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A wire of length $2L,$ is made by joining two wires $\text{A}$ and $\text{B}$ made of the same material having same length but different radii $r$ and $2r$. It is vibrating at a frequency such that the joint of the two wires forms a node. If the number of antinodes in wire $\text{A}$ is $p$ and that in $\text{B}$ is $q$, then the ratio $p:q$ is:

A particle is performing S.H.M with time period $T$. Find the shortest time taken by the particle to go from the mean position to an extreme position.

For the case of an ideal gas, find the equation of the process in which the molar heat capacity varies as $C=C_v+\alpha T^2$, where $\alpha$ is constant. Take $V_0$ to be the initial volume of the gas.

A particle is moving with speed $6m/s$ along the direction of $\overrightarrow{A}=2\hat{i}+2\hat{j}-\hat{k}$, find the velocity (in $m/s$).

The current in a coil of self-inductance $2.0\text{H}$ is increasing according to equation $I=2\sin t^2\text{A}$. Find the amount of energy spent during the period when the current changes from $0$ to $2\text{A}$.

Two identical straight wires are stretched so as to produce 6 beats per second when vibrating simultaneously. On changing the tension slightly in one of them, the beat frequency remains unchanged. Denoting the higher and the lower initial tension in the strings as T₁,T₂ respectively, then it could be said that while making the above changes in tension :

A photocell is illuminated by a small bright source placed 1 m away. When the same source of light is placed $\frac{1}{2}m$ away, the number of electrons emitted by photo cathode would

The maximum speed and acceleration of a particle executing simple harmonic motion are $10cm{s}^{-1}and50cm{s}^{-2}$. Find the position(s) of the particle when the speed is $8c{m}^{-1}$.

A river $5\sqrt{3}km$ wide flows with a velocity of $5kmph$. A boat which can move with a velocity of $10kmph$ in still water crosses the river, without any drift. For this, the boat has to tilt its velocity by $\theta$ with the straight path in the right direction (see figiure). If the boat's velocity is tilted by the same angle $\theta$ in the wrong direction, the drift will be

Consider the following statements about sound passing through a gas.

(A) The pressure of the gas at a point oscillates over time.

(B) The position of a small layer of the gas oscillates over time.

A system consists of Block $A$ and $B$ of masses $5\text{kg}$ each and connected to a pulley as shown in figure. Find the minimum value of force $F$ at which they will start moving? (Assume $g=10{\text{m/s}}^2$).

Three identical spheres each of mass $1\text{kg}$ are kept as shown in the figure. They are kept in such a way that they are touching each other with their respective centres on a straight line . If their centres are marked $P,Q,R$ respectively, then the distance of the centre of mass of the system from $P$ is:

A stone is tied to one end of a string 50 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 10 revolutions in 20 s, what is the magnitude of acceleration of the stone

A point object $O$ is placed at a distance of $0.3m$ from a convex lens $($focal length $0.2m)$ cut into two halves each of which is displaced by $0.0005m$ as shown in the figure.







If this arrangement will generate more than one image, then what will be the total number of images?

A ring of radius 1m has moment of inertia of $10kg-m^2$ about an axis through its centre and perpendicular to its plane. If a constant force of 50N is applied to it tangentially as shown, what will be its angular acceleration?

A smaller disc is removed from a uniform circular lamina as shown in the figure. Find the position of $COM$ of the uniform lamina, if diameter of the smaller disc is $12\text{m}$.

A motor cyclist moving with a velocity of 72 km/hour on a flat road takes a turn on the road at a point where the radius of curvature of the road is 20 meters. The acceleration due to gravity is 10 $m/se{c}^2$. In order to avoid skidding, he must not bend with respect to the vertical plane by an angle greater than

A projectile is fired horizontally with a velocity of $98\text{m/s}$ from the top of a hill $490\text{m}$ high. Find the time taken by the projectile to reach the ground.

If $\overrightarrow{C}=\overrightarrow{A}+\overrightarrow{B}$, then the magnitude of $\overrightarrow{B}$ is

In a two block system shown in figure, match the following.







$\begin{array}{cc}\text{Table -1}& \text{Table -2}\\ \text{(A) Velocity of centre of}& \text{(P) Keep on changing all}\\ \text{mass}& \text{the time}\\ \text{(B) Momentum of centre of}& \text{(Q) First decreases then}\\ \text{mass}& \text{becomes zero}\\ \text{(C) Momentum of 1 kg block with respect to 2 kg block}& \text{(R) zero}\\ & \text{(S) constant}\end{array}$

A trolley ($100\text{kg}$) moving forward with acceleration of $1{\text{m/s}}^2$ has block of $1\text{kg}$ mass present on its floor. Find pseudo force acting on block ${}^{\prime }{A}^{\prime }$ with respect to a reference frame attached to trolley

A furnace is made of a red brick wall of thickness $0.5\text{m}$ and conductivity $0.7\text{W/mK}$. For the same rate of heat conduction and the same temperature drop, this can be replaced by a diatomite wall of conductivity $0.14\text{W/mK}$ and the same cross-sectional area. Find the thickness of the diatomite wall.

Venus looks brighter than other planets because

A ball thrown vertically upwards with a speed of $19.6{\text{ms}}^{-1}$ from the top of a tower returns to the earth in $6\text{s}$. Find the height of the tower.

(Take $g=9.8{\text{m/s}}^2$)

A cylinder of mass $M$, radius $R$ is resting on a horizontal platform (which is parallel to $xy-$plane) with its axis fixed along the $y-$axis and only free to rotate about its axis. The platform is given a motion in $x-$direction given by $x=A\cos \left(\omega t\right)$. There is no slipping between the cylinder and platform. The maximum torque acting on the cylinder during its motion is

Find $\overrightarrow{A}\times \overrightarrow{B},$ if $\overrightarrow{A}=\hat{i}+4\hat{k}$ and $\overrightarrow{B}=-\hat{j}$

There is a hole in the bottom of tank having water. If total pressure at bottom is 3 atm (1 atm = ${10}^5$N/$m^2$) then the velocity of water flowing from hole is

The value of $C_v$ for one mole of neon gas is

The variation in velocity of a particle moving along a straight line is shown in the figure. If the particle is moving along x-axis with initial position $x=10\text{m}$, find the final position of particle.

A person walks up a stationary escalator in $t_1$ second. If he remains stationary on the escalator, then it can take him up in $t_2$ second. If the length of the escalator is L, then how much time would it take him to walk up the moving escalator?

A thin circular plate of radius $1\text{m}$ has its density varying as $\rho \left(r\right)=8r$ where $r$ is the distance from its geometrical centre. Find the moment of inertia of the circular plate about an axis passing through its edge and perpendicular to the plate.

A small stone of man $0.2\text{kg}$ tied to a massless inextensible string is rotated in a vertical circle of radius of $2\text{m}$. If the particle is just able to complete a vertical circle, what is the speed at the highest point of the circular path? Also, calculate speed if the mass of stone is increased by $50$% (Take $g=10m/s^2)$

The potential differences across the resistance, capacitance and inductance are $80V,40V$ and $100V$ respectively in an L-C-R circuit. The power factor of this circuit is

If $S=4x^2-\sin x$, find $\frac{d^{2}S}{d{x}^2}$

Slope of the curve $y=x^3-e^x+\cos x$ is

Electric charges are distributed in a small volume. The flux of the electric field through a spherical surface of radius $10\text{cm}$ containing the total charge is $25\text{V-m}$. The flux through a concentric sphere of radius $20\text{cm}$ will be?

In the given figure, $a=15m/s^2$ represents the total acceleration of a particle moving in the clockwise direction in a circle of radius $R=2.5m$ at a given instant of time. The speed of the particle is

A point charge $Q$ is located on the axis of a disc of radius $R$ at a distance $b$ from the plane of the disc. If one-fourth of the total electric flux from the charge passes through the disc, then choose the correct option.

A uniform metal rod is used as a bar pendulum. If the room temperature rises by ${10}^{\circ }C$, and the coefficient of linear expansion of the metal of the rod is 2 ´ ${10}^{-6}pe{r}^{\circ }C$ , the period of the pendulum will have percentage increase of

In the given arrangement, find the distance travelled by wedge $B$ in $1sec$. System starts from rest. $(\theta ={60}^{\circ })$

A particle retards from a velocity $v_0$ while moving in a straight line. If the magnitude of deceleration is directly proportional to the square root of the speed of the particle, find its average velocity for the total time of its motion.

A cube of side 20cm is floating on liquid with 5cm of the cube outside the liquid. If the density of liquid is 0.8 gm/c.c then the mass of the cube is

The figure shows a charge q placed inside a cavity in an uncharged conductor. Now if an external electric field is switched on:

Two coherent light sources each of wavelength $\lambda$ are separated by a distance $3\lambda$. The total number of minima formed on line AB which runs from $-\infty$ to $+\infty$ is: