Surface tension of a soap bubble is \(2.0 \times 10^{-2}\; Nm ^{-1}\). Work done to increase the radius of soap bubble from \(3.5\; cm\) to \(7\; cm\) will be \(.........\times 10^{-4}\,J\) [Take \(\pi=\frac{22}{7}\) ]

If momentum \((P),\) area \((A)\) and time \((T)\) are taken to be the fundamental quantities then the dimensional formula for energy is :

If the angular momentum of a planet of mass \(m\), moving a round the Sun in a circular orbit its \(L\), about the center of the Sun, its areal velocity is

Two sources of light emit X-rays of wavelength \(1\, nm\) and visible light of wavelength \(500\, nm ,\) respectively. Both the sources emit light of the same power \(200\, W\). The ratio of the number density of photons of X-rays to the number densitty of photons of the visible light of the given wavelengths is

A steel wire having a radius of \(2.0\, mm\), carrying a load of \(4\, kg\), is hanging from a ceiling. Given that \(g = 3.1\pi \,m{s^{ - 2}}\) , what will be the tensile stress that would be developed in the wire?

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time \(t_1\) . On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time \(t_2\). The time taken by her to walk up on the moving escalator will be 

Three conductors of same length having thermal conductivity \(\mathrm{k}_1, \mathrm{k}_2\) and \(\mathrm{k}_3\) are connected as shown in figure.

Area of cross sections of \(1^{\text {st }}\) and \(2^{\text {nd }}\) conductor are same and for \(3^{\text {rd }}\) conductor it is double of the \(1^{\text {st }}\) conductor. The temperatures are given in the figure. In steady state condition, the value of \(\theta\) is _______ \({ }^{\circ} \mathrm{C}\).
(Given : \(\mathrm{k}_1=60 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, \mathrm{k}_2=120 \mathrm{Js}^{-1}\mathrm{~m}^{-1}\) \( \mathrm{~K}^{-1}, \mathrm{k}_3=135 \mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}\) )

For an \(RLC\) circuit driven with voltage of amplitude \(v_m\) and frequency \({\omega _0} = \frac{1}{{\sqrt {LC} }}\) the current exhibits resonance. The quality factor, \(Q\) is given by:

The binding energy for the following nuclear reactions are expressed in MeV .
\({ }_2 He ^3+{ }_0 n ^1 \rightarrow{ }_2 He ^4+20 MeV\)
\({ }_2 He ^4+{ }_0 n ^1 \rightarrow{ }_2 He ^5-0.9 MeV\)
If \(X _3, X _4, X _5\) denote the stability of \({ }_2 He ^3,{ }_2 He ^4\) and \({ }_2 He ^5\), respectively, then the correct order is :

Let \([t]\) denote the greatest integer less than or equal to \(\mathrm{t}\). Let \(\mathrm{f}(\mathrm{x})=\mathrm{x}-[\mathrm{x}], \mathrm{g}(\mathrm{x})=1-\mathrm{x}+[\mathrm{x}]\), and \(h(x)=\min \{f(x), g(x)\}, x \in[-2,2]\). Then \(h\) is :

If a \(\triangle ABC\) has vertices \(A (-1,7), B (-7,1)\) and \(C (5,-5),\) then its orthocentre has coordinates

A metal exposed to light of wavelength \(800\,nm\) and emits photoelectrons with a certain kinetic energy. The maximum kinetic energy of photo-electron doubles when light of wavelength \(500\, nm\) is used. The work function of the metal is \(...eV\) (Take \(hc =1230 \,eV - nm\) ).

Let \(H: \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\) be a hyperbola such that the distance between its foci is \(6\) and the distance between its directrices is \(\dfrac{8}{3}\). If the line \(x=\alpha\) intersects the hyperbola \(H\) at the points \(A\) and \(B\) such that the area of the triangle \(AOB\) is \(4\sqrt{15}\), where \(O\) is the origin, then \(\alpha^2\) equals