A ball is projected from the ground with a speed \(15 \,ms ^{-1}\) at an angle \(\theta\) with horizontal so that its range and maximum height are equal, then \(tan\,\theta\) will be equal to 

In the given arrangement of a doubly inclined plane two blocks of masses \(\mathrm{M}\) and \(\mathrm{m}\) are placed. The blocks are connected by a light string passing over an ideal pulley as shown. The coefficient of friction between the surface of the plane and the blocks is \(0.25\) . The value of \(\mathrm{m}\), for which \(\mathrm{M}=10\) \(\mathrm{kg}\) will move down with an acceleration of \(2 \mathrm{~m} / \mathrm{s}^2\), is _______. (take \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) and \(\left.\tan 37^{\circ}=3 / 4\right)\)

An upright object is placed at a distance of \(40\, cm\) in front of a convergent lens of focal length \(20\, cm\). A convergent mirror of focal length \(10\, cm\) is placed at a distance of \(60\, cm\) on the other side of the lens. The position and size of the final image will be

The root mean square speed of smoke particles of mass \(5 \times 10^{-17}\,kg\) in their Brownian motion in air at NTP is approximately \(.......\,mm\,s ^{-1}\) [Given \(k =1.38 \times 10^{-23}\,J\,K ^{-1}\) ]

Two billiard balls of mass \(0.05\,kg\) each moving in opposite directions with \(10\,ms ^{-1}\) collide and rebound with the same speed. If the time duration of contact is \(t=0.005\,s\), then \(\dots N\)is the force exerted on the ball due to each other.

Given below are two statements: Statement \((I)\) : When currents vary with time, Newton's third law is valid only if momentum carried by the electromagnetic field is taken into account. Statement \((II)\) : Ampere's circuital law does not depend on Biot-Savart's law. In the light of the above statements, choose the correct answer from the options given below:

An ac current is represented as
\(\mathrm{i}=5 \sqrt{2}+10 \cos \left(650 \pi \mathrm{t}+\frac{\pi}{6}\right) \mathrm{Amp}\)
The r.m.s value of the current is

Given figure shows few data points in a photo electric effect experiment for a certain metal. The minimum energy for ejection of electron from its surfface is\(.......eV\) (Plancks constant \(\left. h =6.62 \times 10^{-34}\, J . s \right)\)

An oscillator of mass \(M\)  is at rest in its equilibrium position in a potential \(V\, = \,\frac{1}{2}\,k{(x - X)^2}.\) A particle of mass \(m\)  comes from right with speed \(u\)  and collides completely inelastically with \(M\) and sticks to it . This process repeats every time the oscillator crosses its equilibrium position .The amplitude of oscillations after \(13\)  collisions is: \((M = 10,\, m = 5,\, u = 1,\, k = 1 ).\) 

The stopping potential for photoelectrons emitted from a surface illuminated by light of wavelength \(6630 \; \mathring A\) is \(0.42 \; V\). If the threshold frequency is \(x \times\) \(10^{13}  / s\), where \(x\) is\(\dots\) (nearest integer). (Given, speed light \(=3 \times 10^{8} \; m / s\), Planck's constant \(=6.63 \times 10^{-34} \; Js\) )

An insulated cylinder of volume \( 60\,cm^{3} \) is filled with a gas at \( 27^{\circ}C \) and 2 atmospheric pressure. Then the gas is compressed making the final volume as \( 20\,cm^{3} \) while allowing the temperature to rise to \( 77^{\circ}C \). The final pressure is __________atmospheric pressure.

When \(5\ V\) potential difference is applied across a wire of length \(0.1\ m\), the drift speed of electrons is \(2.5 \times 10^{-4}\) \(ms^{-1}\) . If the electron density in the wire is \(8 \times 10^{28}\) \(m^{-3}\) the resistivity of the material is close to

Two particles, of masses \(M\) and \(2M\), moving as shown, with speeds of \(10\, m/s\) and \(5\, m/s\), collide elastically at the origin. After the collision, they move along the indicated directions with speeds \(v_1\) and \(v_2\) respectively. The value of \(v_1\) and \(v_2\) are nearly