A body floats with \(\frac{1}{\mathrm{n}}\) of its volume keeping outside of water. If the body has been taken to height h inside water and released, it will come to the surface after time t. Then

Let \(v_{n}\) and \(E_{n}\) be the respective speed and energy of an electron in the \(n\) th orbit of radius \(r_{n}\), in a hydrogen atom, as predicted by Bohr's model. Then.

The de-Broglie wavelength of a moving bus with speed is \(v\) is \(\lambda\). Some passengers left the bus at a stoppage. Now when the bus moves with twice of its initial speed. Its kinetic energy is found to be twice of its value. What is the de-Broglie wavelength of the bus now?

An equilateral triangle is made by uniform wires \(A B, B C, C A .\) A current \(I\) enters at \(A\) and leaves from the mid point of \(B C\). If the lengths of each side of the triangle is \(L,\) the magnetic field \(B\) at the centroid \(O\) of the triangle is

A container \(\mathrm{AB}\) in the shape of a rectangular parallelopiped of length \(5 \mathrm{~m}\) is divided internally by a movable partition \(\mathrm{P}\) as shown in the figure. The left compartment is filled with a given mass of an ideal gas of molar mass 32 while the right compartment is filled with an equal mass of another ideal gas of molar mass 18 at same temperature. What will be the distance of P from the left wall A when equilibrium is established?

A glass slab consists of thin uniform layers of progressively decreasing refractive indices RI (see figure) such that the RI of any layer is \(\mu-m \Delta \mu\). Here, \(\mu\) and \(\Delta \mu\) denote the \(\mathrm{RI}\) of 0 th layer and the difference in RI between any two consecutive layers, respectively. The integer \(m=0,1,2,3, \ldots\) denotes the numbers of the successive layers. A ray of light from the oth layer enters the 1st layer at an angle of incidence of \(30^{\circ}\). After undergoing the mth refraction, the ray emerges parallel to the interface. If \(\mu=1.5\) and \(\Delta \mu=0.015,\) the value of \(m\) is

A uniform solid spherical ball is rolling down a smooth inclined plane from a height \(h\). The velocity attained by the ball when it reaches the bottom of the inclined plane is v. If the ball is now thrown vertically upwards with the same velocity \(v\), the maximum height to which the ball will rise is

The quadratic equation \(x^2+15|x|+14=0\) has

A car moving at a velocity of \(17 \mathrm{ms}^{-1}\) towards an approaching bus that blows a horn at a frequency of \(640 \mathrm{Hz}\) on a straight track. The frequency of this horn appears to be \(680 \mathrm{Hz}\) to the car driver. If the velocity of sound in air is \(340 \mathrm{ms}^{-1}\), then the velocity of the approaching bus is

Let \(f(x)=\left\{\begin{array}{cc}-2 \sin x, & \text { if } x \leq-\frac{\pi}{2} \\ A \sin x+B, & \text { if }-\frac{\pi}{2} < x < \frac{\pi}{2} \text { . Then, } \\ \cos x, & \text { if } x \geq \frac{\pi}{2}\end{array}\right.\)

A frictionless piston-cylinder based enclosure contains some amount of gas at a pressure of \(400 \mathrm{kPa}\). Then heat is transferred fo the gas at constant pressure in a quasi-static process. The piston moves up slowly through a height of \(10 \mathrm{cm} .\) If the piston has a cross-sectional area of \(0.3 \mathrm{m}^{2},\) the work done by the gas in this process is

If the four points with position vectors \(-2\hat i+\hat j + \hat k, \hat i+\hat j+\hat k,\hat j- \hat k\) and \(\lambda \hat j + \hat k\) are coplanar, then \(\lambda\) is equal to

Let \(f(x)=\int_{\sin x}^{\cos x} e^{-t^2} d t\). Then \(f^{\prime}\left(\frac{\pi}{4}\right)\) equals