If the plane $3x - 2y - z - 18 = 0$ meets the coordinate axes at $A$, $B$, and $C$, then the centroid of $\triangle ABC$ is:

Two resistances of $400\,\Omega$ and $800\,\Omega$ are connected in series with a $6\,V$ battery of negligible internal resistance. $A$ voltmeter of resistance $10,000\,\Omega$ is used to measure the potential difference across the $400\,\Omega$ resistor. The error in the measurement of potential difference (in volts) is approximately:

$A$ monoatomic ideal gas,initially at temperature $T_1$,is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature $T_2$ by releasing the piston suddenly. If $L_1$ and $L_2$ are the lengths of the gas column before and after expansion respectively,then $\frac{T_1}{T_2}$ is given by:

Observe the following statements regarding the purification of bauxite:
$I$. During Hall's process,silica is removed as $Si$ (vapour).
$II$. Bauxite ore contaminated with $Fe_2O_3$ is purified in Baeyer's process.
$III$. During Serpeck's process,$AlN$ is formed.
The correct answer is:

Particles of masses $m, 2m, 3m, \ldots, nm$ grams are placed on the same line at distances $l, 2l, 3l, \ldots, nl$ $cm$ from a fixed point. The distance of the centre of mass of the particles from the fixed point in centimetres is

$A$ uniform beam of weight $W$ is attached to a vertical wall by a hinge $H$. The beam is held horizontal by a rope as shown below. Which one of the following best shows the direction of the reaction force $R$ at the hinge?