A sinusoidal voltage of peak value \(250\, V\) is applied to a series \(LCR\) circuit, in which \(R =8 \Omega, L =24\, mH\) and \(C =60 \mu F\). The value of power dissipated at resonant condition is \('x'\, kW\). The value of \(x\) to the nearest integer is .............

In a perfectly inelastic collision, two spheres made of the same material with masses 15 kg and 25 kg , moving in opposite directions with speeds of \(10 m / s\) and \(30 m / s\), respectively, strike each other and stick together. The rise in temperature (in \({ }^{\circ} C\) ), if all the heat produced during the collision is retained by these spheres, is :
(specific heat of sphere material \(31 cal / kg .{ }^{\circ} C\) and \(1 cal =4.2 J)\)

The gravitational field in a region is given by \(\overrightarrow g  = 5\,N/kg\hat i\, + \,12\,N/kg\hat j\). The change in the gravitational potential energy of a particle of mass \(1\, kg\) when it is taken from the origin to a point \((7\, m, - 3\, m)\) is .......  \(J\)

A letter \('A'\) is constructed of a uniform wire with resistance \(1.0\,\Omega \) per \(cm\) , The sides of the letter are \(20\, cm\) and the cross piece in the middle is \(10\, cm\) long. The apex angle is \(60\) . The resistance between the ends of the legs is close to ................ \(\Omega\)

Ratio of thermal energy released in two resistor \(R\) and \(3 R\) connected in parallel in an electric circuit is :

Let \(\vec A\, = \,(\hat i\, + \,\hat j)\,\)  and \(\vec B\, = \,(2\hat i\, - \,\hat j)\,.\)  The magnitude of a coplanar vector \(\vec C\) such that  \(\vec A\cdot \vec C\, = \,\vec B\cdot \vec C\, = \vec A\cdot \vec B\) is given by

Consider two \(npn\) transistors as shown in figure. If \(0\,Volts\) corresponds to false and \(5\, Volts\) correspond to true then the output at \(C\) corresponds to

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 _______.

The volume of an ideal gas increases 8 times and temperature becomes (1/4) of initial temperature during a reversible change. If there is no exchange of heat in this process (\( \Delta Q=0 \)) then identify the gas from the following options:

The diameter of the objective lens of microscope makes an angle \(\beta \) at the focus of the microscope. Further, the medium between the object and the lens is an oil of refractive index \(n\). Then the resolving power of the microscope