If mass is written as \(\mathrm{m}=\mathrm{kc}^{\mathrm{p}} \mathrm{G}^{-1 / 2} \mathrm{~h}^{1 / 2}\) then the value of \(P\) will be _______. (Constants have their usual meaning with \(\mathrm{k}\) a dimensionless constant)

The width of one of the two slits in a Young's double slit experiment is three times the other slit. If the amplitude of the light coming from a slit is proportional to the slit-width, the ratio of minimum to maximum intensity in the interference pattern is \(x: 4\) where \(x\) is ..... .

Identify the pair of physical quantities which have different dimensions 

\(A\) and \(B\) are two sources generating sound waves. A listener is situated at \(C\) . The frequency of the source at \(A\) is \(500\,Hz\) . \(A,\) now, moves towards \(C\) with a speed \(4\,m/s.\) The number of beats heard at \(C\) is \(6.\) When \(A\) moves away from \(C\) with speed \(4\,m/s,\) the number of beats heard at \(C\) is \(18.\) The speed of sound is \(340\,m/s.\) The frequency of the source at \(B\) is ..... \(Hz\)

The source of time varying magnetic field may be \((A)\) a permanent magnet \((B)\) an electric field changing linearly with time \((C)\) direct current \((D)\) a decelerating charge particle \((E)\) an antenna fed with a digital signal Choose the correct answer from the options given below:

A capacitor of capacitance \(\mathrm{C}\) and potential \(\mathrm{V}\) has energy \(E\). It is connected to another capacitor of capacitance \(2 \mathrm{C}\) and potential \(2 \mathrm{~V}\). Then the loss of energy is \(\frac{x}{3} E\), where \(\mathrm{x}\) is _______.

The equivalent resistance between \(A\) and \(B\) as shown in figure is \(........\,k\Omega\)

Suppose there is a uniform circular disc of mass M kg and radius r m shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis A of the disc is given by \(\frac{ x }{256} Mr ^2\). The value of x is ___________.

Suppose \(f(x)=\frac{\left(2^x+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^2-x+1\right)}}{\left(7 x^2+3 x+1\right)^3}\). Then the value of \(f^{\prime}(0)\) is equal to

If P is a point on the circle \( x^{2}+y^{2}=4 \), Q is a point on the straight line \( 5x+y+2=0 \) and \( x-y+1=0 \) is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such point P is ........... .

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 number of real solutions of the equation \(x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0\) is

Muon \((\mu ^{-1})\) is negatively charged \(\left( {\left| q \right| = \left| e \right|} \right)\) with a mass \(m_{\mu}\, = 200\,m_e\), where \(m_e\) is the mass of the electron and \(e\) is the electronic charge. If \(\mu ^{-1}\) is bound to a proton to form a hydrogen like atom, identify the correct statements \((A)\) Radius of the muonic orbit is \(200\) times smaller than that of the electron \((B)\) the speed of the \(\mu ^{-1}\) in the \(n^{th}\) orbit is \(\frac{1}{{200}}\) times that of the election in the \(n^{th}\) orbit \((C)\) The Ionization energy of muonic atom is \(200\) times more than that of an hydrogen atom \((D)\) The momentum of the muon in the \(n^{th}\) orbit is \(200\) times more than that of the electron