As shown in the figure, a block of mass \(\sqrt{3}\, kg\) is kept on a horizontal rough surface of coefficient of friction \(\frac{1}{3 \sqrt{3}}\). The critical force to be applied on the vertical surface as shown at an angle \(60^{\circ}\) with horizontal such that it does not move, will be \(3 x\). The value of \(3x\) will be \(\left[ g =10 m / s ^{2} ; \sin 60^{\circ}=\frac{\sqrt{3}}{2} ; \cos 60^{\circ}=\frac{1}{2}\right]\)

Two electrons are moving with non-relativistic speeds perpendicular to each other. If corresponding de Broglie wavelengths are \({\lambda _1}\) and \({\lambda _2}\), their de Broglie wavelength in the frame of reference attached to their centre of mass is

A particle is projected with velocity \(u\) so that its horizontal range is three times the maximum height attained by it. The horizontal range of the projectile is given as \(\frac{n u^2}{25 g}\), where value of \(n\) is : (Given ' \(g\) ' is the acceleration due to gravity).

An aluminium rod with Young's modulus \(Y =7.0\) \(\times 10^{10} N / m ^2\) undergoes elastic strain of \(0.04 \%\). The energy per unit volume stored in the rod in SI unit is:

An object is placed at a distance of \(12\,cm\) in front of a plane mirror. The virtual and erect image is formed by the mirror. Now the mirror is moved by \(4\,cm\) towards the stationary object. The distance by which the position of image would be shifted, will be:

In the reported figure, there is a cyclic process \(ABCDA\) on a sample of \(1\, {mol}\) of a diatomic gas. The temperature of the gas during the process \({A} \rightarrow {B}\) and \({C} \rightarrow {D}\) are \({T}_{1}\) and \({T}_{2}\left({T}_{1}\,>\,{T}_{2}\right)\) respectively. Choose the correct option out of the following for work done if processes \(B C\) and \(D A\) are adiabatic.

Given below are two statements. One is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion A :Two identical balls \(A\) and \(B\) thrown with same velocity '\(u\) ' at two different angles with horizontal attained the same range \(R\). If \(A\) and \(B\) reached the maximum height \(h_{1}\) and \(h_{2}\) respectively, then \(R =4 \sqrt{ h _{1} h _{2}}\) Reason R: Product of said heights. \(h _{1} h _{2}=\left(\frac{u^{2} \sin ^{2} \theta}{2 g }\right) \cdot\left(\frac{u^{2} \cos ^{2} \theta}{2 g }\right)\) Choose the \(CORRECT\) answer 

Two forces having magnitude \(A\) and \(\frac{ A }{2}\) are perpendicular to each other. The magnitude of their resultant is

Match List - I with List - II.

List - I	List - II
(A) Permeability of free space	(I) \(\left[\mathrm{M} \mathrm{~L}^2 \mathrm{~T}^{-2}\right]\)
(B) Magnetic field	(II) \(\left[\mathrm{M} \mathrm{~T}^{-2} \mathrm{~A}^{-1}\right]\)
(C) Magnetic moment	(III) \(\left[\mathrm{M} \mathrm{~L} \mathrm{~T}^{-2} \mathrm{~A}^{-2}\right]\)
(D) Torsional constant	(IV) \(\left[\mathrm{L}^2 \mathrm{~A}\right]\)

Choose\(\text{ }\)the\(\text{ }\)correct\(\text{ }\)answer\(\text{ }\)from\(\text{ }\)the\(\text{ }\)options\(\text{ }\)given\(\text{ }\)below\(\text{ }\):

Match List - I with List - II.

List - I	List - II
(A) Magnetic induction	(I) Ampere meter2
(B) Magnetic intensity	(II) Weber
(C) Magnetic flux	(III) Gauss
(D) Magnetic moment	(IV) Ampere meter

Choose the correct answer from the options given below :

A \(25\times10^{-3}\, m^3\) volume cylinder is filled with \(1\, mol\) of \(O_2\) gas at room temperature \((300\, K)\). The molecular diameter of \(O_2\), and its root mean square speed, are found to be \(0.3\, nm\) and \(200\, m/s\), respectively. What is the average collision rate (per second) for an \(O_2\) molecule?