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If the distance between the earth and the sun were reduced to half its present value, Then the number of days in one year would have been

The length of a simple pendulum is $32\text{cm}$. It is suspended by the roof of a lift which is moving upwards with an acceleration of $6.2{\text{m/s}}^2$. Find the time period of pendulum.

Two discs have same mass and thickness. Their materials are of densities $d_1$ and $d_2$. The ratio of their moments of inertia about an axis passing through the centre and perpnedicular to the plane is

From what minimum height h must the system be released when a spring is unstretched so that after perfectly inelastic collision (e=0) with ground, B may be lifted off the ground (spring constant =k)

Two blocks of masses $1\text{kg}$ and $2\text{kg}$ are connected by a metal wire going over a smooth pulley as shown in the figure. The breaking stress of the metal is $\left(40/3\pi \right)\times {10}^6{\text{N/m}}^2$. If $g=10{\text{m/s}}^2$, then what should be the minimum radius of the wire used if it is not to break?

Two containers of equal volume contain the same gas at pressures $P_1$ and $P_2$ and absolute temperatures $T_1$ and $T_2$ respectively. On joining the vessels, the gas reaches a common pressure P and common temperature T. The ratio $\frac{P}{T}$ is equal to

A sound wave of frequency n travels horizontally to the right with speed c. It is reflected from a broad wall moving to the left with speed v. The number of beats heard by a stationary observer to the left of the wall is

A homogeneous block having its cross section as a parallelogram of sides $a$ and $b$ as shown, is lying at rest and is in equilibrium on a smooth horizontal surface. Then, find the value of acute angle $\theta$ for which it will be in equilibrium

If a capacitor of $10\mu F$ is charged to 10V and then connected to a new circuit to discharge through a$10\Omega$ resistor, the charge on it after $20\mu s$ is.

From the foot of a tower $90\text{m}$ high, a stone is thrown up so as to reach the top of the tower with zero velocity. Two seconds later another stone is dropped from the top of the tower. Find when will two stones meet for the first time ?

Two particles of masses $m_1$ and $m_2$ are located at $x=0\text{m}$ and $x=6\text{m}$. If the position of their centre of mass is at $x=4\text{m}$, then the ratio of their masses $\frac{m_2}{m_1}$ is

A force of $5\times {10}^3\text{N}$ is applied on the rod of length $1\text{m}$. Find the linear strain if elongation in the rod is $10\text{mm}$.

Periodic time of a satellite revolving above Earth’s surface at a height equal to R, where R the radius of Earth, is [g is acceleration due to gravity at Earth’s surface]

A car of mass $1000\text{kg}$ moves in a circular path with a constant speed of $12\text{m/s}$. It is turned by ${180}^{\circ }$, while travelling a distance of $628\text{m}$ on the road. The centripetal force acting on the car is

Block $A$ and $B$ shown in the figure are connected with a bar of negligible weight. $A$ and $B$ each has mass $170\text{kg}$, the coefficient of friction between $A$ and the plane is $0.2$ and that between $B$ and the plane is $0.4$ ($g=10{\text{ms}}^{-2}$)

What is the total force of friction between the blocks and the plane?



[Hint: ${\tan }^{-1}\left(\frac{8}{15}\right)={28}^{\circ },\cos {28}^{\circ }=0.883,\sin {28}^{\circ }=0.469$]

A body of mass $m$ is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass $m$ is slightly pulled down and released, it oscillates with a time period of $3\text{s}$. When the mass $m$ is increased by $1\text{kg}$, the time period of oscillations becomes $5\text{s}$. The value of $m$ in $\text{kg}$ is

Find the maximum force that should act on the $1\text{kg}$ body for the system to accelerate together with $2{\text{m/s}}^2$. Take $g=10{\text{m/s}}^2$.

In the formation of primary rainbow, the sunlight rays emerge at from a raindrop after

A rod of mass $m$ is placed on two supports as shown. Calculate the normal reaction of each support.

Column-I gives some devices and Column-II gives some processes on which the functioning of these devices depend. Match the devices in Column-I with the processes in Column-II and indicates your answer by darkening appropriate bubbles in the 4 $\times$ 4 matrix given in the ORS.

A particle of mass m moving in the x-direction with speed 2v is hit by another particle of mass 2m moving in the y-direction with speed v. If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to

A logic gate is an electronic circuit which

Two sinusoidal waves each of amplitude $2A$, travel in the same direction in a medium. If the phase difference between the two waves is ${120}^{\circ }$, then find the resultant amplitude of the superimposed wave.

A wire of length L and radius r is rigidly fixed at one end. On stretching the other end of the wire with a force F, the increase in its length is l. If another wire of same material but of length 2L and radius 2r is stretched with a force of 2F, the increase in its length will be

Magnitude of vector which comes on addition of two vectors, $6\hat{i}+7\hat{j}$ and $3\hat{i}+4\hat{j}$ is

A particle is performing SHM. When the displacement is one half of its amplitude, find what fraction of total energy are the kinetic and potential energies of the particle?

A body is projected with a velocity of $30\text{m/s}$ at an angle of ${60}^{\circ }$ to the horizontal. Find the magnitude of velocity when the angle between the velocity vector and acceleration vector is ${120}^{\circ }$.

The resistance of wire at ${20}^{\circ }C$ is $20\Omega$, and at ${300}^{\circ }C$ resistance of wire is $50\Omega$. The temperature at which its resistance will be $40\Omega$ is

A disc of mass $4m$ is resting on a horizontal frictionless surface and pivoted at point $\text{O}$ as shown in the figure, perpendicular to its plane. A point object of mass $m$, moving horizontally with velocity $v$, hits the disc and sticks to it. Find the angular velocity of the system after the hit.

The difference between the apparent frequencies of a source of sound as perceived by a stationary observer during its approach and recession is 2% of the actual frequency of the source. If the speed of sound is 300 m/s, the speed of source is

From a tower of height $H$, a particle is thrown vertically upward with a speed $u$. The time taken by the particle, to hit the ground, is $n$ times that taken by it to reach the highest point of its path. The relation between $H,u$ and $n$ is

If the force constant of a wire is $k$, the work done in increasing the length $L$ of the wire by $l$ is

The position of the particle is given by $x=(e^t-\cos t)$ in $\text{m}$. The velocity $\left(\text{in m/s}\right)$ of the particle as a function of time $t$ is

If $\overrightarrow{A}.\overrightarrow{B}=|\overrightarrow{A}\times \overrightarrow{B}|,$ find angle between $\overrightarrow{A}$ and $\overrightarrow{B}.$

The equation of a travelling wave is y = 60 cos (1800t – 6x) where y is in microns, t in seconds and x in metres. The ratio of maximum particle velocity to velocity of wave propagation is

If $y=e^{4x}sin2x$, then $\frac{dy}{dx}$=?

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?

The bulk modulus of $n$ spherical objects is $B$. If it is subjected to uniform pressure $P$, the fractional decrease in radius is -

A light ray travelling in glass medium is incident on glass-air interface at an angle of incidence $\theta$. The reflected (R) and transmitted (T) intensities, both as function of θ, are plotted. The correct sketch is

Two different coils have self-inductance $L_1$ = 8mH and $L_2$ = 2mH respectively. The current in both the coils is increased at a same rate. At a certain instant of time, the power being supplied to both the coils is equal. At that instant, the current, the induced voltage, and the energy stored in $L_1$ and $L_2$ are $I_1$ and $I_2$, $V_1$ and $V_2$, and $W_1$ and $W_2$ respectively. Then

A box is moved along a straight line by a machine delivering constant power. The distance moved by the body in time t is proportional to

A particle moves along a straight line such that its displacement at any time $t$ is given by $s=(t^3-6t^2+3t+4)\text{m}$. The velocity when the acceleration is zero is

A particle is moved from $(0,0)$ to $(a,a)$ under a force $\overrightarrow{F}=(3\hat{i}+4\hat{j})$ from two paths. Path $1$ is $OP$ and path $2$ is $OQP$. Let $W_1$ and $W_2$ be the work done by this force along respective paths. Then

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.

A bob of mass ${}^{\prime }{m}^{\prime }$ is attached to ceiling of a lift with the help of string as shown. Calculate tension in string when lift is accelerated up with acceleration ${}^{\prime }{a}^{\prime }$.

The Gauss' theorem for gravitational field may be written as