The difference in the number of waves when yellow light propagates through air and vacuum columns of the same thickness is one. The thickness of the air column is \(....\,{mm} .\) [Refractive index of air \(=1.0003\), wavelength of yellow light in vacuum \(=6000 \,\mathring {{A}}\) ]

Two identical thin biconvex lenses of focal length \(15\, cm\) and refractive index \(1.5\) are in contact with each other. The space between the lenses is filled with a liquid of refractive index \(1.25\). The focal length of the combination is \(cm\).

A particle is executing simple harmonic motion. Its amplitude is \(A\) and time period is \(5\) sec. The time required by it to move from \(x = A\) to \(x = \dfrac{A}{\sqrt{2}}\) is _______ sec.

The general displacement of a simple harmonic oscillator is \(x = A \sin \omega t\). Let \(T\) be its time period. The slope of its potential energy (U) - time (t) curve will be maximum when \(t=\frac{T}{\beta}\). The value of \(\beta\) is \(.........\)

If a semiconductor photodiode can detect a photon with a maximum wavelength of \(400\, nm ,\) then its band gap energy is Planck's constant \(h=6.63 \times 10^{-34}\, J . s\) Speed of ligh \(c =3 \times 10^{8}\, m / s\)

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason(R).
Assertion (A) : Magnetic monopoles do not exist.
Reason (R) : Magnetic field lines are continuous and form closed loops.
In the light of the above statements, choose the most appropriate answer from the options given below :

The speed of light in media \('A'\) and \('B'\) are \(2.0 \times\) \(10^{10} \,cm / s\) and \(1.5 \times 10^{10} \,cm / s\) respectively. A ray of light enters from the medium \(B\) to \(A\) at an incident angle ' \(\theta\) '. If the ray suffers total internal reflection, then

A large block of wood of mass \(M =5.99\, kg\) is hanging from two long massless cords. A bullet of mass \(m =10\, g\) is fired into the block and gets embedded in it. The (block \(+\) bullet) then swing upwards, their centre of mass rising a vertical distance \(h =9.8\,cm\) before the (block \(+\) bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before collision is: (Take \(g =9.8\, ms ^{-2}\) ) (in \(m/s\))

As per the given circuit, the value of current through the battery will be \(\dots \; A\).

An inclined plane making an angle of \(30^{\circ}\) with the horizontal is placed in a uniform horizontal electric field \(200 \, \frac{ N }{ C }\) as shown in the figure. A body of mass \(1\, kg\) and charge \(5\, mC\) is allowed to slide down from rest at a height of \(1\, m\). If the coefficient of friction is \(0.2,\) find the time (in \(s\) )taken by the body to reach the bottom. \(\left[ g =9.8 \,m / s ^{2}, \sin 30^{\circ}=\frac{1}{2}\right.\); \(\left.\cos 30^{\circ}=\frac{\sqrt{3}}{2}\right]\)

Two vectors \(\overrightarrow{ A }\) and \(\overrightarrow{ B }\) have equal magnitudes. If magnitude of \(\overrightarrow{ A }+\overrightarrow{ B }\) is equal to two times the magnitude of \(\overrightarrow{ A }-\overrightarrow{ B }\), then the angle between \(\overrightarrow{ A }\) and \(\overrightarrow{ B }\) will be .......................

If a radioactive element having half-life of \(30\,min\) is undergoing beta decay, the fraction of radioactive element remains undecayed after \(90\,min\). will be :

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : The binding energy per nucleon is found to be practically independent of the atomic number A, for nuclei with mass numbers between 30 and 170.
Reason (R): Nuclear force is long range.
In the light of the above statements, choose the correct answer from the options given below :