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Find the integration of $\int \frac{e^{{\tan }^{-1}x}}{1+x^2}.dx$

A band brake consists of a lever attached to one end of the band. The other end of the band is fixed to the ground. The wheel has a radius of $200mm$ and the wrap angle of the band is ${270}^{\circ }$. The braking force applied to the lever is limited to $100N$, and the coefficient of friction between the band and the wheel is $0.5$. No other information is given.

The maximum wheel torque that can be completely braked is

In a Fraunhofer diffraction, at single slit of width $a$ with incident light of wavelength $5500\mathring{\text{A}}$, the first minimum is observed at an angle of ${30}^{\circ }$. The first secondary maximum will be observed at an angle $\theta$, equal to,

Two ${}_{29}C{u}^{64}$ nuclei almost touch each other. The electronics repulsive energy of the system will be $(R_0=1\text{fm})$

A ball of mass $m$ is projected upward from a tower with velocity $30\text{m/s}$ and at the same time, another ball of mass $2m$ is dropped from rest from the same tower. Find the displacement of the $COM$ in $2\text{s}$. [Take $g=10{\text{m/s}}^2$]

A ray of light parallel to $y-$axis is incident on a parabola $y=4x^2$ polished from inside as shown in the figure. Find the co-ordinates $x\text{and}y$ of the point at which the incident ray should fall such that the reflected ray becomes horizontal after reflection.

For what value of displacement, the kinetic energy and potential energy of a simple harmonic oscillation become equal?



$[A$ is amplitude$]$

A ball of mass 1 kg is dropped from 20 m height on ground and it rebounds to height 5m. Find magnitude of change in momentum during its collision with the ground.[ Take g =10m/s^{​​​​​2​​}

Four particles $A$, $B$, $C$ and $D$ with masses $m_a=m,m_B=2m,m_C=3m$ and $m_D=4m$ are at the corners of a square. They have accelerations of equal magnitude with directions as shown in the figure. The acceleration of centre of mass of the particles (in ${\text{ms}}^{-2}$) is :

The current $I$, potential difference $V_L$ across the inductor and potential difference $V_C$ across the capacitor in circuit as shown in the figure are best represented vectorially as

Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red, yellow and violet rays are ${\mu }_r,{\mu }_y$ and ${\mu }_v$ respectively and those for the flint glass are ${\mu }_r^{{}^{\prime }},{\mu }_y^{{}^{\prime }}$ and ${\mu }_v^{{}^{\prime }}$ respectively. Find $\frac{A^{{}^{\prime }}}{A}$ for which, there is no net deviation in the yellow light.

Find the angle of deviation of the particle as it comes out of the magnetic field if the lateral width $d$ of the region is slightly larger than $\frac{mv}{qB}$.

A circular wire loop of radius a carries a total charge Q distributed uniformly over its length. A small length DL of the wire is cut off. Find the electric field at the centre due to the remaining wire.

The figure shows a uniform smooth solid cylinder A of radius 4 m rolling without slipping on the 8 kg plank which in turn is supported by a fixed smooth surface. Block B, is known to accelerate down with $6m/s^2$.
(Take $g=10m/s^2$)

What is the angular acceleration of the cylinder ?

The ratio of minimum wavelengths of Lyman and Balmer series will be

The initial velocity of a projectile of mass 1 kg is given by $u=(\hat{i}+\sqrt{3}\hat{j})m/s$. What is the time of flight?

A gas does 4.5 J of external work during adiabatic expansion. Its temperature falls by 2 K. Its internal energy will

A body falls from a height 'h' on a horizontal surface and rebounds. Then it falls again and again rebounds and so on. If the restitution coefficient is $\frac{1}{3}$, the total distance covered by the body before it comes to rest is

In a common-base mode of a transistor, the collector current is $5.488\text{mA}$ for an emitter current of $5.60\text{mA}$. The value of the current amplification factor $\left(\beta \right)$ will be

Two uniform rods $AB$ and $BC$ have Young's modulii $1.2\times {10}^{11}{\text{N/m}}^2$ and $1.5\times {10}^{11}{\text{N/m}}^2$ respectively. If coefficient of linear expansion of $AB$ is $1.5\times {10}^{-5}/{}^{\circ }\text{C}$ and both have equal area of cross section, then coefficient of linear expansion of $BC$, for which there is no shift of the junction at all temperatures, is

A body of weight $50\text{N}$ is dragged slowly with constant velocity on a horizontal surface as shown in the figure, with a force of $F=28.2\text{N}$ at an angle of ${45}^{\circ }$ with horizontal. The net contact force acting on the body will be

A ring of mass $M$ and radius $R$ lies in the $x-y$ plane with its centre at the origin as shown. The mass distribution of the ring is non-uniform such that at any point $\text{P}$ on the ring, the mass per unit length is given by $\lambda ={\lambda }_0{\cos }^2\theta$ (where ${\lambda }_0$ is a positive constant). Then the moment of inertia of the ring about $z-$ axis is:

The potential due to charged spherical shell of radius 20 cm at a point 5 cm distance from the center is v the potential on the surface of the spherical shell is.

What is the mass of one formula unit of Nickel Fluoride?

One gram of hydrogen is converted into $0.993\text{grams}$ of helium in a thermonuclear reaction. The energy released is

An air bubble of diameter 2 mm rises steadily through a solution of density 1750 kg m^-3 at the rate of 0.35 cm s^-1. Calculate the coefficient of viscosity of the solution. The density of air is negligible.