The initial speed of a projectile fired from ground is $$\mathrm{u}$$. At the highest point during its motion, the speed of projectile is $$\frac{\sqrt{3}}{2} u$$. The time of flight of the projectile is :

A bar magnet with a magnetic moment $$5.0 \mathrm{Am}^{2}$$ is placed in parallel position relative to a magnetic field of $$0.4 \mathrm{~T}$$. The amount of required work done in turning the magnet from parallel to antiparallel position relative to the field direction is _____________.

Spherical insulating ball and a spherical metallic ball of same size and mass are dropped from the same height. Choose the correct statement out of the following

{Assume negligible air friction}

The effect of increase in temperature on the number of electrons in conduction band ($$\mathrm{n_e}$$) and resistance of a semiconductor will be as:

Which of the following correctly represents the variation of electric potential $$(\mathrm{V})$$ of a charged spherical conductor of radius $$(\mathrm{R})$$ with radial distance $$(\mathrm{r})$$ from the center?

The drift velocity of electrons for a conductor connected in an electrical circuit is $$\mathrm{V}_{\mathrm{d}}$$. The conductor in now replaced by another conductor with same material and same length but double the area of cross section. The applied voltage remains same. The new drift velocity of electrons will be

A rod with circular cross-section area $$2 \mathrm{~cm}^{2}$$ and length $$40 \mathrm{~cm}$$ is wound uniformly with 400 turns of an insulated wire. If a current of $$0.4 \mathrm{~A}$$ flows in the wire windings, the total magnetic flux produced inside windings is $$4 \pi \times 10^{-6} \mathrm{~Wb}$$. The relative permeability of the rod is

(Given : Permeability of vacuum $$\mu_{0}=4 \pi \times 10^{-7} \mathrm{NA}^{-2}$$)

At a certain depth "d " below surface of earth, value of acceleration due to gravity becomes four times that of its value at a height $$\mathrm{3 R}$$ above earth surface. Where $$\mathrm{R}$$ is Radius of earth (Take $$\mathrm{R}=6400 \mathrm{~km}$$ ). The depth $$\mathrm{d}$$ is equal to

Two polaroide $$\mathrm{A}$$ and $$\mathrm{B}$$ are placed in such a way that the pass-axis of polaroids are perpendicular to each other. Now, another polaroid $$\mathrm{C}$$ is placed between $$\mathrm{A}$$ and $$\mathrm{B}$$ bisecting angle between them. If intensity of unpolarized light is $$\mathrm{I}_{0}$$ then intensity of transmitted light after passing through polaroid $$\mathrm{B}$$ will be:

100 balls each of mass $$\mathrm{m}$$ moving with speed $$v$$ simultaneously strike a wall normally and reflected back with same speed, in time $$\mathrm{t ~s}$$. The total force exerted by the balls on the wall is

If $$\mathrm{R}, \mathrm{X}_{\mathrm{L}}$$, and $$\mathrm{X}_{\mathrm{C}}$$ represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless :

The maximum potential energy of a block executing simple harmonic motion is $$25 \mathrm{~J}$$. A is amplitude of oscillation. At $$\mathrm{A / 2}$$, the kinetic energy of the block is

If 1000 droplets of water of surface tension $$0.07 \mathrm{~N} / \mathrm{m}$$, having same radius $$1 \mathrm{~mm}$$ each, combine to from a single drop. In the process the released surface energy is :-

$$\left( {\mathrm{Take}\,\pi = {{22} \over 7}} \right)$$

As shown in figure, a $$70 \mathrm{~kg}$$ garden roller is pushed with a force of $$\vec{F}=200 \mathrm{~N}$$ at an angle of $$30^{\circ}$$ with horizontal. The normal reaction on the roller is

(Given $$\mathrm{g=10~m~s^{-2}}$$)

The pressure of a gas changes linearly with volume from $$\mathrm{A}$$ to $$\mathrm{B}$$ as shown in figure. If no heat is supplied to or extracted from the gas then change in the internal energy of the gas will be

The correct relation between $$\gamma = {{{c_p}} \over {{c_v}}}$$ and temperature T is :

If a source of electromagnetic radiation having power $$15 \mathrm{~kW}$$ produces $$10^{16}$$ photons per second, the radiation belongs to a part of spectrum is.

(Take Planck constant $$h=6 \times 10^{-34} \mathrm{Js}$$ )

A lift of mass $$\mathrm{M}=500 \mathrm{~kg}$$ is descending with speed of $$2 \mathrm{~ms}^{-1}$$. Its supporting cable begins to slip thus allowing it to fall with a constant acceleration of $$2 \mathrm{~ms}^{-2}$$. The kinetic energy of the lift at the end of fall through to a distance of $$6 \mathrm{~m}$$ will be _____________ $$\mathrm{kJ}$$.

For hydrogen atom, $$\lambda_{1}$$ and $$\lambda_{2}$$ are the wavelengths corresponding to the transitions 1 and 2 respectively as shown in figure. The ratio of $$\lambda_{1}$$ and $$\lambda_{2}$$ is $$\frac{x}{32}$$. The value of $$x$$ is __________.

In the figure given below, a block of mass $$M=490 \mathrm{~g}$$ placed on a frictionless table is connected with two springs having same spring constant $$\left(\mathrm{K}=2 \mathrm{~N} \mathrm{~m}^{-1}\right)$$. If the block is horizontally displaced through '$$\mathrm{X}$$' $$\mathrm{m}$$ then the number of complete oscillations it will make in $$14 \pi$$ seconds will be _____________.

The speed of a swimmer is $$4 \mathrm{~km} \mathrm{~h}^{-1}$$ in still water. If the swimmer makes his strokes normal to the flow of river of width $$1 \mathrm{~km}$$, he reaches a point $$750 \mathrm{~m}$$ down the stream on the opposite bank.

The speed of the river water is ___________ $$\mathrm{km} ~\mathrm{h}^{-1}$$

A solid sphere of mass $$1 \mathrm{~kg}$$ rolls without slipping on a plane surface. Its kinetic energy is $$7 \times 10^{-3} \mathrm{~J}$$. The speed of the centre of mass of the sphere is __________ $$\operatorname{cm~s}^{-1}$$

Expression for an electric field is given by $$\overrightarrow{\mathrm{E}}=4000 x^{2} \hat{i} \frac{\mathrm{V}}{\mathrm{m}}$$. The electric flux through the cube of side $$20 \mathrm{~cm}$$ when placed in electric field (as shown in the figure) is __________ $$\mathrm{V} \mathrm{~cm}$$.

A thin rod having a length of $$1 \mathrm{~m}$$ and area of cross-section $$3 \times 10^{-6} \mathrm{~m}^{2}$$ is suspended vertically from one end. The rod is cooled from $$210^{\circ} \mathrm{C}$$ to $$160^{\circ} \mathrm{C}$$. After cooling, a mass $$\mathrm{M}$$ is attached at the lower end of the rod such that the length of rod again becomes $$1 \mathrm{~m}$$. Young's modulus and coefficient of linear expansion of the rod are $$2 \times 10^{11} \mathrm{~N} \mathrm{~m}^{-2}$$ and $$2 \times 10^{-5} \mathrm{~K}^{-1}$$, respectively. The value of $$\mathrm{M}$$ is __________ $$\mathrm{kg}$$.

(Take $$\mathrm{g=10~m~s^{-2}}$$)

In a medium the speed of light wave decreases to $$0.2$$ times to its speed in free space The ratio of relative permittivity to the refractive index of the medium is $$x: 1$$. The value of $$x$$ is _________.

(Given speed of light in free space $$=3 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}$$ and for the given medium $$\mu_{\mathrm{r}}=1$$)

Two identical cells, when connected either in parallel or in series gives same current in an external resistance $$5 ~\Omega$$. The internal resistance of each cell will be ___________ $$\Omega$$.

An inductor of $$0.5 ~\mathrm{mH}$$, a capacitor of $$20 ~\mu \mathrm{F}$$ and resistance of $$20 ~\Omega$$ are connected in series with a $$220 \mathrm{~V}$$ ac source. If the current is in phase with the emf, the amplitude of current of the circuit is $$\sqrt{x}$$ A. The value of $$x$$ is ___________