A convex lens of focal length \(40 \mathrm{~cm}\) forms an image of an extended source of light on a photoelectric cell. A current I is produced. The lens is replaced by another convex lens having the same diameter but focal length \(20 \mathrm{~cm}\). The photoelectric current now is:

An object of mass \(0.2 \mathrm{~kg}\) executes simple harmonic motion along \(\mathrm{x}\) axis with frequency of \(\left(\frac{25}{\pi}\right) \mathrm{Hz}\). At the position \(\mathrm{x}=0.04 \mathrm{~m}\) the object has kinetic energy \(0.5 \mathrm{~J}\) and potential energy \(0.4 \mathrm{~J}\) The amplitude of oscillation is _______ cm.

A body is projected from the ground at an angle of \(45^{\circ}\) with the horizontal. Its velocity after \(2s\) is \(20 \,ms ^{-1}\). The maximum height reached by the body during its motion is \(m\). (use \(g =10\, ms ^{-2}\) )

In a Screw Gauge, fifth division of the circular scale coincides with the reference line when the ratchet is closed. There are \(50\) divisions on the circular scale, and the main scale moves by \(0.5 \,{mm}\) on a complete rotation. For a particular observation the reading on the main scale is \(5\, {mm}\) and the \(20^{\text {th }}\) division of the circular scale coincides with reference line. Calculate the true reading. (in \({mm}\))

A string \(A\) of length \(0.314\) m and Young's modulus \(2 \times 10^{10}\) N/m\(^2\) is connected to another string \(B\) of length and Young's modulus both twice of those of \(A\). This series combination of strings is then suspended from a rigid support and its free end is fixed to a load of mass \(0.8\) kg. The net change in length of the combination is _____ mm. (radius of both the strings is \(0.2\) mm and acceleration due to gravity \(= 10\) m/s\(^2\)) (Mass of both strings is to be neglected as compared to the mass of load)

A uniformly charged ring of radius \(3a\) and total charge \(q\) is placed in \(xy-\) plane centered at origin. A point charge \(q\) is moving towards the ring along the \(z-\) axis and has speed \(v\) at \(z = 4a\). The minimum value of \(v\) such that it crosses the origin is

There are two identical chambers, completely thermally insulated from surroundings. Both chambers have a partition wall dividing the chambers in two compartments. Compartment \(1\) is filled with an ideal gas and Compartment \(3\) is filled with a real gas. Compartments \(2\) and \(4\) are vacuum . A small hole (orifice) is made in the partition walls and the gases are allowed to expand in vacuum Statement \(-1\) : No change in the temperature of the gas takes place when ideal gas expands in vacuum. However, the temperature of real gas goes down (cooling) when it expands in vacuum Statement \(-2\) : The internal energy of an ideal gas is only kinetic. The internal energy of a real gas is kinetic as well as potential

Figure shows a circuit that contains four identical resistors with resistance \(R =2.0\, \Omega,\) two identical inductors with inductance \(L =2.0 \,mH\) and an ideal battery with \(e m f E =9 \,V .\) The current \('i'\) just after the switch \('S'\) is closed will be .... \(A\)

The largest value of \(r\) for which the region represented by the set \(\{ \omega \, \in \,C|\,\omega \, - \,4\, - \,i|\, \le \,r\} \) is contained in the region represented by the set \((z\, \in \,c/|\,z\, - \,1\,\,\left|  \le  \right|\,z\, + \,i|),\) is equal to

Let \(f :(0,1) \rightarrow R\) be a function defined by \(f(x)=\frac{1}{1-e^{-x}}\), and \(g(x)=(f(-x)-f(x))\). Consider two statement: \((I)\) \(g\) is an increasing function in \((0,1)\) \((II)\) \(g\) is one-one in \((0,1)\) Then,

The integral \(\int_0^\pi \frac{8 x d x}{4 \cos ^2 x+\sin ^2 x}\) is equal to

The solution curve of the differential equation, \(\left(1+ e ^{- x }\right)\left(1+ y ^{2}\right) \frac{ dy }{ dx }= y ^{2},\) which passes through the point \((0,1),\) is

As shown in the figure, a current of \(2\,A\) flowing in an equilateral triangle of side \(4 \sqrt{3}\,cm\). The magnetic field at the centroid \(O\) of the triangle is: (Neglect the effect of earth's magnetic field.)