A wire having a linear mass density \(9.0 \times 10^{-4} \;{kg} / {m}\) is stretched between two rigid supports with a tension of \(900\; {N}\). The wire resonates at a frequency of \(500\;{Hz}\). The next higher frequency at which the same wire resonates is \(550\; {Hz}\). The length of the wire is \(...... {m}\)

Two moles of an ideal gas with \(\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}=\frac{5}{3}\) are mixed with \(3\) moles of another ideal gas with \(\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}=\frac{4}{3} .\) The value of \(\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}\) for the mixture is

The gravitational field, due to the 'left over part' of a uniform sphere (from which a part as shown, has been 'removed out'), at a very far off point , \(P\), located as shown, would be (nearly)

In normal adjustment, for a refracting telescope, the distance between objective and eye piece is \(30\,cm\). The focal length of the objective, when the angular magnification of the telescope is \(2\) , will be \(.....cm\) 

An ac current is represented as
\(\mathrm{i}=5 \sqrt{2}+10 \cos \left(650 \pi \mathrm{t}+\frac{\pi}{6}\right) \mathrm{Amp}\)
The r.m.s value of the current is

A particle starts from origin at \(t=0\) with a velocity \(5 \hat{i} \mathrm{~m} / \mathrm{s}\) and moves in \(x-y\) plane under action of a force which produces a constant acceleration of \((3 \hat{i}+2 \hat{j}) \mathrm{m} / \mathrm{s}^2\). If the \(x\)-coordinate of the particle at that instant is \(84 \mathrm{~m}\), then the speed of the particle at this time is \(\sqrt{\alpha} \mathrm{m} / \mathrm{s}\). The value of \(\alpha\) is___________.

The angular momentum of an electron in a hydrogen atom is proportional to _______. (Where \(\mathrm{r}\) is the radius of orbit of electron)

The diffraction pattern of a light of wavelength \(400\) \(\mathrm{nm}\) diffracting from a slit of width \(0.2 \mathrm{~mm}\) is focused on the focal plane of a convex lens of focal length \(100 \mathrm{~cm}\). The width of the \(1^{\text {st }}\) secondary maxima will be _______.

A paratrooper jumps from an aeroplane and opens a parachute after 2 s of free fall and starts deaccelerating with \(3 m / s ^2\). At 10 m height from ground, while descending with the help of parachute, the speed of paratrooper is \(5 m / s\). The initial height of the aeroplane is ___________m.
\(\left( g =10 m / s ^2\right)\)

The radius of fifth orbit of the \(L ^{++}\)is \(.........\times 10^{-12}\,m\). Take : radius of hydrogen atom \(=0.51 \mathring A\)