A flask contains argon and oxygen in the ratio of \(3: 2\) in mass and the mixture is kept at \(27^{\circ} C\). The ratio of their average kinetic energy per molecule respectively will be ...........

The magnetic flux \(\phi\) (in weber) linked with a closed circuit of resistance \(8\  \Omega\) varies with time (in seconds) as \(\phi=5 t^2-36 t+1\). The induced current in the circuit at \(t=2 \mathrm{~s}\) is _______ \(\mathrm{A}\).

A set of \(20\) tuning forks is arranged in a series of increasing frequencies. If each fork gives \(4\) beats with respect to the preceding fork and the frequency of the last fork is twice the frequency of the first, then the frequency of last fork is \(\dots \; Hz\).

Two blocks of masses \(3 \,{kg}\) and \(5\, {kg}\) are connected by a metal wire going over a smooth pulley. The breaking stress of the metal is \(\frac{24}{\pi} \times 10^{2}\, {Nm}^{-2}\). What is the minimum radius of the wire? (Take \(\left.g=10\, {ms}^{-2}\right)\) (in \(cm\))

The workdone in an adiabatic change in an ideal gas depends upon only :

Considering the Bohr model of hydrogen like atoms, the ratio of the ratio of the radius \(5^{\text {th}}\) orbit of the electron in \(\mathrm{Li}^{2+}\) and \(\mathrm{He}^{+}\) is _______.

A brass wire of length 2 m and radius 1 mm at \(27^{\circ} C\) is held taut between two rigid supports. Initially it was cooled to a temperature of \(-43^{\circ} C\) creating a tension T in the wire. The temperature to which the wire has to be cooled in order to increase the tension in it to 1.4 T , is _________ \({ }^{\circ} C\)

A body of mass (\(2 {M}\)) splits into four masses \(\{{m}, {M}-{m}, {m}, {M}-{m}\}\), which are rearranged to form a square as shown in the figure. The ratio of \(\frac{{M}}{{m}}\) for which, the gravitational potential energy of the system becomes maximum is \({x}: 1\). The value of \(x\) is ..... .

The area of the region \(\{(x, y): y \leq \pi - |x|, y \leq |x \sin x|, y \geq 0\}\) is:

A ray of light along \(x + \sqrt 3 y = \sqrt 3 \) gets reflected upon reaching \(x- \) axis , the equation of the reflected ray is 

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three units vectors such that \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0} .\) If \(\lambda=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}} \) and \(\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}},\) then the ordered pair \((\lambda, {\mathrm{\vec d}})\) is equal to 

If the sum  \(\frac{3}{1^2} + \frac{5}{{{1^2} + {2^2}}} + \frac{7}{{{1^2} + {2^2} + {3^2}}} + ...... + \) up to \(20\) terms is equal to \(\frac{k}{{21}}\), then \(k\) is equal to

Given two independent events, if the probability that exactly one of them occurs is \(\frac {26}{49}\) and the probability that none of them occurs is \(\frac {15}{49}\) , then the probability of more probable of the two events is