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A ball of mass 4 kg moving with a velocity of 12 m/s impinges directly on another ball of mass 8 kg moving with velocity of 4 m/s in the same direction. Find their velocities after impact (if $e$ = 0.5)

In $YDSE$, if a slab whose refractive index can be varied is placed in front of the upper slit. Then, the variation of resultant intensity $I$, at mid-point of screen with $\mu$ will be best represented by $(\mu \ge 1$ and $I_0$ is the maximum intensity)

A non-conducting square sheet of side 10 m is charged with a uniform surface charge density,$\sigma =-60\frac{\mu C}{m^2}$ . Find the magnitude and orientation of electric field vector due to the sheet at a point which is d = 0.02 mm away from the midpoint of the sheet.

An ideal gas has a molar heat capacity at constant volume $C_V$. Find the molar heat capacity of this gas as a function of its volume $V$, if it undergoes a process $T=T_o{e}^{\alpha V^2}$, where $T_o$ and $\alpha$ are constants.

Two bodies A and B, of different materials are kept on a horizontal surface. A graph of limiting friction (F) versus normal force (N) is as shown. Which one has a smoother surface in contact with the plane?

A bullet of mass $20g$ hits a block of $1.98\text{kg}$ suspended from massless string of length $100\text{cm}$ and sticks to it. The bullet flies down at an angle of ${30}^{\circ }$ to the horizontal with a velocity of $200\text{m/s}$. Through what height will the block rise ?

A
student forgot to add the reaction mixture to the round bottomed
flask at 27 °C but instead he/she placed the flask on the flame.
After a lapse of time, he realized his mistake, and using a pyrometer
he found the temperature of the flask was 477 °C. What fraction
of air would have been expelled out?

Two rods, one of aluminum and the other made of steel, having initial lengths $l_1$ and $l_2$ are connected together to form a single rod of length $l_1+l_2$. The co-efficients of linear expansion for aluminum and steel are ${\alpha }_a$ and ${\alpha }_s$ respectively. If the length of each rod increases by the same amount when their temperature are raised by $t^{\circ }C$, then find the ratio $\frac{l_1}{(l_1+l_2)}$

Linear mass density (mass per unit length) of a rod depends on the distance from one end (say A) as ${\lambda }_x=\alpha x+\beta .Here\alpha and\beta$ are constants. The moment of inertia of this rod about an axis passing through A and perpendicular to the rod. (Length of the rod is l) is

A long, vertical wire carrying a current of 10 A in the upward direction is placed in a region where a horizontal magnetic field of magnitude $2.0\times {10}^{-3}T$ exits from south to north. Find the point where the resultant magnetic field is zero.

The first minima due to a single slit diffraction is at $\theta ={30}^{\circ }$ for a light of wavelength $5000\stackrel{\circ }{A}$. The width of slit is

Minimum thickness of a mica sheet having $\mu =\frac{3}{2}$ which should be placed in front of one of the slits in $YDSE$, such that the intensity at the centre of the screen reduces to half of maximum intensity is (Consider an equal intensity of light from each slit)

A $\text{PNP}$ in common emitter configuration used as an amplifier, its current gain is $50$. If input resistance is $1k\Omega$ and input voltage is $5\text{V}$ , then output current will be-

A system is shown in the figure. A man standing on the block is pulling the rope. If the man pulls $2m$ of the rope in each second, then the velocity of the block will be

[Assume that the block does not rotate and rope is inextensible.]

A rod is fixed between two points at ${20}^{\circ }$. The coefficient of linear expansion of material of rod is $\frac{1.1\times {10}^{-5}}{{}^{\circ }C}$ and Young's modulus is $1.2\times {10}^{11}\frac{N}{m^2}$. Find the stress developed in the rod if temperature of rod becomes ${10}^{\circ }C$

A cylinder of mass M has length L that is $\sqrt{3}$ times its radius. What is the ratio of its moment of ineria about its own axis and that about an axis passing through its centre and perpendicular to its axis ?

Four charges $2q,q,–q$ and $–q$ are placed at corners of a square of side length $a.$ The electrostatic potential at the centre of the square will be equal to



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A string of length $L$ vibrates with a frequency of $7.5\text{Hz}$ to form a standing wave with three antinodes as shown below.





What is the oscillating frequency of this string, if it produces the standing wave pattern shown below?

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A frame of reference that is accelerated with respect to an inertial frame of reference is called a non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis with a constant angular velocity $\omega$ is an example of a non-inertial frame of reference. The relationship between the force ${\overrightarrow{F}}_{\text{rot}}$ experience by a particle of mass $m$ moving on the rotating disc and the force ${\overrightarrow{F}}_{\text{in}}$ experienced by the particle in an inertial frame of reference is

${\overrightarrow{F}}_{\text{rot}}={\overrightarrow{F}}_{\text{in}}+2m({\overrightarrow{v}}_{\text{rot}}\times \overrightarrow{\omega })+m(\overrightarrow{\omega }\times \overrightarrow{r})\times \overrightarrow{\omega },$



where ${\overrightarrow{v}}_{\text{rot}}$ is the velocity of the particle in the rotating frame of reference and $\overrightarrow{r}$ is the position vector of the particle with respect to the centre of the disc. Now consider a smooth slot along a diameter of a disc of radius $R$ rotating counterclockwise with a constant angular speed $\omega$ about its vertical axis through its center. We assign a coordinate system with the origin at the center of the disc, the $x-$axis along the slot, the $y-$ axis perpendicular to the slot and the $z-$ axis along the rotation axis $\overrightarrow{\omega =\omega \overrightarrow{k}}$. A small block of mass $m$ is gently placed in the slot at $\overrightarrow{r}=\left(R/2\right)\overrightarrow{i}$ at $t=0$ and is constrained to move only along the slot.





The distance $r$ of the block at time $t$ is

Two simple harmonic motions of an angular frequencies 100 and 100radS^-1 have the same amplitude.Find the ratio of their maximum acceleration.

Find the angle of incidence so that the incident light ray retraces its path after third reflection.

A hybrid stepping motor has eight poles which have been castellated to have six teeth each. If the rotor has 60 teeth, calculate the stepping angle.

An ideal solenoid carries a current 4 A. Number of turns per unit length is 1500. An electron is projected in solenoid along direction perpendicular to axis, find frequency of electron in $rev/s$.

An ideal gas is initially at a temperature $T$ and volume $V$. Its volume is increased by $\Delta V$ due to an increase in temperature $\Delta T$, pressure remaining constant. The quantity $\delta =\Delta V/\left(V\Delta T\right)$ varies with temperature as

A particle A suffers an oblique elastic collision with a particle B that is at rest initially. If their masses are the same, then, after the collision

What is the maximum value of the force $F$ such that the block shown in the arrangement, does not move?

A plank is moving with a velocity of $4\text{m/s}$. A disc of radius $1\text{m}$ rolls without slipping on it with an angular velocity of $3\text{rad/s}$ as shown in figure. Find out the velocity of the centre of the disc.

When two charged particles are placed at the certain distance apart in vacuum, then the force experienced by one of the charges is equal to $F.$ If vacuum is replaced by a medium of dielectric constant $6,$ then the net force experienced by the charged particle will be equal to



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A wire of resistance R is bent into a circle as shown. Find the effective resistance between A and B which are situated diametrically opposite.

The frequency of $K_{\alpha }$ line of a source of atomic number Z is proportional to

a stone dropped from a height h reaches at the earth surface in 1 second. If the same stone is taken to the moon and drop freely from a height h then in what time will it reach the ground?

A marble block of mass $2kg$ lying on ice when given a velocity of $6m/s$ is stopped by friction in $10s$. Then the coefficient of friction is