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The instantaneous values of alternating current and voltages in a circuit are given as,

$i=\frac{1}{\sqrt{2}}\sin \left(100\pi t\right)\text{A}$ and

$E=\frac{1}{\sqrt{2}}\sin (100\pi t+\frac{\pi }{3})\text{V}$

The average power (in watts) consumed in the circuit is,

A block of mass $10\text{kg}$ is slowly slid up a smooth inclined plane of inclination ${37}^{\circ }$ by a person. Calculate the work done by the person in moving the block through a distance of $20\text{m}$ along the incline, if the person applies the force parallel to the inclined plane. Take $g=10{\text{m/s}}^2$.

A rod $\text{AB}$ rests with the end $\text{A}$ on rough horizontal ground such that end $\text{A}$ is prevented from sliding to the right. And the end $\text{B}$ against a smooth vertical wall. The rod is uniform and of weight $W$. If the rod is in a stationary position as shown in figure, the number of forces acting in the FBD of rod is/are ?

The currect through a wire depends on time as,
$i=(10+4t)$
Here, $i$ is in ampere and $t$ in seconds. Find the charge crossed through a section in time interval between $t=0$ to $t=10\text{s}$.

In the system below, the moment of inertia of the combination of thin rod of mass $m$ and solid spherical ball of mass $2m$ about the axis of rotation $A{A}^{\prime }$ is:

An aeroplane moves $400\text{m}$ towards north, $300\text{m}$ towards west and then $1200\text{m}$ vertically upward, then its displacement from the initial position is

A smooth inclined plane having an angle of inclination of ${30}^{\circ }$ with the horizontal has a $2.5\text{kg}$ mass held by a spring which is fixed at the upper end. If the mass is taken $2.5\text{cm}$ up along the surface of the inclined plane, the tension in the spring reduces to zero. If the mass is now released, the angular frequency of oscillation is:

Two blocks A and B of mass 10 kg and 20 kg respectively are placed as shown in figure. Coefficient of friction between all the surfaces is 0.2 (g = 10 m/s²). Then

A wire $\text{A}$, in the shape of an arc of a circle, carrying a current of $2\text{A}$ and having a radius of $2\text{cm}$ and another wire $\text{B}$, be also bent in the shape of an arc of a circle, carrying a current of $3\text{A}$ and having radius of $4\text{cm},$ are placed as shown in the figure. The ratio of the magnetic field due to the wires $\text{A}$ and $\text{B}$ at common centre $\text{O}$ is:

A light string fixed at one end to a clamp on ground passes over a fixed pulley and hangs at the other side. It makes an angle of ${30}^{\circ }$ with the ground. A monkey of mass $5\text{kg}$ climbs up the rope. The clamp can tolerate a vertical force of $40\text{N}$ only. The maximum acceleration in upward direction with which the monkey can climb safely is (Neglect friction and take $g=10{\text{ms}}^{-2})$

A copper wire has a cross sectional area of $1m{m}^2$. The electron density of copper is $8.5\times {10}^{28}{m}^3$. Calculate the mean drift velocity of the electrons through the wire when the current is 3 A.

The graph shows how the magnification m produced by a convex thin lens varies with image distance v. What was the focal length of the used
[DPMT 1995]

A sound wave of wavelength $40\text{cm}$ travels in air. If the difference between the maximum and minimum pressure at a given point is $1\times {10}^{-3}{\text{Nm}}^{-2}$, find the amplitude of vibration of the particles of the medium. The bulk modulus of air is $1.4\times {10}^5{\text{Nm}}^{-2}$.

A long straight wire of circular cross-section is made of a non-magnetic material. The wire is of radius $25\text{cm}$. The wire carries a current of $10\text{A}$ which is uniformly distributed over its cross-section. The energy stored per unit length in the magnetic field contained within the wire is :

A stone of mass $3\text{kg}$ tied to a light string of length $l=\frac{7}{10}\text{m}$ is whirled in a circular path in a vertical plane. If the ratio of the maximum to minimum tension in the string is $9$, find the speed of the stone at the highest point.

A wheel of radius $r=1\text{m}$ is rolling with an angular velocity $\omega =\frac{\pi }{3}\text{rad/s}$ about its center and velocity of its center of mass is $v_{com}=2\text{m/s}$. Point $P(0,0)$ is in contact with ground initially, find its height (in $\text{m}$) from the surface after time $t=1\text{s}$.

A thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle ${60}^o$ with vertical? [$g$ is the acceleration due to gravity]

Given that $h\left(t\right)=10e^{-10t}u\left(t\right)andf\left(t\right)=\sin \left(10t\right).u\left(t\right),$ the Laplace transform of the signal, $y\left(t\right)={\int }_0^{t}f\left(\tau \right).h(t-\tau ).d\tau$ is

A certain charge $Q$ is divided at first into two parts, $\left(q\right)$ and $(Q-q)$. Later on the chrages are placed at a certain distance. If the force of interaction between the two charges is maximum, then

The equation of state for a gas is given by $PV=nRT+\alpha V$, where $n$ is the number of moles and $\alpha$ is a positive constant. The initial pressure and temperature of 1 mole of the gas contained in a cylinder are $P_0$ and $T_0$ respectively. The work done by the gas when its temperature doubles isobarically is

Find the integral of the given function w.r.t x

$y=sin6x+10se{c}^{2}x-cosecxcotx$

A transmitting antenna at the top of a tower has a height of $50\text{m}$ and the height of the receiving antenna is $80\text{m}$. What is the range of communication for Line of Sight (LoS) mode?

[Use radius of earth $=6400\text{km}$]

The equivalent resistance between points A and B with switch S open and closed are respectively:

A $1{\text{MW/m}}^2$ continuously operating laser, is used as in "light rocket", to propel a $100\text{kg}$ spacecraft, through gravity free space. The cross-sectional area of the beam is $3{\text{m}}^2$. What is the acceleration of the spacecraft?

In the given figure $\left(a\right)$, a cuboidal block is gradually released face-down into a liquid. The block has height $h$ and on the bottom and top, the face area is $A=5.67{\text{cm}}^2$. Figure $\left(b\right)$ gives the apparent weight $\left(W_{\text{app}}\right)$ of the block as a function of the depth $\left(d\right)$ of its lower face. The scale on the vertical axis is set by $W_s=0.25\text{N}$. What is the density of the liquid ?