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Two blocks A and B of mass m and 2m are connected by a massless spring of force constant k. They are placed on a smooth horizontal plane. Spring is stretched by an amount x and then released. The relative velocity of the blocks when the spring comes to its natural length is

(a) [√(3k/2m)]x (b) [√(2k/3m)]x

(c) √(2kx/m) (d) √(3km/x)

If either

or,
then.
Is the converse true? Justify your answer with an example.

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two unit vectors such that $\overrightarrow{a}$$+2\overrightarrow{b}$ and $5\overrightarrow{a}$$-4\overrightarrow{b}$ are perpendicular to each other, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is

Air is filled at ${60}^{\circ }\text{C}$ in a vessel of open mouth. The vessel is heated to a temperature $T$, so that $\frac{1}{4}\text{th}$ part of the air escapes. Assuming the volume of the vessel remains constant, the value of $T$ is

Consider an ideal op-amp differential amplifier shown in the figure below:



$\text{then the value of}\frac{V_{\text{out}}}{V_{{\text{in}}_2}-V_{{\text{in}}_1}}\text{is equal to}$

A wire carrying a $10\text{A}$ current is bent to pass through various sides of a cube of side $10\text{cm}$ as shown in the figure. A magnetic field $\overrightarrow{B}=(2\hat{i}-3\hat{j}+\hat{k})\text{T}$ is present in the region. Find the net torque on the loop.

A PN- junction has a thickness of the order of ?

In the arrangement shown, neglect the mass of the ropes and pulley. What must be the value of $m$ to keep the system in equilibrium? There is no friction anywhere.

Two equal charges of magnitude Q each are placed at a distance d apart. Their electrostatic energy is E. A third charge $-\frac{Q}{2}$ is brought midway between these two charges. The electrostatic energy of the system is now

Two Carnot engines $A$ and $B$ are operated in series. The first engine $A$, receives heat at $800\text{K}$and rejects to a reservoir at temperature $T$. The second engine $B$ receives the heat rejected by the first engine and in turn rejects to a heat reservoir at $300\text{K}$. Find the value of $T$, when the work outputs of the engines are equal.

A 12 HP motor has to be operated 8 hours/day. How much will it cost at the rate of 50 paisa/kWh in 10 days

At a place the speed of sound of frequency 300Hz is V. At the same place the speed of 150 Hz sound will be

Two simple harmonic motions are represented by the equations $y_1=0.1sin(100\pi t+\frac{\pi }{3})$ and $y_2=0.1cos\pi t$The phase difference of the velocity of particle 1 with respect to the velocity of particle 2 at t= 0 is

Four particles, each of mass m, are placed at the corners of a square of side a in the x-y plane. If the origin of the coordinate system is taken at the point of intersection of the diagonals of the square, the coordinates of the centre of mass of the system are

A parallel plate capacitor has area $6{\text{cm}}^2$ and separation $3\text{mm}$. The gap between the plates is filled with three dielectric materials of equal thickness as shown. The equivalent dielectric constant of a material which would give the same capacitance when fully inserted in the capacitor, would be

A person with his hands in his pockets is skating on ice at the velocity of 10 m/s and describes a circle of radius 50 m. What is his inclination with vertical

The displacement x of a particle varies with time according to the relation $x=\frac{a}{b}(1-e^{-bt})$. Which of the following statements is correct?

A T.V. transmission tower has height of $80m$. Radius of earth is $6400km$. What is the coverage range of tower?

If a car travelling at $40\text{m/s}$ is to be brought to rest in a distance of $100\text{m}$, then its retardation should be

A cubical block is floating in a liquid with one fourth of its volume immersed in the liquid. If the whole of the system accelerates upwards with acceleration $g/4$, the fraction of volume immersed in the liquid will be

Mass A is connected with a string which passes through a fixed pulley. The other end of the string is connected with a movable pulley N. The block B is connected with another string which passes through the pulley N as shown in figure. Find the relation between acceleration of A & B.

A train of mass ${10}^4\text{kg}$ rounds a curve of radius $100\text{m}$ at a speed of $10\text{m/s}$. At what angle the track be banked in order to have no thrust on the rails? ($g=10{\text{m/s}}^2$)

A ray of light passes through an equilateral prism such that the angle of incidence is equal to $\frac{3}{4}th$ the angle of emergence and the latter is equal to 3/4th of the angle of prism. The angle of deviation is

Find the integral of the given function w.r.t - x

$y=e^{2x}+\frac{1}{x^2}$

A PMC system uses a uniform quantizer followed by an 8 bit encoder. If the bit rate of the system is ${10}^5$ kbps, then the maximum bandwidth of the low-pass message signal for which the system operates satisfactorily is