The percentage decrease in the weight of a rocket, when taken to a height of \(32 km\) above the surface of earth will, be\(.....\%\) (Radius of earth \(=6400\,km\) )

In a closed organ pipe, the frequency of fundamental note is \(30 \mathrm{~Hz}\). A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to \(110 \mathrm{~Hz}\). If the organ pipe has a cross-sectional area of \(2 \mathrm{~cm}^2\), the amount of water poured in the organ tube is _______ \(g.\) (Take speed of sound in air is \(330 \mathrm{~m} / \mathrm{s}\) )

For an electromagnetic wave travelling in free space, the relation between average energy densities due to electric \(\left( U _{ e }\right)\) and magnetic \(\left( U _{ m }\right)\) fields is

A square loop of side \(2\, a ,\) and carrying current I, is kept in \(XZ\) plane with its centre at origin. A long wire carrying the same current \(I\) is placed parallel to the \(z-\)axis and passing through the point \((0, b, 0),(b>>a)\). The magnitude of the torque on the loop about \(z-\)axis is given by:

Two waves are simultaneously passing through a string and their equations are : \({y}_{1}={A}_{1} \sin {k}({x}-v {t}), {y}_{2}={A}_{2} \sin {k}\left({x}-{vt}+{x}_{0}\right) .\) Given amplitudes \({A}_{1}=12\, {mm}\) and \({A}_{2}=5\, {mm}\) \({x}_{0}=3.5\, {cm}\) and wave number \({k}=6.28\, {cm}^{-1}\). The amplitude of resulting wave will be \(......\,{mm}\)

Two equal resistances when connected in series to a battery, consume electric power of \(60\,W.\) If these resistances are now connected in parallel combination to the same battery, the electric power consumed will be .............. \(W\)

A particle of charge \(16\times10^{-16}\, C\) moving with velocity \(10\, ms^{-1}\) along \(x-\) axis enters a region where magnetic field of induction \(\vec B\) is along the \(y-\) axis and an electric field of magnitude \(10^4\, Vm^{-1}\) is along the negative \(z-\) axis. If the charged particle continues moving along \(x-\) axis, the magnitude of \(\vec B\) is

A convex lens of focal length 30 cm is placed in contact with a concave lens of focal length 20 cm. An object is placed at 20 cm to the left of this lens system. The distance of the image from the lens in cm is ________

Figure shows charge \((q)\) versus voltage \((V)\) graph for series and parallel combination of to given capacitors. The capacitances are

Let \(\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=12 \vec{a}+4 \vec{b}\), and \(\overrightarrow{O C}=\vec{b}\), where \(O\) is the origin. If \(S\) is the parallelogram with adjacent sides \(O A\) and \(O C\), then find the value of \(\frac{\text { area of quadrilateral } O A B C}{\text { area of } S} .\)

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) :


Three identical spheres of same mass undergo one dimensional motion as shown in figure with initial velocities \(v_{\mathrm{A}}=5 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}, v_{\mathrm{C}}=4 \mathrm{~m} / \mathrm{s}\). If we wait sufficiently long for elastic collision to happen, then \(v_{\mathrm{A}}=4 \mathrm{~m} / \mathrm{s}, v_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}\), \(v_{\mathrm{C}}=5 \mathrm{~m} / \mathrm{s}\) will be the final velocities.

Reason (R): In an elastic collision between identical masses, two objects exchange their velocities.
In the light of the above statements, choose the correct answer from the options given below :

A nucleus with mass number \(242\) and binding energy per nucleon as \(7.6\,MeV\) breaks into fragment each with mass number \(121\) . If each fragment nucleus has binding energy per nucleon as \(8.1\,MeV\), the total gain in binding energy is \(........MeV\)

The area (in sq. units) of the region enclosed by the curves \(y=x^{2}-1\) and \(y=1-x^{2}\) is equal to