Which of the following combinations has the dimension of electrical resistance ( \({ \varepsilon _0}\) is the permittivity of vacuum and \({\mu _0}\) is the permeability of vacuum) ?

A particle is moving in a straight line. The variation of position ' \(x\) ' as a function of time ' \(t\) ' is given as \(x=\left(t^3-6 t^2+20 t+15\right) m\). The velocity of the body when its acceleration becomes zero is _______.

A galvanometer having coil resistance \(10 \ \Omega\) shows a full scale deflection for a current of \(3 \mathrm{~mA}\). For it to measure a current of \(8 \mathrm{~A}\), the value of the shunt should be:

A Laser light of wavelength \(660\,nm\) is used to weld Retina detachment. If a Laser pulse of width \(60\, ms\) and power \(0.5\, kW\) is used the approximate number of photons in the pulse are : [Take Planck's constant \(h\, = 6.62\times10^{- 34}\, Js\)]

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) :


Three identical spheres of same mass undergo one dimensional motion as shown in figure with initial velocities \(v_{\mathrm{A}}=5 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}, v_{\mathrm{C}}=4 \mathrm{~m} / \mathrm{s}\). If we wait sufficiently long for elastic collision to happen, then \(v_{\mathrm{A}}=4 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}\), \(v_{\mathrm{C}}=5 \mathrm{~m} / \mathrm{s}\) will be the final velocities.

Reason (R): In an elastic collision between identical masses, two objects exchange their velocities.
In the light of the above statements, choose the correct answer from the options given below :

A convex lens, of focal length \(30\,cm,\)  a concave lens of focal length \(120\,cm,\) and a plane mirror are arranged as shown. For an object kept at a distance of \(60\,cm\)  from the convex lens, the final image, formed by the combination, is a real image, at a distance of

A sonometer wire of length \(114\, cm\) is fixed at both the ends. Where should the two bridges be placed so as to divide the wire into three segments whose fundamental frequencies are in the ratio \(1 : 3 : 4\) ?

A plane electromagnetic wave propagates along the \(+x\) direction in free space. The components of the electric field, \(\vec{E}\) and magnetic field, \(\vec{B}\) vectors associated with the wave in Cartesian frame are

Let a triangle be bounded by the lines \(L _{1}: 2 x +5 y =10\); \(L _{2}:-4 x +3 y =12\) and the line \(L _{3}\), which passes through the point \(P (2,3)\), intersect \(L _{2}\) at \(A\) and \(L _{1}\) at \(B\). If the point \(P\) divides the line-segment \(A B\), internally in the ratio \(1: 3\), then the area of the triangle is equal to

Which of the following relations is true for two unit vectors \(\hat{ A }\) and \(\hat{ B }\) making an angle \(\theta\) to each other\(?\)

An aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at \(30^{\circ} C\). The coefficient of linear expansion of aluminium and steel are \(24 \times 10^{-6} /{ }^{\circ} C\) and \(1.2 \times 10^{-5} /{ }^{\circ} C\), respectively. The length of this composite rod when its temperature is raised to \(100^{\circ} C\), is _________ cm .

A square loop \(\mathrm{PQRS}\) having \(10\) turns, area \(3.6 \times\) \(10^{-3} \mathrm{~m}^2\) and resistance \(100 \Omega\) is slowly and uniformly being pulled out of a uniform magnetic field of magnitude \(\mathrm{B}=0.5 \mathrm{~T}\) as shown. Work done in pulling the loop out of the field in \(1.0 \mathrm{~s}\) is _______ \(\times 10^{-6} \mathrm{~J}\).

If the wavelength of the first member of Lyman series of hydrogen is \(\lambda\). The wavelength of the second member will be _______.