-2}$)