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The magnitudes of vectors $\overrightarrow{A},\overrightarrow{B}and\overrightarrow{C}$ are 3, 4 and 5 units respectively. If $\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{C}$, the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ is

The coefficient of thermal conductivity of copper, mercury and glass are respectively $K_C,K_m$ and $K_g$ such that $K_c>K_m>K_g$. If the same quantity of heat is to flow per second per unit area of each and corresponding temperature gradients are $X_C,X_m$ and $X_g$, then

A point moves along a circle of radius $4\text{m}$. The distance $x$ is related to time by $x=c{t}^3$. What should be the value of $c$, so that the tangential acceleration is equal to the normal acceleration when its linear velocity is $4\text{m/s}$?

A boy standing on a long railroad car throws a ball straight upwards with a speed of $10\text{m/s}$ at $t=0$ when the car starts moving on a horizontal road with an acceleration of $1{\text{m/s}}^2$. How far behind the boy will the ball fall on the car? (Take $g=10{\text{m/s}}^2$).

The diagram shows the displacement-time graph for a particle moving in a straight line. The average velocity for the interval $t=0s$ to $t=5s$ is (Displacement is in $m$ and time is in $s$)

There are $n_1$ photons of frequency ${\gamma }_1$ in a beam of light. In an equally energetic beam, there are $n_2$ photons of frequency ${\gamma }_2$ . Then the correct relation is

The half-life of cobalt-60 is 5.25 yrs. After how long does its activity reduce to about one eight of its original value?

A boy of mass $M$ stands on a platform of radius $R$ capable to rotate about its axis. The moment of inertia of the platform is $I$. The system is at rest. The friend of the boy throws a ball of mass $m$ with a velocity $v$ horizontally. The boy on the platform catches it. Find the angular velocity of the system in the process.

A liquid flows through a horizontal tube. The velocities of the liquid in the two sections, which have areas of cross-section $A_1$ and $A_2$ are $v_1$ and $v_2$ respectively. The difference in the levels of the liquid in the two vertical tubes is $h$.

A) The volume of the liquid flowing through the tube in unit time is $A_1{v}_1$

B) $v_2-v_1$ = $\sqrt{2gh}$

C) $v_2^2-v_1^2$ = 2gh

D) The energy per unit mass of the liquid is the same in both section of the tube.

A gun is aimed at a target in a line of its barrel. The target is released and allowed to fall under gravity at the same instant the gun is fired. The bullet will

A coil having n turns and resistance R $\Omega$ is connected with a galvanometer of resistance 4R$\Omega$ . This combination is moved in time t seconds from a magnetic field $W_1$ weber to $W_2$ weber. The induced current in the circuit is

A car is moving in a rainstorm at a speed of $36\text{km/hr}$. The rain drops fall vertically w.r.t the ground with a constant speed of $20\text{m/s}$. The speed of the raindrops w.r.t the car is

If $\overrightarrow{a}=\hat{i}+\hat{j}$ and $\overrightarrow{b}=-\hat{j}+2\hat{k}$. Find $(\overrightarrow{a}-\overrightarrow{b}).(\overrightarrow{a}+2\overrightarrow{b})$.

A wire fixed at one end is stretched by constant force $F$. If elongation in wire is $l$, then energy stored in wire will be

A circular race-car track of radius $300\text{m}$ is banked at an angle of ${15}^{\circ }$. If the coefficient of static friction between the wheels of the race-car and the road is $0.2$, what is the maximum permissible speed to avoid slipping? Take $\tan {15}^{\circ }=0.27,g=10{\text{m/s}}^2$.

An insect crawls up a hemispherical surface very slowly as shown in the figure. The coefficient of static friction between the insect and the surface is $\frac{1}{3}$. If the line joining the centre of the hemispherical surface to the insect makes an angle $\alpha$ with the vertical, then the maximum possible value of $\alpha$ is given by

Heat required to convert one gram of ice at $0^{\circ }C$ into steam at ${100}^{\circ }C$ is (given $L_{stream}$ = 536 cal/gm)

A body of mass m accelerates uniformly from rest to velocity $v_1$ in time $t_1$. The instantaneous power delivered to the body as a function of time t is

A thin wire of uniform density is bend in the form of an arc of a circle which subtends an angle $\alpha$ at the centre of the circle as shown in the figure. Where is the centre of mass of this circular arc located?

Two particles execute SHM of the same amplitude and frequency along the same straight line. They pass one another when going in opposite directions each time their displacement is half their amplitude. What is the phase difference between them?

A ball of mass m is moving towards a batsman at a speed v. The batsman strikes the ball and deflects it by an angle $\theta$ without changing its speed. The impulse imparted to the ball is given by

The displacement-time$(x-t)$ graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at $t=\frac{4}{3}\text{s}$ is

A block of mass $2\text{kg}$ is placed on the top of a bigger block of mass $10\text{kg}$ as shown in figure. The system is released from rest. Assume all the surfaces are frictionless. Find the distance moved by the bigger block at the instant when the smaller block reaches the ground.

A mercury-in-glass thermometer has a stem of internal diameter 0.06 cm and contains 43 g of mercury. The mercury thread expands by 10 cm when the temperature changes from $0^{\circ }C$ to ${50}^{\circ }C$. Find the coefficient of cubical expansion of mercury. Relative density of mercury = 13.6 and ${\alpha }_{glass}=9\times {10}^{-6}{K}^{-1}$

The motion of a particle along a straight line is described by the function $x=6+4t^2–t^4$ where $x$ is in meters and $t$ is in seconds. What is the maximum velocity attained by the particle ?

Density ratio of $O_2$ and $H_2$ is $16:1$. The ratio of their rate of effusion under same conditions is

The displacement of a particle in string stretched in X direction is represented by y. Among the following expressions for y, those describing wave motions are

An object is placed beyond the principal focus at a distance of $9\text{cm}$ from it in front of a concave mirror on the principal axis. The image is formed beyond the principal focus at a distance of $16\text{cm}$ from the principal focus. The focal length of the mirror is

An ideal gas is taken through a cyclic thermodynamic process through four steps. 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 $W_1=2200\text{J}$, $W_2=-825\text{J},W_3=-1100\text{J}$ and $W_4$ respectively . Find the efficiency of the cycle.

A dielectric slab of dielectric constant $k=5$ is placed between plates of parallel plate capacitor, thickness of slab is $30$% of plate spacing & its area is same as that of the plates, new capacitance is $C^{\prime }$ & old capacitance was $C$. The ratio $\frac{C^{\prime }}{C}$ is approximately equal to

The temperature of water at the surface of a deep lake is $2^{\circ }\text{C}$. The temperature expected at the bottom is _____.

A road is banked at an angle of $\theta ={30}^{\circ }$ and provides a coefficient of static friction ${\mu }_s=0.4$. The road gets seriously damaged if it experiences a normal reaction force of more than $10000\text{N}$. What should be the maximum mass of the vehicle allowed on this road?

A body of mass 400g slides on a rough horizontal surface. If the frictional force is 3.0 N, find (a) the angle made by the contact force on the body with the vertical and (b) the magnitude of the contact force. Take g = 10$\frac{m}{s^2}$

A glass flask of volume one litre at $0^{\circ }C$ is filled, level full of mercury at this temperature. The flask and mercury are now heated to ${100}^{\circ }C$. How much mercury will spill out, if coefficient of volume expansion of mercury is 1.82×${10}^{-4}{/}^{\circ }C$ and linear expansion of glass is 0.1×${10}^{-4}{/}^{\circ }C$ respectively

If $y=\sin x$ and $x=3t$, then $\frac{dy}{dt}$ will be

Two discs of mass $2\text{kg}$ and $3\text{kg}$ of radius $1\text{m}$ and $2\text{m}$ respectively placed together are rotated about an axis through their common centre as shown in figure. Find the net moment of inertia of the system about given axis.

A particle of mass $m$ is attached to a spring of natural length $L$ and spring constant $k$. If it is rotated in a horizontal plane with an angular speed $\omega$, then what will be the new length of the spring ?

Two Charges $Q_1$ and $Q_2$ are as shown in the figure. A third charge $Q_3$ is moved from points A to B along a circular path. Change in potential energy of the system is

In the nuclear fusion reaction ${}_1^{2}H{+}_1^{3}H{\to }_2^{4}He+n$ given that the repulsive potential energy between the two nuclei is $7.7\times {10}^{-14}J$, the temperature at which the gases must be heated to initiate the reaction is nearly [Boltzmann's constant $k=1.38\times {10}^{-23}J/K$ ]

A glass slab $(\mu =\frac{3}{2})$ is placed infront of a concave mirror as shown in the figure. At what distance $x$ should an object be placed to the left of the slab, such that the final image forms on object itself.

The angle between $\overrightarrow{A}$ and resultant of $(\overrightarrow{A}+\overrightarrow{B})$ and $(\overrightarrow{A}-\overrightarrow{B})$ is

If the speed of the earth’s rotation is increased such that the apparent weight at the equator is one third the initial value, then what will be the length of the day in this case? (Take radius of Earth =$6.4\times {\text{1}0}^6\text{m and}g=10{\text{m/s}}^2$)

As shown in the figure, the $2\text{kg}$ block is subjected to an external force of $42\text{N}$. What is the value of normal reaction ($N$) acting on the $3\text{kg}$ block (exerted on it by the surface), when spring reaches maximum extension? (Take $g=10{\text{m/s}}^2$)

A tank is filled with water of density 1 g per $c{m}^3$ and oil of density 0.9 g $c{m}^{-3}$. The height of water layer is 100 cm and of the oil layer is 400 cm. If

g = 980 $c{m}^{-2}$ then the velocity of efflux from an opening in the bottom of the tank is

A $100\text{m}$ long train crosses a man travelling at $5\text{km/h}$ in the opposite direction in $7.2\text{s}$. Then the velocity of train is

A block of mass $m_1=0.5\text{kg}$ is moving with the speed of $u_1=2\text{m/s}$ on a smooth surface. It strikes another mass of $m_2=1\text{kg}$ initially at rest and then they move together as a single body. The energy lost during the collision is

Two bodies are held separated by $9.8\text{m}$ vertically one above the other. They are released simultaneously to fall freely under gravity. After $2\text{s}$ the relative distance between them is