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If pressure at half depth of a lake is equal to $\frac{2}{3}rd$ pressure at the bottom of the lake, then what is the depth of the lake?

(Take atmospheric pressure as ${10}^5\text{Pa}$, density of water as $1000{\text{kg/m}}^3$ and $g=10{\text{m/s}}^2$)

A particle starting from rest moves with constant tangential acceleration on a circular path of radius $\frac{10}{\pi }\text{m}$. After $2.5$ rotation the velocity of the particle is $50\text{m/s}$, its tangential acceleration is

When a rubber band is stretched by a distance x, it exerts a restoring force of magnitude $F=ax+b{x}^2$, where a and b are constants. The work done in stretching the upstretched rubber band by L is

Find the minimum value of the function $y=5x^2-2x+1$.

Three liquids with masses $m_1,m_2,m_3$ are thoroughly mixed, If their specific heats are $c_1,c_2,c_3$ and their temperatures $T_1,T_2,T_3$ respectively, then the temperature of the mixture is

What is the smallest radius of a circular path on which a cyclist can travel with uniform speed of $36km{h}^{-1}$, with angle of inclination (from vertical, for cyclist) ${45}^{\circ }$ and $g=10m{s}^{-2}$ ?

Statement 1: EMF of a battery can never be equal to potential difference across its terminals.

Statement 2: EMF and Potential Difference of a battery are completely independent.

A ball is thrown upwards at an angle of ${60}^{\circ }$ to the horizontal. It falls on the ground at a distance of 90 m. If the ball is thrown with the same initial velocity at an angle ${30}^{\circ }$, it will fall on the ground at a distance of

Flux is a __ (Scalar/vector) quantity.

The equation of state of a gas is given by $P(V-b)=nRT$. If $1\text{mole}$ of a gas is isothermally expanded from volume $V$ and $2V$, the work done during the process is

A boat is rowed perpendicular to river flow at the rate of $5\text{km/hr}$. The river flows at the rate of $2.5\text{km/hr}$. The speed of boat approximately in $\text{m/s}$ is

A car is moving in a rainstorm at a speed of $60\text{km/h}$. The rain drops fall vertically w.r.t the ground with a constant speed of $15\text{m/s}$. The angle with the vertical at which raindrops strike the car is

Anushka starts walking straight towards east. After walking $75$ metres, she turns to the left and walks $25$ metres straight. Again she turns to the left, walks a distance of $40$ metres straight, again she turns to the left and walks a distance of $25$ metres. How far is she from the starting point?

The dimension of coefficient of thermal conductivity is

$(\frac{dQ}{dt}=-KA\frac{dT}{dx})$, where $dQ$ is heat transferred, $dt$ is time taken, $K$ is coefficient of thermal conductivity, $A$ is cross-sectional area of conductor, $dT$ is change in temperature and $dx$ is length of conductor.

Vectors $\overrightarrow{A}=a\hat{i}+a\hat{j}+3\hat{k}$ and $\overrightarrow{B}=a\hat{i}-2\hat{j}-\hat{k}$ are perpendicular to each other, then the positive value of $a$ is

When a battery is connected across a series combination of self inductance L and resistance R, the variation in the current i with time t is best represented by

A cylinder is released from rest from the top of an inclined plane of inclination $\theta$ and length 'l'. If the cylinder rolls without slipping, what will be its speed when it reaches the bottom?

Two bodies move toward each other from 10 m apart with a constant velocity they meet at C after 2s. What is the displacement of A w.r.t. B?

A rod AB of mass M and length 4L rests on a smooth horizontal plane. A small particle P moving with a velocity V strikes the rod elastically in a direction normal to the length, at a point distant L from the centre C of the rod. Immediately after the collision, if P comes to rest, then its mass should be

A circular disc of radius ${}^{\prime }{R}^{\prime }$ and mass ${}^{\prime }{m}^{\prime }$ is hinged at the centre about horizontal axis. A particle of same mass is attached at the top of the disc. Now the system is released. Calculate the angular velocity of the disc when the particle reaches bottom.

Two earth-satellites are revolving in the same circular orbit round the centre of the earth. They must have the same

A charge particle having charge q is accelerated through a potential difference V and then it enters a perpendicular magnetic field in which it experiences a force F. If V is increased to 5V, the particle will experience a force

Find the amplitude of the electric field in a parallel beam of light of intensity $2.0W{m}^{-2}$

A motorbike is being chased by a car. The motorbike moves with the speed of $10m/s$ and has acceleration of $5m/s^2$. Whereas, the car has speed of $20m/s$ and accelerates with $3m/s^2$. If the initial separation between motorbike and car is $60m$, find the distance by which car will miss the motorbike.

Which of the following conditions hold true if the ray (shown in figure) always undergo total internal reflection at the surface $\text{AC}$. The critical angle between prism and air is $i_c$.

Find the force exerted at the support $S$ in terms of ‘$m$’ if the system shown is in equilibrium.

A table with smooth horizontal surface is placed in a cabin which moves in a circle of a large radius R with $\omega$. A smooth pulley of small radius is fastened to the table. Two masses m and 2m placed on the table are connected through a string going over the pulley. Initially the masses are held by a person with the strings along the outward radius and then the system is released from rest (with respect to the cabin). Find the magnitude of the initial acceleration of the masses as seen from the cabin.

The potential energy of a $1\text{kg}$ particle free to move along the $x-$ axis is given by $V=(\frac{x^4}{4}-\frac{x^2}{2})\text{J}$

The total mechanical energy of the particle is $2\text{J}$. Then the maximum speed is

If $c_p$ and $c_v$ denotes the specific heat of nitrogen at constant pressure and constant volume respectively, then

Write the dimensions of $(a\times b)$ in the relation $E=\frac{b-x^2}{at}$, where $E$ is the energy, $x$ is the displacement and $t$ is time.

Four particles of masses $1\text{g},2\text{g},3\text{g}$ and $4\text{g}$ are kept at points $(0,0),(0,3\text{cm}),(3\text{cm},3\text{cm}),(3\text{cm},0\text{cm})$ respectively. Find the moment of inertia of the system of particles (in ${\text{g-cm}}^2$) about $x$ axis.

The retarding potential for having zero photo-electron current

A force $F=-K(y\hat{i}+x\hat{j})$, where K is a positive constant, acts on a particle moving in the x-y plane. Starting from the origin, the particle is taken along the positive x-axis to the point (a, 0) and then parallel to the y-axis to the point (a, a). The total work done by the force F on the particle is

A metallic sphere of radius $r$ and specific heat $S$ is rotated about an axis passing through its centre of mass at an angular speed of $n$ rotations per second. It is suddenly stopped and $50\%$ of its energy is used in increasing its temperature. Then, the rise in temperature of the sphere is

Find the projection of $\overrightarrow{A}=10\hat{i}+8\hat{j}-6\hat{k}$ along $\overrightarrow{r}=5\hat{i}+6\hat{j}+9\hat{k}$

A ball of mass $1\text{kg}$ is tied to one end of a string of length $1\text{m}$, pivoting the other end of string, ball is rotated at an angular speed of $2\text{rev/min}$. The tension in the string is

A parallel-plate capacitor is constructed using three different dielectric materials as shown in the figure. What is the capacitance across P and Q?

A wire carrying current i, is present in three quarter-circle sections on xy, yz and zx plane. Find $\overrightarrow{M}$

A $1\text{kg}$ particle strikes a wall with a velocity $1\text{m/s}$ at an angle ${30}^{\circ }$ and reflects at the same angle in $0.1$ second, then the force applied by the wall on the particle will be

The wall with a cavity consists of two layers of brick separated by a layer of air. All three layers have the same thickness, and the thermal conductivity of the brick is much greater than that of air. The left layer is at a higher temperature than the right layer and steady state condition exists. Which of the following graphs predicts correctly the variation of temperature $T$ with distance $d$ inside the cavity ?

A particle starts from rest with a time varying acceleration $a=(2t-4)$. Here $t$ is in $s$ and $a$ in $m/s^2$. At what time, particle will have maximum velocity in negative direction?

A liquid flows in the tube, left to right as shown in figure. $d_1$ and $d_2$ are the diameter of tube as shown in figure. Find the ratio of speed $\frac{V_1}{V_2}$.

The frequency of the fourth harmonic of an organ pipe is $33\text{Hz}$ less than the frequency of the fifth harmonic of the same organ pipe. Find the difference in wavelengths for the given two cases. Neglect end corrections.

[speed of sound in air $=330\text{m/s}$]

A constant force $F_1=10\sqrt{3}N$ applied at an angle ${30}^{\circ }$ with horizontal on body $(m=5\text{kg})$ at rest for $8\text{second}$. In this time interval, body moves from position A to B. To move from position B to A, another constant force $F_2=20\text{N}$ is applied on body . If work done by this force is $960\text{J}$, then find the angle at which the force is acting on the body with horizontal.

A circular plate of uniform thickness has a diameter of $28\text{cm}$. A circular portion of diameter $21\text{cm}$ is removed from the plate as shown. $O$ is the centre of mass of the complete plate. The position of centre of mass of the remaining portion will shift towards left from $O$ by

Each side of the cube is $L$ then the magnitude of magnetic dipole moment of the loop $oabcdeo$ carrying current $i$ as shown in figure is

Charges Q₁ and Q₂ are near each other. We have no information of their position, nature and magnitude, all we know is Q₂ experiences a force of 5N to the right. What can we infer for sure from this?

If the speed of the particle is $v=4t^3-3t^2+3\text{m/s}$, the distance of the particle as a function of time $t$ and the average speed in $2\text{sec}$ respectively are

Assume that a tunnel is dug across the earth (radius = R) passing through its centre. Find the time a particle takes to cover the length of the tunnel if it is projected into the tunnel with a speed of $\sqrt{gR}$