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What is the minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass $M$ and radius $R$ in a circular orbit at an altitude of $2R$ ?

Tap the bubbles having trigonometric ratios whose values are "not defined".

A solid sphere of radius $r=2\text{m}$ is under a combination of rotational and translational motion as shown in figure.





If acceleration of COM $\left(O\right)$ is $a_o=2{\text{m/s}}^2$ and horizontal component of acceleration at point $A$ is $a_A=6{\text{m/s}}^2$, then find angular acceleration $\alpha$.

The polar form of $(i^{25}{)}^3$ is

A particle moving with kinetic energy $E$, has de-Broglie wavelength $\lambda$. If energy $\Delta E$ is added to it, the wavelength become $\lambda /2$. The value of $\Delta E$ is,

A cubical block of mass m and side L rests on a rough horizontal surface with coefficient of friction $\mu$ A horizontal force F is applied on the block as shown. If the coefficient of friction is sufficiently high so that the block does not slide before toppling, the minimum force required to topple the block is

A capacitance of $2\mu F$ is required in an electrical circuit across a potential difference of $1.0kV$. A large number of $1\mu F$ capacitors are available which can withstand a potential difference of not more than $300V$. The minimum number of capacitors required to achieve this is

A bus of mass $M$ hits a man of mass $m$ standing on the road. If $M>>m$ and initially bus was moving with $15\text{m/s}$. What will be the speed of man just after the collision? Assuming collision is perfectly elastic.

For a hydrodynamically and thermally fully developed laminar flow through a circular pipe of constant cross section, the Nusselt number at constant wall heat flux $\left(N{u}_q\right)$ and that at constant wall temperature $\left(N{u}_T\right)$ are related as

Two men with weights in the ratio 5 : 3 run up a staircase in times in the ratio 11 : 9. The ratio of power of first to that of second is

1. 151.95

If ${\lambda }_0$ is the de-Broglie wavelength for a proton accelerated through a potential difference of $100\text{V}$, then the de-Broglie wavelength for $\alpha$ particle accelerated through the same potential difference is -

Ten one-rupee coins are put on top of each other on a table. Each coin has a mass m. Give the magnitude and direction of:
a) the force on the $7^{th}$ coin (counted from the bottom) due to all the coins on its top,
b) the force on the $7^{th}$ coin by the eighth coin,
c) the reaction of the 6th coin on the $7^{th}$ coin.

The maximum kinetic energy of photoelectron liberated from the surface of lithium (work function = 2.35eV) by electromagnetic radiation whose electric component varies with time as $E=a[1+cos(2\pi f_{1}t\left)\right]cos\left(2\pi f_{2}t\right)$ where ‘a’ is a constant, $f_1=1.2\times {10}^{15}Hz$ and $f_2=3.6\times {10}^{15}Hz$ is [Take: $h=6.6\times {10}^{-34}Js$ ]

1. 5

1. 0.17

A particle is fired with velocity $u$ making angle $\theta$ with the horizontal. What is the magnitude of change in velocity when it is at the highest point?

A ball of mass $100\text{g}$ and another ball of mass $120\text{g}$ moves towards each other with speeds $6\text{m/s}$ and $5\text{m/s}$ respectively. If they stick to each other after colliding, what would be the velocity of the combined mass after the collision?

A cylindrical tube, open at both ends, has a fundamental frequency $f_0$, in air. The tube is dipped vertically into water such that half of its length is inside water. The fundamental frequency of the air column now is

If $x^{3}dy+xydx=x^{2}dy+2ydx;y\left(2\right)=e$ and $x>1$, then $y\left(4\right)$ is equal to:

An aluminium piece of mass 50g initially at ${300}^{\circ }C$ is dipped quickly and taken out of 1kg of water, initially at ${30}^{\circ }C$. If the temperature of the aluminium piece be $160\circ C$, what is the temperature of the water then (Specific heat capacities of aluminium and water are 900 $JK{g}^{-1}{K}^{-1}$ and 4200 $Jk{g}^{-1}{k}^{-1}$, respectively)

A gas expands $0.25m^3$ at constant pressure ${10}^{3}M/m^2$, the work done is

Eight spherical drops of equal size are falling vertically through air with a terminal velocity $0.1{\text{ms}}^{-1}$. If the drops coalesce to form a large spherical drop, its terminal velocity would be

The magnetic field due to a long current carrying wire at a perpendicular distance $R$ is $B,$ then the magnetic field at perpendicular distance $2R$ will be equal to



$[$0.77 Mark$]$

The masses of Blocks $\text{A}$ , $\text{B}$ and $\text{C}$ are in the ratio $1:1:4$ . Block $\text{A}$ is given an initial velocity $v_0$ towards $\text{B}$ due to which it collides with $\text{B}$ perfectly inelastically. The combined mass collide with $\text{C}$ , also perfectly inelastically . If the loss in kinetic energy is $n$ times the initial kinetic energy , then the value of $n$ is

A network of resistances is constructed with $R_1$ and $R_2$ as shown in the figure. The potential at the points 1, 2, 3 … n are $V_1,V_2,V_3,\dots V_n$ respectively, each having a potential k times smaller than the previous one.

The ratio $\frac{R_2}{R_3}$ is

A charged particle is projected with velocity υ₀ along positive $x-$axis. The magnetic field $B$ is directed along negative $z-$axis between $x=0$ and $x=L$. The particle direction of projection. Find the velocity with which the same particle is projected (at $x=0$) along positive $x-$ axis so that when it emerges out (at $x=L$), the angle made by it is ${30}^{\circ }$with the direction of projection

In an NPN transistor circuit, the collector current is 10 mA. If 90% of the electrons emitted reach the collector, the emitter current (i_E) and base current (i_B) are given by

A current $I$ flows in a circuit shaped like an isoceles trapezium. The ratio of the bases is $2$. The length of the smaller base is $l$. Calculate the magnetic induction at a point $P$ located in the plane of the trapezium, but at a distance $a$ from the midpoint of the smaller base.