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A TV tower has a height of $100m$. How much population is covered by the TV broadcast if the average population density around the tower is $1000k{m}^{-2}$?

River of width $d$ is flowing with velocity $v_0$ as shown. A swimmer swims (with respect to water) perpendicular to the current with acceleration $a=2t$ (where $t$ is time) starting from rest from the origin $O$ at $t=0$. The equation of trajectory of the path followed by the swimmer is

The atmosphere is held to the earth by_________.

Find the derivative of $x^5$

A horizontal force of $10\text{N}$ is necessary to just hold a block stationary against a wall as shown in the figure. The weight of the block is $2\text{N}$. The coefficient of friction $\mu$, between the block and the wall is

Three liquids of densities $d,2d$ and $3d$ are mixed in equal proportions of weights. What is the relative density of the mixture, if $d$ is the density of water?

A small body of mass $m=0.10kg$ moves in the reference frame rotating about a stationary axis with constant angular velocity $\omega =5.0rad/s$. What work does the centrifugal force of inertia perform during the transfer of this body along an arbitrary path from point $1$ to point $2$ which are located at the distance $r_1=30cm$ and $r_2=50cm$ from the axis of rotation?

A point source of electromagnetic radiation has an average power output of 800 W. The maximum value of electric field at a distance 4.0 m from the source is

A particle with a specific charge $s$ is fired with a speed $v$ towards a wall at a distance $d$ perpendicular to the wall. What minimum magnetic field must exist in this region for the particle to not to hit the wall?

Three blocks $A$,$B$ and $C$ weighing $1,8$ and $27\text{kg}$ respectively are connected as shown in the figure with an inextensible string and are moving on a smooth surface. If $T_3$ is equal to $36\text{N}$, then $T_2$ is

What is the value of $\sin \left(\frac{31\pi }{3}\right)$?

Consider the four processes $A,B,C,D$ shown in the graph given below



List I shows process & List II gives value of index ${}^{\prime }{n}^{\prime }$ for the equation $P{V}^n=\text{const}$

Match List I with list II.

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ A.& A& 1.& 0\\ B& B& 2& 1\\ C& C& 3.& 1.4\\ D& D& 4.& \infty \end{array}$

The maximum and minimum magnitude of the resultant of two vectors are $23$ units and $7$ units respectively. Then the magnitude of the resultant vector when they act perpendicular to each other is

Two satellites of masses $m_1$ and $m_2(m_1>m_2)$ are revolving round the earth in circular orbits of radius $r_1$ and $r_2(r_1>r_2)$ respectively. Which of the following statements is true regarding their speeds $v_1$ and $v_2$?

Find the value of $F$, if the tension in the string $T_1=36\text{N}.(g=10{\text{m/s}}^2$)

A wave pulse is traveling on a string at 2 m/s. Displacement y of the particle at x = 0 at any time t is given by $\frac{2}{(t^2+1)}$. Find the expression of the function y = (x, t), i.e., displacement of the particle at position x and time t.

The percentage error in the measurement of mass and speed are $2\%$ and $3\%$ respectively. How much will be the maximum error in the estimation of the kinetic energy obtained by measuring mass and speed

If the speedometer of a bike riding along a straight road varies with time as given $v\left(t\right)=(t+3{)}^2+t^2$ What will be its accelaration at time t = 1s.

A particle executes simple harmonic motion with a frequency f. The frequency with which the kinetic energy oscillates is

[IIT JEE 1987]

What is equation of a line passing through the points $(4,2)$ and $(15,-4)$?

A car of mass $2000\text{Kg}$ is moving with a velocity of $20\text{m/s}$. If the velocity-time graph for this car is a horizontal line parallel to the time axis, then the velocity of the car at the end of $52\text{sec}$ will be

A car is moving on a circular path with a constant speed of $10m/s$. What is the magnitude of change in velocity when it moves through ${120}^{\circ }$?

Width of the base of the accelerated container shown in the figure is: [Assume $g=10{\text{m/s}}^2$]

For a square sheet of side $1\text{m}$ having uniform surface density, the position of $\text{COM}$ is at $x_1$. On the other hand, if surface density is varying as $\sigma =2x{\text{kg/m}}^2$, $\text{COM}$ is observed to be at position $x_2$. The distance between $x_2$ and $x_1$ is :

A tuning fork of frequency $300\text{Hz}$ is vibrated on arm-$1$, then the air column vibrates in fundamental mode. If the same tuning fork is vibrated on arm-$2$, the air column vibrates in first overtone. Both the arms are of length $1\text{m}$. If velocity of sound is $300\text{m/s},g=10{\text{m/s}}^2$ and density of water is ${10}^3{\text{kg/m}}^3$ then the density of the unknown liquid is

$($Neglect surface tension and end corrections$)$

A copper and a tungsten plate having a thickness $\delta$ = 2mm each are riveted together so that at $0^{\circ }C$ they form a flat bimetallic plate. Find the average radius of the curvature of this plate at $t$ = ${200}^{\circ }C$. The coefficient of linear expansion for copper and tungsten are ${\alpha }_{Cu}$ = $1.7\times {10}^{-5}/K$ and ${\alpha }_W$ = $0.4\times {10}^{-5}/K$.

Between the two stations, a train accelerates from rest uniformly at first, then moves with constant velocity and finally retards uniformly to come to rest. If the ratio of the time taken be $1:8:1$ and the maximum speed attained be $60\text{km/h}$, then, what is the average speed over the whole journey?

Potential energy of a particle is given by $U=(2-3x-4x^2)\text{J}$ where '$x$' is in '$m$'. Find the magnitude of force at $x=4\text{m}$, if the particle is free to move along $x$-direction.

The dimensions of the coefficient of thermal conductivity are

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities $d_1$ and $d_2$ are filled in the tube. Each liquid subtends ${90}^{\circ }$ angle at centre. Radius joining their interface makes an angle $\alpha$ with vertical.

Ratio $\frac{d_1}{d_2}$ is:

Two putty balls of equal masses moving with equal speeds in mutually perpendicular direction, stick together after collision. If the balls were initially moving with a speed of $45\sqrt{2}m{s}^{-1}$ each, the speed of their combined mass after collision is.

A graph of mass ejected vs time is shown in the figure below. [Given $\ln \left(\frac{50}{49}\right)=0.02$]





Find the velocity of the rocket after $5\text{s}$, if initial launch velocity is $200\text{m/s}$ and velocity of ejected fuel w.r.t the rocket is $20\text{m/s}$. [Take $g=10{\text{m/s}}^2$ and initial mass of the rocket is $500\text{kg}$ ].

If points A and B have a potential 3V and 5V respectively, and a dipole is brought between the two points, which way will its dipole moment align?

Given: Dipole is an arrangement of two equal and opposite charges separated by a very small fixed distance and Dipole moment is a vector quantity which points from negative to the positive charge of the dipole arrangement.

The period of a body under $\text{SHM}$ is represented by $T=P^a{D}^b{S}^c$, where $P$ is pressure, $D$ is density and $S$ is surface tension. The value of $a,b$ and $c$ are

A roller coaster is travelling on a flat track with a minimum radius of curvature of $25\text{m}$ and the train car has a maximum velocity of $15\text{m/s}$. What should be the friction coefficient between the train car and the track for the ride to be safe?

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

The brakes applied to a car produce an acceleration of $6{\text{m/s}}^2$ in the opposite direction to the motion. If the car takes $2\text{s}$ to stop after the application of brakes, calculate the distance it travels during this time.

A wheel having a radius of $10\text{cm}$ is coupled by a belt to another wheel of radius $30\text{cm}$. If the 1st wheel increases its angular speed from rest at a uniform rate of $1.57{\text{rad/s}}^2$, the time for the second wheel to reach a rotational speed of $100\text{rev/min}$ is (Assume that the belt does not slip)

An rms voltage of 110 V is applied across a RLC series circuit having a resistance 11W and an impedance 22W. The power consumed is

An aircraft executes a horizontal loop of radius $1\text{km}$ with a steady speed of $900{\text{kmh}}^{-1}$. Find its centripetal acceleration.

Consider a source of sound S, and an observer/detector D. The source emits a sound wave of frequency $f_0$. The frequency observed by D is found to be

Two uniformly charged spheres with radii $r_1$ & $r_2$ and charges $Q_1$ & $Q_2$ are separated by distance d (from the periphery of the two spheres). What is the magnitude of force on $Q_2$ due to $Q_1$?

A circuit contains an ammeter, a battery of $30V$ and a resistance of $40.8$ ohm all connected in series. If the ammeter has a coil of resistance $480$ ohm and a shunt of resistance $20$ ohm, the reading in the ammeter will be

A particle of mass $m$ is projected from the ground with initial linear momentum $p$ (magnitude) such that to have maximum possible range. Its minimum kinetic energy will be

A projectile has the same range $R$ for two angles of projection. If $T_1$ and $T_2$ be the times of flight in the two cases, then $R$ is

The figure shows the displacement-time graph of a body moving in a straight line. Which of the following shows the corresponding velocity-time graph?