A very long (length \(L\) ) cylindrical galaxy is made of uniformly distributed mass and has radius \(R(R < < L).\) A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of star is \(T\) and its distance from the galaxy's axis is \(r,\) then

A person pushes a box on a rough horizontal platform surface. He applies a force of \(200\, N\) over a distance of \(15\, m .\) Thereafter, he gets progressively tired and his applied force reduces linearly with distance of \(100\, N\). The total distance through which the box has been moved is \(30\, m\). What is the work done by the person during the total movement of the box\(........J\)\(?\)

For which of the following particles will it be most difficult to experimentally verify the deBroglie relationship?

A parallel beam of light of wavelength \(900\,nm\) and intensity \(100\,Wm ^{-2}\) is incident on a surface perpendicular to the beam. Tire number of photons crossing \(1\,cm ^{2}\) area perpendicular to the beam in one second is :

In the given figure an ammeter \(A\) consists of a \(240 \Omega\) coil connected in parallel to a \(10 \Omega\) shunt. The reading of the ammeter is _______ \(\mathrm{mA}\).

A block of mass \(40 \,kg\) slides over a surface, when a mass of \(4 \,kg\) is suspended through an inextensible massless string passing over frictionless pulley as shown below. The coefficient of kinetic friction between the surface and block is \(0.02\). The acceleration of block is ............ \(ms ^{-2}\) (Given \(g =10 \,ms ^{-2}\).)

Two infinite identical charged sheets and a charged spherical body of charge density ' \(\rho\) ' are arranged as shown in figure. Then the correct relation between the electrical fields at \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D points is :

Let \(B\) be the centre of the circle \(x^{2}+y^{2}-2 x+4 y+1=0\) Let the tangents at two points \(\mathrm{P}\) and \(\mathrm{Q}\) on the circle intersect at the point \(\mathrm{A}(3,1)\). Then \(8.\) \(\left(\frac{\text { area } \triangle \mathrm{APQ}}{\text { area } \triangle \mathrm{BPQ}}\right)\) is equal to .... .

Consider the function \(f : R \rightarrow R\) defined by \(f(x)=\left\{\begin{array}{cc}\left(2-\sin \left(\frac{1}{x}\right)\right)|x|, x \neq 0 \\ 0 & , x=0\end{array} .\right.\) Then \(f\) is

Let \({S_n} = 1 + q + {q^2} + ..... + {q^n}\) and \({T_n} = 1 + \left( {\frac{{q + 1}}{2}} \right) + {\left( {\frac{{q + 1}}{2}} \right)^2} + ...... + {\left( {\frac{{q + 1}}{2}} \right)^n}\) where \(q\) is a real number and \(q \ne 1\). If \({}^{101}{C_1} + {}^{101}{C_2}.{S_1} + ...... + {}^{101}{C_{101}}.{S_{100}} = \alpha\, {T_{100}}\) then \(\alpha \) is equal to

An electromagnetic wave of frequency \(3\, GHz\) enters a dielectric medium of relative electric permittivity \(2.25\) from vacuum. The wavelength of this wave in that medium will be \(.......\,\times 10^{-2} \, cm\)

Out of \(60 \%\) female and \(40 \%\) male candidates appearing in an exam, \(60\%\) candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is.

Let a unit vector \(\hat{\mathrm{u}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}\) make angles \(\frac{\pi}{2}, \frac{\pi}{3}\) and \(\frac{2 \pi}{3}\) with the vectors \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}, \frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\) and \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}\) respectively. If \(\overrightarrow{\mathrm{v}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\), then \(|\hat{\mathrm{u}}-\overrightarrow{\mathrm{v}}|^2\) is equal to