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A rectangular vessel when full of water, takes $10\text{min}$ to be emptied through an orifice at its bottom. How much time will it take to be emptied when it is half filled ?

Two bodies $A$ & $B$ rotate about an axis such that angle ${\theta }_A$ (in radians) covered by first body is proportional to square of time, & ${\theta }_B$ (in radians) covered by second body varies linearly. At $t=0,{\theta }_A={\theta }_B=0$. If $A$ completes its first revolution in $\sqrt{\pi }\text{s}$ & $B$ needs $4\pi \text{s}$ to complete half revolution, then angular velocities ${\omega }_A$ & ${\omega }_B$ at $t=5\text{s}$ are in the ratio

A long straight wire of radius $a$ carries a steady current $i$. The current is uniformly distributed across its cross section. The ratio of the magnetic field at $\frac{a}{2}$ and $2a$ from its axis is

A small
pin fixed on a table top is viewed from above from a distance of 50
cm. By what distance would the pin appear to be raised if it is
viewed from the same point through a 15 cm thick glass slab held
parallel to the table? Refractive index of glass = 1.5. Does the
answer depend on the location of the slab?

What are the merits of Dalton’s atomic theory?

A skater is moving on ice at a velocity of $15\text{m/s}$ in a circle of radius $22.5\text{m}$. What is the angle the person makes with the ground?

A hemispherical bowl just floats without sinking in a liquid of density 1.2 $\times$ ${10}^{3}kg/m^3$. If outer diameter and the density of the bowl are 1 m and 2 $\times$ 10${}^4$ kg/m${}^3$ respectively, then the inner diameter of the bowl will be

A photon of wavelength $4400\stackrel{\circ }{A}$ is passing through vacuum. The effective mass and momentum of the photon are respectively

The output of a strain gauge bridge using two active strain gauge used in poisson's configuration is $(\nu \to poisso{n}^{\prime }sratio)$

A stationary particle explodes into two particles of masses x and y which move in opposite directions with velocity v1 and v2. The ratio of their kinetic energy is

A string of length $1\text{m}$ is fixed at one end with a bob of mass $100\text{g}$ and the string makes $\left(\frac{2}{\pi }\right)\text{rev/s}$ around a vertical axis through a fixed point. The tension in the string is

A particle moves in a circle of radius 0.5 m with a linear speed of 2 m/s. Its angular speed is

A uniform metre scale balances horizontally on a knife edge placed at $55\text{cm}$ mark when a mass of $25\text{g}$ is suspended from one end, then mass of the scale is

The figure shows a container filled with a liquid of density $\rho$. Four points $A,B,C$ and $D$ lie on the vertices of a vertical square. Points $A$ and $C$ lie on a vertical line and points $B$ and $D$ lies on a horizontal line. Choose the correct statement (s) about the pressure at the four points.

Two springs of spring constants 1500 N/m and 3000 N/m respectively are stretched with the same force. They will have potential energy in the ratio

A solid sphere of radius $r$ is gently placed on a rough horizontal surface with an initial angular speed ${\omega }_0$ but no linear velocity. If the coefficient of friction is $\mu$, then the time $t$ when the slipping will stop is

Velocity of a car as a function of time is given by $v\left(t\right)=(3t^3-\sin t)$ in $\text{m/s}$. The displacement in $\left(m\right)$ of the car as a function of time $t$ is

[Take $x=0$ at $t=0$]

The focal length of convex mirror $M_1$ is $20\text{cm}$ and that of concave mirror $M_2$ is $30\text{cm}$. For the object placed at $O$ as shown in the figure, find the distance of the second image $I_2$ from the pole of the mirror $M_2$. Consider first reflection to be at $M_1$.

${}^{57}\text{Co}$ decays to ${}^{57}\text{Fe}$ by ${\beta }^{+}$ emission. The resulting ${}^{57}\text{Fe}$ is in its excited state and comes to the ground state by emitting $\gamma$ rays. The half-life of ${\beta }^{+}$ decay is 270 days and that of the $\gamma$ emission is ${10}^{-6}$ s. A sample of ${}^{57}\text{Co}$ gives 5.0$\times {10}^9$ gamma rays per second. How much time will elapse before the emission rate of gamma rays drops to 2.5 $\times {10}^9$

A long solenoid is formed by winding $20\text{turns/cm}$. The current necessary to produce a magnetic field of $20\text{millitesla}$ inside the solenoid will be approximately

$(\frac{{\mu }_0}{4\pi }={10}^{-7}\text{tesla-metre/ampere})$

If the momentum of a particle is increased by $20\%$, then the kinetic energy increases by

A boat goes across a river in shortest possible time with a velocity $12km/h$ . The magnitude of its resultant speed in flowing water is $13km/h$. The magnitude of velocity of water flow in the river is

A $2kg$ object is subjected to three forces that give it an acceleration $\overrightarrow{a}=-8m/s^2\hat{i}$. If two of the three forces are $F_1=\left(30.0N\right)\hat{i}+16.0N)\hat{j}$ and $F_2=-\left(12N\right)\hat{i}+\left(8N\right)\hat{j}$, find the third force.