A capacitor of capacitance \(100\  \mu \mathrm{F}\) is charged to a potential of \(12  \mathrm{~V}\) and connected to a \(6.4 \ \mathrm{mH}\) inductor to produce oscillations. The maximum current in the circuit would be :

A particle executes SHM of amplitude A. The distance from the mean position when its's kinetic energy becomes equal to its potential energy is on

Light source having wavelength \(331\) nm is used to generate photo-electrons whose stopping potential is \(0.2\) V. The work function of the used metal in the experiment is \(\alpha \times 10^{-19}\) J. The value of \(\alpha\) is _____.
(\(h = 6.62 \times 10^{-34}\) J s, \(e = 1.6 \times 10^{-19}\) C and \(c = 3 \times 10^8\) m/s)

The least count of the main scale of a vernier callipers is \(1\, mm\). Its vernier scale is divided into \(10\) divisions and coincide with \(9\) divisions of the main scale. When jaws are touching each other, the \(7\) th division of vernier scale coincides with a division of main scale and the zero of vernier scale is lying right side of the zero of main scale. When this vernier is used to measure length of a cylinder the zero of the vernier scale between \(3.1\, cm\) and \(3.2\, cm\) and \(4^{th}\) \(VSD\) coincides with a main scale division.The length of the cylinder is \(.....cm\) (\(VSD\) is vernier scale division)

A square loop \(\mathrm{PQRS}\) having \(10\) turns, area \(3.6 \times\) \(10^{-3} \mathrm{~m}^2\) and resistance \(100 \Omega\) is slowly and uniformly being pulled out of a uniform magnetic field of magnitude \(\mathrm{B}=0.5 \mathrm{~T}\) as shown. Work done in pulling the loop out of the field in \(1.0 \mathrm{~s}\) is _______ \(\times 10^{-6} \mathrm{~J}\).

A Carnot engine whose heat \(\operatorname{sinks}\) at \(27\,^{\circ} C\), has an efficiency of \(25 \%\). By how many degrees should the temperature of the source be changed to increase the efficiency by \(100 \%\) of the original efficiency \(?\)

The electric field of an electromagnetic wave travelling through a medium is given by \(\overline{E}(x,t) = 25\ sin(2.0 \times 10^{15}t - 10^{7}x)\hat{n}\) then the refractive index of the medium is ___________.
(All given measurement are in SI units)

The sum of all values of \(\theta \in[0,2 \pi]\) satisfying \(2 \sin ^2 \theta=\cos 2 \theta\) and \(2 \cos ^2 \theta=3 \sin \theta\) is

If \(f(x)\, = {x^2} - x + 5,\,\,x > \frac{1}{2},\) and \(g(x)\) is its inverse function, then \(g'(7)\) equals

Statement \(-1\) : The equation \(x\, log\, x = 2 - x\) is satisfied by at least one value of \(x\) lying between \(1\) and \(2\) Statement \(-2\) : The function \(f(x) = x\, log\, x\) is an increasing function in \([1, 2]\) and \(g (x) = 2 -x\) is a decreasing function in \([ 1 , 2]\) and the graphs represented by these functions intersect at a point in \([ 1 , 2]\)

A capacitor of reactance \(4 \sqrt{3} \Omega\) and a resistor of resistance \(4 \Omega\) are connected in series with an ac source of peak value \(8 \sqrt{2} \mathrm{~V}\). The power dissipation in the circuit is _______ \(W\).

If the mean and the variance of \(6,4, a, 8, b, 12,10\), 13 are 9 and 9.25 respectively, then \(a+b+a b\) is equal to :

In a Young's double slit experiment \(15\) fringes are observed on a small portion of the screen when light of wavelength \(500 \;\mathrm{nm}\) is used. Ten fringes are observed on the same section of the screen when another light source of wavelength \(\lambda\) is used. Then the value of \(\lambda\) is (in \(\mathrm{nm})\)