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IIT JAM 2015

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Brijmohan
Banaras Hindu University (BHU), Varanasi
PhD Condensed matter Physics
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fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES JAM JOINT ADMISSION TEST FOR MSc 2015 SECTION A: MCQ Q1. A proton from outer space is moving towards earth with velocity 0.99 c as measured in earth s frame. A spaceship, traveling parallel to the proton, measures proton s velocity to be 0.97 c . The approximate velocity of the spaceship in the earth s frame, is (a) 0.2 c Q2. (b) 0.3 c (c) 0.4 c Temperature dependence of resistivity of a metal can be described by (a) (b) R R T T (c) R (d) R T Q3. (d) 0.5 c T The trace of a 2 2 matrix is 4 and its determinant is 8 . If one of the eigenvalues is 2 1 i , the other eigenvalue is (a) 2 1 i Q4. (b) 2 1 i (c) 1 2i (d) 1 2i At room temperature, the speed of sound in air is 340 m/sec. An organ pipe with both ends open has a length L 29 cm . An extra hole is created at the position L / 2 . The lowest frequency of sound produced is (a) 293 Hz (b) 586 Hz (c) 1172 Hz Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com (d) 2344 Hz Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 1 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q5. The electric field of a light wave is given by E E0 i sin t kz j sin t kz . 4 The polarization state of the wave is (a) Left handed circular (c) Left handed elliptical Q6. (b) Right handed circular (d) Right handed elliptical A particle with energy E is incident on a potential given by 0, V x V0 , x 0 x 0 . The wave function of the particle for E V0 in the region x 0 (in terms of positive constants A, B and k ) is (a) Aekx Be kx Q7. (b) Ae kx (c) Aeikx Be ikx (d) Zero A mass m , lying on a horizontal, frictionless surface, is connected to one end of a spring. The other end of the spring is connected to a wall, as shown in the figure. At t 0 , the mass is given an impulse. m Impulse The time dependence of the displacement and the velocity of the mass (in terms of nonzero constants A and B ) are given by (a) x t A sin t , v t B cos t (c) x t A cos t , v t B sin t Q8. (b) x t A sin t , v t B sin t (d) x t A cos t , v t B cos t Consider the coordinate transformation x x y 2 , y x y 2 . The relation between the area elements dx dy and dxdy is given by dx dy jdxdy . The value of j is (a) 2 (b) 1 (c) 1 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com (d) 2 Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 2 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q9. A charge q is at the center of two concentric spheres. The outward electric flux through the inner sphere is while that through the outer sphere is 2 . The amount of charge contained in the region between the two spheres is (a) 2q Q10. (c) q (b) q (d) 2q A system consists of N number of particles, N 1 . Each particle can have only one of the two energies E1 or E1 0 . If the system is in equilibrium at a temperature T , the average number of particles with energy E1 is (a) Q11. N 2 N (b) e / kT 1 (c) N e / kT 1 (d) Ne / kT Vibrations of diatomic molecules can be represented as those of harmonic oscillators. Two halogen molecules X2 and Y2 have fundamental vibrational frequencies v X 16.7 1012 Hz and vY 26.8 1012 Hz , respectively. The respective force constants are K X 325 N / m and K Y 446 N / m . The atomic masses of F, Cl and Br are 19.0, 35.5 and 79.9 atomic mass unit respectively. The halogen molecules X 2 and Y2 are (a) X 2 F2 and Y2 Cl2 (c) X 2 Br2 and Y2 F2 Q12. (b) X 2 Cl 2 and Y2 F2 (d) X 2 F2 and Y2 Br2 A rigid triangular molecule consists of three non-collinear atoms joined by rigid rods. The constant pressure molar specific heat C p of an ideal gas consisting of such molecules is (a) 6R Q13. (b) 5R (c) 4R (d) 3R A system comprises of three electrons. There are three single particle energy levels accessible to each of these electrons. The number of possible configurations for this system is (a) 1 (b) 3 (c) 6 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com (d) 7 Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 3 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q14. The variation of binding energy per nucleon with respect to the mass number of nuclei is shown in the figure. Average binding energy per nucleon (MeV) 9 8 7 6 5 4 3 2 1 0 20 40 60 80 100 120 140 160 180 200 220 240 Number of nucleons in nucleus, A Consider the following reactions: (i) 238 92 206 U 82 Pb 10 P 22n (ii) 238 92 U Pb 8 24He 6e 206 82 Which one of the following statements is true for the given decay modes of 238 92 U? (a) Both (i) and (ii) are allowed (c) (i) is forbidden and (ii) is allowed Q15. (b) Both (i) and (ii) are forbidden (d) (i) is allowed and (ii) is forbidden A Zener regulator has an input voltage in the range 15V 20V and a load current in the range of 5 mA 20 mA . If the Zener voltage is 6.8V , the value of the series resistor RS should be V0 15 20V (a) 390 (b) 420 6 .8 V (c) 440 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com (d) 460 Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 4 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q16. A hollow, conducting spherical shell of inner radius R1 and outer radius R 2 encloses a charge q inside, which is located at a R1 distance d R1 from the centre of the spheres. The potential at q d the centre of the shell is (b) (c) Q17. 1 q q 4 0 d R1 1 q 4 0 d (d) (a) Zero R2 1 q q q 4 0 d R2 R2 A satellite moves around the earth in a circular orbit of radius R centered at the earth. A second satellite moves in an elliptic orbit of major axis 8 R , with the earth at one of the foci. If the former takes 1 day to complete a revolution, the latter would take (a) 21.6 days Q18. (b) 8 days (c) 3 hours (d) 1.1 hour A positively charged particle, with a charge q , enters a region in which there is a uniform electric field E and a uniform magnetic field B , both directed parallel to the positive y -axis. At t 0 , the particle is at the origin and has a speed v 0 directed along the positive x - axis. The orbit of the particle, projected on the x- z plane, is a circle. Let T be the time taken to complete one revolution of this circle. The y -coordinate of the particle at t T is given by (a) Q19. 2 mE 2qB 2 (b) 2 2 mE qB 2 (c) 2 mE qB 2 v0 m qB (d) 2 mv 0 qB A rigid and thermally isolated tank is divided into two compartments of equal volume V , separated by a thin membrane. One compartment contains one mole of an ideal gas A and the other compartment contains one mole of a different ideal gas B . The two gases are in thermal equilibrium at a temperature T . If the membrane ruptures, the two gases mix. Assume that the gases are chemically inert. The change in the total entropy of the gases on mixing is (a) 0 (b) R ln 2 (c) 3 R ln 2 2 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com (d) 2R ln 2 Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 5 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q20. Doppler effect can be used to measure the speed of blood through vessels. Sound of frequency 1.0522 MHz is sent through the vessels along the direction of blood flow. The reflected sound generates a beat signal of frequency 100 Hz. The speed of sound in blood is 1545 m/sec. The speed of blood through the vessel, in m/sec, is (a) 14.68 Q21. (b) 1.468 (c) 0.1468 (d) 0.01468 A conducting wire is in the shape of a regular hexagon, which is I inscribed inside an imaginary circle of radius R , as shown. A current I flows through the wire The magnitude of the magnetic field at the R C center of the circle is (a) Q22. 3 0 I 2 R (b) 0 I 2 3 R (c) 3 0 I R (d) 3 0 I 2 R . Let C be the circle x 2 y 2 4 on the xz 3 j zyk Consider a vector field F yi plane z 2 , oriented counter-clockwise. The value of the contour integral F d r is C (a) 28 Q23. (b) 4 (c) 4 (d) 28 An observer is located on a horizontal, circular turntable which rotates about a vertical axis passing through its center, with a uniform angular speed of 2 rad/sec . A mass of 10 grams is sliding without friction on the turntable. At an instant when the mass is at a distance of 8 cm from the axis it is observed to move towards the center with a speed of 6 cm/sec. The net force on the mass, as seen by the observer at that instant, is (a) 0.0024 N Q24. (b) 0.0032N (c) 0.004N (d) 0.006N Miller indicates of a plane in cubic structure that contains all the directions 100 , 011 and 111 are (a) 011 (b) 101 (c) 100 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com (d) 110 Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 6 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q25. The Fourier series for an arbitrary periodic function with period 2L , is given by f x a0 n x n x n 1 a n cos n 1 bn sin . For the particular periodic function 2 L L shown in the figure the value of a 0 is f x 1 1/ 2 2 (a) 0 Q26. (b) 0.5 1 0 1 (c) 1 2 x (d) 2 Seven uniform disks, each of mass m and radius r , are inscribed inside a regular hexagon as shown. The moment of inertia of this system of seven disks, about an axis passing through the central disk and perpendicular to the plane of the disks, is (a) (c) Q27. 7 2 mr 2 13 mr2 2 (b) 7mr 2 (d) 55 mr2 2 A nucleus has a size of 10 15 m . Consider an electron bound within a nucleus. The estimated energy of this electron is of the order of (a) 1 MeV Q28. (b) 10 2 MeV Consider the equation (c) 10 4 MeV (d) 10 6 MeV dy y 2 with the boundary condition y 1 1 . Out of the dx x following the range of x in which y is real and finite is (a) x 3 (c) 0 x 3 Q29. (b) 3 x 0 (d) 3 x The phase of the complex number 1 i i in the polar representation is (a) 4 (b) 2 (c) 3 4 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com (d) 5 4 Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 7 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q30. Which of the following circuits represent the Boolean expression S P QR Q P (a) P Q (b) P Q S (c) P Q S (d) P Q S R S R SECTION B: MSQ Q1. 4 j units. The edges A unit cube made of a dielectric material has a polarization P 3i of the cube are parallel to the Cartesian axes. Which of the following statements are true? (a) The cube carries a volume bound charge of magnitude 5 units (b) There is a charge of magnitude 3 units on both the surfaces parallel to the y z plane (c) There is a charge of magnitude 4 units on both the surfaces parallel to the x z plane (d) There is a net non-zero induced charge on the cube Q2. Muons are elementary particles produced in the upper atmosphere. They have a life time of 2.2 s . Consider muons which are traveling vertically towards the earth s surface at a speed of 0.998c . For an observer on earth, the height of the atmosphere above the surface of the earth is 10.4 km . Which of the following statements are true? (a) The muons can never reach earth s surface (b) The apparent thickness of earth s atmosphere in muon s frame of reference is 0.96 km (c) The lifetime of muons in earth s frame of reference is 34.8 s (d) Muons traveling at a speed greater than 0.998 c reach the earth s surface Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 8 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q3. A rod is hanging vertically from a pivot. A partic1e traveling in horizontal direction, collides with the rod as shown in the figure. For the rod-particle system, consider the linear momentum and the angular momentum about the pivot .Which of the following statements are NOT true? (a) Both linear momentum and angular momentum are conserved (b) Linear momentum is conserved but angular momentum is not (c) Linear momentum is not conserved but angular momentum is conserved (d) Neither linear momentum nor annular momentum are conserved Q4. The following figure shows a double slit Fraunhofer diffraction pattern produced by two slits, each of width a separated by a distance b, a b . Secondarymaxima Primarymaxima Which of the following statements are correct? (a) Reducing a increases the separation between consecutive primary maxima (b) Reducing a increases the separation between consecutive secondary maxima (c) Reducing b increases the separation between consecutive primary maxima (d) Reducing b increases the separation between consecutive secondary maxima Q5. For an electromagnetic wave traveling in free space, the electric field is given by V E 100 cos 108 t kx j . Which of the following statements are true? m (a) The wavelength of the wave in meter is 6 (b) The corresponding magnetic field is directed along the positive z direction (c) The Poynting vector is directed along the positive z direction (d)The wave is linearly polarized Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 9 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q6. A particle of mass m is moving in x y plane. At any given time t , its position vector is given by r t A cos t i B sin t j where A, B and are constants with A B . Which of the following statements are true? (a) Orbit of the particle is an ellipse (b) Speed of the particle is constant (c) At any given time t the particle experiences a force towards origin (d) The angular momentum of the particle is m AB k Q7. In an ideal Op-Amp circuit shown below R1 3k , R2 1k and Vi 0.5sin t (in Volt). Which of the following statements are true? Vi V0 P R2 R1 (a) The current through R1 The current through R2 R (b) The potential at P is V0 2 R1 (c) The amplitude of V 0 is 2V (d) The output voltage V 0 is in phase with V i Q8. A particle is moving in a two dimensional potential well V x, y 0, 0 x L, 0 y 2 L , elsewhere which of the following statements about the ground state energy E1 and ground state eigenfunction 0 are true? (a) E1 (c) 0 2 2 mL2 (b) E1 2 x y sin sin L L 2L 5 2 2 8mL2 (d) 0 2 x y cos cos L L 2L Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 10 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q9. Consider the circuit, consisting of an AC function generator V t V0 sin 2 vt with V0 5V an inductor L 8.0mH , resistor R 5 and a capacitor C 100 F . Which of the following statements are true if we vary the frequency? L C R (a) The current in the circuit would be maximum at 178Hz (b) The capacitive reactance increases with frequency (c) At resonance, the impedance of the circuit is equal to the resistance in the circuit (d) At resonance, the current in the circuit is out of phase with the source voltage Q10. As shown in the P V diagram AB and CD are two isotherms at temperatures T1 and T 2 , respectively T1 T2 . AC and BD are two reversible adiabats. In this Carnot cycle, which of the following statements are true? P Q Q (a) 1 2 T1 T2 Q1 A B T1 (b) The entropy of the source decreases (c) The entropy of the system increases (d) Work done by the system W Q1 Q2 C Q2 D T2 V SECTION C: NAT Q1. In an experiment on charging of an initially uncharged capacitor, an RC circuit is made with the resistance R 10k and the capacitor C 1000 F along with a voltage source of 6V . The magnitude of the displacement current through the capacitor (in A ), 5 seconds after the charging has started, is _______________ Q2. The power radiated by sun is 3.8 10 26 W and its radius is 7 10 5 km . The magnitude of the Poynting vector (in W ) at the surface of the sun is _________________ cm 2 Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 11 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q3. Unpolarized light is incident on a calcite plate at an angle of incidence 50 o as shown in the figure. Take n0 1.6584 and ne 1.4864 for calcite. The angular separation ( in degrees) between the two emerging rays within the plate is Air 500 Optic axis Q4. Calcite X rays of wavelength 0.24 nm are Compton scattered and the scattered beam is observed at an angle of 60 o relative to the incident beam. The Compton wavelength of the electron is 0.00243nm . The kinetic energy of scattered elections in eV is___________ Q5. GaAs has a diamond structure. The number of Ga-As bonds per atom which have to be broken to fracture the crystal in the 001 plane is _______ Q6. In the hydrogen atom spectrum. the ratio of the longest wavelength in the Lyman series (final state n 1) to that in the Balmer series (final State n 2 ) is ____________ Q7. A rod is moving with a speed of 0.8c in a direction at 60 o to its own length. The percentage contraction in the length of the rod is __________ Q8. In the given circuit VCC 10V and 100 for n p n transistor. The collector voltage VCC VC (in volts) is __________. 1K 100 K VC 5V Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 12 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q9. A particle is in a state which is a superposition of the ground state 0 and the first excited state 1 of a one-dimensional quantum harmonic oscillator. The state is given by 1 5 0 2 5 1 . The expectation value of the energy of the particle in this state (in units of , being the frequency of the oscillator) is ________ Q10. A diode at room temperature kT 0.025 eV with a current of 1 A has a forward bias voltage VF 0.4V . For VF 0.5V , the value of the diode current (in A ) is _________ Q11. A uniform disk of mass m and radius R rolls, without slipping, down a fixed plane inclined at an angle 30 o to the horizontal. The linear acceleration of the disk (in m / sec2 ) is _____________. Q12. In a region of space, a time dependent magnetic field B t 0.4t Tesla points vertically upwards. Consider a horizontal, circular loop of radius 2 cm in this region. The magnitude of the electric field (in mV / m ) induced in the loop is ___________. Q13. For the arrangement given in the following figure, the coherent light sources A, B and C have individual intensities of 2 mW / m 2, 2 mW / m 2 and 5 mW / m2 respectively at point P . The wavelength of each of the sources is 600 nm . The resultant intensity at point P P (in mW / m2 ) is ___________. 15 mm A 3.22 mm B 2.04 mm Q14. 1m C A nozzle is in the shape of a truncated cone, as shown in the figure. The area at the wide end is 25 cm 2 and the narrow end has an area of 1 cm 2 . Water enters the wider end at a rate of 500 gm / sec . The 50 cm height of the nozzle is 50 cm and it is kept vertical with the wider end at the bottom. The magnitude of the pressure difference in kPa ( 1 kPa 10 3 N / m 2 ) between the two ends of the nozzle is __________ Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 13 fiziks Institute for NET/JRF, GATE, IIT-JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES Q15. A plane electromagnetic wave of frequency 5 1014 Hz and amplitude 103 V / m traveling in a homogeneous dielectric medium of dielectric constant 1.69, is incident normally at the interface with a second dielectric medium of dielectric constant 2.25. The ratio of the amplitude of the transmitted wave to that of the incident wave is __________. Q16. A block of mass 2 kg is at rest on a horizontal table The coefficient of friction between the block and the table is 0.1. A horizontal force 3 N is applied to the block The speed of the block (in m/s) after it has moved a distance 10 m is ________________. Q17. One gram of ice at 0 o C is melted and heated to water at 39 o C . Assume that the specific heat remains constant over the entire process. The latent heat of fusion of ice is 80 Calories/gm. The entropy change in the process (in Calories per degree) is _________. Q18. In the thermodynamic cycle shown in the figure, one mole of a monatomic ideal gas is taken through a cycle. AB is a reversible isothermal expansion at a temperature of 800K reduced to 300K . CA is a constant volume process in which Pressure in which the volume of the gas is doubled. BC is an isobaric the pressure and temperature return to their initial values. The P1 net amount of heat (in Joules) absorbed by the gas in one P2 contraction to the original volume in which the temperature is A C B complete cycle is _____________ Q19. V 2V Volume Consider a 20 m diameter p n junction fabricated in silicon. The donor density is 1016 per cm3 . The charge developed on the n side is 1.6 10 13 C . Then the width (in m ) of the depletion region on the n side of the p n junction is _________. Q20. A homogeneous semi-circular plate of radius R 3m is shown in the figure. The distance of the center of mass of the p1ate (in meter) from the point O is _______. 0 3m Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi-16 Phone: 011-26865455/+91-9871145498 Website: www.physicsbyfiziks.com Email: fiziks.physics@gmail.com Branch office Anand Institute of Mathematics, 28-B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi-16 14

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