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The speed of a projectile at its maximum height is $\frac{\sqrt{3}}{2}$ times of its initial speed. If range of projectile is $p$ times the maximum height reached by it, then find the value of $p$.

Two blocks $A$ and $B$ are arranged as shown in the figure. The mass of block $A$ is $10\text{kg}$. The coefficient of friction between block $A$ and the horizontal plane is $0.2$. The minimum mass of block $B$ to start motion will be

An isolated particle of mass $m$ is moving in horizontal plane $x-y$, along the x-axis, at a certain height above ground. It suddenly explodes into two fragments of masses $\frac{m}{4}$ and $\frac{3m}{4}$ . An instant later, the smaller fragment is at $y=+15\text{cm}$. The larger fragment at this instant is at

The displacement of a body from a reference point, is given by $\sqrt{x}=2t+3$ where x is in metres and t in seconds. This shows that the body is

One mole of an ideal gas is heated slowly so that it goes from the $P-V$ state $(P_i,V_i)$ to $(3P_i,3V_i)$ in such a way that the pressure of the gas is directly proportional to the volume. How much work is done on the gas in the process?

A rocket of mass $3500\text{kg}$ has to achieve a velocity of $1000\text{m/s}$ over a launch time of $10$ minutes. The nozzle of the rocket is designed to let out exhaust gases at a velocity of $300\text{m/s}$ w.r.t the rocket. What is the amount of fuel that is consumed during the launch?

The kinetic energy of a body is increased by $21\%$. What is the percentage increase in the magnitude of linear momentum of the body.

Moment of inertia of a uniform disc of internal radius $r$ and external radius $R$ and mass $M$ about an axis through its centre and perpendicular to its plane is

A wire of length $l$, mass $m$ and resistance $R$ slides without any friction, down the parallel conducting rails of negligible resistance (see figure). The rails are connected to each other at the bottom by a resistanceless rail parallel to the wire so that the wire and the rails form a closed rectangular conducting loop. The plane of the rails makes an angle $\theta$ with the horizontal and a uniform vertical magnetic field of induction B exists throughout the region. Find the steady state velocity of the wire.

Figure shows a container filled with a liquid of density $\rho$. Four points $A,B,C\text{and}D$ lie on the diametrically opposite points of a circle as shown. Points $A$ and $C$ lie on a vertical line and points $B$ and $D$ lie on a horizontal line. $P_A,P_B,P_C,P_D$ are the absolute pressures at the respective points. Then, the correct statement(s) is/are :

Two wires A and B of same length and of the same material have the respective radii $r_{1}and{r}_2$. Their one end is fixed with a rigid support, and at the other ends equal twisting couple is applied. Then the ratio of the angle of twist at the end of A and The angle of twist at the end of B will be

Two particles of equal masses moving the same speed collide perfectly in-elastically. After the collision the combined mass moves with half of the speed of the individual masses. The angle between the initial momenta of individual particle is

If velocity of a particle is $v\left(t\right)=(3t^2-2t+4)\text{m/s}$, the magnitude of average velocity of the particle in $10\text{sec}$ 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 thin uniform rod of mass $m$ and length $l$ is kept on a smooth horizontal surface such that it can move freely. At what distance $x$ from the centre of the rod should a particle of mass $m$ strike on the rod such that the point $P$ at a distance $\frac{l}{3}$ from the end of the rod is instantaneously at rest just after the elastic collision ?

In a car race, car $A$ takes $20\text{s}$ less than car $B$ to finish. It passes the finishing point with speed $v$ more than that of car $B$. Assuming that both the cars start from rest and travel with a constant acceleration of $50{\text{m/s}}^2$ and $40{\text{m/s}}^2$ respectively, what is the value of $v$ ?

Two cars P and Q move with velocities ${\overrightarrow{v}}_P$ and ${\overrightarrow{v}}_Q$ towards north and east respectively.

Velocity of car P w.r.t. Q (i.e., velocity of car P for an observer on cars Q) i.e., ${\overrightarrow{v}}_{PQ}$ will be in which direction?

If $x=a{t}^4,y=b{t}^3,then\frac{dy}{dx}=?$

A body of mass $m$ is projected horizontally with a velocity $v$ from the top of a tower of height $h$ and it reaches the ground at a distance $x$ from the foot of the tower. If a second body of mass $2m$ is projected horizontally from the top of a tower of height $2h$, it reaches the ground at a distance $2x$ from the foot of the tower. The horizontal velocity of the second body is

A particle of mass $1\text{kg}$ is fastened to one end of a string of length $2\text{m}$ and a mass $2\text{kg}$ is attached to the middle point, the other end of the string is fastened to a fixed point on a smooth horizontal table. The particles are then projected so that the two portions of the string are always in the same straight line and describes horizontal circles. Then the ratio of tensions $\left(\frac{T_1}{T_2}\right)$ in the two parts of the string as shown in diagram is

Kinetic energy with which the electrons are emitted from the metal surface due to photoelectric effect is

The capacitance of a cylindrical capacitor of length $72\text{mm}$ is $2\text{pF}$. If the ratio of radii of its outer cylindrical conductor to that of its inner cylindrical conductor is $e^x$, then $x$ is

The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If the length of open organ pipe is $80\text{cm}$ then the length of the closed organ pipe will be

The displacement - time $(x-t)$ graph of a particle executing simple harmonic motion is shown in figure. The correct variation of net force $F$ acting on the particle as a function of time is

Figure shows a platform on which a man of mass $M$ is standing and holding a string passing over a system of ideal pulleys. Another mass $m$ is hanging as shown in figure. Consider the entire system to be in equilibrium. Take $M=5m$, find the force exerted by the platform on man.

A bimetallic strip is formed out of two identical length strips of which one is copper and the other is brass. The coefficient of linear expansion of the two metals are ${\alpha }_C$ and ${\alpha }_B$ . On heating, the temperature of the strip goes up by $\Delta T$ and the strip bends to form an arc of radius of curvature $R$, then which of the following is/are correct ?

A force $F=20+10y$ acts on a particle in y-direction, where $F$ is in Newtons and $y$ is in metres. Work done by the force to move the particle from $y=0$ to $y=1m$ is

A current I flows in a long thin walled cylinder of radius R. What pressure do the walls of the cylinder experience?

A cavity of radius $r$ is made inside a solid sphere. The volume charge density of the remaining sphere is $\rho$. An electron (charge $e$ , mass $m$) is released inside the cavity from point $P$ as shown in figure. The center of sphere and center of cavity are seperated by a distance $a$. The time after which the electron again touches the sphere is

The refractive index of a piece of transparent quartz is the greatest for

[MP PET 1985, 94]

As the position of an object (u) reflected from a concave mirror is varied, the position of the image (v) also varies. By letting the u changes from 0 to + $\infty$ the graph between v versus u will be

A stone tied to a string is rotated in a circle. If the string is cut, the stone flies away from the circle because

A copper wire of length 1.0 m and a steel wire of length 0.5 m having equal cross-sectional areas are joined end to end. The composite wire is stretched by a certain load which stretches the copper wire by 1 mm. If the Young's moduli of copper and steel are $1.0\times {10}^{11}N{m}^{-2}$ and $2.0\times {10}^{11}N{m}^{-2}$, the total extension of the composite wire is

If the binding energy per nucleon in

$\begin{array}{c}7\\ 3\end{array}Li$ and $\begin{array}{c}4\\ 2\end{array}He$ nuclei are $5.60\text{MeV}$ and $7.06\text{MeV}$ respectively, then in the reaction $p+\begin{array}{c}7\\ 3\end{array}Li\to 2\begin{array}{c}4\\ 2\end{array}He$, energy of proton must be

Given $\overrightarrow{A}=0.3\hat{i}+0.4\hat{j}+c\hat{k}$. Calculate the value of c if A is a unit vector.

In a direct impact, loss in kinetic energy is given by $△K=\frac{M_1{M}_2}{2(M_1+M_2)}.(V_1-V_2{)}^2(1-K^2)$ with usual

notations(except k). The quality k will have dimensional formula:

If I drop a ball from a height of 5 m then how long will it take to reach the ground?

An armored car $2\text{m}$ long and $3\text{m}$ wide is moving at $10\text{m/s}$ when a bullet hits it in a direction making an angle ${\tan }^{-1}\left(\frac{3}{4}\right)$ with the car as seen from the street. The bullet enters one corner of the car and passes out of the diagonally opposite corner. Neglecting any interaction between the bullet and the car, find the time for bullet to cross through the car.

A resistance of 20 ohms is connected to a source of an alternating potential $V=220sin\left(100\pi t\right)$. The time taken by the current to change from its peak value to r.m.s value is

A ball is projected upwards from the foot of a tower. The ball crosses the top of the tower twice after an interval of $6\text{s}$ and the ball reaches the ground after $12\text{s}$. The height of the tower is ($g=10m/s^2$)

Which of these is a wave/pulse equation?

How many forces are acting on $M$ when its $FBD$ is drawn? (Given all surfaces are smooth)

Calculate the area of the triangle determined by the two vectors $\overrightarrow{A}=3\hat{i}+4\hat{j}and\overrightarrow{B}=-3\hat{i}+7\hat{j}$

If we take units of mass, length and time as $2\text{kg},2\text{m}$ and $2\text{s}$ respectively then the unit of energy will be

In $v-t$ graph shown in fig. the distance travelled by the body in $5$ seconds will be

If the coefficient of friction between all surfaces is $0.4$ then find the minimum force $F$ applied to have system in equilibrium.