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A long solenoid has $1000$ turns per meter and its radius is $R=2.5\text{cm}$. Current in the solenoid increases at a rate of $\frac{200}{\pi }\text{A/s}$. Find the magnitude of induced electric field at a point outside the solenoid at a distance $4\text{cm}$ from its axis.

$co{t}^{-1}\left({2.1}^2\right)+co{t}^{-1}\left({2.2}^2\right)+co{t}^{-1}\left({2.3}^2\right)+\cdots +$ up to $\infty$ is equal to

The value of the definite integral $\underset{\pi /24}{\overset{5\pi /24}{\int }}\frac{dx}{1+\sqrt[3]{3\tan 2x}}$ is

In the figure shown, find the velocity acquired by the block on reaching the ground and the force exerted by the inclined plane on the block if takes 5 seconds for the block to reach the ground. Assume all surfaces to be smooth. (Take $g=10m/s^2)$

Th half-life of ${}^{198}\text{Au}$ is 2.7 days. (a) Find the activity of a sample containing 1.00 $\mu g$ of ${}^{198}\text{Au}$. (b) What will be the activity after 7 days ? Take the atomic weight of ${}^{198}\text{Au}$ to be 198 g mo$l^{-1}$

Velocity-time equation of a particle moving in a straight line is, $v=(10+2t+3t^2)$. Find displacement of particle from the origin at time $t=1\text{s}$, if it is given that displacement is $20\text{m}$ at time $t=0$

A steel wire of length $116\text{cm}$ is connected to an aluminum wire of length $150\text{cm}$ and stretched between two fixed supports. The tension produced is $52\text{N}$. The cross-section of each wire is $2{\text{mm}}^2$. If a transverse wave is set up in the wire, find the lowest frequency for which standing waves with a node at the joint are produced. (density of aluminum $\left({\rho }_{al}\right)=2.6{\text{g/cm}}^3$ and density of steel $\left({\rho }_s\right)=7.8{\text{g/cm}}^3)$.

A cylindrical pipe of length $L$ has two layers of material of conductivity $k_1$ and $k_2$ (see figure). If the innermost surface of the cylinder is maintained at $T_1$ and the outermost surface is at $T_2(<T_1)$, calculate the radial rate of heat flow.

The armature winding and field winding resistance of a 220 V, 100 A, 1500 rpm dc series motor are $0.05\Omega$ each. The motor is operated under dynamic braking at twice the rated torque and at 1000 rpm. The value of braking resistance is______

A uniform rod of length l is suspended by an end and is made to undergo small oscillations. Find the length of the simple pendulum having the time period equal to that of the rod.

The two nearest harmonics of a tube closed at one end and open at other end are $220Hz$ and $260Hz$. What is the fundamental frequency of the system?

Two particles of mass m each are attached to a light rod of length d, one at its centre and the other at a free end. The rod is fixed at the other end and is rotated in a plane at an angular speed $\omega$. Calculate the angular momentum of the particle at the end with respect to the particle at the centre (i.e. angular momentum of B with respect to A).

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 $\mu$.

A coil of self inductance $10\text{mH}$ and resistance $0.1\Omega$ is connected through a switch to a battery of internal resistance $0.9\Omega$. After the switch is closed, the time taken for the current to attain $80\%$ of the saturation value is

$[\text{take}\ln 5=1.6]$

A boat crosses a river from port $A$ to port $B$, which are directly opposite to each other. The speed of the water is $v_w$ and that of boat is $V_b$ relative to water. Assume $V_b=2V_w$. The time taken by the boat, if it has to cross the river directly on the $AB$ line is $(\text{given}AB=D)$

A copper rod $ab$ of length $l$, pivoted at one end $a$, rotates at constant angular velocity $\omega$, at right angle to a uniform magnetic field $B$. $c$ is the midpoint of the rod. What will be the ratio of induced emf across $ac$ to $cb$?

The displacement of a particle executing simple harmonic motion is given by

$y=A_0+A\sin \omega t+B\cos \omega t$. Then, the amplitude of its oscillation is given by

In a Young’s
double-slit experiment, the slits are separated by 0.28 mm and the
screen is placed 1.4 m away. The distance between the central bright
fringe and the fourth bright fringe is measured to be 1.2 cm.
Determine the wavelength of light used in the experiment.

When a PN junction diode is reverse biased

A ball of mass 1 kg hangs in equillibrium from a two strings OA and OB as shown in figure. What are the tensions in strings OA and OB? $(Takeg=10m/s^2)$

A boat moves relative to water with a velocity 'v' which is n times less than the river flow velocity u. At what angle to the north direction must the boat move to minimize drifting if the river flows from west to east?

Are these two forces equal ? [1 MARK]

A wheel rotates around a stationary axis passing through its centre so that the angle of rotation $\varphi$ varies with time as $\varphi =k{t}^2$ where $k=\sqrt{2}{\text{rad/s}}^2$. Find the magnitude of total acceleration of the point at the rim at time $t=1\text{s}$, if the tangential velocity of the point at that moment is $2\text{m/s}$.

A massless platform is kept on a light elastic spring, as shown in the figure. When a particle of mass $0.1\text{kg}$ is dropped on the pan from a height of $0.24\text{m}$, the spring gets compressed by $0.01\text{m}$. From what height should the same particle be dropped to cause a compression of $0.04\text{m}$ in the spring?
Neglect the effect of collision.