Two tubes of radii \(r_1\) and \(r_2\), and lengths \(l_1\) and \(l_2\), respectively, are connected in series and a liquid flows through each of them in streamline conditions. \(P_1\) and \(P_2\) are pressure differences across the two tubes. If \(P_2\) is \(4P_1\) and \(l_2\) is \( \frac{l_1}{4}\) , then the radius \(r_2\) will be equal to

A thin prism of angle \(6^{\circ}\) and refractive index for yellow light \(\left(n_{Y}\right) 1.5\) is combined with another prism of angle \(5^{\circ}\) and \(n _{ Y }=1.55\). The combination produces no dispersion. The net average deviation ( \(\delta\) ) produced by the combination is \(\left(\frac{1}{x}\right)^{\circ}\). The value of \(x\) is.......

A particle having electric charge \(3 \times 10^{-19} C\) and mass \(6 \times 10^{-27} kg\) is accelerated by applying an electric potential of 1.21 V . Wavelength of the matter wave associated with the particle is \(\alpha \times 10^{-12} m\). The value of \(\alpha\) is _________ .
(Take Planck's constant \(\left.=6.6 \times 10^{-34} J . s \right)\)

The radius in kilometer to which the present radius of earth \(( R =6400\, km )\) to be compressed so that the escape velocity is increased \(10\) time is ............\(km\)

How many alpha and beta particles are emitted when Uranium \({ }_{92} U ^{238}\) decays to lead \({ }_{82} Pb ^{206}\) ?

Two long parallel conductors \(S_{1}\) and \(S_{2}\) are separated by a distance \(10 \,cm\) and carrying currents of \(4\, A\) and \(2 \,A\) respectively. The conductors are placed along \(x\)-axis in \(X - Y\) plane. There is a point \(P\) located between the conductors (as shown in figure). A charge particle of \(3 \pi\) coulomb is passing through the point \(P\) with velocity \(\overrightarrow{ v }=(2 \hat{ i }+3 \hat{ j }) \,m / s\); where \(\hat{i}\) and \(\hat{j} \quad\) represents unit vector along \(x\) and \(y\) axis respectively. The force acting on the charge particle is \(4 \pi \times 10^{-5}(-x \hat{i}+2 \hat{j}) \,N\). The value of \(x\) is

In a meter bridge experiment to determine the value of unknown resistance, first the resistances \(2 \Omega\) and \(3 \Omega\) are connected in the left and right gaps of the bridge and the null point is obtained at a distance \(l cm\) from the left. Now when an unknown resistance \(x \Omega\) is connected in parallel to \(3 \Omega\) resistance, the null point is shifted by 10 cm to the right of wire. The value of unknown resistance \(x\) is _________ \(\Omega\).

A thin strip \(10\, cm\) long is on a \(U\) shaped wire of negligible resistance and it is connected to a spring of spring constant \(0.5\,Nm^{-1}\) (see figure). The assembly is kept in a uniform magnetic field of \(0.1\, T\). If the strip is pulled from its equilibrium position and released, the number of oscillations it performs before its amplitude decreases by a factor of \(e\) is \(N\). If the mass of the strip is \(50\, grams\), its resistance \(10\,\Omega \) and air drag negligible, \(N\) will be close to

A wire of length \(30\,cm\), stretched between rigid supports, has it's \(n^{\text {th}}\) and \((n+1)^{\text {th}}\) harmonics at \(400\,Hz\) and \(450\; Hz\), respectively. If tension in the string is \(2700\,N\), it's linear mass density is.........\(kg/m\).

A hydraulic automobile lift is designed to lift vehicles of mass \(5000\,kg\). The area of cross section of the cylinder carrying the load is \(250\,cm ^2\). The maximum pressure the smaller piston would have to bear is [Assume \(g=10\,m / s ^2\)]

Two sitar strings, \(A\)  and \(B,\)  playing the note \('Dha'\)  are slightly out of tune and produce beats and frequency  \(5\,Hz.\)  The tension of the string \(B\) is slightly increased and the beat frequency is found to decrease by  \(3\,Hz\) . If the frequency of \(A\) is \(425\,Hz,\)  the original frequency of \(B\) is  ... \(Hz\)

A short straight object of height \(100\, cm\) lies before the central axis of a spherical mirror whose focal length has absolute value \(| f |=40\, cm .\) The image of object produced by the mirror is of height \(25\, cm\) and has the same orientation of the object. One may conclude from the information