\(n\) polarizing sheets are arranged such that each makes an angle \(45^{\circ}\) with the proceeding sheet. An unpolarized light of intensity \(I\) is incident into this arrangement. The output intensity is found to be \(\frac{I}{64}\). The value of \(n\) will be:

A given object takes \(n\) times the time to slide down \(45^{\circ}\) rough inclined plane as it takes the time to slide down an identical perfectly smooth \(45^{\circ}\) inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is _______.

Two projectiles are fired from ground with same initial speeds from same point at angles \(\left(45^{\circ}+\alpha\right)\) and \(\left(45^{\circ}-\alpha\right)\) with horizontal direction. The ratio of their times of flights is

A vertical electric field of magnitude \(4.9 \times 10^{5} N / C\) just prevents a water droplet of a mass \(0.1\, g\) from falling. The value of charge on the droplet will be  ........ \(\times 10^{-9} \;C\) \(\left(\right.\) Given \(\left.g =9.8 m / s ^{2}\right)\)

A helicopter is flying horizontally with a speed \('v'\) at an altitude \('{h}'\) has to drop a food packet for a man on the ground. What is the distance of helicopter from the man when the food packet is dropped?

A cylindrical wire of radius \(0.5 \,mm\) and conductivity \(5 \times 10^{7}\, S / m\) is subjected to an electric field of \(10 \,mV / m .\) The expected value of current in the wire will be \(x ^{3} \,\pi mA\). The value of \(x\) is ......

Given below are two statements: Statement \(I:\) If heat is added to a system, its temperature must increase. Statement \(II:\) If positive work is done by a system in a thermodynamic process, its volume must increase. In the light of the above statements, choose the correct answer from the options given below

A rod of mass \(M\) and length \(L\) is lying on a horizontal frictionless surface. A particle of mass ' \(m\) ' travelling along the surface hits at one end of the rod with velocity \('u'\) in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses \(\left(\frac{m}{M}\right)\) is \(\frac{1}{x}\). The value of ' \(x\) ' will be ..... .

The \(rms\) speeds of the molecules of Hydrogen, Oxygen and Carbondioxide at the same temperature are \({V}_{{H}}, {V}_{0}\) and \({V}_{{C}}\) respectively then