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The maximum particle velocity is $3$ times the wave velocity of a progressive wave. If the amplitude of the particle is $a$, the phase difference between the two particles separated by a distance of ${}^{\prime }{x}^{\prime }$ is

A ring of radius a and resistance per unit length $\lambda$ slides along two long parallel rails connected by means of a vasistance R at one end. Find the current through the resistance.

An object is placed at $O$ and the image obtained at $O^{\prime }$ as shown below.





If the refractive index of glass slab is $1.5$, then the thickness of the slab is

Two blocks of same mass m are connected to each other by spring of spring constant k. Initially spring is compressed by $x_0$ and blocks are connected by string. Whole system is placed on a smooth horizontal table. Now both blocks are given same horizontal velocity v perpendicular to spring and string is cut simultaneously. During subsequent motion of blocks.

Find the speed of blocks when spring is in its natural length position.

The electric field in the region between two parallel plates is shown below. The displacement current in the region is $i_d$. Then -

A plot of potential energy and kinetic energy versus displacement x for a block attached to a spring obeying Hooke's law is

Two beams $A$ and $B$ of plane polarized light with mutually perpendicular planes of polarization are through a polaroid. From the position when the beam $A$ has maximum intensity (and beam $B$ has zero intensity), a rotation of polaroid through ${30}^{\circ }$ makes the two beams appear equally bright. If the initial intensities is of the two beams are $I_A$ and $I_B$ respectively, then $\frac{I_A}{I_B}$ is equal to,

In the circuit shown, a four-wire potentiometer is made of a $400\text{cm}$ long wire, which extends between $A$ and $B$. The resistance per unit length of the potentiometer wire is $r=0.01\Omega /\text{cm}$. If an ideal voltmeter is connected as shown with jockey $J$ at $50\text{cm}$ from end $A$, the expected reading of the voltmeter will be:

Two prism $\text{A}$ and $\text{B}$ in contact with each other. The dispersive power of $\text{A}$ and $\text{B}$ is $0.04$ and $0.03$ respectively. The angular dispersion produced by $\text{A}$ is $8^{\circ }.$ The angular dispersion produced by the combination, if the combination does not produce a net deviation is-

The distance between charges $5\times {10}^{-11}C$ is 0.2m. The distance at which a third charge should be placed in order that it will not experience any force along the line joining the two charges is (the distance from smaller charge)

A $30\text{kg}$ block rests on a rough horizontal surface. A horizontal force of $200\text{N}$ is applied on the body. The block acquires a speed of $4\text{m/sec}$, starting from rest, in $2\text{sec}$. What is the value of coefficient of friction?

Let G = (V,E) be a weighted undirected graph and let T be a Minimum Spanning Tree (MST) of G maintained using adjacency lists. Suppose a new weighted edge ( u, v) $\in$ V x V is added to G. The worst case time complexity of determining if T is still an MST of the resultant graph is

A charge Q is placed at the centre of uncharged, hollow metallic sphere of radius a. (a) Find surface charge density on the inner surface and on the outer surface. (b) If a charge q is put on the sphere, what would be the surface charge densities on the inner and the outer surfaces ? (c) Find the electric field inside the sphere at a distance x form the centre in the situations (a) and (b).

Three resistances are connected as shown in diagram through the resistance 5 $\Omega$, a current of 1 A is flowing :

(i) What is the current through the other two resistors?



(ii) What is the potential difference (p.d.) across AB and across AC ?



(iii) What is the total resistance ? [5 MARKS]

A chain of mass $m$ and length $l$ is placed on a table with one-sixth of it hanging freely from the table edge. The amount of work done to pull the chain on to the table is

A uniform rod of length $10\text{m}$ is placed along $x$ axis as shown in figure. The distance of the centre of mass of the rod from the origin is

Match the column I with column II :
$$\begin{array}{cc}column1& column2\\ a)Whennonconservativeforcesareacting& p)workdonebyforcemaybenegative\\ b)Inthecaseofapendulum,workdonebythetension& q)Networkdoneiszero\\ c)Whenthereisfriction& r)Theonlyforceactingisgravity\\ d)InProjectilemotion& s)Workdonebyforceinaroundtripisnotequaltozero\end{array}$$

Consider a uniform rod of mass $M=4m$ and length $L$ pivoted about its centre. A mass $m$ moving with a velocity $V$ making an angle $\theta =\frac{\pi }{4}$ to the rod's long axis collides with one end of the rod, and sticks to it. The angular speed of the rod-mass system just after the collision is:

A disc is rotating with an angular velocity ${\omega }_o$. A constant retarding torque is applied on it to stop the disc. The angular velocity becomes $\frac{{\omega }_o}{2}$ after $n$ rotations. How many more rotations will it make before coming to rest?

1. 79.562

Two identical springs of spring constant $k$ each are connecetd in series and parallel as shown in figure. A mass $M$ is suspended from them. The ratio of their frequencies of vertical oscillation will be

A ball is dropped from a height of $20\text{m}$. At the same instant, another ball is thrown upward from the ground with a speed of $20\text{m/s}$ along the same line. After what time will the two balls meet?

A gas found to obey the law $p^{2}V$ constant. The initial temperature and volume are $T_{0}and{V}_0$. If the gas expands to a volume $2V_0$, its final temperature becomes