Assume that protons and neutrons have equal masses. Mass of a nucleon is \(1.6 \times 10^{-27}\,kg\) and radius of nucleus is \(1.5 \times 10^{-15} A ^{1 / 3}\,m\). The approximate ratio of the nuclear density and water density is \(n \times 10^{13}\). The value of \(n\) is \(.................\)

A block is fastened to a horizontal spring. The block is pulled to a distance \(x =10\,cm\) from its equilibrium position (at \(x =0\) ) on a frictionless surface from rest. The energy of the block at \(x =5\) \(cm\) is \(0.25\,J\). The spring constant of the spring is \(.........Nm ^{-1}\)

An object is kept at rest at a distance of 3 R above the earth's surface where R is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is :
(Assume \(\mathrm{M}=\) mass of earth, \(\mathrm{G}=\) Universal gravitational constant)

Consider two infinitely large plane parallel conducting plates as shown below. The plates are uniformly charged with a surface charge density \(+\sigma\) and \(-2 \sigma\). The force experienced by a point charge +q placed at the mid point between two plates will be :

A travelling wave is described by the equation \(y(x, t)=[0.05 \sin (8 x-4 t] \;m\) The velocity of the wave is \(.....ms ^{-1}\) [all the quantities are in SI unit]

Identify the operation performed by the circuit given below

A thin uniform bar of length \(L\) and mass \(8\,m\) lies on a smooth horizontal table. Two point masses \(m\) and \(2\,m\) moving in the same horizontal plane from opposite sides of the bar with speeds \(2v\) and \(v\) respectively. The masses stick to the bar after collision at a distance \(\frac{L}{3}\) and \(\frac{L}{6}\) respectively from the centre of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be

The stopping potential for photoelectrons emitted from a surface illuminated by light of wavelength \(6630 \; \mathring A\) is \(0.42 \; V\). If the threshold frequency is \(x \times\) \(10^{13}  / s\), where \(x\) is\(\dots\) (nearest integer). (Given, speed light \(=3 \times 10^{8} \; m / s\), Planck's constant \(=6.63 \times 10^{-34} \; Js\) )

The number density of molecules of a gas depends on their distance \(r\) from the origin as, \(n\left( r \right) = {n_0}{e^{ - \alpha {r^4}}}\). Then the total number of molecules is proportional to

Statement \(I :\) The ferromagnetic property depends on temperature. At high temperature, ferromagnet becomes paramagnet. Statement \(II :\) At high temperature, the domain wall area of a ferromagnetic substance increases. In the light of the above statements, choose the most appropriate answer from the options given below :

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A :\) Earth has atmosphere whereas moon doesn't have any atmosphere. Reason \(R\) : The escape velocity on moon is very small as compared to that on earth. In the light of the above statement, choose the correct answer from the options given below :

A radioactive material decays by simultaneous emissions of two particles with half lives of \(1400\, years\) and \(700\, years\) respectively. What will be the time after which one third of the material remains? (Take In \(3=1.1\) ) (In \(years\))

A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is \(\left(\frac{a-1}{a}\right)\) then the value of \(a\) is _______.