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MH SET 2025: 15 June, Paper II, Physics

  • Date: 15 June 2025
  • Paper code: JUN-32225
  • Series: A
  • Language: Bilingual
  • 32 pages
MH SET 2025 question paper, 15 June 2025, Paper II, Physics, page 1 of 32
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Test Booklet Code & Serial No.
                                                                                                                               A
                                                                             Paper-II
                                                             PHYSICAL SCIENCE
       Signature and Name of Invigilator                                                          Seat No.
       1. (Signature) .........................................                                                      (In figures as in Admit Card)
             (Name) ................................................                      Seat No. ..............................................................
       2. (Signature) .........................................                                                (In words)

             (Name) ................................................                  OMR Sheet No.
       JUN - 32225                                                                                                 (To be filled by the Candidate)
       Time Allowed : 2 Hours]                                                                                  [Maximum Marks : 200
       Number of Pages in this Booklet : 32                                                      Number of Questions in this Booklet : 100
                              Instructions for the Candidates
       1.          Write your Seat No. and OMR Sheet No. in the space provided              1.
                   on the top of this page.
       2.          This paper consists of 100 objective type questions. Each question
                   will carry two marks. All questions of Paper II will be compulsory.      2.
       3.          At the commencement of examination, the question booklet
                   will be given to the student. In the first 5 minutes, you are
                   requested to open the booklet and compulsorily examine it as             3.
                   follows :
                   (i)      To have access to the Question Booklet, tear off the                    (i)
                            paper seal on the edge of this cover page. Do not accept
                            a booklet without sticker-seal or open booklet.
                   (ii)     Tally the number of pages and number of questions in                    (ii)
                            the booklet with the information printed on the cover
                            page. Faulty booklets due to missing pages/questions
                            or questions repeated or not in serial order or any
                            other discrepancy should not be accepted and correct
                            booklet should be obtained from the invigilator within
                            the period of 5 minutes. Afterwards, neither the Question
                            Booklet will be replaced nor any extra time will be                     (iii)
                            given. The same may please be noted.
                   (iii)    After this verification is over, the OMR Sheet Number
                            should be entered on this Test Booklet.                         4.                     (A), (B), (C)   (D)
       4.          Each question has four alternative responses marked (A), (B),
                   (C) and (D). You have to darken the circle as indicated below on
                   the correct response against each item.
                   Example : where (B) is the correct response.                                              (B)

                                A                  C        D                                                        A              C       D
                   Following wrong methods should not be used as they are not                                                                 (Digitized)
                   recognised by scanning machine in digitized assessment.
                   Candidate using such method will be responsible for their loss.

CORRECT METHOD                          WRONG METHODS
                                                                           CORRECT METHOD                                WRONG METHODS
        C      D        A B C D        A B C D         A B C D      A B C D
                                                                                   C D                      A B C D      A B C D    A B C D      A B C D
       5.          Your responses to the items are to be indicated in the OMR
                   Sheet given inside the Booklet only. If you mark at any place            5.
                   other than in the circle in the OMR Sheet, it will not be evaluated.
       6.          Read instructions given inside carefully.                                6.
       7.          Rough Work is to be done at the end of this booklet.                     7.
       8.          If you write your Name, Seat Number, Phone Number or put
                   any mark on any part of the OMR Sheet, except for the space              8.
                   allotted for the relevant entries, which may disclose your
                   identity, or use abusive language or employ any other unfair
                   means, you will render yourself liable to disqualification.
       9.          You have to return original OMR Sheet to the invigilator at the
                   end of the examination compulsorily and must not carry it with           9.
                   you outside the Examination Hall. You are, however, allowed
                   to carry the Test Booklet and duplicate copy of OMR Sheet on
                   conclusion of examination.                                               10.
       10.         Use only Blue/Black Ball point pen.
       11.         Use of any calculator or log table, etc., is prohibited.                 11.
       12.         There is no negative marking for incorrect answers.                      12.
MH SET 2025 question paper, 15 June 2025, Paper II, Physics, page 2 of 32
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JUN - 32225/II—A




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MH SET 2025 question paper, 15 June 2025, Paper II, Physics, page 3 of 32
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JUN - 32225/II—A
                                  Physical Science
                                        Paper II
Time Allowed : 120 Minutes]                          [Maximum Marks : 200
Note : This paper contains Hundred (100) multiple choice questions. Each question
       carrying Two (2) marks. Attempt All questions.
                                                                              
1.   The    value    of the       line integral    I =  A . dr     where       r  xiˆ  yjˆ   and
                                                 2
     A  ( x  y)iˆ  ( y  x) ˆj along the path y = x from the point (1, 1) to the point
     (4, 2) is :
     (A) 10.33                                    (B)   11.33

     (C) 12.33                                    (D)   13.33

                                                         e1/ z
2.   It is given that the residue of the complex function n at an isolated singular
                                                          z
                      1
     point z = 0 is      . The value of n must be equal to :
                      9!
     (A) n = 1                                    (B)   n = –9
     (C) n = –1                                   (D)   n = –8
3.   A set V of complex numbers forms a two-dimensional real vector space under
     the usual addition of complex numbers and multiplication by real numbers.
     Let T : V  V be a linear transformation defined as T(z) = z . Eigenvalues
     of T are :
     (A) 1, 1                                     (B)   0, 0
     (C) 2, –2                                    (D)   1, –1
4.   The function f(x) = exp(2 x) sin(2 x) can be expanded as the Taylor series :
                3n                                              n
                    n  x                                           n  x
                             n                                           n

     (A)  2 2 sin                              (B)       2 2 sin     
         n0        4  n!                             n 0          4  n!
                3
                    n  x                                          n  x
                         n                                               n

     (C)  2 2 sin                              (D)       2n sin     
         n 0       4  n!                             n 0         4  n!

                                            3
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                                                                       
                                                                            e2 ix sin x
5.   The Cauchy principal value of the integral  dx                                    is :
                                                                       
                                                                                  x
     (A) 2i                                                 (B)    i
     (C) 0                                                   (D)   1
                                                               2
                                                           2s
6.   The inverse Laplace transform of                                is :
                                                        s  5s 2  4
                                                         4


           4           2                                            2           4
     (A)     sin(2 x)  sin( x), x  0                       (B)      sin(2 x)  sin( x), x  0
           3           3                                            3           3

           2           4                                            4           2
     (C)     sin(2 x)  sin( x), x  0                       (D)      sin(2 x)  sin( x), x  0
           3           3                                            3           3

                    
                       2n  1
7.   If (1  x)            Pn ( x) where P (x) are Legendre polynomials and  is usual
                                             n
                   n0   2

     Dirac-delta function, then  (1 + x) can be expressed in terms of Pn(x) as :
             
                2 n  1                                               
                                                                                    2 n  1
     (A)   
           n 0    2
                         Pn ( x)                             (B)     (1)
                                                                    n 0
                                                                                n

                                                                                       2
                                                                                             Pn ( x)

             
                        2n  1                                         
                                                                                    2n  1
     (C)    (1)
           n 0
                    n

                          2
                               Pn ( x)                       (D)     (1)
                                                                    n 0
                                                                                n

                                                                                      2
                                                                                           Pn ( x)

8.   Let X be a random variable. Define the random variable Y = X –                                    X . Let

      X    =  and          ( X  )2   2 . Choose the correct alternative :

     (A)     Y   and (Y  Y )2  2                        (B)     Y  0 and (Y  Y )2  2

     (C)     Y  0 and (Y  Y )2  0                         (D)     Y  0 and (Y  Y )2  2   2

9.   The Wronskian of two solutions of the differential equation,

                                         d 2 y( x)        dy( x) 8
                                   x4         2
                                                    2 x3        x y( x)  0
                                          dx               dx
     is (upto a multiplicative constant) :
     (A) x                                                   (B)   x2
     (C) x3                                                  (D)   x4
                                                        4
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10. Consider the differential operator

                                                  d3       d2      d
                                            La      3
                                                        b    2
                                                                c
                                                  dx       dx      dx

    acting on the space of functions {ui ( x)(1  x  2121satisfying the boundary conditions
    ui(1) = 0 = ui(2). If L = L†, then :

    (A) a = –1, b = 1, c = i                               (B)    a = i, b = 1, c = i

    (C) a = 1, b = 1, c = i                                (D)    a = –1, b = 1, c = 1
11. Let T be any 2 × 2 matrix with eigenvalues –1, 3. For each positive integer
    n, the numbers an and bn are such that
                                        Tn+1 = anT + bnII
    where II is 2 × 2 identity matrix. Then an and bn satisfy the recurrence relations :

    (A) an1  2an  3bn ; bn1  3an  2bn                (B)     an1  2an  bn ; bn1  3an

    (C) an1  3bn ; bn1  3an  bn                       (D)     an1  an  bn ; bn1  3an

12. A general solution of the partial differential equation

                                             2 f ( x, y)  2 f ( x, y)
                                                                       0
                                                x2           y2
    can be expressed in terms of arbitrary functions u and v as :

    (A) f ( x, y)  u( x  y)  v( x  y)                  (B)     f ( x, y)  u( x  iy)  v( x  iy)

    (C) f ( x, y)  u(2 x  y)  v(2 x  y)                (D)     f ( x, y)  u( x  2 y)  v( x  2 y)

13. The order of magnitude of the error in estimating the integral
     1

      ( x  2 x  4 x  9) dx using Simpson's rule with step size 0.1 is :
         3     2

     0

    (A) 0                                                  (B)    10–4
    (C) 10–5                                               (D)    10–3

                                                      5
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14. Consider S' a rotating frame rotating with constant angular velocity  with
    respect to stationary reference frame S (S and S' are having a common origin).
                                                                                  d2 A
    If A is any vector represented in S' frame, then                                   in S frame is given
                                                                                  dt2
    by :

           d2 A      d              dA                           d2 A      d              dA
    (A)                    A  2           A       (B)                   A  2           A
           dt 2 S ' dt S '           dt S '                       dt 2 S ' dt S '           dt S '


           d2 A      d              dA                           d2 A      d              dA
    (C)                    A  2           A       (D)                   A  2           A
           dt 2 S ' dt S '           dt S '                       dt 2 S ' dt S '           dt S '

15. If the potential U in the Lagrangian contains velocity dependent terms, the
    canonical momentum corresponding to a coordinate of rotation  of the entire
    system is no longer the mechanical angular momentum (l ) , but given by .......

    (where  is the gradient operator in velocity space, n̂ is the unit vector along
    the direction of rotation, l c is the canonical angular momentum)

    (A) l c  l  nˆ . r  U                            (B)   l c  l  nˆ . r  U

    (C) l c  l  nˆ . r  U                            (D)   l c  l . nˆ . r  U

16. A particle initially at rest at point A(0, 0) slides down from a frictionless wire
    in a vertical plane to another point B(y0, z0) under the influence of gravity.
    Then according to principle of variation, the minimum time () taken by particle
    can be evaluated using the expression :

                           dz
    (where, z '              )
                           dy


             1        z0     1  z '2                               1        z0    1  z '2
    (A)
             2g
                  z 0
                                  z
                                        dy                  (B)
                                                                    2g
                                                                         z0
                                                                                      z
                                                                                              dy



             1        z0     1  z '2                               1        z0    1  z '2
    (C)
             2g
                  z 0
                                  z'
                                        dy                  (D)
                                                                    2g
                                                                         z0
                                                                                     zz '
                                                                                              dy

                                                        6
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17. A number of degrees of freedom for an astronaut constrained to move on surface
    of a planet (moving freely in space is) :

     (A) 6                                       (B)    5

    (C) 4                                      (D) 2
18. Let, I1, I2, I3 and 1 , 2 , 3 , represent the moment of inertias and angular
    velocities in the respective principle directions. A rigid body which is symmetric
    about an a 3rd-axis and has one point fixed on this axis. If A is the component
    of angular velocity in the direction of the axis of symmetry such that I1 = I2,
    then the angular velocity vector  precesses about the angular momentum
    vector  with a frequency of :
     (Given : Euler's equation for the system are
                                   I1 1  ( I3  I1 )23  0
                                   I1 2  ( I1  I3 )31  0
                                   I3 3  0)
           1 ( I3  I1 (                                 1 ( I2  I3 (
     (A)                 A                       (B)                   A
           2     I1                                     2    I2
           1 ( I1  I3 (                                 1 ( I3  I1 (
     (C)                 A                       (D)                   A
           2     I2                                     2     I1

19. Two masses m1 and m2 are attached to the extreme ends of a massless spring
    of force constant k as shown in figure below. The system is at rest on a horizontal
     surface. The vibrational frequency w of the system is :




             k(m1  m2 )                                    k(m1  m2 )
     (A)       m1m2                              (B)          m1 m2

               m1m2                                           m1 m2
     (C)     k(m1  m2 )                         (D)        k(m1  m2 )
                                            7
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20. A rod AOB (shown in figure below) rotates in a vertical plane [y-z plane] about
    a horizontal axis about through O perpendicular to this plane [x-axis] with
    constant angular velocity  . The equation of motion in r for a particle P with
    mass m and constrain to move along the rod, will be :


    (assuming no frictional forces)




            d2r                                             d2r
    (A) m         m2   mg sin(t)            (B)    m         m2   mg sin(t)
            dt 2                                            dt 2


            d2r                                             d2r
    (C) m         m2  mg sin(t)              (D)    m         m2  mg sin(t)
            dt 2                                            dt 2
21. In a non-relativistic regime, time 't' is considered to be a parameter by changing
    which we change q(t), q (t) and therefore L( q, q , t) as a whole. Now we consider
    't' (time) to be an independent co-ordinate itself, and  is the parameter varying

                              dq                                                       dt 
    which we vary q(), q '()  , t() and therefore L  q(), q '(), t(), t '()    , .
                              d                                                       d 
    If the action (I) remains invariant under this re-parameterisation, then :
                    q'                                        q' 
    (A) L  t ' L  q, , t                     (B)    L  L  q, , t 
                    t'                                        t' 
               t '  q'                                       q'  q' 
    (C) L       L  q, , t                    (D)    L      L  q, , t 
               q'  t '                                       t'  t' 

                                             8
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22. The in order to have following transformation to be canonical

                                  Q  q1/  cos( p)

                                  P  q1/  sin ( p)

    what should be the value of  and  ?
                                                          1
    (A)   1,   1                           (B)       , 2
                                                          2
                        1
    (C)   2,                               (D)        2,   2
                        2

                                                      p2
23. The Hamiltonian of a free particle is H             . The phase space will be :
                                                     2m

    (Where, m, p and E are mass, momentum and energy of the particle, respectively)

    (A) A straight line parallel to q-axis, intersecting the p-axis

    (B) A straight line parallel to p-axis, intersecting the q-axis

    (C) A straight line with slope     2mE

    (D) A straight line with slope – 2mE

24. Which identity of the following does not hold true for Poisson bracket ?

    (Where all the symbols have their usual meaning)

    (A) [ xi , p j ]  ij                     (B)      [ Li , L j ]  ij

    (C) [ Li , p j ] ijk pk                  (D)      [ xi , L j ] ijk xk

25. A planet moving with a velocity of 0.6 c, where c is the velocity of light. An
    astronaut performing an observes a particle moving with a velocity of 0.5 c
    in the same direction as that of the motion of the planet. A stationary observer
    on the earth would observe the particle to have the velocity .......... .

    (A) 0.6 c                                  (B)      0.5 c

    (C) 0.45 c                                 (D)      0.27 c
                                          9
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26. Consider a particle P of mass m moving on XY plane (shown in Fig.). Two forces
    are acting on the particle. F1 acts towards the origin and F2 acts parallel to x-
    axis. The classical equation of motion of the particle under these two forces in
    (r, ) coordinates will be :




     (A) m(r  r 2 )  ( F1  F2 sin ) and m(2r   r
                                                            )  F2 cos 
     (B) m(r  r 2 )  ( F  F sin ) and m(2r  r
                           1    2                           )  F cos 
                                                                 2

     (C) m(r  r 2 )  ( F1  F2 sin ) and m(2r  r
                                                           )  F2 sin 
     (D) m(r  r )  ( F1  F2 cos ) and m(2r  r
                     2
                                                           )  F2 cos 
                                                                                     
                                                                                    r       r / r0
27. An electric charge distribution produces an electric field, E = E0 (1  e     ) 3.
                                                                                    r
    The net charge within a sphere of radius r0 centered at the origin is ...............

                    1                                                   1
     (A) 4  0 E0  1                                (B)    4  0 E0  1  
                       e                                                   e

                   1                                                   1
     (C) 4 0 E0  e                                 (D)    4 0 E0  e  
                      e                                                   e
28. Two infinite planes, both positively charged, with surface charge density 
    each are placed parallel at x = –a and x = a respectively. Which of the following
    statements is correct ?
            
                               
                                  
     (A) E  0 for ( x( a and E  xˆ for( x( a
                                  0
                                                     
                                                               
     (B) E  0 for ( x( a ; E     xˆ for 0 < x < a and E   xˆ for –a < x < 0
                                 0                           0
                             
                                                          
                                                                
     (C) E  0 for ( x( a ; E   xˆ for –a < x < 0 and E   xˆ for 0 < x < a
            

                                  0                             0
                                  
                                                        
                                                              
     (D) E  0 for ( x( a and E   xˆ for x > a and E   xˆ for x < –a
            

                                        0                                    0

                                                  10
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29. A uniform surface current is flowing in the positive y-direction over an infinite
    sheet lying in x-y plane. The direction of the magnetic field is ........

    (A) along x̂ for z > 0 and along – x̂ for z < 0

    (B) along ẑ for z > 0 and along – ẑ for z < 0

    (C) along – x̂ for z > 0 and along x̂ for z < 0

    (D) along – ẑ for z > 0 and along ẑ for z < 0
                                                
30. The induced surface current density K on a conducting sphere, in a uniform
                                                
                                       3  ( B  nˆ )
                                       
    background magnetic field, is K                 . Where n̂ is the outward normal
                                       2  0
    on the surface of the sphere. Now consider a current I flows in a long, straight
    wire placed along the z-axis. A conducting sphere of radius R is placed at a
    distance d from the wire (R << d). The induced surface current density on the
    sphere is proportional to ..........
          I                                               I
    (A)                                             (B)
          R                                               R2
          I                                               I
    (C)                                             (D)
          d                                               d2
31. You cannot separate North pole and South pole of a magnet because if you
    could,......
                                                                  
    (A) it would have violated Maxwell equation,  . B  0 .

                                                                    
                                                                        
                                                                  
                                                                     E
    (B) it would have violated Maxwell equation,  B  0  J  0     .
                                                                    t 
                                                                       
                                                     
                                                           
    (C) it would have violated charge conservation,  . J   0 .
                                                           t
    (D) none of the Maxwell equations or charge conservation would have been
          violated but it is not possible for different reasons.

                                           11
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32. Which of the following electric field and magnetic field represent a propagating
    electromagnetic wave ? ( k̂ is a unit vector in the direction of the propagation).
                                                                 
                                                       E                 ˆ 
    (A) E  E0 (kˆ . r  ct)2 e ( k . r  ct ) , B  kˆ  0 (kˆ . r  ct)2 e ( k . r  ct )
                                                                                              2
                                            ˆ     2


                                                                   c
                                        
    (B) E  E0 sin(kˆ . r  ct), B  0
                                                             
                                                     E                   
    (C) E  E0 sin(kˆ . r )cos(ct), B  kˆ  0 sin( kˆ . r ) cos(ct)
                                                              c
                                                 
                                         E                 ˆ 
    (D) E  (kˆ . r ) e i( kˆ . r  ct ) , B  0 ( kˆ . r ) e i( k . r  ct )
                                                   c
33. For plane monochromatic electromagnetic wave incident from a denser medium
    onto a rarer medium (refractive indices of the media are n1 and n2 respectively),
    the Brewster's angle and the critical angle are denoted by  B and C respectively.
    Which of the following statements is correct ?
                    n1                                              n1
    (A) tan  B                                   (B)   sin C 
                    n2                                              n2
    (C)  B  C                                   (D)    B  C
34. An oscillating magnetic dipole is formed by making sinusoidal current flow
    through a circular loop. Consider this dipole as a perfect dipole. If peak value
    of current flowing is doubled, then time averaged power radiated by this dipole
    becomes ............. times the initial power.
                                                         1
    (A) 4                                          (B)
                                                         4
                                                          1
    (C) 16                                         (D)
                                                         16
35. Let q and a denote the charge and the acceleration of a moving point charge.
    Larmor formula states that total power radiated by the point charge is
    proportional to q a , where ..............
    (A)   1 and   1                            (B)     1 and   2
    (C)   2 and   1                            (D)     2 and   2
                                              12
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36. A relativistic particle of charge q and rest mass m0 moves with velocity
                                                  
    v  vx xˆ  vy ˆy in a uniform magnetic field, B  B0 zˆ . The relativistic Larmor radius
     R (the radius of the circular path the charged particle takes) is given by ..............

               m0 (vx2  vy2 )                              m0 (vx2  vy2 )
     (A) R                                       (B)   R
                     qB0                                          qB0

               m0 (vx2  vy2 )                                2m0 (vx2  vy2 )
     (C) R                                       (D)   R
                    qB0                                           qB0
                                                                               
37. For a localized time varying current distribution, if the Poynting vector, S
                   1
    varies as ( S( n , where r is the distance from the center of the current
                     r
    distribution, then ..........
     (A) n  2 because surface integral of Poynting vector has to be non-zero;
         electromagnetic fields created by this configuration are radiating fields.
     (B) n > 2 because the localized time varying current distribution creates a
          radiating electromagnetic fields.
     (C) n > 2 and the electromagnetic fields are not radiating.
     (D) one cannot say anything about n without more information about the current
          distribution.
38. If in terms of 4-vector potential, all the Maxwell equations can be written as
              1  2 A
      2 A  2     2
                         0 J  , then which of the following statements is true ?
             c t
     (A) It is not true because only inhomogeneous Maxwell equations can be written
          as above.
     (B) In any gauge it is true.

                                                     A
     (C) It is true when Lorentz gauge condition,         0 is satisfied.
                                                     x
                                          
     (D) It is true under the condition,  . A  0 .

                                             13
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39. Consider a rectangular wave guide with dimensions 2.28 cm × 1.01 cm. What
    TE modes will propagate, if the driving frequency is 17 GHz ? And, to excite
    only one TE mode, what range of frequencies could be used ?
     (A) 3.3 GHz   6.6 GHz                             (B)       6.6 GHz    13.2 GHz
     (C) 66 GHz   132 GHz                              (D)       1.62 GHz   6.6 GHz
40. A one-dimensional box contains three identical particles in the ground state
    of the system. Find the ratio of the total energies of these particles if they
    were spin-1/2 fermions, to that if they were bosons.
     (A) 1                                                (B)   14/3
     (C) 2                                                (D)   1/3
                       d      d       
41. The operator           x      x  is equivalent to :
                       dx      dx     
           2
         d        2                                      d2
    (A)        x                                  (B)        x2  1
        dx 2                                            dx2
         d2         d 2                                  d2        d
    (C)     2
               x     x   1                       (D)     2
                                                              2x      x2
        dx         dx                                   dx        dx
42. The normalized wave function of a particle can be written as
                                                                n
                                                      
                                                           1 
                                            ( x)  N         n ( x)
                                                      n0  7 


     where n ( x) are the normalized energy eigenfunctions of a given Hamiltonian.
     The value of N is :
             6                                                        1
     (A)                                                  (B)
             7                                                        7
             3                                                       13
     (C)                                                  (D)
             7                                                        7
43. Consider a spin-1 particle in the state ( Sz , 1 . Let P1x , P0x and Px1 denote the
     probabilities that the measurement of Sx gives the values  , 0 and –  ,
     respectively. Then :
     (A) P1x  1 / 2, P0x  1 / 4, Px1  1 / 4           (B)       P1x  1 / 3, P0x  1 / 3, Px1  1 / 3
     (C) P1x  1 / 4, P0x  1 / 2, Px1  1 / 4           (D)       P1x  1 / 4, P0x  1 / 4, Px1  1 / 2

                                                    14
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44. Three mutually non-interacting electrons are confined in a hollow spherical
    cavity of radius R with impenetrable walls. The average pressure exerted by
    the electrons on the walls of cavity when the system is in the lowest energy
    state is given by :

          32                                        32
    (A)                                        (B)
          4mR5                                       2mR5
        72                                         32
    (C)                                        (D)
        2mR5                                         4mR3
45. Two identical spin-1/2 fermions are placed in a 1-dimensional square potential
    well with infinitely high walls, V = 0 for 0  x  L , otherwise V  . The normalized

                                                       2      nx 
    single particle energy eigenstates are un(x)        sin       ; n = 1, 2, 3, ... . Let
                                                       L      L 
                                                                   
    s be the quantum number corresponding to total spin S . The ground state
    energy of the system is given by :

        5 2  2                                      2 2
    (A)          if s = 1                      (B)          if s = 1
        4mL2                                         mL2

           2 2                                     5 2 2
    (C)          if s = 0                      (D)           if s = 0
          mL2                                        2mL2
46. Consider an isotropic harmonic oscillator in two-dimensions. The Hamiltonian
    is given by
                                        2
                                   Px2 Py mw2 2
                                H          ( x  y2 )
                                   2m 2m    2
    A time independent perturbation of the form H' = mw2 xy (where  << 1) is
    introduced in the system. First order energy shift for the ground state is given
    by :
                                                 
    (A) w                                 (B)    w
                                                 2
         
    (C)    w                               (D) 0
         4
                                          15
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47. A particle with incoming wave vector k, after being scattered by the potential
             c
    V (r )  2 , c real, goes out with wave vector k'. The differential scattering cross-
            r
    section, calculated in the first Born approximation, depends on q = (k – k'(,
     as

     (A) 1/q2                                   (B)   1/q4

     (C) 1/q                                    (D)   1/q3/2
                                                                                           
48. A particle of mass m in three dimensions is subjected to a potential V ( r ) =
                           
     A ln(r/ro) where r ( r ( and A and ro are constants. Then, in some normalized
                                                                          
     stationary state say ( r ) , the mean square velocity, < v 2 >, is equal to :

     (A) 2A/m                                   (B)   A/2m

     (C) 3A/m                                   (D)   A/m

49. Consider a spinless particle in a two-dimensional infinite well V(x, y) =  for
     0  x  L.0  y  L and V(x, y) = 0 otherwise. Add a potential of the form
     V'(x, y) = xy;0  x  L, 0  y  L . Taking this as a weak perturbation, the ground
     state energy shift at first order is given by :

         L3                                          L3
     (A)                                        (B)
          8                                            4
           L3
     (C)                                        (D)   L3
            2

                                                            e r / ao
50. A hydrogen atom is in its ground state : 100 ( r )                 . The Coulomb potential
                                                               ao3
        
     V (r )   e2 / r of hydrogen atom is perturbed by adding H' = x 2 where  is
     a constant parameter. The first order ground state energy shift is :

     (A) 2ao2                                  (B)   ao2

     (C) ao2 /2                                (D)       2ao2
                                           16
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51. A 1-dimensional Harmonic oscillator of mass m and classical angular frequency
    w is described by the Hamiltonian (written in terms of annihilation and creation
    operators) H  w(a† a  1 / 2). The unperturbed spectra is given by
     H ( p  w( p  1 / 2); p = 0, 1, 2, ... . For t < 0 it is in the state ( m . At time
    t = 0 the harmonic oscillator potential is perturbed by adding time dependent
    potential V(t) of the form
                               ( / 2mw)3/ 2[ a  a† ]3 , for 0  t  T
                      V ( t)  
                               0, for t  T
     Under this the system makes a transition from the state (m to a state (n .
     At the first order in perturbation the integer n may be related to m as
     [Hint : You may use            n(( a  a† )( m  mm,m1  m  1n,m1 ]
    (A) n = m – 2                          (B) n = m + 4
    (C) n = m + 3                          (D) n = m – 2 OR n = m + 4
52. The WKB quantization condition for the energy of a particle in a radial potential
    V(r) with turning points r1 and r2 is given by :
            r2
     (A)    dr 2m  E  V (r)   (n  1 / 2)
            r1               n


            r2                    l(l  1)2 
     (B) r1 dr 2m   E
                    n  V ( r )             
                                    2mr 2   (n  1 / 2)

            r2                    l(l  1) 2 
     (C) r1 dr 2m   E
                    n  V ( r )              
                                    2mr 2   (n  1 / 2)
            r2
     (D)    dr r
                    2
                        2m  En  V (r )   (n  1 / 2)
            r1

53. Two identical random walkers have started walking randomly from the origin
    on a 1 d lane. They are taking steps simultaneously and of equal length l. They
    travelled for 10 steps. The probability of meeting together at 4th step is ......
                                                                              3
         10!  1 
                        10
                                                               10!  1 10 
     (A)                                                (B)             
         4!6!  2                                            4!6!  2  
                             2                                                    2
          10!  1 10                                      10! 2  1 10 
     (C)                                            (D)              
           4!6!  2                                        4!6!   2  

                                                   17
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54. The system consists of two indistinguishable particles, each of which can be
    in either of the two energy states, 0 or  . The system is in thermal equilibrium
    with a heat bath at temperature T. There is also an interaction energy term
    that increases the energy by an amount  if the two particles are in the same
    energy state. The average energy of the system is <E> = ..
         e  (2  )e(2    )  e              (  )e  (2  )e(2  )  e
     (A)                                              (B)
              e  e(2   )  e                            e  e(2   )  e
              (2  )  
    (C) e  e(2   )  e           (D) (2  3) / 3
55. Due to gravitational force acting on the atmospheric gas particles, atmospheric
    pressure P(h) decreases with the height h. Consider atmospheric gases behaving
    like a perfect gas and by balancing the forces acting on a slab of air of thickness
    dh, the variation in atmospheric pressure as a function of height can be expressed
    as P(h) = ........
    (A) 29.92 exp (–mgh/kBT) inch of Hg
    (B) 760 exp (gh / kBT ) mm of Hg
    (C) 1013 exp (–mg/kBT)millibars
    (D) 103 exp (–mgh/kBT)kPa
56. Consider Na (Avogadro number) number of classical one-dimensional oscillators
    each having energy given by E = p2/2m + 0.5kx2 + x4 (here k and  are positive
    constants). The classical specific heat at temperature T = 400 K is Cv = ..
    (A) 3R(R is gas constant)                (B) 2.5 R
    (C) R                                    (D) 1.25 R
57. A solid with heat capacity CA at temperature TA is placed in contact with the
    heat bath having P heat capacity CB at temperature TB such that CB >> CA.
    The change in entropy of the combined system after a solid reached a thermal
     equilibrium with a heat bath is S  .....
                   TB        T                                   T T         
     (A) C A ln        CB ln B                       (B)   C A  ln B  A  1
                   TA        TA                                  TA TB 
              TB TB                                                TB         T      
     (C) C A  ln   1                              (D)   C A ln       CB ln  A  1 
              TA TA                                                TA          TB    
                                                 18
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58. The temperature of an ideal monoatomic gas is changed from T1 to T2, keeping
    (i) pressure constant, and (ii) keeping its volume constant. Then the ratio of
    the change in entropy (S) p / (S)V is :
     (A) 1/2                                    (B)   1.0
     (C) 5/3                                    (D)   2/3
59. The thermodynamic identity for the ideal rubber band is dU = Tds + dL
    where T is the temperature,  is the tension, U is the internal energy of the
    rubber band, S is its entropy, and L is its length. Which of the following Maxwell's
     relation is correct for this rubber band system is ....

            S                                   S    
     (A)                                  (B)    L    T 
            L T  T  L                               T      L

            T                                    S   L 
     (C)                                  (D)            
            L  S  S  L                             T  T 
60. The chemical potential  for an indistinguishable ideal monoatomic gas molecule
    is  = ..... (where,  = 1/(kBT) :

                V 3 2m                                    V 3 2m 
     (A)  kBT  ln  ln 2
                            1                 (B)    kBT  ln  ln 2 
                N 2 h                                     N 2 h 

                V 3 2m                                          3 2m 
     (C)  kBT  ln  ln 2
                            5 / 2             (D)    kBT  ln V  ln     1
                N 2 h                                           2  h2    
61. A distant star has its surface temperature of about 2898 K. The wavelengths
     of light radiation which has maximum intensity emitted from these stars i.e.
      max 
     (A) 289.8 nm                               (B)   1000 nm
     (C) 10000 nm                               (D)   550 nm
62. The ratio of pressure exerted by photon gas to that of the ideal gas particles
     is ...... to ratio of their energy densities.
     (A) twice                                  (B)   thrice
     (C) equal                                  (D)   half

                                           19
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63. In an ideal Fermi system, if the number density of fermions is decreased to
    half then the ratio of there single particle momenta corresponding to the Fermi
    energies is approximately :
    (A) 2.0                                   (B) 3.2
    (C) 0.666                                 (D) 1.58
64. The external magnetic field Bz is applied to the paramagnetic material containing
    N number of spin 1/2 particles each having a magnetic dipole moment  . The
    magnetisation <M> of a system is measured at low magnetic field and at
    sufficiently high temperate. Which of the following statements is correct ?
                                N 2
    (A) magnetization <M>           Bz and susceptibility   T
                                kBT

                                N 2
    (B) magnetization <M>           Bz and susceptibility   1 / T
                                kBT

                            N 2
    (C) magnetization <M>  k T Bz and susceptibility   1 / T
                                 B


                                kBT
    (D) magnetization <M>           Bz and susceptibility   T
                                N 2
65. The density of ideal Bose gas in the condensate state is denoted by 0 (T ) and
     be the density of Bose gas in non-condensate state. At the temperature T
    below a Bose temperature Tb (T < Tb) the value of 0 (T ) = .....
                   3/ 2                                          3/ 2
           T                                         T 
    (A)  1                                (B)    1  b 
           Tb                                        T
                   1/ 2
           T                                       T
    (C)  1                            (D)   1  T 
           Tb                                       b 

66. A barrier potential of a P-N junction depends on :
    I. Type of semiconductor
    II. Concentration of doping element
    III. Temperature
    (A) I and II only                     (B) II only
    (C) II and III only                   (D) I, II and III
                                         20
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67. The current 'I' supplied by the power supply in the given circuit is ...............




     (A) 1 mA                                  (B)   0.05 A

     (C) 0.1 A                                 (D)   5 mA

68. From the circuit of the following logic gates the basic logic gate
     obtained is ................




     (A) NOT gate                              (B)   AND gate

     (C) OR gate                               (D)   NAND gate

69. Find the concentration of holes (pp) and electronics (np) in a type germanium at
     300 K, if the conductivity is 100  cm–1, hole mobility ( p ) = 1800 cm2/V sec,
     intrinsic charges carrier concentration (ni) = 2.25 × 1013/cm3, charge on electron
     e = 1.6 × 10–19 C. Assume that the conductivity due to electron is negligible
     as compared to that due to holes.

     (A) pp = 1.83 × 105/cm3, np = 1.83 × 109/cm3

     (B) pp = 3.41 × 1017/cm3, np = 1.83 × 109/cm3

     (C) pp = 1.83 × 1019/cm3, np = 1.83 × 109/cm3

     (D) pp = 3.41 × 1017/cm3, np = 3.41 × 1017/cm3

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70. The co-ordinates of the operating point (Q) for the following JFET circuit are ........




     (A) VDS = 15 mV,            ID = 5 mA.

     (B) VDS = 5 mV,            ID = 15 mA.

     (C) VDS = 30 mV,            ID = 10 mA.

     (D) VDS = 15 V,           ID = 5 mA.

71. Consider a 8-bit serial input-parallel output shift registrar operated at clock frequency
     of 10 MHz. The time taken by the registrar to shift 8 bit number is ..........

     (A) 800 ns                                     (B)   100 ms

     (C) 100 ns                                     (D)   10 s

72. Consider R-2R ladder network based on inverting operational amplifier, where
     Do indicates Least Significant Digit (LSB), and D3 as Most Significant Digit
     (MSB). If the reference voltage is + 5V, then the output voltage for binary
     input 1010 is ....................

     (A) – 5.625 V                                  (B)   – 6.25 V

     (C) – 3.125 V                                  (D)   5.0 V

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73. A displacement transducer with a shaft stroke of 100 mm is applied to the
    circuit shown below. The total resistance is 10 k  and voltage is 5 V. If the
    position of wiper is at a distance of 40 mm from the end A then the value
    of output voltage V0 = ..................




     (A) 3V                                    (B)   2V
    (C) 1.5V                              (D) 2.5V
74. The output waveform for the following operational amplifier circuit is .............




     (A)                                       (B)




     (C)                                       (D)


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75. For a given full wave bridge rectifier the input line voltage has nominal value
    of 120V. The actual voltage coming out of the power outlet vary from 105 to
    125V and corresponding load voltage as 8.8V and 12.8V respectively. Therefore,
    the line regulation is .................
    (A) 4.5%                                 (B) 45%
    (C) 22.5%                                (D) 2.25%
76. The output voltage for the following series regulator circuit is .............




    (A) 7.98V                              (B) 6V
    (C) 3.99V                              (D) 15.96V
77. For a given unijunction transistor (UJT) circuit peak voltage (Vp) is ..........




    Symbol carrier usual meaning.
    (A) 3.4V                                  (B) 1.4V
    (C) 1.0V                                  (D) 2.4V
78. In the filtered half wave rectifier circuit as shown below, the value of the DC
     output voltage is ............ .




     (A) 12V                                   (B)   16V
     (C) 18V                                   (D)   17V
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79. The fine structure correction to hydrogen atom Bohr energies (including spin-
    orbit interaction correction and relativistic correction) is given by :
                                                 
                                   En  2  n    3
                               E  2           
                                    n  j1 4
                                             2   
     where n is the principal quantum number, En is the energy of the nth Bohr
     energy level,  is the fine structure constant, and j is the total angular
     momentum quantum number. For n = 2, the energy corrections E are :
           3 2 E2      5 2 E2                       3 2 E2      7 2 E2
     (A) 0,        and                        (B) 0,          and
              16          16                           16             16
            2          2                                      2
        5 E2         E2                                  7 E2
    (C)          and                          (D) 0 and
          16           16                                   16
80. The expression for the weak field Zeeman effect correction Ez to hydrogen atom
    energies and its values for the ground state energy are (  B is the Bohr magneton,
    gJ the Lande g-factor, mj the total angular momentum magnetic quantum
    number, B external magnetic field) :
     (A) Ez   B gJ Bm j , for ground state Ez = 0
     (B) Ez   B gJ Bm j , for ground state Ez =   B B
             3                                      3
    (C) Ez   B gJ Bm j , for ground state Ez =   B B
             2                                      2
                                                  3
    (D) Ez   B gJ Bm j , for ground state Ez =   B B
                                                  2
81. In an electron spin resonance (ESR) experiment, sodium atoms (11Na), in their
     ground state, are observed to have a resonance frequency  = 1.0 × 1010 Hz.
     Given that for 11Na ground state, Lange g-factor gJ = 2 and total angular
     momentum magnetic quantum number mJ   1 / 2 . What is the external
     magnetic field applied to obtain the resonance ?
     (Bohr magneton  B = 9.274 × 10–24 J/T, Planck's constant h = 6.626 × 10–34 Js)
     (A) 0.71 Tesla                            (B)    0.18 Tesla
     (C) 1.2 Tesla                             (D)    0.36 Tesla

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82. An HCl molecule has force constant 516 N/m. Using harmonic oscillator
    approximation, the vibrational frequency  and the wave number  of this
    molecule are :

    (Given : Mass of H atom is 1.674 × 10 –27 kg, mass of Cl atom is
    5.887 × 10–26 kg, Planck's constant h = 6.626 × 10–34 Js, speed of light
    c = 3 × 108 m/s.)

    (A)  = 89.6 × 1012 Hz,  = 1890 cm–1

    (B)  = 179.2 × 1012 Hz,  = 5974 cm–1

    (C)  = 89.6 × 1012 Hz,  = 2987 cm–1

    (D)  = 44.8 × 1012 Hz,  = 1493.5 cm–1

83. For the carbon monoxide (which can be considered a rigid diatomic molecule)
    for the transition J = 0 to J = 1 (where J is the rotational quantum number),
    the wave number  = 3.842 cm–1. The length of CO bond is :

    (Given : Mass of carbon atom is 19.921 × 10–27 kg, mass of oxygen atom is
    26.561 × 10–27 kg, Planck's constant h = 6.626 × 10–34 Js, speed of light
    c = 3 × 108 m/s.)

    (A) 1.483 Å                                (B)   0.956 Å

    (C) 1.131 Å                                (D)   1.762 Å

84. A hypothetical material used was producing a laser beam was consisting of
    atoms having a three-level system with energies E1, E2 and E3 such that
    E1 < E2 < E3. The number of atoms in these energy levels were N1 in E1,
    N2 in E2 and N3 in E3, where N2 > N1 > N3. Laser emission will be due
    to transitions between the levels :

    (A) E2  E1 and E1  E3                    (B)   E2  E1

    (C) E1  E3                                (D)   E2  E1 and E3  E1

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85. In the Born-Oppenheimer approximation for molecular Hamiltonian, the
    following terms are included (choose the correct alternatives from below) :
    (A) Only kinetic energy of electrons, Coulombic potential energy due to
         attractions between electrons and nuclei, Coulombic potential energy due
         to electron-electron repulsions, Coulombic potential energy due to nucleus-
         nucleus repulsions.
    (B) Only kinetic energy of nuclei, Coulombic potential energy due to attractions
         between electrons and nuclei, Coulombic potential energy due to electron-
         electron repulsions, Coulombic potential energy due to nucleus-nucleus
         repulsions.
    (C) Only Coulombic potential energy due to attractions between electrons and
         nuclei, Coulombic potential energy due to electron-electron repulsions,
         Coulombic potential energy due to nucleus-nucleus repulsions.
    (D) Only kinetic energy of nuclei, Coulombic potential energy due to attractions
         between electrons and nuclei, and Coulombic potential energy due to nucleus-
         nucleus repulsions.
86. Given a two-dimensional material, in which atoms (denoted by dots) are arranged
                                                                             
    in a honeycomb lattice with lattice constant a. As shown in the figure, a1 and
     
     a2 are two lattice vectors. Which one of the following is the area of the first
    Brillouin zone for this lattice ?




          82                                       4 2
    (A)                                       (B)
           3a2                                       3a2

           8 2                                      4 2
    (C)                                       (D)
          3 3a2                                     3 3a2
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87. A semiconductor with donor impurities can be thought in terms of a filled valence
    band, a filled donor level and an empty conduction band at T = 0, as shown in
    the figure below.




     If the band gap between donor level and conduction band is 1 and that between
     conduction and valence band is 2 where 2 >> 1 , which of the following figures
     depict the qualitative features of the resistance (R) s temperature (T) graph of
     the semi-conductor ? (Assume temperature-independent scattering rates and a flat
     density of states for the bands.)



                                                (B)
     (A)




     (C)                                        (D)


88. Consider a 2-D square lattice of lattice constant a. The ratio of the kinetic energy
    of a free electron at a corner of the first Brillouin zone (Ec) to that of an electron
    at the midpoint of a side face of the same zone (Em) is Ec/Em =
    (A) 1/2                                     (B) 2
    (C) 2                                       (D) 1
89. Second sound in superfluid helium is :
    (A) the wave motion of entropy and temperature
    (B) the pressure wave
    (C) the density wave
    (D) wave motion of the normal liquid helium
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90. Frenkel defect is a defect in solid where :
    (A) oppositely charged ions leave their lattice sites creating vacancy defects
    (B) dislocations are created
    (C) disclinations are created
    (D) smaller ion is displaced from its lattice position to an interstitial site creating
        vacancy defect and interstitial defect
91. Which of the following statement is true ?
    (A) Quasi crystals are a phase of matter with long range order having symmetry
        elements forbidden to periodic crystals
    (B) The liquid-crystalline state has more order than the three-dimensional
        structure of the solid state
    (C) Orientational order refers to translational and rotational symmetry of liquid-
        crystalline state
    (D) A liquid-crystalline pattern can continuously fill all available space, but
        it lacks translational symmetry
92. X-rays of wavelength  , when incident on the (101) plane of a cubic lattice
    with lattice constant a produce a first-order Bragg's reflection at  = 30º ( 
    is measured from the lattice plane). Suppose this cubic lattice is compressed
    along the z-axis such that its lattice parameters along the x- and y-axis remain
                                                        1
    the same while that along the z-axis becomes           a (see figure).
                                                         3




    The first-order reflection for the (101) plane of the compressed lattice occurs at :
    (A)  = 45º                                (B)  = 15º
    (C)  = 30º                                (D)  = 60º
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                  ( 157 N)
93. The ratio               , where  denotes the pairing term in the Semi-empirical
                 ( 256
                    100 Fm)


     mass formula (Weizsäcker formula) for binding energy, is :
     (A) –8                                     (B)   8
     (C) –4                                     (D)   0
94. The Q value of 197
                    79 Au  6 C  85 At  ?
                           12    206
                                            in MeV is :
     (Given : 1 u = 931.5 MeV/c2, M ( 11 H) = 1.007825 u, Mn = 1.008665 u, M ( 126 C)
     = 12.000000 u, M ( 197
                         79 Au)
                                = 196.966543 u, M ( 206
                                                     85 At)
                                                            = 205.986667 u)
     (A) –30                                    (B)   30
     (C) –43                                    (D)   43
95. A radioactive source with half-life (1/ 2 ) of 69.3 sec has an activity of 1 Ci .
     How many radioactive nuclei will remain after time t = 69.3 sec ?
     (A) 3.7 × 106                              (B)   3.7 × 105
     (C) 1.85 × 106                             (D)   1.85 × 105

96. An alpha particle with an energy of 6.17 MeV is emitted from a natural
     radioactive source 251
                         98 Cf
                               through a process of quantum mechanical tunnelling
     process. This process is classically forbidden. Estimate the classical barrier range
     through which the alpha particle escape through a tunnelling process

                              Cf  247
                            251
                             98     96 Cm    Q(6.17 MeV)


     (Given : R0 = 1.2 fm where, R0 is constant of proportionality in a relation of
     radius of a nuclei and mass number A)

     (A) 35 fm                                  (B)   8 fm

     (C) 45 fm                                  (D)   60 fm

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97. According to the scattering experiments of electrons from various nuclei, the
    nuclear charge density  ch as a function of radial distance r is approximated
    by relation .............
    (Where 0 is constant nucleon density in the core, a is the diffusiveness parameter
    and R is the average radius of particular nucleus.)

                         0                                   0
     (A) ch (r)                        (B) ch (r) 
                           rR                                r R
                  1  exp                            1  exp      
                           a                                  a 
                         0                                   0
    (C) ch (r)                         (D) ch (r) 
                           r  R                              rR
                  1  exp                            1  exp      
                           a                                  a 
98. The only di-nucleon n-p bound system is observed in nature, but not the
    n-n or p-p di-nucleon bound system, this is because :
     (A) Nuclear forces are mainly central
     (B) Nuclear forces are strongly spin-dependent
     (C) Nuclear forces are charge independent
     (D) Nuclear forces are charge dependent
99. In a particular decay process 20
                                  41
                                     Ca converts to 19
                                                    41
                                                       K , via beta-decay process.
     From the given information, conclude which type of decay mode is energetically
     possible :
     [Given : M( 20
                 41
                    Ca) = 40.962278 u, M( 19
                                          41
                                             K) = 40.961825 u, M( 11 H) = 1.007825 u,
     Mn = 1.008665 u, Me = 0.00055 u and 1 u = 931.5 MeV/c2]
     (A)   decay                             (B)     decay
     (C) Electron capture                      (D)   Both   and   decay
100. For the given reaction, n  p    0 which of the conservation statement is
     true ?
     (A) Baryon number (B) is not conserved, the reaction is forbidden
     (B) Electron lepton number (Le) is not conserved, the reaction is forbidden
     (C) Charge (Q) is not conserved, the reaction is forbidden
     (D) All conservations hold true, the reaction is allowed
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ROUGH WORK




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Source: Savitribai Phule Pune University. Question paper published by Savitribai Phule Pune University.