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Let f(x, y) = x^{3} − 2y^{3} . The curve along which ∇^{2}f = 0 is
A curve is given by The unit vector of the tangent to the curve at t =1 is
There are three planets in circular orbits around a star at distances a, 4a and 9a, respectively. Attime t = t_{0}, the star and the three planets are in a straight line. The period of revolution of theclosest planet is T. How long after t_{0} will they again be in the same straight line?
A current I is flowing through the sides of an equilateral triangle of side a. The magnitude of the magnetic field at the centroid of the triangle is
Two vehicles A and B are approaching an observer O at rest with equal speed as shown in the figure. Both vehicles have identical sirens blowing at a frequency fs . The observer hears these sirens at frequency f_{A} and f_{B}, respectively from the two vehicles. Which one of the following is correct?
Three infinite plane sheets carrying uniform charge densities −σ, 2σ, 3σ are placed parallel to the
xz plane at y = a, 3a, 4a , respectively. The electric field at the point (0, 2a, 0) is
Two boxes A and B contain an equal number of molecules of the same gas. If the volumes are V_{A}
and V_{B}, and λ_{A} and λ_{B} denote respective mean free paths, then
Let T_{g} and T_{e} be the kinetic energies of the electron in the ground and the third excited states of ahydrogen atom, respectively. According to the Bohr model, the ratio T_{g}/ T_{e }is
Which one of the following arrangements of optical components can be used to distinguish between
an unpolarised light and a circularly polarised light?
Which one of the following graphs shows the correct variation of v_{o} with v_{i}? Here, v_{d} is the voltage
drop across the diode and the OpAmp is assumed to be ideal.
Which one of the figures correctly represents the TS diagram of a Carnot engine?
The plane of polarisation of a plane polarized light rotates by 60^{° }after passing through a waveplate. The passaxis of the wave plate is at an angle α with respect to the plane of polarisation of theincident light. The wave plate and α are
A rectangular loop of dimensions l and w moves with a constant speed of v through a region containing a uniform magnetic field B directed into the paper and extending a distance of 4w. Which of the following figures correctly represents the variation of emf (ε) with the position (x) of the front end of the loop?
The equation of state for one mole of a nonideal gas is given by PV = A where the coefficients A and B are temperature dependent. If the volume changes from V_{1} to V_{2} in an isothermal process, the work done by the gas is
An ideal gas consists of three dimensional polyatomic molecules. The temperature is such that only
one vibrational mode is excited. If R denotes the gas constant, then the specific heat at constant
volume of one mole of the gas at this temperature is
A long solenoid is carrying a time dependent current such that the magnetic field inside has the form is along the axis of the solenoid. The displacement current at the
point P on a circle of radius r in a plane perpendicular to the axis
Consider an ensemble of thermodynamic systems, each of which is characterized by the samenumber of particles, pressure and temperature. The thermodynamic function describing theensemble is
Given a spherically symmetric charge density ρ(r) (k being a constant), the electric field for r < R is (take the total charge as Q)
An infinitely long solenoid, with its axis along , carries a current I. In addition there is a uniform line charge density λ along the axis. If is the energy flux, in cylindrical coordinates then
Consider two waves y_{1} = a cos(ωt − kz) and y_{2} = a cos[(ω + Δω)t − (k + Δk)z]. The group velocity of the superposed wave will be (Δω ≪ ω and Δk ≪ k)
Consider a convex lens of focal length f. A point object moves towards the lens along its axis between 2f and f. If the speed of the object is V_{O}, then its image would move with speed V_{I}. Whichof the following is correct?
A disc of radius R_{1} having uniform surface density has a concentric hole of radius R_{2} < R_{1}. If its
mass is M, the principal moments of inertia are
The function is expanded as a Fourier series of the form
Which one of the following curves correctly represents (schematically) the solution for the equation
The mean momentum of a nucleon in a nucleus of mass number A and atomic number Z depends on A, Z as
Consider the transformation to a new set of coordinatesfrom rectangular Cartesian coordinates (x, y), where In the coordinate system, the area element dxdy is
A particle of mass m is in a one dimensional potential
At some instant its wave function is given by
are the ground and the first excited states, respectively. Identify the correct statement.
A raindrop falls under gravity and captures water molecules from atmosphere. Its mass changes atthe rate λm(t), where λ is a positive constant and m(t) is the instantaneous mass. Assume thatacceleration due to gravity is constant and water molecules are at rest with respect to earth beforecapture. Which of the following statements is correct?
A particle P of mass m is constrained to move on the surface of a cylinder under a force as
shown in figure (k is the positive constant). Which of the following statements is correct? (Neglect
friction.)
Let f(x) = 3x^{6} − 2x^{2} − 8. Which of the following statements is (are) true?
Two projectiles of identical mass are projected from the ground with same initial angle (α) with respect to earth surface and same initial velocity (u) in the same plane. They collide at the highest point of their trajectories and stick to each other. Which of the following statements is (are) correct?
Two beams of light in the visible range (400 nm  700 nm) interfere with each other at a point. The optical path difference between them is 5000 nm. Which of the following wavelengths will interfere constructively at the given point?
Which of the following relations is (are) true for thermodynamic variables?
Consider a convex lens of focal length f. The lens is cut along a diameter in two parts. The two lens parts and an object are kept as shown in the figure. The images are formed at following distances from the object:
Let the electric field in some region R be given by From this we may conclude that
In presence of a magnetic field and an electric field and an electric field a particle moves undeflected. Which of the following statements is (are) correct?
In a pn junction, dopant concentration on the p side is higher than that on the nside. Which of the following statements is (are) correct, when the junction is unbiased?
Which of the combinations of crystal structure and their coordination number is (are) correct?
The coefficient of x^{3} in the Taylor expansion of sin(sin x) around x = 0 is ______. (Specify your answer upto two digits after the decimal point.)
A particle of mass m is moving along the positive x direction under a potential (k and λ are positive constants). If the particle is slightly displaced from its equilibrium position, it oscillates with an angular frequency (ω) __________.
(Specify your answer in units of as an integer.)
A planet has average density same as that of the earth but it has only 1/8 of the mass of the earth. If the acceleration due to gravity at the surface is g_{p} and g_{e} for the planet and earth, respectively, then
In a grating with grating constant d =a + b, where a is the slit width and b is the separation between the slits, the diffraction pattern has the fourth order missing. The value of b/a is ________.
(Specify your answer as an integer.)
Consider an electromagnetic plane wave where λ is the wavelength, c is the speed of light and b is a constant. The value of b is ________.
(Specify your answer upto two digits after the decimal point.)
Consider a monoatomic ideal gas operating in a closed cycle as shown in the PV diagram given below. The ratio
(Specify your answer upto two digits after the decimal point.)
Consider the first order phase transition of the sublimation of zinc. Assume the vapor to be an ideal gas and the molar volume of solid to be negligible. Experimentally, it is found that log_{10} (p) =where P is the vapor pressure in Pascal, T is in K, C_{1} = 6790 K and C_{2} = 9. The latent heat of sublimation of zinc from the Clausius  Clapeyron equation is __________ kJ/mole. (R = 8.314 J/mole.K)
(Specify your answer in kJ/mole upto one digit after the decimal point.)
A system of 8 noninteracting electrons is confined by a three dimensional potential The ground state energy of the system in units of ℏ is _______________.
(Specify your answer as an integer.)
For the given circuit, value of the base current (I_{b}) of the npn transistor will be _______ mA. (β is the current gain and assume OpAmp as ideal.)
(Specify your answer in mA upto two digits after the decimal point.)
The lattice constant of unit cell of NaCl crystal is 0.563 nm. Xrays of wavelength 0.141 nm are diffracted by this crystal. The angle at which the first order maximum occurs is ________degrees. (Specify your answer in degrees upto two digits after the decimal point.)
For the following circuit, the collector voltage with respect to ground will be ________V. (Emitter diode voltage is 0.7 V and β_{DC} of the transistor is large.) (Specify your answer in volts upto one digit after the decimal point.)
A body of mass 1 kg is moving under a central force in an elliptic orbit with semi major axis 1000 m and semi minor axis 100 m. The orbital angular momentum of the body is 100 kg m^{2} s^{1}. The time period of motion of the body is __________ hours.
(Specify your answer in hours upto two digits after the decimal point.)
The moon moves around the earth in a circular orbit with a period of 27 days. The radius of the earth (R) is 6.4 × 10^{6} m and the acceleration due to gravity on the earth surface is 9.8 ms^{2}. If D is the distance of the moon from the center of the earth, the value of D/R will be _____.
(Specify your answer upto one digit after the decimal point.)
A syringe is used to exert 1.5 atmospheric pressure to release water horizontally. The speed of water immediately after ejection is ________. (take 1 atmospheric pressure = 10^{5} Pascal, density of water = 10^{3} kg m^{3}) (Specify your answer in ms^{1} as an integer.)
Consider a slit of width 18 µm which is being illuminated simultaneously with light of orange color (wavelength 600 nm) and of blue color (wavelength 450 nm). The diffraction pattern is observed on a screen kept at a distance in front of the slit. The smallest angle at which only the orange color is observed is θ_{1}, and the smallest angle at which only the blue color is observed is θ_{2}. The angular difference θ_{2} − θ_{1} (in degrees) is _____. (Specify your answers upto two digits after the decimal point.)
A particle of mass m is moving in a circular orbit given by x = R cos(ωt); y = R sin(ωt), as observed in an inertial frame S_{1}. Another inertial frame S_{2} moves with uniform velocity with respect to S_{1}. S_{1} and S_{2} are related by Galilean transformation, such that the origins coincide at t = 0. The magnitude of the angular momentum of the particle at as observed in S_{2} about its origin, is expressed as (mR^{2} ω)x. Then x is _______. (Specify your answer upto two digits after the decimal point.)
Rod R_{1} has a rest length 1m and rod R_{2} has a rest length of 2m. R_{1} and R_{2} are moving with respect to the laboratory frame with velocities,respectively. If R_{2} has a length of 1m in the rest frame of R_{1} v/c is given by ____________________.
(Specify your answer upto two digits after the decimal point.)
Two events E_{1 }and E_{2} take place in an inertial frame S with respective timespace coordinates (in SI units): and E_{2} (t_{2} = 0, x_{2} = 10^{8} , y_{2} = 0, z_{2} = 0). Another inertial frame S′ is moving with respect to S with a velocity The time difference (t′_{2}− t′_{1} )as observed in S′ is ____s.(c = 3×10^{8} ms^{1})
(Specify your answer in seconds upto two digits after the decimal point.)
In the following circuit, the time constant RC is much greater than the period of the input signal. Assume diode as ideal and resistance R to be large. The dc output voltage across resistance R will be _______V.
(Specify your answer in volts upto one digit after the decimal point.)
For a metal, the electron density is 6.4 × 10^{28} m^{3}. The Fermi energy is ________eV. (h = 6.626 × 10^{34} J s , m_{e} = 9.11 × 10^{31} kg, 1eV = 1.6 × 10^{19} J)
(Specify your answer in electron volts (eV) upto one digit after the decimal point.)
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