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Component of $3\hat{i}+4\hat{j}$ perpendicular to $\hat{i}+\hat{j}$ and in the same plane as that of $3\hat{i}+4\hat{j}$ is

If $\int \frac{x^4+1}{x^6+1}dx={\tan }^{-1}f\left(x\right)-\frac{2}{3}{\tan }^{-1}g\left(x\right)+C$ for constant of integration $C,$ then $f\left(x\right)-g\left(x\right)$ is

A large plane charged sheet having surface charge density $\sigma =+2\times {10}^{-6}{\text{C/m}}^2$ lies in the $\text{x-y}$ plane. Find the flux of the electric field through a circular area of radius $1\text{cm}$ lying completely in the region where $\text{x, y, z}$ all are positive and with its normal making an angle of ${60}^{\circ }$ with the $\text{z-}$axis.

Consider a uniform square plate fo side a and mass M. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

A wire of resistance x ohm is drawn out, so that its length is increased to twice its original length,and its new resistance becomes 20 Ω then x will be

The magnetic moment $\left(\mu \right)$ of an electron revolving around the nucleus, varies with principal quantum number $n$ as,

When a ray of light of frequency $6\times {10}^{14}\text{Hz}$ travels from water of refractive index $\frac{4}{3}$ to glass of refractive index $\frac{8}{5}$, its

A man can row a boat with $4km/hr$ in still water. If he is crossing a river where the current is $2km/hr$, how long will it take him to row $2km$ upstream and then back to his starting point?

If energy, planck's constant and speed of light are choosen as fundamental quantities, then dimensional formula of area will be

1. 393.5

An electron emitted by a heated cathode and accelerated through a potential difference of $2\text{kV}$ enters a region with uniform magnetic field of $0.15\text{T}$. If the magnetic field makes an angle of ${30}^{\circ }$ with the initial velocity, the radius of the hellical trajectory of electron will be: $(m_e=9.0\times {10}^{-31}\text{kg})$

A river $400m$ wide is flowing at a rate of $2.0m/s$. A boat is sailing at a velocity of $10.0m/s$ with respect to the water, in a direction perpendicular to the river. How far from the point directly opposite to the starting point does the boat reach the opposite bank?

If a satellite is orbiting the earth very close to its surface, then the orbital velocity mainly depends on

A boy of mass $50\text{kg}$ jumps to a height of $0.8\text{m}$ from the ground, then momentum transferred by the ground to the boy is -

1. 0.8

A point object $S$ is placed midway between two concave mirrors having equal focal length $f$ as shown in the figure. Find the possible value(s) of $d$ for which only one image is formed of the object.

The reading of a centigrade thermometer coincides with that of Fahrenheit thermometer for a liquid. The temperature of the liquid $\left(\text{in}{C}^{\circ }\right)$ is

A force varying with distance is given as $F=a{e}^{-bx}$ acts on a particle of mass $m$ moving in a straight line. Find the work done on the particle if it is displaced by a distance $d$ from the origin.

Five uniform rings, each of diameter $5\text{cm}$ and mass $10\text{g}$ are arranged as shown in the figure. Using the origin shown, the $y-$ co-ordinate of $COM$ of the five ring system is

A charge $Q$ is placed at a distance $a/2$ above the centre of a horizontal, square surface of edge $a$ as shown in figure. Find the flux of the electric field through the square surface.

The total energy density in an EM wave is:

The $P-V$ diagram of a system undergoing thermodynamic transformation is shown in figure. The work done by the system and heat given to the system in going from $A\to B\to C$ is $30\text{J}$ and $40\text{J}$ respectively. The change in internal energy between $A$ and $C$ is

A particle is placed at the lowest point of a smooth wire frame in the shape of a parabola, lying in the vertical $xy-$ plane having equation $x^2=5y$, where $(x,y)$ are in $\text{metres}$. After slight displacement, the particle is set free to move. Find the angular frequency of oscillation $\text{in rad/s}$.
[Take $g=10{\text{m/s}}^2)$]

1. 2

If an alpha particle and a deuteron move with velocities of $v$ and $2v$ respectively, the ratio of their de-Broglie wavelengths will be

Molar heat capacity at constant pressure for a non-linear triatomic gas is

A steel wire is rigidly fixed at both ends. Its length, mass and cross-sectional area are $1\text{m, 0.1 kg and}{10}^{-6}{\text{m}}^2$respectively. Tension in the wire is produced by lowering the temperature by ${20}^{\circ }\text{C}$. If transverse waves are set up by plucking the wire at $0.25\text{m}$ from one end, and assuming that the wire vibrates with minimum number of loops possible for such a case. The frequency of vibration (in $Hz$) is found to be $11\times \text{K}$. Find the value of $K$.$(\text{Given}\alpha =1.21\times {10}^{-5\circ }{C}^{-1},Y=2\times {10}^{11}N/m^2)$

A rod of negligible mass having length $l=2\text{m}$ is pivoted at its center and two masses, $6\text{kg}$ and $4\text{kg}$ are hung from the ends as shown in the figure. Find the initial angular acceleration of the rod if it is suddenly released from the horizontal position.







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

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A copper rod of length l is rotated about one end perpendicular to the magnetic field B with constant angular velocity. The induced e.m.f. between the two ends is

Four identical rods of mass $M=6\text{kg}$ each are welded at their ends to form a square and then welded to a massive ring having mass $m=4\text{kg}$ and radius $R=1\text{m}$. If the system is allowed to roll down the incline of inclination $\theta ={30}^{\circ }$.





The moment of inertia of the system about the centre of ring will be