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From a circular ring of mass $M$ and radius $R$, an arc corresponding to a ${90}^{\circ }$ sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the center of the ring and perpendicular to the plane of the ring is $K$ times $M{R}^2$. Then the value of $K$ is -

Formula of di electric constant

1. 0.6326

What is the speed of electron in the second Bohr's orbit of Hydrogen atom? $(Take{v}_0=2.2\times {10}^{6}m/s)$.

Three rings are placed in the same plane as shown. Calculate the MI w.r.t the given axis.

The current through a wire depends on time as $I=(2+3t)\text{A}$. Calculate the charge crossed through a cross-section of the wire in $10\text{s}$.

What does 2x - y equal if x = -4 and y = 5?

1. 300

The rope shown at an instant is carrying a wave travelling towards right, created by a source vibrating at frequency n Hz. Here a, b, c, d and e are x co-ordinates. Choose the correct statement(s)

Earthquakes
generate sound waves inside the earth. Unlike a gas, the earth can
experience both transverse (S)
and longitudinal (P)
sound waves. Typically the speed of S
wave is about 4.0 km
s^–1,
and that of P wave
is 8.0 km s^–1.
A seismograph records P
and S
waves from an
earthquake. The first P
wave arrives 4 min
before the first S
wave. Assuming the
waves travel in straight line, at what distance does the earthquake
occur?

In which of the following cases, the flux crossing the surface is zero?

Two spheres P and Q of equal radii have densities ${\rho }_1$ and ${\rho }_2$ repectively. The spheres are connected by a massless string and placed in liquids $L_1$ and $L_2$ of densities ${\sigma }_1$ and ${\sigma }_2$ and viscosities ${\eta }_1$ and ${\eta }_2$ respectively. They float in equilibrium with the sphere P in $L_1$ and Q in $L_2$ and the string being taut (see figure). If sphere P alone in $L_2$ has terminal velocity ${\overrightarrow{V}}_P$ and Q alone in $L_1$ has terminal velocity ${\overrightarrow{V}}_Q$,
(Terminal velocity of a sphere having radius $r$ and density $\rho$ thrown in a liquid having density $\sigma$ and viscoscity $\eta$ is given by $=\frac{2r^2}{9\eta }(\rho -\sigma )$). Then which of the following are correct ?

Light waves from two coherent sources superimposes at a point. The waves at this point can be expressed as $y_1=a\sin \left({10}^{15}\pi t\right)$ and $y_2=2a\sin ({10}^{15}\pi t+\varphi )$. Here all quantities are in SI units.

Find the resultant amplitude $\left(\text{in m}\right)$, and frequency $\left(\text{in Hz}\right)$, of the resultant wave, if phase difference $\varphi$ is equal to $\pi /3$.

Figure shows part of a circuit. Energy stored in $2\mu F$ capacitor at steady state is:

If $y=co{s}^{2}x,then\frac{dy}{dx}=?$

If n identical water droplets falling under gravity with terminal velocity v coalesce to form a single drop which has the terminal velocity 4v. Find the number n.

Two identical sheets of a metallic foil are separated by d and capacitance of the system is C and charged to a potential difference E. Keeping the charge constant, the separation is increased by l. Then the new capacitance and potential difference will be:

A gas is taken through a cyclic process ABCA as shown in figure. If 2.4 cal of heat is given in the process, what is the value of J ?

An ideal gas is taken through a cyclic thermodynamic process through four steps $1,2,3,4$ respectively. The amounts of heat involved in these steps are $Q_1=5960\text{J},Q_2=5585\text{J},Q_3=-2980\text{J}$ and $Q_4=3645\text{J}$ respectively. The corresponding quantities of work involved are $W_1=2200\text{J},W_2=8250\text{J},W_3=-1100\text{J}$ and $W_4$ respectively. The value of $W_4$ is

If the speed of revolution of a particle on the circumference of a circle and in a separate case, the speed gained in falling through a distance equal to the radius are equal, then the centripetal acceleration will be

Two shots are fired simultaneously from the bottom and top of a vertical tower $AB$ at angles $\gamma$ and $\beta$ with horizontal respectively. Both strike the same point $C$ on the ground at a distance $S$ from the bottom of the tower. What is the height of the tower?

Starting from the origin at time $t=0$, with initial velocity $5\hat{j}{\text{ms}}^{-1},$ a particle moves in the $xy-$ plane with a constant acceleration of $10\hat{i}+4\hat{j}{\text{ms}}^{-2}.$ At time $t$, its coordinates are $20\text{m},y_0\text{m}.$ The values of $t$ and $y_0$ are, respectively-

Some positive charge is spread uniformly over the surface of a thin spherical shell. Radius of sphere is $100cm$ and electric field on the surface is $90\times {10}^{3}N/C$. A point $A$ is at $50cm$ from the centre & point $B$ is at $150cm$ from the centre. Field at $A$ & $B$ are $E_A$ & $E_B$ respectively, then

The equation of the line passing through the points $(3,-2)$ and $(-3,2)$ is

Seven identical circular planar disks, each of mass $M$ and radius $R$ are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point P is :

An object, a convex lens (focal length $f=20\text{cm}$) and a plane mirror are arranged as shown in the figure. How far behind the mirror is the second image formed?

A single conservative force $F\left(x\right)$ acts on a $2.5\text{kg}$ particle that moves along $x-$axis. The potential energy $U\left(x\right)$ is given by $U\left(x\right)=(10+(x-4{)}^2)$ where $x$ is in meter. At $x=6\text{m}$, The particle has kinetic energy of $20\text{J}$. What is mechanical energy of the system?

Suppose the loop in
Exercise 6.4 is stationary but the current feeding the electromagnet
that produces the magnetic field is gradually reduced so that the
field decreases from its initial value of 0.3 T at the rate of 0.02 T
s⁻¹. If the cut is joined and the loop has a
resistance of 1.6 Ω how
much power is dissipated by the loop as heat? What is the source of
this power?

The angle between two lines whose direction ratios are 1,2,1 and 2,-3,4 is:

Two electric dipoles of moment P and 64 P are placed in opposite direction on a line at a distance of 25 cm. The electric field will be zero at point between the dipoles whose distance from the dipole of moment P is

A black hole is an object whose gravitational field is so strong that even light cannot escape from it. To what approximate radius would earth $(\text{mass}=6\times {10}^{24}\text{kg})$ have to be compressed to be a black hole?

If you start at the origin, then move 3 units right and 4 units up, what coordinate do you land on?

A beam of length L is carrying a uniformly distributed load w per unit length. The flexural rigidity of the beam is EI.The reaction at the simple support at the right end is

A small point mass carrying some positive charge on it, is released from the edge of a table. There is a uniform electric field in this region in the horizontal direction. Which of the following options then correctly describe the trajectory of the mass? (Curves are drawn schematically and are not to scale).