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A string fixed at both ends has consecutive standing wave modes for which the distances between adjacent nodes are $18\text{cm}$ and $16\text{cm}$ respectively. The minimum possible length of the string is:

The equation $\varphi (x,t)=jsin\left(\frac{2\pi }{\lambda }vt\right)cos\left(\frac{2\pi }{\lambda }x\right)$ represents

[MNR 1994]

Find the derivative of $x^5$

Two identical blocks $A$ and $B$, connected through a massless string are placed on a frictionless horizontal surface. A bullet having mass $2$ times that of a single block, moving with velocity $u$ strikes block $B$ from behind as shown in the figure. If the bullet comes to rest after striking the block, find the impulse on block $A$ due to tension in the string.

A mass of 1 kg is dropped from a height of 2m on a horizontal spring board. The vertical spring supporting the board has a spring constant of 87.5 N/m .The maximum distance by which the mass compress the spring is close to..?

An uniform lamina in the shape of a right angled triangle is shown in the figure. Find the coordinates of $\text{COM}$ of the lamina. Assume the origin to be at the intersection of x and y axes.

A particle is executing SHM along a straight line. Its velocities at distances $2\text{cm}$ and $4\text{cm}$ from the mean position are $3\text{cm/s}$ and $2\text{cm/s}$ respectively. Find the time period of SHM.

How many external forces would appear to act on $(m_2+m_3)$ if they are considered as a system and $m_1$ is considered as another system in the figure shown?

Choose the correct answer
If $\theta$ is the angle between any two vectors a and b, then $|a\cdot b|=|a\times b|$, when $\theta$ is equal to
a) Zero
b) $\frac{\pi }{4}$
c) $\frac{\pi }{2}$
d) $\pi$

A wet porous substance in the open air loses its moisture at a rate proportional to the moisture content. If a sheet hung in the wind loses half its moisture during the first hour, then the time when it would have lost $99.9\%$ of its moisture is: (weather condition remaining same) [Take ${\log }_{10}2=0.301$ ]

In arrangement given in figure, if the block of mass m is displaced, the frequency is given by

A rocket of mass 1000 kg exhausts gases at a rate of 4 kg/sec with a velocity 3000 m/s. The thrust developed on the rocket is

A particle is projected so as to graze the top of two walls each of height $20m$. The walls are at distances of $30m$ and $170m$ respectively from the point of projection. Find the angle of projection.

Mass density of a solid sphere is ρ. Radius of the sphere is R. The gravitational field at distance r from the centre of the sphere inside it is

Two magnets of equal mass are joined at right angles to each other as shown the magnet 1 has a magnetic moment 3 times that of magnet 2. This arrangement is pivoted so that it is free to rotate in the horizontal plane. In equilibrium what angle will the magnet 1 subtend with the magnetic meridian

A train $75\text{m}$ long moving at a speed of $54\text{km/h}$ starts overtaking another train of length $125\text{m}$ moving at a speed of $36\text{km/h}$ on a parallel track. How long will the first train take to completely overtake the second train?

The electric field in a certain region is acting radially outward and is given by E = Ar. A charge contained in a sphere of radius= a centred at the origin of the field, will be given by:

In the circuit shown the battery is ideal. The current from C to D is

A thin uniform rod of mass $m$ and length $L$ is hinged at its upper end and released from rest in horizontal position. The tension at a point located at a distance $L/3$ from the hinge point , when the rod becomes vertical, will be -

Find the time period of small oscillations for the block of mass $m$ as shown in the figure. [All springs are identical]

The resultant amplitude due to superposition of two waves
$Y_1=5\sin (\omega t-kx)$ and $Y_2=-5\cos (\omega t-kx-{150}^{\circ })$

The diode used in the circuit shown in the figure has a constant voltage drop at 0.5 V at all currents and a maximum power rating of 100 milli watts.
What should be the value of the resistor R, connected in series with diode, for obtaining maximum current?

Each situation in column I gives graph of a particle moving in circular path. The variables and t represent angular speed (at any time t ), angular displacement (in time t) and time respectively. Column II gives certain resulting interpretation. Match the graphs in column I with statements in column II.

The motion of a particle along a straight line is described by equation $x=8+12t-t^3$, where, x is in meter and t is in second. The retardation of the particle when its velocity becomes zero, is

A wooden block of mass $M$ rests on a horizontal surface. A bullet of mass $m$ moving in the horizontal direction strikes and gets embedded in it. The combined system covers a distance $x$ on the surface. If the coefficient of friction between wood and the surface is $\mu$, the speed of the bullet at the time of striking the block is (where $m$ is mass of the bullet)

How much energy is needed to move a body of ${10}^{3}kg$ mass from Earth's surface to a height of $2R_E$?$(R_E=radiusofEarth=6400km)$

Figure shows two parallel and coaxial loops. The smaller loop (radius $r$) is above the larger loop (radius $R$), by distance $x$ ($x>>R$). The magnetic field due to current $i$ in the larger loop is nearly constant throughout the smaller loop. Suppose that $x$ is increasing at a constant rate of $\frac{dx}{dt}=v$. The induced emf in the smaller loop is

If $24$ cells each of internal resistance $I$ $\Omega$ are to be connected in series and parallel combination to get the maximum power output across a load of $10$ $\Omega$. If $x$ represents the number of rows and $y$ represents the number of cells in each row. Which of the following relation between $x$ and $y$ is correct?

$Abodyhangsfromaspringbalancesupportedfromthefromtheceilingofanelevator.Iftheelevatorhasanupwardaccelerationof2.5m/s^{2}andthebalancereads50N,whatisthetrueweightofthebody?$

The Kinetic energy 'K' of a particle moving along a circle of radius R depends on the distance covered 'S' as $K=a{S}^2$. The force acting on the particle is

Match the following columns. Here, $I$ represent intensity, $A$ amplitude and $r$ distance from the source.
Column $\text{I}$ Column $\text{II}$
$a.$ $I$ due to point source $p.$ $\propto r^{-1/2}$
$b.$ $A$ due to point source $q.$ $\propto r^{-1}$
$c.$ $I$ due to line source $r.$ $\propto r^{-2}$
$d.$ $A$ due to line source $s.$ $\propto r^{-4}$

If ${\lambda }_1$ and ${\lambda }_2$ denote the wavelength of de Broglie waves for electrons in Bohr’s first and second orbits in the hydrogen atom, then $\frac{{\lambda }_1}{{\lambda }_2}$ will be:

The magnetic flux $\varphi$ (in weber) linked with a coil of resistance $10\Omega$ varies with time t (in second) as $\varphi =8t^2-4t+1$. The current induced in the coil at t = 0.1 sec is

The $x-y$ plane is the boundary between two transparent media. Medium 1 with $z\ge 0$ has a refractive index $\sqrt{2}$ and medium 2 with $z\le 0$ has a refractive index $\sqrt{3}$. A ray of light in medium 1 given by the vector $\overrightarrow{A}=6\sqrt{3}\hat{i}+8\sqrt{3}\hat{j}-10\hat{k}$ is incident on the plane of separation. The angle between vector $\overrightarrow{A}$ and the positive $z$ - direction is

In the circuit shown in figure, the ratio of charges on $5\mu F$ and $4\mu F$ capacitor is -

A mass of $5\text{kg}$ is suspended from a spring of stiffness constant $4000\text{N/m}$. The system is fitted with a damper with a damping coefficient of $0.2$. The mass is pulled down by $50\text{mm}$ and then released. Calculate the damped frequency.

Given X-ray spectrum is for a Coolidge tube having accelerating potential V. If accelerating potential is decreased to V/4, then $\Delta \lambda =(\lambda -{\lambda }_c)$ becomes four times with changes in anode element atomic no. from (neglect screening effect).

In the circuit given, the value of y is [consider $G_1,G_{2}and{G}_{3}asNANDgate$]

Two men who can swim with a speed $v_1$ in still water start from the middle of a river of width d and move in opposite directions always swimming at an angle $\theta$ with the banks. What is the distance between them along the river when they reach the opposite banks, if the velocity of the river is $v_2$

Find the force of interaction between $2kg$ and $1kg$ bodies for the figure shown. (Take $g=10m/s^2$)

A bar magnet of length $l$ and magnetic dipole moment $M$ is bent in the form of an arc as shown in figure. The new magnetic dipole moment will be

A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The mass is undergoing circular motion in the x - y plane with centre at O and constant angular speed $\omega$ . If the angular momentum of the system, calculated about O and P are denoted by $\overrightarrow{L_O}$ and $\overrightarrow{L_P}$ respectively, then