Particles of masses \(m, 2 m, 3 m, \ldots n m\) grams are placed on the same line at distance \(l, 2 l, 3 l \ldots n l\) \(\mathrm{cm}\) from a fixed point. The distance of centre of mass of the particles from the fixed point in centimetre is

0. For the given concentration, if the ratio of the diameters of the molecules of two gases is \(1: 2\), then the ratio of their mean free paths is

A vessel has $6 \mathrm{g}$ of oxygen at pressure $P$ and temperature $400 \mathrm{K}$. A small hole is made in it so that oxygen leaks out. How much oxygen leaks out if the final pressure is $\frac{P}{2}$ and temperature is $300 \mathrm{K}$?

A body is moving with a uniform speed of \(20 \mathrm{~ms}^{-1}\) on a horizontal circle. The change in velocity of the body in half revolution is

A rectangular metal plate \(8 \mathrm{~cm} \times 4 \mathrm{~cm}\) at \(127^{\circ} \mathrm{C}\) emits \(E \mathrm{Js}^{-1}\). If both length and breadth are halved and the temperature is raised to \(327^{\circ} \mathrm{C}\), the rate of emission is

A resistance \(R\) is connected to ' \(n\) ' identical cells. If the current in the resistance is same whether the cells are connected either in series or in parallel, then the internal resistance (r) of each cell is

The pressure required to decrease the volume of \(4000 \mathrm{cc}\) water by \(0.05 \%\) is
(Bulk modulus of water \(=2.2 \times 10^9 \mathrm{Nm}^{-2}\) )

The vector equation of any plane passing through the line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_1=\mathrm{q}_1\) and \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_2=\mathrm{q}_2\) is given by \(\overrightarrow{\mathrm{r}}\left(\overrightarrow{\mathrm{m}}_1+\lambda \overrightarrow{\mathrm{m}}_2\right)=\mathrm{q}_1+\lambda \mathrm{q}_2\) for \(\lambda \in \mathbb{R}\). The vector equation of a plane passing through the point \(2 \hat{i}-3 \hat{j}+\hat{k}\) and the line of intersection of the planes r. \((\hat{i}-2 \hat{j}+3 \hat{k})=5\) and \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

A given mass of a gas is compressed isothermally until its pressure is doubled. It is then allowed to expand adiabatically until its original volume is restored and its pressure is then found to be 0.75 of its initial pressure. The ratio of the specific heats of the gas is approximately :

Which of the following molecules has T-shaped geometry?

What are \(\mathrm{X}\) and \(\mathrm{Y}\) in the following reaction?
\(\mathrm{R}-\mathrm{OH}+\mathrm{PCl}_5 \rightarrow \mathrm{X}+\mathrm{Y}+\mathrm{HCl}\)

In a triangle $ABC$, if $a\ne \mathrm{b},\frac{a\cos A-b\cos B}{a\cos B-b\cos A}+\cos C=$

When a number of charged liquid drops coalesce, which of the following quantity does not change?