A piece of bone of an animal from a ruin is found to have \(^{14}C\) activity of \(12\) disintegrations per minute per gm of its carbon content. The \(^{14}C\) activity of a living animal is \(16\) disintegrations per minute per gm. How long ago nearly did the animal die? ............\(years\) (Given halflife of \(^{14}C\) is \(t_{1/2} = 5760\,years\) )

If one wants to remove all the mass of the earth to infinity in order to break it up completely. The amount of energy that needs to be supplied will be \(\frac{x}{5}\, \frac{ GM ^{2}}{ R }\) where \(x\) is \(\quad\,..........\) (Round off to the Nearest Integer) (\(M\) is the mass of earth, \(R\) is the radius of earth, \(G\) is the gravitational constant)

Hydrogen \((_1H^1),\) Deuterium \((_1H^2)\),singly ionised Helium \((_2He^4)^+\) and doubly ionised lithium \({\left( {{}_3^6Li} \right)^{ + + }}\), all have one electron around the nucleus. Consider an electron transition from \(n = 2\) to \(n = 1\). If the wavelengths of emitted radiation are \(λ_1,λ_2,λ_3\) and \(λ_4\) respectively then approximately which one of the following is correct?

With reference to the observations in photo-electric effect, identify the correct statements from below: \(A.\) The square of maximum velocity of photoelectrons varies linearly with frequency of incident light. \(B.\) The value of saturation current increases on moving the source of light away from the metal surface. \(C.\) The maximum kinetic energy of photo-electrons decreases on decreasing the power of LED (light emitting diode) source of light. \(D.\) The immediate emission of photo-electrons out of metal surface can not be explained by particle nature of light/electromagnetic waves. \(E.\) Existence of threshold wavelength can not be explained by wave nature of light/electromagnetic waves. Choose the correct answer from the options given below:

A pole is vertically submerged in swimming pool, such that it gives a length of shadow \(2.15 \,m\) within water when sunlight is incident at an angle of \(30^{\circ}\) with the surface of water. If swimming pool is filled to a height of \(1.5\,m\), then the height of the pole above the water surface in centimetres is \(\left(\eta_W=4 / 3\right)........\)

A tennis ball (treated as hollow spherical shell) starting from \(O\) rolls down a hill. At point \(A\) the ball becomes air borne leaving at an angle of \(30^o\) with the horizontal. The ball strikes the ground at \(B\). What is the value of the distance \(AB\) ? (Moment of inertia of a spherical shell of mass \(m\) and radius \(R\) about its diameter .......... \(m\). \(= \frac {2}{3}\,mR^2\))

A particle of mass \(m\) with an initial velocity \(u\hat i\) collides perfectly elastically with a mass \(3m\) at rest. It moves with a velocity \(v\hat j\) after collision, then, \(v\) is given by

Three solid spheres each of mass \(\mathrm{m}\) and diameter \(d\) are stuck together such that the lines connecting the centres form an equilateral triangle of side of length \(d\). The ratio \(\mathrm{I}_{0} / \mathrm{I}_{\mathrm{A}}\) of moment of inertia \(\mathrm{I}_{0}\) of the system about an axis passing the centroid and about center of any of the spheres \(\mathrm{I}_{\mathrm{A}}\) and perpendicular to the plane of the triangle is

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R)
Assertion (A) : Refractive index of glass is higher than that of air.
Reason ( \(\mathrm{R}\)) : Optical density of a medium is directly proportionate to its mass density which results in a proportionate refractive index.
In the light of the above statements, choose the most appropriate answer from the options given below :

A body of mass \(m\) is moving in a circular orbit of radius \(R\) about a planet of mass \(M\). At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius \(\frac{R}{2}\) , and the other mass, in a circular orbit of radius \(\frac{3R}{2}\). The difference between the final and initial total energies is

A circular disc reaches from top to bottom of an inclined plane of length \('L'.\) When it slips down the plane, it takes time \('t_{1}'\). when it rolls down the plane, it takes time \(t_{2}\). The value of \(\frac{t_{2}}{t_{1}}\) is \(\sqrt{\frac{3}{x}}\). The value of \(x\) will be .... .

A beam of electrons of energy \(E\) scatters from a target having atomic spacing of \(1\, A\). The first maximum intensity occurs at \(\theta=60^{\circ} .\) Then \(E\) (in \(eV )\) is\(......\) (Planck constant \(h =6.64 \times 10^{-34}\, Js\) \(1 \,eV =1.6 \times 10^{-19}\,J\), electron mass \(\left. m =9.1 \times 10^{-31}\, kg \right)\)