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Consider $\overrightarrow{r}=a\hat{i}+b\hat{j}+c\hat{k}$. If $\alpha$, $\beta$ and $\gamma$ be the projection of $\overrightarrow{r}$ along $\frac{2\hat{i}+\hat{j}}{\sqrt{5}}$, $\frac{-2\hat{i}+\hat{j}}{\sqrt{5}}$ and $\hat{k}$ respectively such that $|\alpha |=|\beta |=|\gamma |$ and $|\overrightarrow{r}|=1,$ the number of such ordered triplets $(a,b,c)$ is less than

A wide vessel of uniform cross-section with a small hole at the bottom is filled with a $40\text{cm}$ thick layer of water and $30\text{cm}$ thick layer of kerosene. The relative density of kerosene is $0.8$. The initial velocity of flow of the water streaming out of the hole is -

80 gm of water at ${30}^{\circ }C$ are poured on a large block of ice at $0^{\circ }C$. The mass of ice that melts is

In a compound microscope, the focal length of the objective lens is $0.5\text{cm}$ and that of the eyepiece is $5\text{cm}$. The intermediate real image of the object is formed at a distance of $15.5\text{cm}$ from the objective lens. If the final image is formed at a distance of $25\text{cm}$ from the eyepiece, what is the magnifying power of the compound microscope?

A cyclist goes round a circular path of circumference 34.3 m in $\sqrt{22}$ sec. the angle made by him, with the vertical, will be

Two point charges each of +1$\mu$C are at a distance of 3 m apart. The work done in bringing the two charges closer to a distance 1 m apart is

If ${ϵ}_{0}and{\mu }_0$ are the permittivity and permeability of free space and $ϵand\mu$ are the corresponding quantities for a medium, Then refractive index of the medium is

An incompressible liquid flows through a horizontal tube as shown in the figure. Then the velocity $v$ of the liquid coming out of the tube of area $1.5A$ is:

Two balls, each of mass $m$, moving in the same direction with speeds $v$ and $3v$, collide with each other. If the collision is perfectly inelastic, find out the loss in kinetic energy.

In the circuit shown, the potential difference between $A$ and $B$ is :

A particle of mass $4M$, initially at rest, disintegrates into two particles of masses $M$ and $3M$, respectively, having non-zero velocities. The ratio of de-Broglie wavelength of the particle of mass $M$, to that of mass $3M$, will be,

A target is made of two plates, one of wood and the other of iron. The thickness of the wooden plate is $4\text{cm}$ and that of iron plates is $2\text{cm}$. A bullet fired goes through the wood first and then penetrates $1\text{cm}$ into the iron. A similar bullet fired with the same velocity from the opposite direction goes through the iron first and then penetrates $2\text{cm}$ into the wood. If $a_1$ and $a_2$ be the retardation offered to the bullet by the wood and iron plates respectively, then

Two masses 90 kg and 160 kg are at a distance 5 m apart. Compute the magnitude of intensity of the gravitational field at a point distance 3m from the 90 kg and 4m from the 160 kg mass. G=6.67 $\times$ ${10}^{-11}$SI units..

An airplane flying at a height of $1960\text{m}$ from the ground with constant horizontal velocity of $600\text{km/h}$ releases a bomb to hit a target on the ground. The time of flight and range of the bomb respectively are

(a) Show
that the normal component of electrostatic field has a discontinuity
from one side of a charged surface to another given by

Where

is
a unit vector normal to the surface at a point and σ is the
surface charge density at that point. (The direction of
is
from side 1 to side 2.) Hence show that just outside a conductor, the
electric field is σ

(b) Show
that the tangential component of electrostatic field is continuous
from one side of a charged surface to another. [Hint: For (a), use
Gauss’s law. For, (b) use the fact that work done by
electrostatic field on a closed loop is zero.]

In a photoelectric experiment anode potential is plotted against plate current

(IIT-JEE-2004)

A car moving at a speed of 36 km/hr is taking a turn on a circular road of radius 50 m. A small wooden plate is kept on the seat with its plane perpendicular to the radius of the circular road (figure 7-E4). A small block of mass 100 g is kept on the seat which rests against the plate. The friction coefficient between the block and the plate is $\mu =\mathrm{0.58.}$ (a) Find the normal contact force exerted by the plate on the block. (b) The plate is slowly turned so that the angle between the normal to the plate and the radius of the road slowly increases. Find the angle at which the block will just start sliding on the plate.

A cotton reel of mass $m$ and moment of inertia $2{\text{kgm}}^2$ is kept at rest on a smooth horizontal surface. The reel has inner radius $0.2\text{m}$ and outer radius $0.5\text{m}$. A horizontal force $F=200\text{N}$ starts acting as shown is the figure. Find the angular acceleration of the reel.

Four particles of masses $m,2m,3m$ and $4m$ are arranged at the corners of a parallelogram (shown in figure) with each side equal to $a$ and one of the angles between two adjacent sides is ${60}^{\circ }$. The parallelogram lies in the $\text{x- y}$ plane with mass$m$ at the origin and $4m$ on the x - axis. The centre of mass of arrangement will be located at

A container made up of insulating material contains 4 moles of an ideal diatomic gas at temperature $T$. Heat $Q$ is supplied to this gas, due to which 2 moles of the gas are dissociated into atoms, but the temperature of the gas remains constant. Find the option that correctly represents the relation between $Q$ and $T$.

In an isothermal reversible expansion, if the volume of $96\text{g}$ of oxygen at ${27}^{\circ }\text{C}$ is increased from $70\text{litres}$ to $140\text{litres}$, then the work done by the gas will be

A tuning fork is used to set a stretch string into vibration in two different occasion. Tension being the only parameter that is varied in two experiments, it is found that in first occasion under tension $T_1$ the string oscillates in fundamental mode while in second occasion under tension $T_2$, the string oscillates in its second harmonic mode. Then

A long wire is bent into the shape shown in figure without cross contact at $\text{P}$. The radius of the circular section is $R.$ If current $I$ flows in the wire, then the magnetic field at the center $\text{C}$ is-

A boy riding on his bike is going towards east at a speed of $4\sqrt{2}m{s}^{-1}$. At a certain point he produces a sound pulse of frequency 1650 Hz that travels in air at a speed of 334$m{s}^{-1}$. A second boy stands on the ground 45${}^{\circ }C$ south of east from him. Find the frequency of the pulse as received by the second boy.

The maximum velocity (in $m{s}^{–1}$) with which a car driver must traverse a flat curve of radius 150 m and coefficient of friction 0.6 to avoid skidding is

A block of mass $m$ is placed on a surface with a vertical cross - section given by $y=\frac{x^3}{6}$. If the co-efficient of friction is $0.5$, the maximum height above the ground at which the block can be placed without slipping is

The figure shows spherical conducting shells with inner sphere earthed. If $a=8\text{cm}$ and $b=9\text{cm}$, then the capacitance of the system is