The angular resolution of a $10\,cm$ diameter telescope at a wavelength of $5000\,\mathring{A}$ is of the order of

Let $\overrightarrow{a}=\hat{i}-2 \hat{j}+3 \hat{k}$,$\overrightarrow{b}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+\hat{j}+(2 \lambda-1) \hat{k}$. If $\overrightarrow{c}$ is parallel to the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$,then $\lambda$ is equal to

$A$ gas at the temperature $250\, K$ is contained in a closed vessel. If the gas is heated through $1\, K$, then the percentage increase in its pressure will be .... $\%$

$A$ small ball of mass $m$ is released at a height $R$ above the Earth's surface,as shown in the figure. The ball enters a narrow groove and reaches a maximum depth of $R/2$ inside the Earth before coming to rest momentarily. The groove contains an ideal spring of spring constant $K$ and natural length $R$. Find the value of $K$. ($R$ is the radius of the Earth and $M$ is the mass of the Earth).

The oxidation state of $Xe$ in $XeO_3$ and the bond angle in it respectively are:

At $27^{\circ} C$,a closed vessel contains a mixture of equal weights of helium (mol. wt. = $4$),methane (mol. wt. = $16$) and sulphur dioxide (mol. wt. = $64$). The pressure exerted by the mixture is $210 \ mm$. If the partial pressures of helium,methane and sulphur dioxide are $p_1, p_2$ and $p_3$ respectively,which one of the following is correct?