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The position vector of the centre of mass $r_{\text{cm}}$ of an asymmetric uniform bar of negligible area of cross - section as shown in figure is:

Two point charges $q_1$ and $q_2$ are placed at a distance of $50\text{m}$ from each other in air, interact with a certain electrostatic force. If the medium is replaced with oil whose relative permittivity is $5$, then the interaction force between them is still the same. Now the separation between them is

(I) The braking system of the tourist bus carrying 40 passengers failed while negotiating a stiff climb on a hilly road

(II) The tourist bus fell into the gorge killing atleast ten passengers and seriously injuring all the remaining

If the pulley is pulled down with a force of $120\text{N}$ as shown in figure. The acceleration of $8\text{kg}$ block would be:

(All pulleys are light and frictionless, string is massless and inextensible)

(Take $g=10{\text{m/s}}^2$)

A block is placed on a rough floor and a horizontal external force F is applied on it. The force of friction f by the floor on the block is measured for different values of F and a graph is plotted between them. (taking f on Y and F on X axis)

A $10cm$ long perfectly conducting wire PQ is moving with a velocity $1cm/s$ on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor $L=1mH$ and a resistance $R=1\Omega$ as shown in figure. The horizontal rail, L and R lie in the same plane with a uniform magnetic field $B=1T$ perpendicular to the plane. If the key $S$ is closed at certain instant , the current in the circuit after 1 millisecond is $x\times {10}^{-3}A$, where the value of $x$ is .

[Assume the velocity of wire PQ remains constant $\left(1cm/s\right)$ after key S is closed. Given : $e^{-1}=0.37$, where e is base of the natural logarithm]

A capacitor of capacitance $2\mu \text{F}$ is charged with a $10\text{V}$ battery. Now the battery is disconnected and a dielectric of dielectric constant $K=4$ is inserted between the plates. What is the potential difference between the plates of the capacitor ?

The vapour pressure of a solvent is decreased by $10\text{mm}$ of Hg, when a non volatile solute was added to the solvent. The mole fraction of the solute in the solution is $0.2$, what should be the mole fraction of the solvent if the decrease in the vapour pressure is to be $20$ mm of Hg?

A physical quantity is measured and its value is found to be equal to $nu$, where $n$ is the numerical value and $u$ is the unit. Which of the following relations is true?

A capacitor of capacitance $C$ is charged with a battery of emf $E$. What is the flux of electric field through the Gaussian surface highlighted with dotted line?

A particle of charge of $q$ and mass $m$ starts moving from the origin under the action of an electric field $\overline{E}=E_0\hat{i}$ and $\overline{B}=B_0\hat{i}$ with a velocity $\overline{v}=v_0\hat{j}.$ The speed of the particle will become $2v_0$ after a time

A particle of mass $0.01\text{kg}$ and charge ${10}^{-6}\text{C}$ is placed at rest in a uniform electric field $2\times {10}^6\text{N/C}$ and then released. The kinetic energy attained by the particle after moving a distance of $2\text{m}$ is

[Assume gravity free space]

A body of mass $m$ was slowly hauled up the hill by a force which at each point was directed along the trajectory. Find the work performed by this force, if the height of this hill is $h$, the length of its base is $l$ and the coefficient of friction is $\mu$.

Four charges are arranged at the corners of a square ABCD, as shown. The force on a positive charge kept at the center of the square is

A bucket full of hot water cools from ${75}^{\circ }C$ to ${70}^{\circ }C$ in time $T_1$ , from ${70}^{\circ }C$ to ${65}^{\circ }C$ in time $T_2$ and from ${65}^{\circ }C$ to ${60}^{\circ }C$ in time $T_3$, then

A turn table having moment of inertia $10kg-m^2$ rotates at a constant rate of $120rpm$. If a block is placed such that the new moment of inertia of the system is $20kg-m^2$, the new angular speed of the system is

A bar is subjected to an axial tension. Axial strain in bar is $ϵ$. While volumetric strain in bar is $0.4ϵ$. What is Poisson's ratio of the material ?

A body is projected at an angle $\theta$ to the horizontal with kinetic energy $E_k$. The potential energy of the body at the highest point of the trajectory is

In the figure shown, the coefficient of friction between all surfaces in contact is $0.4$. Taking $g=10m/s^2$, find the tension $T$ in the string.

A uniform disc of radius r is to be suspended through a small hole made in the disc. Find the minimum possible time period of the disc for small oscillations. What should be the distance of the hole from the centre for its to have minimum time period ?

The electric filed intensity at all point in space is given by $\overrightarrow{E}=(\sqrt{3}\hat{i}-\hat{j})\text{V/m}$. The equipotential lines in $xy$ plane will be represented by :

The minimum wavelength of X-rays produced in an X-ray tube is $\lambda$ when the operating voltage is V. What is the minimum wavelength of the X-rays when the operating voltage is $\frac{V}{2}$?

The range of a projectile, which is launched at an angle of ${15}^o$ with the horizontal, is $150m$. What is the range for an angle of projection of ${45}^o$ to the horizontal?

A series RLC circuit has a bandwidth of $300\text{rad/s}$ at a resonance frequency of $3000\text{rad/s}$ when excited by a voltage source of $100\text{V}$. The inductance of the coil is $0.1\text{H}$. The value of $R$ is -

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Relative permittivity and permeability of a material is ${ϵ}_r$ and ${\mu }_r$, respectively. Which of the following values of these quantities are allowed for a diamagnetic material ?

A disc of moment of inertia $2{\text{kg-m}}^2$ revolving with $8\text{rad/s}$ is placed gently on another disc of moment of inertia $4{\text{kg-m}}^2$ revolving with $4\text{rad/s}$, in the same direction. What is the angular velocity of the composite disc?

An $\text{L-shaped}$ object, made of thin rods of uniform mass density, is suspended with a string as shown in the figure. If $AB=BC$, and the angle made by $AB$ with downward vertical is $\theta$, then:

The antenna current of an AM transmitter is 8$A$ when only carrier is sent. It is increased to 8.96$A$ when the carrier is sinusoidally modulated. The percentage modulation is :

A particle of mass m is attached to a light string of length l, the other end of which is fixed. Initially the string is kept horizontal and the particle is given an upward velocity v. The particle is just able to complete a circle.

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Two spherical balls of radii $r_1\text{and}{r}_2(<r_1)$ and density ${\rho }_s$ are tied up with a long string and released in a long viscous liquid column of lesser density $\rho$ with the string just taut as shown. The tension of the string when the terminal velocity is attained is

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