The aperture of the objective is \(24.4\,cm\). The resolving power of this telescope. If a light of wavelength \(2440 \mathring A\) is used to see the object will be

A pendulum is suspended by a string of length \(250\,cm\). The mass of the bob of the pendulum is \(200\,g\). The bob is pulled aside until the string is at \(60^{\circ}\) with vertical as shown in the figure. After releasing the bob. the maximum velocity attained by the bob will be________ \(ms ^{-1}\). (if \(g=10\,m / s ^{2}\) )

A bi-convex lens has radius of curvature of both the surfaces same as \(1 / 6 \mathrm{~cm}\). If this lens is required to be replaced by another convex lens having different radii of curvatures on both sides \(\left(R_1 \neq R_2\right)\), without any change in lens power then possible combination of \(R_1\) and \(R_2\) is :

A parallel plate capacitor has capacitance C , when there is vacuum within the parallel plates.
A sheet having thickness \(\left(\frac{1}{3}\right)^{ rd }\) of the separation between the plates and relative permittivity K is introduced between the plates. The new capacitance of the system is :

Two cells of emfs \(1\) V and \(2\) V and internal resistance \(2\,\Omega\) and \(1\,\Omega\), respectively connected in parallel, gave a current of \(1\) A through an external resistance. If the polarity of one cell is reversed, then value of current through the external resistance will be \(\dfrac{\alpha}{5}\) A. The value of \(\alpha\) is __________.

Find the equivalent resistance between two ends of the following circuit

Three masses \(M =100\,kg , m _{1}=10\,kg\) and \(m_{2}=20\,kg\) are arranged in a system as shown in figure. All the surfaces are frictionless and strings are inextensible and weightless. The pulleys are also weightless and frictionless. \(A\) force \(F\) is applied on the system so that the mass \(m_{2}\) moves upward with an acceleration of \(2\,ms ^{-2}\). The value of \(F\) is \(......N\) \(\left(\right.\) Take \(\left.g =10\,ms ^{-2}\right)\)

In the circuit shown in the figure, the ratio of the quality factor and the band width is \(.............\) s.

A body of mass \(5\, kg\) under the action of constant force \(\overrightarrow F  = {F_x}\hat i + {F_y}\hat j\) has velocity at \(t\,= 0\, s\) as \(\overrightarrow v  = \left( {6\hat i - 2\hat j\,m/s} \right)\)  and at \(t\, = 10\,s\) as \(\overrightarrow v  = +6\hat j\,m/s\). The force \(\overrightarrow F \) is 

The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal place: \((i)\) a ring of radius \(R,\,(ii)\) a solid cylinder of radius \(\frac {R}{2}\)  nd \((iii)\) a solid sphere of radius \(\frac {R}{4}.\) If, in each case, the speed of the center of mass at bottom of the incline is same, the ratio of the maximum heights they climb is

A particle of mass \({m}\) is suspended from a ceiling through a string of length \(L\). The particle moves in a horizontal circle of radius \(r\) such that \({r}=\frac{{L}}{\sqrt{2}}\). The speed of particle will be:

Two iron solid discs of negligible thickness have radius \(R_1\) and \(R_2\) and moment of intertia \(I_1\) and \(I_2\), respectively. For \(R_2=2 R_1\), the ratio of \(I_1\) and \(I_2\) would be \(1 / x\), where \(\mathrm{x}=\) ________ -

The \(rms\) speeds of the molecules of Hydrogen, Oxygen and Carbondioxide at the same temperature are \({V}_{{H}}, {V}_{0}\) and \({V}_{{C}}\) respectively then