The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants \(G, h\) and \(c\) . Which of the following correctly gives the Planck length?

Mass per unit area of a circular disc of radius \(\mathrm{a}\) depends on the distance \(\mathrm{r}\) from its centre as \(\sigma(\mathrm{r})=\mathrm{A}+\mathrm{Br} .\) The moment of inertia of the disc about the the axis, perpendicular to the plane and passing through its centre is

A capillary tube of radius 0.1 mm is partly dipped in water (surface tension \(70 \mathrm{dyn} / \mathrm{cm}\) and glass water contact angle \(\simeq 0^{\circ}\)) with \(30^{\circ}\) inclined with vertical. The length of water risen in the capillary is ________ cm.
\(\left(\right.\) Take \(\left.\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^2\right)\)

A convex lens of focal length \(20\,cm\) is placed in front of convex mirror with principal axis coinciding each other. The distance between the lens and mirror is \(10\,cm\). A point object is placed on principal axis at a distance of \(60\,cm\) from the convex lens. The image formed by combination coincides the object itself. The focal length of the convex mirror is \(...cm\).

\(7\) mole of certain monoatomic ideal gas undergoes a temperature increase of \(40 K\) at constant pressure. The increase in the internal energy of the gas in this process is\(....J\) (Given \(R =8.3 JK ^{-1} mol ^{-1}\) )

Two Carnot engines \(A\) and \(B\) are operated in series. Engine \(A\) receives heat from a reservoir at \(600\,K\) and rejects heat to a reservoir at temperature \(T\). Engine \(B\) receives; heat rejected by engine \(A\) and in turn rejects it to a reservoir at \(100\,K\). If the efficiencies of the two engines \(A\) and \(B\) are represented by \({\eta _A}\) and \({\eta _B}\) respectively, then what is the value of \(\frac{{{\eta _A}}}{{{\eta _B}}}\)

Given below are two statements: one is labelled as Assertion \(\mathrm{A}\) and the other is labelled as Reason \(\mathrm{R}\) Assertion A : The kinetic energy needed to project a body of mass m from earth surface to infinity is \(\frac{1}{2} \mathrm{mgR}\), where R is the radius of earth.
Reason \(\mathrm{R}\) : The maximum potential energy of a body is zero when it is projected to infinity from earth surface.
In the light of the above statements, choose the correct answer from the option given below

The longest wavelength associated with Paschen series is _______. (Given \(R_H=1.097 \times 10^7 \mathrm{SI}\) unit)

Given below are two statement : Statement \(I :\) If the number of turns in the coil of a moving coil galvanometer is doubled then the current sensitivity becomes double. Statement \(II :\) Increasing current sensitivity of a moving coil galvanometer by only increasing the number of turns in the coil will also increase its voltage sensitivity in the same ratio: In the light of the above statement, choose the correct answer from the options given below :

charge \(Q\) is uniformly distributed over a long rod \(AB\) of length \(L\) as shown in the figure. The electric potential at the point \(O\) lying at distance \(L\) from the end \(A\) is

The time dependence of the position of a particle of mass \(m = 2\) is given by \(\vec r\,(t)\, = \,2t\,\hat i\, - 3{t^2}\hat j\) Its angular momentum with respect to the origin at time \(t = 2\) is