1

SECTION: VECTOR ALGEBRA & VECTOR CALCULUS

G G
1. Let aG and b be two distinct three-dimensional vectors. Then the component of b that is
perpendicular to a is given by : [CSIR June 2011]
G G G G G G GG G GG G
au b ua b u aub a.b b b .a a
(a) (b) (c) (d)
a2 b2 b2 a2

2. The equation of the plane that is tangent to the surface xyz 8 at the point (1, 2, 4) is :
[CSIR Dec. 2011]
(a) x  2 y  4 z 12 (b) 4 x  2 y  z 12

(c) x  4 y  2 z 12 (d) x  y  z 7

3. A vector perpendicular to any vector that lies on the plane defined by x  y  z 5 is:
[CSIR June 2012]
G G G G G G G G G G
(a) i  j (b) j  k (c) i  j  k (d) 2i  3 j  5k

§ a b c ·
4. The unit normal vector at the point ¨ , , ¸ on the surface of the ellipsoid
© 3 3 3¹

x2 y2 z2
  1 , is [CSIR Dec. 2012]
a2 b2 c2

bciˆ  cajˆ  abkˆ aiˆ  bjˆ  ckˆ

(a) (b)
a 2b 2  b 2 c 2  c 2 a 2 a 2  b2  c 2

biˆ  cjˆ  akˆ iˆ  ˆj  kˆ

(c) (d)
a 2  b2  c 2 3

5. A unit vector n̂ on the xy-plane is at an angle of 120º with respect to iˆ . The angle
between the vectors u aiˆ  bnˆ and vG anˆ  biˆ will be 60º if : [CSIR June 2013]

(a) b 3a / 2 (d) b 2a / 3 (c) b a / 2 (d) b a

G G
6.
G
If A yziˆ  xyjˆ  xykˆ , then the integral v³ A.dl (where C is along the perimeter of a
C

rectangular area bounded by x 0, x a and y 0, y b ) is : [CSIR Dec. 2013]

1 3 3 2 2 3 3
(a) a b (b) S ab  a b (c) S a  b (d) 0
2
G
7. If A yziˆ  zxjˆ  xykˆ and C is the circle of unit radius in the plane defined by z = 1, with
G G
the centre on the z-axis, then the value of the integral v³ A.dl
is : [CSIR June 2014]
C

S S
(a) (b) S (c) (d) 0
2 4

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G
8. Let rG be the position vector of any point in three dimensional space and r r . Then
[CSIR Dec. 2014]
G G G
G r G G G
(a) ’.rG 0 and ’ ur (b) ’.rG 0 and ’2 r 0
r
G 2G G
G r G G
(c) ’.rG 3 and ’ r (d) ’.rG 3 and ’ u rG 0
r2
9. If S is the closed surface enclosing a volume V and n̂ is the unit normal vector to the
G
G
surface and r is the positive vector, then the value of the following integral ³³ ˆ
r .ndS
is:
S

[GATE 2001]
(a) V (b) 2V (c) 0 (d) 3V
1 1 1
10. Consider the set of vectors 1,1,0 , 0,1,1 and 1,0,1 : [GATE 2001]
2 2 2
(a) The three vectors are orthonormal
(b) The three vectors are linearly independent
(c) The three vectors cannot form a basis in a three-dimensional real vector space

1 1 1
(d) 1,1,0 can be written as the linear combinational of 0,1,1 and 1,0,1
2 2 2
G G
11. If A xeˆx  yeˆ y  zeˆz , then ’2 A equals : [GATE 2001]

(a) 1 (b) 3 (c) 0 (d) –3

G
12. A vector A 5 x  2 y iˆ  3 y  z ˆj  2 x  az kˆ is solenoidal if the constant a has a
value: [GATE 2002]
(a) 4 (b) –4 (c) 8 (d) –8

13. Which of the following vectors is orthogonal to the vector aiˆ  bjˆ where a and a z b
are constants, and iˆ and ĵ are unit orthogonal vectors? [GATE 2002]

(a) biˆ  ajˆ (b)  aiˆ  bjˆ (c)  aiˆ  bjˆ (d) biˆ  ajˆ

14. The unit vector normal to the surface 3 x 2  4 y z at the ponit (1, 1, 7) is:[GATE 2002]

(a) 6iˆ  4 ˆj  kˆ / 53 (b) 4iˆ  6 ˆj  kˆ / 53

(c) 6iˆ  4 ˆj  kˆ / 53 (d) 4iˆ  6 ˆj  kˆ / 53

Common data for Q.15 and Q.16 [GATE 2003]

G G
r
Consider the vector V
r3

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15. The surface integral of this vector over the surface of a cube of size and centered at the
origin.

(a) 0 (b) 2S (c) 2 S a 3 (d) 4S

16. Which one of the following is NOT correct?

(a) Value of line integral of this vector around any closed curve is zero
(b) This vector can be written as the gradient of some scalar function
(c) The line integral of this vector from point P to point Q is independent of the path taken
(d) This vector can represent the magnetic field of some current distribution
G
17. The curl of the vector A ziˆ  xjˆ  ykˆ is given by : [GATE 2003]

(a) iˆ  ˆj  kˆ (b) iˆ  ˆj  kˆ (c) iˆ  ˆj  kˆ (d) iˆ  ˆj  kˆ

G G
18. The two vectors P iˆ, q iˆ  ˆj / 2 are : [GATE 2003]

(a) related by a rotation (b) related bya reflection through the xy-p lane
(c) related by an inversion (d) not linearly independent
G
19. For the function I x 2 y  xy the value of ’I at x y 1 is : [GATE 2004]

(a) 5 (b) 5 (c) 13 (d) 13

G G G
20. If a vector field F xiˆ  2 yjˆ  3 zkˆ , then ’ u ’ u F is : [GATE 2005]

(a) 0 (b) iˆ (c) 2 ˆj (d) 3kˆ

§1· § 3· § 2· § 3 ·
¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸
21. Given the four vectors,
u1 ¨ 2 ¸ , u2 ¨ 5 ¸ , u3 ¨ 4 ¸ , u4 ¨ 6 ¸ , the linearly dependent
¨1¸ ¨1¸ ¨ 8 ¸ ¨ 12 ¸
© ¹ © ¹ © ¹ © ¹
pair is : [GATE 2005]
(a) u1 , u2 (b) u1 , u3 (c) u1 , u4 (d) u3 , u4

22. The unit normal to the curve x 3 y 2  xy 17 at the point (2, 0) is : [GATE 2005]

iˆ  ˆj
(a) (b) iˆ (c)  ĵ (d) ĵ
2

G y2 ˆ G
23. A vector field is defined everywhere as F i  zkˆ . The net flux of F associated
L
with a cube of side L, with one vertex at the origin and sides along the positive X, Y and
Z axes, is: [GATE 2007]

(a) L3 (b) 4L3 (c) 8L3 (d) 10L3

G
24. If r xiˆ  yjˆ , then : [GATE 2007]

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G G G G G G G G
(a) ’.rG 0 and ’ r r (b) ’.rG 2 and ’ r r

G G G G G G
G r G r
(c) ’.rG 2 and ’ r (d) ’.rG 3 and ’ r
r r
G ˆ
25. Consider a vector p 2iˆ  3 ˆj  2kˆ in the coordinate system iˆ, ˆj, k . The axes are rotated
G
anti-clockwise about the Y axis by an angle 60°. The vector p in the rotated coordinate

system iˆ ', ˆj ', kˆ ' is : [GATE 2007]

(a) 1  3 iˆ ' 3 ˆj ' 1  3 kˆ ' (b) 1  3 iˆ ' 3 ˆj ' 1  3 kˆ '

(c) 1  3 iˆ ' 3  3 ˆj ' 2kˆ ' (d) 1  3 iˆ ' 3  3 ˆj ' 2 kˆ '

G G
26. The curl of the vector field is F is 2xˆ . Identity the appropriate vector field F from the
choices given below: [GATE 2003]
G G
(a) F 2 zxˆ  3zyˆ  5 yzˆ (b) F 3 zyˆ  5 yzˆ
G G
(c) F 3 xyˆ  5 yzˆ (d) F 2 xyˆ  5 yzˆ

G G
27. The value of the contour integral ³ u d T , for a circle C o f radius r with centre at the
r
C

origin is : [GATE 2009]

r2
(a) 2 Sr (b) (c) Sr 2 (d) r
2
G
28. An electrostatic field E exists in a given region R. Choose the wrong statement:
G
(a) Circulation of E is zero [GATE 2009]
G
(b) E can always be expressed as the gradient of a scalar field
(c) The potential difference between any two arbitary points in the region R is zero
(d) The work done in a closed path lying entirely in R is zero

29. Consider the set of vectors in three-dimensional real vector space R 3 , S 1,1,1 , 1, 1,1

1,1, 1 . Whch of the following statement is true? [GATE 2009]

(a) S is not a linearly independent set (b) S is a basis for R3

(c) The vectors in S are orthogonal
(d) An orthogonal set of vectors cannot be generated from S
G
30. If a force F is derivable from a potential function V r , where r is the distance from the
origin of the coordinate system, it follows that : [GATE 2011]

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G G G G G
(a) ’ u F 0 (b) ’.F 0 (c) ’V 0 (d) ’2V 0

31. The unit vector normal to the surface x 2  y 2  z 1 at the point P(1, 1, 1) is :

[GATE 2011]

iˆ  ˆj  kˆ 2iˆ  ˆj  kˆ iˆ  2 ˆj  kˆ 2iˆ  2 ˆj  kˆ
(a) (b) (c) (d)
3 6 6 3
32. Consider a cylinder of height h and radius a, closed at both ends, centered at the origin.
G ˆ ˆ ˆ be the position vector and a unit vector normal to the surface. The
Let r ix  jy  kz
G
surface integral ³
ˆ
r .nds
over the closed surface of the cylinder is: [GATE 2011]
s

(a) 2Sa 2 a  h (b) 3Sa 2 h (c) 2Sa 2 h (d) Zero

G G
33. Identify the CORRECT statement for the following vectors a 3iˆ  2 ˆj and b iˆ  2 ˆj .
G
(a) The vectors aG and b are linearly independent [GATE 2012]
G
(b) The vectors aG and b are linearly dependent
G
(c) The vectors aG and b are orthogonal
G
(d) The vectors aG and b are normalized
G G G
34. Given F  rG u B , where B B0 iˆ  ˆj  k is a constant vector and r is the position vector..
G G
The value of ³
v F .dr
, where C is a circle of unit radius centered at origin is :
C

[GATE 2012]

(a) 0 (b) 2SB0 (c) 2SB0 (d) 1

G G G G G G
35. If A and B are constant vectors, then ’ ª¬ A B u r º¼ is : [GATE 2013]

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G G G G
(a) A.B (b) A u B (c) rG (d) zero

36. The unit vcctor perpendicular to the surface x 2  y 2  z 2 3 at the point (1, 1, 1) is
[GATE 2014]

xˆ  yˆ  zˆ xˆ  yˆ  zˆ xˆ  yˆ  zˆ xˆ  yˆ  zˆ
(a) (b) (c) (d)
3 3 3 3
37. Four forces arc Riven below in Cartesian and spherical polar coordinates. [GATE 2015]

G § r 2 · G
(i) F1 K exp ¨ 2 ¸ rˆ (b) F2 K x3 yˆ  y 3 zˆ
¨ R ¸
© ¹

G G § Iˆ ·
(c) F3 K x 3 xˆ  y 3 yˆ F
(d) 4 K ¨¨ ¸¸
©r¹
where K is a constant. Identify the correct option.
(a) (iii) and (iv) are conservative but (i) and (ii) are not
(b) (i) and (ii) are conservative but (iii) and (iv) are not
(c) (ii) and (iii) are conservative but (i) and (iv) are not
(d) (i) and (iii) are conservative but (ii) and (iv) are not

R G G
38. Given that magnetic flux through the closed loop PQRSP is I . If ³ A ˜ dl I1 along
P

R G G
PQR, the value of ³ A ˜ dl along PSR is [GATE 2015]
P

(a) I  I1 (b) I1  I (c) I1 (d) I1

G 1 2 1
39. The direction of ’f for a scalar field f x, y, z x  xy  z 2 at the point P(1, 1, 2)
2 2
is : [GATE 2016]

 ˆj  2kˆ  ˆj  2kˆ ˆj  2kˆ ˆj  2kˆ

(a) (b) (c) (d)
5 5 5 5
G G
40. A two-dimensional vector A t is given by A t iˆ sin 2t  ˆj cos 3t

Which of the following graphs besecribes the locus of the tip of the vector, as t is varied
from 0 to 2S ? [TIFR 2013]

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(a) (b)

(c) (d)

41. Consider the surface corresponding to the equation 4 x 2  y 2  z 0 . A possible unit

tangent to this surface at the point (1, 2, –8) is : [TIFR 2013]

1 ˆ 2 ˆ 1ˆ 4ˆ
(a) i j (b) j k
2 5 5 5

4ˆ 8 ˆ 1 ˆ 1 ˆ 3 ˆ 4 ˆ
(c) i  j k (d)  i j k
9 9 9 5 5 5

42. Which of the following vectors is parallel to the surface x 2 y  2 xz 4 at the point (2, –
2, 3)? [TIFR 2015]

(a) 6iˆ  2 ˆj  5kˆ (b) 6iˆ  2 ˆj  5kˆ (c) 6iˆ  2 ˆj  5kˆ (d) 6iˆ  2 ˆj  5kˆ

43. The vector field xziˆ  yjˆ in cylinclerical polar coordinates is : [JEST 2013]

2 2
(a) U z cos I  sin I eˆU  U sin I cos I 1  z eˆI

2 2
(b) U z cos I  sin I eˆU  U sin I cos I 1  z eˆI

2 2
(c) U z sin I  cos I eˆU  U sin I cos I 1  z eˆI

2 2
(d) U z sin I  cos I eˆU  U sin I cos I 1  z eˆI

44. Given that the coordinates of particle are y t A cos 2Zt and x t sin Zt the
trajectory of the particle is a :
(a) Circle (b) Ellipse (c) Hyperbola (d) Parabola
45. A point particle is moving in the (x, y) plane on a trajectory given in polar coordinates by
§ S·
the equation r sin ¨ T  ¸ 5 . The trajectory of the particle is :
© 4¹

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(a) a parabola (b) a straight line (c) a circle (d) a hyperbola

46. A point particle is moving the (x, y) plane on a trajectory given in polar coordinates by the
equation

§ S·
r 2  2r sin ¨ T  ¸  3 0
© 4¹
The trajectory of the particle is :
(a) a parabola (b) a straight line (c) a circle (d) a hyperbola

S
47. The normal to the surface given by the equation z cos x cosh y at the point x and
2
y 0 lies in
(a) (x, y) plane (b) (x, z) plane

S
(c) (y, z) plane (d) On the plane given by the equation x  y  z 1
2

48. A particle is moved quasi-statically from the point (–3, 0) to (3, 0), along a path y x 2  9
G
in an external force field given by F yi  3 yj . Give that all physical quantities are in SI
units, the magnitude of the work done on the particle is given by :
(a) 36 J (b) 18 J (c) 9 J (d) 0
49. A point particle is moving in the (x, y) plane on a trajectory given in polar coordinates by
the equation

25  r 2 cos 2T 0
The trajectory of the particle is a :
(a) parabola (b) circle (c) ellipse (d) hyperbola
50. The value of the line integral

xdy  ydx
v³ x2  y2

along a circle of radius 3 centered at the origin tri the counter clockwise direction is given
by :

3
(a) 0 (b) (c) 2S (d) 6S
2S

§1·
51. ’ 2 ¨ ¸ is :
©r¹

(a) 0 (b) G r (c) 4SG r (d) 4SG r

2§1·
52. ³³³ ’ ¨ ¸ dV r z 0 is :
©r¹

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(a) 0 (b) 4S (c) 4S (d) 1

53. The value oft for which three vectors :

ª¬ 1  t ,0,0 ¼º , ¬ª1, 1  t ,0¼º & ¬ª1,1, 1  t ¼º are linearly dependent is :

(a) 1 (b) 0 (c) 2 (d) –1

G JJG G
54. The value of v³ A.d A along a square loop of side L in a uniform field A is :

(a) 0 (b) 2LA (c) 4LA (d) L2 A

G G
r ˜ dS
55. The value of v³ r3
, where rG is the position vector and S is a closed surface enclosing
S
the origin, is :
(a) 0 (b) S (c) 4S (d) 8S
G
56. If A t is a vector o f constant magnitude, which of the following is true?
G G G G
dA d2A dA G dA G
(a) 0 (b) 0 (c) <A 0 (d) uA 0
dt dt 2 dt dt

57. The function r n rˆ r ! 0 , where r refers to spherical coordinate system, is:

(a) an irrotationat vector for n 2 (b) a solenoidal vector for n 2

(c) an irrotational vector for n 3 (d) a solenoidal vector for n 3
58. Two sets of vectors, one with m elements and the other with n elements (m < n) span the
same linear vector space. If k is the dimension of the vector space, then :
(a) k d m (b) k t m (c) k mn (d) k n
59. Identify the vector field given below which has a finite curl :

(a) (b)

(c) (d)

G G G G G G
60. Given any three non-zero vectors A, B and C , their triple product A. B u C vanishes if

(a) They are perpendicular to each other (b) Any two of them are perpendicular
(c) Any twn of them are parallel (d) They are non-coplanar

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G G
61. The necessary and sufficient condition that v³ A.dr 0
, for an v closed curve C is
c
G G G G G G G G G G
(a) ’. A 0 (b) ’ u A 0 (c) ’.’ u A 0 (d) ’ u ’ u A 0
62. Any arbitrary vector in a three dimnensional Cartesian space can be expressed as a
linear combination of the following number of linearly independent vectors :
(a) Arbitrary number (b) 1 (c) 2 (d) 3
G G
63. The line integral of A vanishes about every closed path. Then A must be equal to :
(a) Curl of a vector function (b) Gradient of a vector function
(c) Gradient of a vector function (d) Zero
64. When the fluid is incompressible, the equal of continuity can be reduced to
wU GG
0 U ’.v . Since the density U is constant in this case. Here vG is the velocity of a
wt
typical particle of the fluid. Further, if the flow is irrotational, then the equation can be
rewritten as :
G G G G G
(a) ’.’I 0 (b) ’ u ’I 0 (c) ’I 0 (d) vG constant
G JJG
65.
G
Given the vector A y,  x, 0 , the line integral v³ .dl
A
, where C is a circle of radius 5 units
c
with itscentre at the origin, (correct to the first decimal place) is :
(a) 172.8 (b) 157.1 (c) –146.3 (d) 62.8
G G
66. The value of integral 1 ³s r .ds where S is the surface enclosing volume V is :

(a) 3 (b) V (c) 3V (d) 0

1.(a) 2.(b) 3.(c) 4.(a) 5.(c) 6.(d) 7.(d)

8.(d) 9.(d) 10.(b) 11.(c) 12.(c) 13.(a) 14.(c)
15.(d) 16.(d) 17.(a) 18.(a) 19.(d) 20.(a) 21.(d)
22.(d) 23.(a) 24.(c) 25.(a) 26.(b) 27.(a) 28.(c)
29.(b) 30.(d) 31.(b) 32.(b) 33.(a) 34.(c) 35.(b)
36.(d) 37.(d) 38.(b) 39.(b) 40.(c) 41.(a) 42.(c)
43.(a) 44.(d) 45.(b) 46.(c) 47.(b) 48.(a) 49.(d)
50.(c) 51.(c) 52.(a) 53.(a) 54.(a) 55.(c) 56.(c)
57.(b) 58.(b) 59.(c) 60.(c) 61.(b) 62.(d) 63.(b)
64.(b) 65.(b) 66.(c)

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SECTION: VECTOR ALGEBRA

§ 1 1 1·
¨ ¸
M ¨ 1 1 1¸
Consider the matrix [CSIR June 2011]
¨ ¸
© 1 1 1¹

1. The eigenvalues of M are

(a) 0,1,2 (b) 0, 0, 1 (c) 1, 1, 1 (d) 1, 1, 3
2. The exponential of Msimplifies (I is the-3 × 3 identity matrix) :

M
§ e3  1 · M2
1 ¨
(a) e ¨ 3 ¸¸ (b) e M
M 1 M 
© ¹ 2!

(c) e M 1  33 M (d) eM e 1 M

3. A 3 × 3 matrix M has Tr M 6, Tr ª M 2 º 26, Tr ª M 3 º 90 . Which of the following

¬ ¼ ¬ ¼
can be possible set of eigenvalues of M? [CSIR June 2011]
(a) {1, 1, 4} (b) {–1, 0, 7} (c) {–1, 3, 4} (d) {2, 2, 2}

§1 2 3·
¨ ¸
A ¨2 4 6¸
4. The eigenvalues of the matrix [CSIR June 2012]
¨3 6 9¸
© ¹

(a) (1, 4, 9) (b) (0, 7, 7) (c) (0, 1, 13) (d) (0, 0, 14)

§ 0  n3 n2 ·
¨ ¸
A ¨ n3 0  n1 ¸
5. The eiegenvalues of the anti-symmetric matrix , where n1 , n2 , n3
¨ n n1 0 ¹¸
© 2
are components of an unit vector, are [CSIR June 2012]
(a) 0, –i, i (b) 0, 1, –1 (c) 0, 1 +i, (d) 0, 0, 0

6. A 2 × 2 matrix 'A' has eigenvalues eiS/5 and eiS/6 . The smallest value of 'n' such that

An I: [CSIR June 2012]

(a) 20 (b) 30 (c) 60 (d) 120

7. Given a 2 × 2 unitary matrix satisfying U 'U UU ' 1 with det U eiM , one can construct
a unitary matrix V V 'V VV ' 1 with det V 1 from it by [CSIR Dec. 2012]

(a) Multiplying U by eiM /2

(b) Multiplying a single element of U by eiM

(c) Multiplying any row or column by eiM /2

(d) Multiplying U by eiM

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8. Consider an n u n n ! 1 matrix A, in which Aij is the product of the indices i and j

(namely Aij ij ). The matrix A : [CSIR Dec. 2013]

(a) has one degenerate eigenvalue with degeneracy n  1

(b) has two degenerate eigenvalues with degeneracies 2 and n  2

(c) has one degenerate eigenvalues with degeneracy n

(d) does no t have any degenerate eigenvalues
9. Consider the matrix

§ 0 2i 3i ·
¨ ¸
M ¨ 2i 0 6i ¸
¨ 3i 6i 0 ¸
© ¹

The eigenvalues of M are [CSIR June 2014]

(a) –5, –2, 7 (b) –7, 0, 7 (c) –4i, 2i, 2i (d) 2, 3, 6
10. The matrices

ª 0 1 0 º ª0 0 1 º ª0 0 0º
A ««1 0 0 »» , B «0 0 0 » , C
« »
«0 0 1 »
« »
«¬ 0 0 0 »¼ «¬ 0 0 0 »¼ «¬ 0 0 0 »¼

satisfy the commutation relation : [CSIR June 2014]

(a) A, B B  C , B, C 0, C , A BC (b) A, B C , B, C A, C , A B

(c) A, B B, B , C 0, C , A A (d) A, B C , B, C 0, C , A B

§a· §0 0 1·
¨ ¸ ¨ ¸
b A ¨0 1 0¸
11. The column vector ¨ ¸ is a simultaneous eigenvcctor of and B
¨a¸ ¨1 0 0¸
© ¹ © ¹

§0 1 1·
¨ ¸
¨ 1 0 1 ¸ if :
¨1 1 0¸
© ¹

(a) b 0 and a 0 (b) b a and b  2a

(c) b 2a and b a (d) b a / 2 and b  a / 2

§1 3 2·
¨ ¸
12. The matrix
M ¨ 3 1 0 ¸ satisfies the equation : [CSIR Dec. 2016]
¨0 0 1¸
© ¹

(a) M 3  M 2  10 M  12 I 0 (b) M 3  M 2  12 M  10 I 0

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(c) M 3  M 2  10 M  10 I 0 (d) M 3  M 2  10M  10 I 0


13. For any of)erator A, i A  A is : [GATE 2001]

(a) Hermitian (b) Anti-Hermitian (c) Unitary (d) Orthogonal

14. If two matrices A and B can be diagonalized simultaneously, which of the following is
true? [GATE 2002]
(a) A2 B B2 A (b) A2 B 2 B2 A
(c) AB BA (d) AB 2 AB BABA2

§ 1 1 ·
15. Which one of the following matrices is the inverse ofthe matrix ¨ ¸ [GATE 2002]
© 0 1¹

§ 1 1· §1 0 · § 1 1· § 1 1 ·
(a) ¨ ¸ (b) ¨ ¸ (c) ¨ ¸ (d) ¨ ¸
© 1 1 ¹ ©1 1 ¹ © 0 1¹ © 0 1¹
16. A 3 × 3 matrix has eigenvalues 0, 2 +i and 2 –i. Which one of following statements is
correct? [GATE 2003]
(a) The matrix is Hermitian (b) The matrix is unitary
(c) The inverse of the matrix (d) The determinant of the matrix is zero
17. A real traceless 4 × 4 unitary matrix is has two eigen values –1 and 1. The other eigenvalues
are : [GATE 2004]
(a) zero and +2 (b) –1 and +1 (c) zero and +1 (d) +1 and +1

§ 1 i·
18. The eigenvalues of the matrix ¨ ¸ are : [GATE 2004]
© i 1 ¹
(a) +1 and +1 (b) zero and +1 (c) zero and +2 (d) –1 and +1

§0 1 00 ·
¨ ¸
¨1 0 0 0 ¸
19. Eigen values of the matrix ¨ 0 0 0 2i ¸ are : [GATE 2005]
¨ ¸
©0 0 2i 0 ¹

(a) –2, –1, 1, 2 (b) –1, 1, 0, 2 (c) 1, 0, 2, 3 (d) –1, 1, 0, 3

20. The determinant of a 3 × 3 real symmetric matrix is 36. If two of its eigen values are 2
and 3 then the third eigenvalue is : [GATE 2005]
(a) 4 (b) 6 (c) 8 (d) 9
§ x1 ·
¨ ¸ §x x ·
T ¨ x2 ¸ ¨ 1 2 ¸
, transforrn a vector xG from a
¨ x ¸ © x2  x3 ¹
21. A linear transformation T, de6ned as
© 3¹
three-dimensional real space to a two-dimensional real space. The transformation matrix
T is: [GATE 2006]

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§1 1 0 · §1 0 0· § 1 1 1· §1 0 0·
(a) ¨ ¸ (b) ¨ ¸ (c) ¨ ¸ (d) ¨ ¸
© 0 1 1¹ © 0 1 0¹ © 1 1 1 ¹ © 0 0 1¹
Common data for Q. 22 and Q. 23 :

§ 2 3 0·
¨ ¸
3 2 0¸
One of the eigen values of the matrix ¨ is 5. [GATE 2006]
¨0 0 1¸
© ¹

22. The other two eigen values are:

(a) 0 and 0 (b) 1 and 1 (c) 1 and –1 (d) –1 and –1
23. The normalized eigen vector corresponding to the eigen value 5 is:

§0· § 1 · §1· §1·

1 ¨ ¸ 1 ¨ ¸ 1 ¨ ¸ 1 ¨ ¸
1 1 0 1
(a) 2 ¨ ¸ (b) 2 ¨ ¸ (c) 2 ¨ ¸ (d) 2 ¨ ¸
¨1¸ ¨0¸ ¨ 1 ¸ ¨0¸
© ¹ © ¹ © ¹ © ¹

24. The eigenvalues of a matrix are i,–2i and 3i. The matrix is : [GATE 2007]
(a) unitary (b) anti-unitary (c) hermitian (d) anti-hermitian

ª5 4 º
25. The eigenvalues and eigenvectois:of the matrix « » are : [GATE 2007]
¬1 2 ¼

ª4º ª 1 º ª4º ª 1 º
(a) 6, 1 and « » , « » (b) 2, 5 and « » , « »
¬1 ¼ ¬ 1¼ ¬1 ¼ ¬ 1¼

ª4º ª 1 º ª4º ª 1 º
(c) 6, 1 and « » , « » (d) 2, 5 and « » , « »
¬1 ¼ ¬ 1¼ ¬1 ¼ ¬ 1¼
26. For arbitary matrices E, F, G and H, if EF  FE 0 , then Trace (EFGH) is equal to
[GATE 2008]
(a) Trace (HGFE) (b)Trace (E).Trace(F).Trace (G).Trace (H)
(c) Trace (GFEH) (d) Trace (EGHF)

ª aeiD bº
27. An unitary matrix « iE » is given, where a, b, c, d, D and E are real. The inverse of
«¬ ce d »¼
the matrix is : [GATE 2008]

ª aeiD ceiE º ª aeiD ceiE º ª ae iD bº ª aeiD ceiE º

(a) « » (b) « » (c) « iE » (d) « »
¬« b d ¼» ¬« b d ¼» «¬ ce d »¼ ¬« b d ¼»

ª cos T sin T º
28. The eigenvalues of the matrix « » are : [GATE 2008]
¬  sin T cos T ¼

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1 1
(a) 3 r i when T 45q (b) 3 r i when T 30q
2 2

1
(c) r1 since, the matrix is unitary (d) 1 r i when T 30q
2

ª0 i º
29. The eigenvaluess ofthe matrix A « » are : [GATE 2009]
¬ i 0¼
(a) real and distinct (b) complex and distinct
(c) complex and coinciding (d) real and coinciding
§ 2 3 0·
¨ ¸
3 2 0¸
30. The eigen values olthe matrix ¨ are : [GATE 2010]
¨0 0 1¸
© ¹
(a) 5, 2, –2 (b) –5, 1, –1 (c) 5, 1, –1 (d) –5, 1, 1
31. Two matrices A and B are said to be similar if B P 1 AP for some invertible matrix P..
Which ofthe following statements is NOT TRUE? [GATE 2011]
(a) Det A = Det B (b) Trate of A = Trice of B
(c) A and B have the same eigenyectors (d) A and B have the same eigenvalues
32. A 3 × 3 matrix has elements such that its trace is 11 and its determinant is 36. The
eigenvalues of the matrix are all known to be positive integers. The largest eigenvalue of
the matrix is : [GATE 2011]
(a) 18 (b) 12 (c) 9 (d) 6
§0 1 0·
¨ ¸
1 0 1¸
33. The eigenvalues of the matrix ¨ are : [GATE 2012]
¨0 1 0¸
© ¹
1 1
(a) 0, 1, 1 (b) 0,  2, 2 (c) , ,0, (d) 2,  2,0
2 2

§ 4  1 1 ·
¨ ¸
34. The degenerate eigenvalues of the matrix
M ¨ 1 4 1¸ is :
¨  1 1 4 ¸
© ¹
(your answer should be an integer) [GATE 2013]

1 ª 1 1  iº
35. The matrix A « » is : [GATE 2012]
3 ¬1  i 1 ¼

(a) orthogonal (b) symmetric (c) anti-symmetric (d) unitary

§ 1 1 0·
¨ ¸
1 1 1¸
36. The matrix ¨ [GATE 2010]
¨0 1 1¸
© ¹

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can be relateed by a similarity transformation to the matrix.

§1 1 1 · §2 1 0 · § 1 1 0 · § 1 1 0 ·
¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸
1 1 1 ¸ 1 1 1¸ 1 1 1 ¸ 1 1 1 ¸
(a) ¨ (b) ¨ (c) ¨ (d) ¨
¨1 1 0 ¸ ¨ ¸ ¨ ¸ ¨ ¸
© ¹ © 0 1 2 ¹ © 0 1 1¹ © 0 1 1¹

§1 0 0 ·
¨ ¸
37. Consider the matrix,
M ¨ 0 0 1 ¸
¨ 0 1 0 ¸
© ¹

A 3-dimensional basis formed by eigenvectors of M is : [TIFR 2011]

§ 2 ·§ 1 · §0· § 1 ·§ 0 · §0·
¨ ¸¨ ¸ ¨ ¸ ¨ ¸¨ ¸ ¨ ¸
1 1 1 1 1 1
(a) ¨ ¸¨ ¸ and ¨ ¸ (b) ¨ ¸¨ ¸ and ¨ ¸
¨ 1¸¨ 1 ¸ ¨ 1 ¸ ¨ 1¸¨ 1 ¸ ¨ 1 ¸
© ¹© ¹ © ¹ © ¹© ¹ © ¹

§ 0 ·§ 0 · §0· § 2 ·§ 0 · § 1 ·
¨ ¸¨ ¸ ¨ ¸ ¨ ¸¨ ¸ ¨ ¸
0 1 1 1 1 1
(c) ¨ ¸¨ ¸ and ¨ ¸ (d) ¨ ¸¨ ¸ and ¨ ¸
¨ 0 ¸¨ 1 ¸ ¨ 1¸ ¨ 1¸¨ 1 ¸ ¨1¸
© ¹© ¹ © ¹ © ¹© ¹ © ¹

§ 0 0 0 S / 4·
¨ ¸
0 0 S/4 0 ¸
A ¨
38. This trace of the real 4 × 4 matrix U exp A , where ¨ 0 S/4 0 0 ¸ is
¨ ¸
©S / 4 0 0 0 ¹
equal to : [TIFR 2011]

(a) 2 2 (b) S / 4 (c) exp iM for M 0, S (d) zero

39. Two different 2 × 2 matrices A and B are found to have the same eigenvalues. It is then
correct to state that A SBS 1 where S can be a : [TIFR 2012]
(a) traceless 2 × 2 matrix (b) Hermitian 2 × 2 matrix
(c) unitary 2 × 2 matrix (d) arbitrary 2 × 2 matrix
40. The product MN of two Hermitian matrices M and N is anti-Hermitian. It follows that :
[TIFR 2012]

(a) M , N 0 (b) M , N 0 (c) M † N (d) M † N 1

41. If the eigenvalues of a symmetric 3 × 3 matrix A are 0, 1, 3 and the corresponding
eigenvectors can be written as : [TIFR 2016]
ª1º ª 1 º ª 1 º
«1» , « 0 » , « 2 »
«» « » « »
«¬1»¼ ¬« 1¼» ¬« 1 ¼»
respectively, then the matrix A4 is

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ª 41 81 40 º ª 82 81 79 º

« 81 0 81» « 81 81 81»
(a) « » (b) « »
«¬ 40 81 41 »¼ «¬ 79 81 83 »¼

ª 14 27 13 º ª 14 13 27 º
« 27 54 27 » « 13 54 13»
(c) « » (d) « »
«¬ 13 27 14 »¼ «¬ 27 13 14 »¼

42. Denote the commutator of two matrices A and B by A, B AB  BA and the anti-
commutator by A, B AB  BA . If A, B 0 , we can write A, BC [TIFR 2017]

(a) B A, C (b) B A, C (c)  B A, C (d) A, C B

§100 2 x 0 ·
¨ ¸
x x ¸
The matrix ¨
0
43. , where x > 0, is known to have-two equal eigenvalues.
¨ ¸
© 0 x 100 2 ¹
Find the valit of x. [TIFR 2017]

44. A unitary matrix U is expanded in terms of a Hermitian matrix H, such that U eiSH /2 .
If we know that :

§ · 3
¨ 12 0 ¸ 2
¨ ¸
H ¨ 0 1 0 ¸
¨ 1¸
¨ 23 0  ¸
© 2¹

then U must be : [TIFR 2017]

§ i 1 3· § i  3· § 2i 3·
¨ 2 ¸ 0 § 1 0 3· 1
2 ¨ 2 2 ¸ ¨ 2 ¸
¨ 1 1 ¸ ¨ 0 ¨ ¸
i i 0 ¸ ¨ 1 2i 0 ¸
(c) ¨
0 2 0 ¸
(a) ¨ 2 2 ¸ (b) ¨ ¸ (d) ¨ ¸
¨¨ 3 ¸ ¨i 3 ¨ ¸
1 i ¹¸ © 2 0  2i ¹¸ © 3 0 1 ¹ ¨ 3
© 2 0 2i ¹¸
© 2 2

45. For an N × N matrix consisting of all ones,

(a) all eigenvalues = 1 (b) all eigenvalues = 0
(c) the eigenvalues are 1, 2, ....., N (d) one eigenvalue = N, the others = 0
46. The coordinate transformation [JEST 2013
x ' 0.8 x  0.6 y , y ' 0.6 x  0.8 y
represents
(a) a translation (b) a proper rotation (c) a reflection (d) none of the above

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§2 1·
47. Given a matrix M ¨ ¸ , which of the following represents cos SM 6 .[JEST 2016]
©1 2¹

1 §1 2· 3 § 1 1 · 3 §1 1 · 1§ 1 3·
(a) ¨ ¸ (b) ¨ ¸ (c) ¨ ¸ (d) 2 ¨¨ ¸
2 © 2 1¹ 4 © 1 1 ¹ 4 ©1 1 ¹ © 3 1 ¹¸

ª 0 16 º
48. The eigenvalues of the matrix « » are :
¬16 0 ¼
(a) 1 and –1 (b) 16 and 16 (c) 16 and –16 (d) 1 and 256

§0 1 0 0·
¨ ¸
0 0 1 0¸
A ¨
49. Consider the matrix, ¨0 0 0 1¸
¨ ¸
©1 0 0 0¹

the eigenvalues of A are, i 1

(a) 1, i, 1, i (b) 1, 1,0,0 (c) 1,1,1,1 (d) 1,1, i, i

50. Let P be a n × n diagonalizable matrix which satisfies the equations :

P2 P, Tr P n 1

Det (P) is :
(a) n (b) 0 (c) 1 (d) n–1
51. Let M be a 3 × 3 Hermitian matrix which satisfies the matrix equation :

M2 5M  6 I 0
Where I refers to the identity matrix. Which of the following are possible eigenvalues of
M
(a) {1, 2, 3} (b) {2, 2, 3} (c) {2, 3, 5} (d) {5, 5, 6}
52. Given the three matrices

§0 1· § 0 i · §1 0 ·
V1 ¨ ¸ , V2 ¨ ¸ , V3 ¨ ¸ and ª¬Vi , V j º¼ Vi V j  V
©1 0¹ ©i 0 ¹ © 0 1 ¹

then ª¬V1 , V2 , V3 º¼  ª¬V2 , V3 , V1 º¼  ª¬V3 , V1 , V2 º¼ is :

(a) V12  V22  V32 (b) V1  V 2  V3 (c) 0 (d) Identity

53. Given the three matrices

§0 1· § 0 i· §1 0 ·
Vi ¨ ¸ , V2 ¨ ¸ , V3 ¨ ¸
©1 0¹ © i 0 ¹ © 0 1 ¹

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Which of the following statements is true for all positive integers n and i = 1, 2, 3 ?

(a) Vin I (b) Vin Vi (c) Vi2n I (d) Vi2n Vi

54. The trace of a 2 × 2 matrix is 1 and its determinate is 1. Which ofthe following has to be
true ?
(a) One of the eigenvalues is 0 (b) One of the eigenvalue is 1
(c) Both of the eigenvalues are 1 (d) Neither of the eigenvalues is 1
55. Let M by a 3 × 3 Hermitain matrix which satisfies the matrix equation :

M 2  7 M  12I 0
where I refers to the identity matrix. What is the determinant of the matrix M given that
the trace is 10 ?
(a) 27 (b) 36 (c) 48 (d) 64
56. Consider a 3 × 3 matrix of the form :

§ a12 a1a2 a1a1 ·

¨ ¸
¨ a1a2 a22 a2 a3 ¸
¨ ¸
¨ a1a3 a2 a3 a32 ¸
© ¹

The number of zero eigenvalues for this matrix is :

(a) 0 (b) 1 (c) 2 (d) 3
57. Which one of the following is not hermitian ?

ª1 0 º ª0 i º ª0 1 º ª i 0º
(a) « » (b) « » (c) « » (d) « »
¬0 1 ¼ ¬ i 0¼ ¬1 0 ¼ ¬0 1 ¼

58. If A is an antisymmetric matrix and AT be the transpose of this matrix, then which one of
the following relations does not hold good?
(a) AT A (b) AAT AT A
(c) A2 is an antisymmetric matrix (d) A2 is a syrrgnetire matrix

§0 1 0·
¨ ¸
0 0 1¸
59. The eigenvalues of the matrix ¨ are given by O1 , O 2 and O 3 . Which one of the
¨1 0 0¸
© ¹
following statement is NOT TRUE?
(a) O1  O 2  O 3 0 (b) O1O 2  O 2 O 3  O3 O1 0
(c)All eigenvalues are real (d) Product of the eigenvalues is 1
§ 0 1 0 ·
¨ ¸
1 0 0¸
60. The matrix ¨ is :
¨0 0 1¸
© ¹

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(a) orthogonal (b) hermitian (c) symmetric (d) anti-symmetric

61. A given (n × n) nilpotent matrix A satisfies the equation Ak 0 for 1  k  n . Therefore

(a) Exactly k eigenvalues of A must be zero
(b) Exactly (n – k) eigenvalue-of A must be zero
(c) Every eigenvalue of A is zero
(d) A can have (n – 1) non-zero eigenvalues

62. If L(x) is a linear differential operator and y1 x , y2 x are two arbitrary functions. Then

(a) L x y1  y2 Ly1  Ly2 (b) L x y1  y2 y1Ly2  y2 Ly1

(c) L x y1 y2 Ly1  Ly2 (d) L x y1  y2 Ly1  Ly2

§ 0 1 1·
¨ ¸
M ¨ 1 3 5 ¸ , does
63. For which of the following values of a, the inverse of the matrix,
¨0 4 a ¸
© ¹
not exist?
(a) 4 (b) 4 (c) 0 (d) 1
64. Which one of the following matrices is orthogonal?

§ cos T sin T · § cos T  sin T ·

(a) ¨ ¸ (b) ¨ ¸
© sin T cos T ¹ © sin T cos T ¹

§  cos T sin T · § cos T sin T ·

(c) ¨ ¸ (d) ¨ ¸
©  sin T cos T ¹ ©  sin T  cos T ¹
65. The number of independent real parameters of a most general hermit ian matrix of order
4 is:
(a) 4 (b) 8 (c) 16 (d) 32

ª 0 0 i º
«0 2 0 »
66. The eigen values of the matrix « » are :
«¬ i 0 0 »¼

(a) Purely imaginary (b) Complex (c) Real (d) Zero

67. If a matrix A satisfies the condition A2 I and Tr A 0 (where I is a n × n identity

matrix), then
(a) The determinant of A must be 0 (b) The determinant of A must be +1
(c) A must be odd dimensional (d) A must be even dimensional
§1 0 0·
¨ ¸
A ¨0 a b¸
68. The matrix A, defined by is orthogonal if :
¨ 0 b a ¸
© ¹

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(a) a 1, b 1 (b) a 1/ 2, b 1 / 2

(c) a 1/ 2, b i / 2 (d) a 1, b 1

69. If A and B are two n × n matrices, then the trace of C A, B is :

(a) Tr AB (b) Tr A Tr B (c) 0 (d) Tr A  Tr B

§a b·
70. Consider the matrix M ¨ ¸ where a and c are real and b may be complex. If M can
©b c¹
be diagonalized by a matrix S, then S is :
(a) diagonal (b) symmetric (c) orthogonal (d) unitary
71. The trace and the determinant of a 2 × 2 matrix are denoted by T and D. The cigenvalues
of the matrix are given by :

T r T 2  4D T 2  4D T rD T
(a) (b) r (c) (d) r
2 2 2 2

1.(b) 2.(a) 3.(c) 4.(d) 5.(a) 6.(c) 7.(a)

8.(a) 9.(b) 10.(d) 11.(b) 12.(c) 13.(a) 14.(c)
15.(c) 16.(d) 17.(b) 18.(c) 19.(a) 20.(b) 21.(a)
22.(c) 23.(d) 24.(d) 25.(a) 26.(a) 27.(d) 28.(b)
29.(b) 30.(c) 31.(c) 32.(d) 33.(b) 34.(5) 35.(d)
36.(d) 37.(b) 38.(a) 39.(c) 40.(a) 41.(c) 42.(c)
43.() 44.(b) 45.(d) 46.(c) 47.(b) 48.(c) 49.(a)
50.(b) 51.(b) 52.(c) 53.(c) 54.(d) 55.(b) 56.(c)
57.(d) 58.(c) 59.(c) 60.(a) 61.(c) 62.(a) 63.(b)
64.(b) 65.(a) 66.(c) 67.(d) 68.(b) 69.(c) 70.(d)
71.(a)

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SECTION : COMPLEX ANALYSIS

³ dz z
2 z
e
1. The value of the integral , where C is an open contour in the complex z-plane as
C
shown in figure below: [CSIR June 2011]

5 5 5 5
(a) e (b) e  (c) e (d)   e
e e e e
2. Which of the following is an analytic function of the complex variable z x  iy in the
domain z  2 ? [CSIR June 2011]

7 4 3
(a) 3  x  iy (b) 1  x  iy 7  x  iy

4 3
(c) 1  2 x  iy 3  x  iy (d) x  iy  1 1/2

3. The first few terms in the Taylor series expansion of the function f x sin x around

S
x are : [CSIR Dec. 2011]
4

1 ª § º
2 3
S· 1 § S· § S·
(a) «1  ¨ x  ¸  ¨ x  ¸  ¨ x  ¸  ......»
2 «¬ © 4 ¹ 2! © 4¹ 3! © 4¹ »¼

1 ª § º
2 3
S· 1 § S· § S·
(b) «1  ¨ x  ¸  ¨ x  ¸  ¨ x  ¸  ......»
2 «¬ © 4 ¹ 2! © 4¹ 3! © 4¹ »¼

ª§ S·  § S·
3 º
«
(c) ¨ x  ¸  ¨ x  ¸  ......»
«¬© 4 ¹ 3! © 4¹ »¼

1 ª x 2 x3 º
(d) «1  x   ......»
2 ¬« 2! 3! ¼»

1
4. The first few terms in the Laurent series for z  1 z  2 in the region 1 d z d 2 and

around z 1 is : [CSIR Dec. 2011]

1ª ª 2 3 º
1  z  z 2
 ...º «1  z  z  z  ...» 1
 z  1 z
2
 1 z
3
 ...
(a) 2 ¬ ¼« 2 4 (b)
¬ 8 »¼ 1 z

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1 ª 1 1 ºª 2 4 º
(c) 2 «
1   2  ...» «1   2  ...» (d) 2 1  z  5 1  z 2
 7 1 z
3
 ...
z ¬ z z ¼¬ z z ¼

f
1 § St ·
5. The value of the integral ³ 2
cos ¨ 2 ¸ dt
© 2R ¹
[CSIR June 2012]
f t  R

2S S S 2S
(a)  (b)  (c) (d)
R R R R

1 2
6. Let u x, y x x  y 2 be the real part of an analytic function f z of the complex
2
variable z x  iy the imaginary part of f z is : [CSIR June 2012]

(a) y | xy (b) xy (c) y (d) y 2  x 2

7. The taylor series expansion of the function ln cosh x , where x is real, about point x = 0
starts with the following terms : [CSIR Dec. 2012]

1 2 1 4 1 2 1 4 1 2 1 4 1 2 1 4
(a)  x  x  ... (b) x  x  ... (c)  x  x  ... (d) x  x  ...
2 12 2 12 2 6 2 6

z2
8. The value of the integrall v³ z 2  5z  6 dz , where C is closed contour defined by the
C

equation 2 z  5 0 , traversed in the anti-clockwise direction is, [CSIR Dec. 2012]

(a) 16Si (b) 16 Si (c) 8Si (d) 2Si

9. With z x  iy , which of the following functions f x, y is NOT a complex analytic

funetion of z? [CSIR June 2013]

3 7
(a) x  iy  8 4  x 2  y 2  2ixy (b) x  iy 7
1  x  iy
3

5 4 6
(c) x 2  y 2  2ixy  3 (d) 1  x  iy 2  x  iy

10. Which of the following functions cannot be the real part of a complex analytic function of
z  x  iy ? [CSIR Dee 2013]

(a) x 2 y (b) x 2  y 2 (c) x 2  3 xy 2 (d) 3x 2 y  y  y 3

f f
dx S dx
11. Given that the integral ³ y 2  x2 ,
2 y the value of
³ y 2  x2 is : [CSIR Dee 2013]
0 0

S S S S
(a) 3 (b) 3 (c) 3 (d)
y 4y 8y 2 y3

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1 dz
12. If C is the contour defined by z
2
, the value of the integral v³ C sin 2 z is :

[CSIR Dee 2013]

(a) f (b) 2Si (c) 0 (d) Si

f
sin 2x
13. The principal value of the integral ³ x3
dx is : [CSIR June 2014]
f

(a) 2S (b) S (c) S (d) 2S

14. The Laurent series expansion of the function f z e z  e1/ S about z = 0 is :

[CSIR June 2014]

f f
zn § 1 ·1
(a) ¦ for all z  f (b) ¦ ¨© z n  z n ¹¸ n! only if 0  z  1
n f n ! n 0

f f
§ 1 ·1 zn
(c) ¦ ¨© z n  z n ¹¸ n! for all 0  z  f (d) ¦ only if z  1
n 0 n f n !

1
15. Consider the function f z ln 1  z of a complex variable
2

z reiT r t 0, f  T  f . The singularities of f z are as follows:[CSIR Dee. 2014]

(a) branch points at z 1 and z f ; and a pole at z 0 only for 0 d T  2S

(b) branch points at z 1 and z f ; and a pole at z 0 only for T , other than 0 d T  2 S
(c) branch points at z 1 and z f ; and a pole at z 0 only for T
(d) branch points at z 0, z 1 and z f
f
1
16. The value of the integral ³ 1  x 4 dx is : [CSIR June 2015]
f

(a) S 2 (b) S 2 (c) 2S (d) 2S

z
17. The function , of a complex variable z has : [CSIR Dec. 2015]
sin Sz 2
(a) a simple pole at 0 and poles of order 2 at z r n for n = 1, 2, 3...

(b) a simple pole at 0 and poles of order 2 at z r n and z ri n for n = 1, 2, 3....

(c) poles of order 2 at z r n for n 0, 1, 2, 3....

(d) poles of order 2 at z r n for n = 0, 1, 2, 3....

1
18. The radius of convergence of the Taylor series expansion of the function cosh x around
x = 0, is : [CSIR June 2015]

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(a) f (b) S (c) S / 2 (d) 1

1 e4 z  1
19. The value of the contour integral 2Si v³ cosh z  2sinh z dz around the unit circle C
C
traversed in the anti-clockwise direction, is : [CSIR June 2016]

8 §1·
(a) 0 (b) 2 (c)  (d)  tanh ¨ ¸
3 ©2¹

³z
10
dz
20. The value of the integral , where C is the unit circle with the origin as the centre
C
is: [GATE 2001]

(a) 0 (b) z11 / 11 (c) 2Siz11 / 11 (d) 1/11

sin z
21. The value of the residue of is : [GATE 2001]
z6

1 1 2Si 2Si
(a)  (b) (c) (d) 
5! 5! 5! 5!

22. If a function f z u x, y  iv x, y of the complex variable z x  iy , where x, y , u

and v are real, is analytic in a domain D of z, then which of the following is true?
[GATE 2002]

wu wv wu wv wu wv
(a) (b) wx and 
wx wx wy wy wx

wu wv wu wv w 2u w2v
(c) and wy  (d)
wx wx wy wxwy wxwy

³ dz / z
2
23. The value of the integral , where z is a complex variable and C is the unit circle
C
with the origin as its centre, is: [GATE 2003]
(a) 0 (b) 2Si (c) 4Si (d) 4Si

3  4i
24. The inverse of the complex number is : [GATE 2004]
3  4i

7 24 7 24 7 24 7 24
(a) i (b)  i (c) i (d)  i
25 25 25 25 25 25 25 25

dz
25. The value of ³ z  a 2 , wherc C is a unit circle (anti clockwise) centered at the origin
2
C

in the complex z-plane is :

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1 S 1
(a) S for a 2 (b) zero for a (c) 4S for a 2 (d) and a
2 2 2

dz
26. The value of the integral ³ z  3 , where C is a circle (anticlockwise) with z 4 is :
C

(a) 0 (b) Si (c) 2Si (d) 4Si [GATE 05]

27. All solutions of the equation e z 3 are :

(a) inS ln 3, n r1, r2...... (b) ln 3  i 2n  1 S, n 0, r1, r2......

(c) ln 3  i 2nS, n 0, r1, r2...... (d) i3nS, n r1, r2......

e2 z
28. The value of v³ z 1
4
dz , where C is a circle defined by z 3 is: [GATE 2006]
C

8Si 2 8Si 1 8Si 8Si 2

(a) e (b) e (c) e (d) e
3 3 3 3

dz
29. The contour integral v³ z 2  a 2 , is to be valuated on a circle of radius 2a centered at the
origin. It will have contributions only from the points. [GATE 2007]

1 i 1 i
(a) a and  a (b) ia and ia
2 2

1 i 1 i 1 i 1 i 1 i 1 i
(c) ia , ia, a and  a (d) a a, a and  a
2 2 2 2 2 2

30. If
I v³ ln z dz , where C is the unit circle taken anticlockwise and ln z is the principal
C
branch ofthe logarithmic function, which of the following is correct? [GATE 2008]
(a) I 0 by residue theorem
(b) I is not defined since, ln z is branch cut
(c) I z 0

(d) v³ ln z 2 dz 2I
C

i
3l. The value of ³S z  1 dz is [GATE 2008]
i

(a) 0 (b) 2Si (c) 2Si (d) 1  2i S

ez 3
32. The value of the integral ³ z 2  3z  2 dz , where the contour C is the circle z
2
is :
C

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[GATE 2009]
(a) 2 Sie (b) Sie (c) 2Sie (d) Sie

e z sin z
33. The value of the integral v³ z2
dz , where the contour C is the unit circle: z  2 1,
C
is : [GATE 2010]
(a) 2Si (b) 4Si (c) Si (d) 0

e 2
 e 2
34. For the complex function, f z , which of the following statement is correct?
sin z

[GATE 2010]
(a) z = 0 is a branch point (b) z = 0 is a pole of order one
(c) z = 0 is a removable singularity (d) z = 0 is an essential singularity
Common data for Q. 35 and Q. 36

z sin z
f z
Consider a function zS
2 of a complex variable z. [GATE 2010]

35. Which of the following statements is TRUE for the function f z ?

(a) f z is analytic everywhere. in the complex plane

(b) f z has a zero at z S

(c) f z has a pole oforder 2 at z S

(d) f z has a simple pole at z S

36. Consider a counterclockwise circular contour z 1 about the origin. The integral

v³ f z dz over contour is : [GATE 2012]

(a) iS (b) zero (c) 1  2iS (d) 2iS

v³ e
1/z
dz
37. The value of the integral using the contour C of circle with unit radius z 1 , is:
C

(a) 0 (b) 1 2 Si (c) 1  2Si (d) 2Si [GATE 12]

16 z
f z
38. For the function z  3 z 1
2 , the residue at the pole z 1 ......

(Your answer should be an integer) [GATE 2013]

z2
39. The value of the integral v³ e z  1 dz , where C is the circle z 4 is : [GATE 2014]
C

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(a) 2Si (b) 2S2i (c) 4S3i (d) 4S2i

40. Consider w f z u x, y  iv x, y to be an analytic function in a domain D. Which

one of the following options is NOT correct? [GATE 2015]

(a) u x, y satisfies Laplace equation in D

(b) v x, y satisfies Laplace equation in D

z2

(c) ³f z dz is dependent on the choice of the contour between z and z in D

1 2
z1

(d) f z can be Taylor expanded in D

1
f z
§ 1·
41. Consider a coinptex function z ¨ z  ¸ cos z S . Which nne of the following
© 2¹
statements is correct? (GATE 2015]

1
(a) f z has simple poles at z = 0 and z 
2

1
(b) f z has a second order pole- at z 
2

(c) f z has infinite number of second order poles

(d) f z has all simple poles

42. Which of the following is an analytic function of z everywhere in the complex plane?
[GATE 2016]
2
(a) z 2 (b) z < (c) z 2 (d) z

43. If z x  iy then the function f x, y 1  x  y 1  x  y  a x 2  y 2  1  2iy

1  x  ax where a is real parameter, is analytic in the complex z plane if a is equal to :

(a) –1 (b) +1 (c) 0 (d) i [GATE 2013]

f
dx
44. The integral ³ 4  x4 evalautes to :
0

S S S
(a) S (b) (c) (d)
2 4 8
2S
dT
45. The integral ³ 1  2a cos T  a 2 where 0  a  1 , evaluates to : [TIFR 2015]
0

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2S 2S 4S
(a) 2 (b) 2 (c) 2S (d)
1 a 1 a 1  a2

sin z
46. The value of the integral v³ z6
dz , where C is the circle with centre z = 0 and radius 1
C
unit : [TIFR 2017]

iS iS iS
(a) iS (b) (c) (d) 
120 60 6

f
dx
47. The value of the integral ³ x4  4 , is : [TIFR 2017]
0

S S S
(a) S (b) (c) (d)
2 4 8

f ln x
48.
³
The value of integral 0 2 2
dx
is : [JEST 2012]
x 1

S S S
(a) 0 (b)  (c)  (d)
4 2 2

Re z 2  lm z 2
49. Compute lim [JEST 2013]
z of z2
(a) the limit does not exist (b) 1
(c) –i (d) –1
50. The value of integral

sin z
l v³ c 2 z  S dz
with c a is circle z 2 , is [JEST 2014]

(a) 0 (2) 2Si (c) Si (d) Si

z10  1
51. The value of limit lim is equal to : [JEST 2014]
z oi z6  1
(a) 1 (b) 0 (c) –10/3 (d) 5/3

52. Given an analytic function f x, y I x, y  i\ x, y where I x, y x2  4 x  y 2 

2 y . If C is a constant, then which of the following relations is true? [JEST 2015]

(a) \ x, y x2 y | 4 y  C (b) \ x, y 2 xy  2 x  C

(c) \ x, y 2 xy  4 y  2 x  C (d) \ x, y x2 y  2 x  C

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f
ln x
53. The value of the integral ³ x2  1
dx
is :
0

S2 S2
(a) (b) (c) S2 (d) 0
4 2

54. Given i 1 , then is :

(a) Purely real (b) Purely imaginary
(c) Of the form x  iy with x z 0, y z 0 (d)Not defined

55. Let z1 and z2 be two non-zero complex numbers. If

z1  z2 z1  z2

Which of the following is true?

(a) Re z1z2  0 , and Im z1z2 0 (b) Re z1z2 ! 0 and Im z1z2 0

(c) Re z1z2 ! 0 and Im z1z2 ! 0 (d) Re z1z2  0 and Im z1z2  0

56. If 'z' is a complex number, them we can express sin 1 z as :

2 2
(a) iAn iz r 1  z (b) iAn z r z  1

2 2
(c) An z  z  1 (d) An z  z  1

dz
57. The value of v³ z  3 if C is the circle z  2 5 is :
C

(a) Si (b) 2Si (c) 0 (d) 2S

1
2
58. The complex function f z z 1 has singular point at z I which is :
e

(a) a pole of order 1 (b) a pole o forder 2

(c) an isolated essential singularity (d) a non-isolated essential singularity

59. The function tan 1 / z does not have a pole at z =

(a) S / 2 (b) 2 / S (c) 2 / 3S (d) 2 / 3S

60. The real part ofa complex function f z , analytic in a region is given by u x, y x2  y 2 .
If the function vainshes at z = 0,the imaginary part of the function is :

(a) 2xy (b) –2xy (c) x 2  y 2 (d) y 2  x 2

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z
61. The residue of the function, f z 2 z  1 5  z at z 1 / 2 is :

(a) 1/11 (b) –1/11 (c) 1/22 (d) –1/22

62. Two complex numbers Z1 and Z 2 are given to satisfy Z1 Z 2 necessarily follows that:

(a) Z1 Z2 (b) Z1 e iI Z 2 (c) Z1 Z 2 (d) Z1 Z 2 1

63. Two functions of a complex vatiable z x  iy are given as (i) z and (ii) z 2 , which of
the following statements is correct?
(a) Both (i) and (ii) are analytic (b) (i) is analytic and (ii) is nonanalytic
(c) Both are nonanalytic (d) (i) is nonanalytic and (ii) is analytic

§1·
64. A function f Z of a complex vartiable f Z is given as f Z Z exp ¨ ¸ . At Z 0
©Z¹
(a) It has no singularity
(b) It has a simple pole and residue is 1
(c) It has an essential sigulatity and residue is 1/2
(d) It has a pole of order 2 and residue 1/6

65. A function f Z of a complex variable Z is analytic everywhere in the complex plane

except at Z Z 0 . Two contours C1 and C2 around Z 0 are shown in the figure.

f Z dZ f Z dZ
The values of the integrals I1 v³ Z  Z0
,I 2 v³ Z  Z0 are :
C1 C2

(a) I1 0, I 2 2Sif Z 0 (b) I1 2Sif Z 0 , I 2 0

(c) I1 Sif Z 0 , I 2 Sif Z 0 (d) I1 2Sif Z 0 , I 2 2 Sif Z 0

z3
66. The value o f the integral v³ z 1
2
dz , where C is the circle z 4 is :
C

(a) 6Si (b) 6Si (c) 2Si (d) Zero

1
f z
67. The leading term in Laurent expansion of z 1 z
2 around z = 0 is :

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1 1 1
(a) 3 (b) 2 (c) (d) 1
z z z

e2 z  3
68. The value of v³ z2 where c is a unit circle with centre at origin is :
c

5 2
(a) 5e2 (b) e (c) 0 (d) 10Sie 2
2Si

Z 2  a 2  2aZ
69. If f z 3 then Z a is :
Z a Z b

(a) Pole of order 1 (b) Pole of order 2

(c) Pole of order 3 (d) An essential singularity
70. Which of the of the following functions is not analytic?

(a) f z z (b) f z z2 (c) f z cos z (d) f z sin z

cos Sz
71. The value of the integral v³ z z  1 where C is the circle z  1 3 is :
c

(a) 0 (b) Si (c) 2Si (d) 4Si

1.(c) 2.(b) 3.(b) 4.(b) 5.(b) 6.(a) 7.(b)

8.(a) 9.(d) 10.(a) 11.(b) 12.(c) 13.(a) 14.(c)
15.(b) 16.(a) 17.(b) 18.(c) 19.(c) 20.(a) 21.(b)
22.(b) 23.(a) 24.(d) 25.(b) 26.(c) 27.(b) 28.(a)
29.(b) 30.(a) 31.(b) 32.(c) 33.(d) 34.(c) 35.(d)
36.(b) 37.(d) 38.(3) 39.(c) 40.(c) 41.(b) 42.(a)
43.(a) 44.(d) 45.(a) 46.(c) 47.(d) 48.(b) 49.(a)
50.(c) 51.(d) 52.(c) 53.(d) 54.(a) 55.(b) 56.(a)
57.(b) 58.(c) 59.(a) 60.(a) 61.(d) 62.(b) 63.(d)
64.(c) 65.(b) 66.(a) 67.(c) 68.(c) 69.(a) 70.(a)
71.(d)

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SECTION : DEFERENTIAL EQUATION & SPECIAL FUNCTIONS

1. Let pn x (where n = 0,1,2, .....) be ploynomiat of order 'n' with real co-efficients defined
4

in the interval 2 d x d 4 . If ³ pm x pn x dx Gmn , then [CSIR June 2011]

2

1 3 1
(a) p0 x and p1 x 3  x (b) p0 x and p1 x 3 3 x
2 2 2

1 3 1 3
(c) p0 x and p1 x 3 x (d) p0 x and p1 x 3 x
2 2 2 2

f
2. The genarating function F x, t ¦ Pn x t n for Legendre polynomials P x is:
n
n 0

1/2
F x, t 1  2 xt  t 2 . The value of P3 1 is: [CSIR Dec. 2011]

(a) 5/2 (b) 3/2 (c) 1 (d) –1

3. Let x1 t and x2 t be two linearly independent solutions of the differential equation

d 2x dx dx2 t dx1 t
2
2 f t x 0 . Let, w t x1 t , x2 t . If w 0 1 then
dt dt dt dt
w 0 1 is given by [CSIR Dec. 2011]

(a) 1 (b) e2 (c) 1 / e (d) 1 / e2

4. Let y x be a continuous real function in the range of 0 to 2S , satisfying the

d2y dy § S· dy
inhomogeneous differential equation: sin x 2
 cos x G ¨ x  ¸ . The value of
dx dx © 2¹ dx
S
at the point x . [CSIR June 2012]
2
(a) is continuous (b) has a discontinuity of 3
(c) has a discontinuity of 1/3 (d) has a discontinuity of 1

d2 f
5. A function f x obeys the differential equation  3  4i f 0 and satisfies the
dx 2
condition f 0 1 and f x o 0 as xo f . The value of f S is : [CSIR June 2012]

(a) e2 S (b) e2 S (c) e2 S (d) e2S

6. The graph of the function f x shown below is best described by : [CSIR Dec. 2012]

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(a) The bessel function J 0 x (b) cos x

cos x
(c) e x cos x (d)
x

f
tn
¦ Hn et  2tx , the value of H 0 is
2

7. Given that x [CSIR June 2013]

n! 4
n 0

(a) 12 (b) 6 (c) 24 (d) –6

dx
8. The solution of the differential equation x 2 with the initial condition x 0 1 will
dt
blow up as t tend to : [CSIR June 2013]
(a) 1 (b) 2 (c) 1/2 (d) f
9. Consider the differential equation : [CSIR June 2014]

d 2x dx
2
2 x 0
dt dt

with intial conditions x 0 0 and x 0 1 . The solution x t attains its maximum value
when 't' is :
(a) 1/2 (b) 1 (c) 2 (d) f
f 1/2
10. Given ¦ Pn x tn 1  2 xt  t 2 for t  1 , the value of P5 1 is : [CSIR Dec.
n 0

(a) 0.26 (b) 1 (c) 0.5 (d) 1 2014]

f n 2 n1
1 § x·
11. The function f x ¦ n! n  1 ! 2 ¨© 2 ¹¸ satisfies the differential equation
n 0

[CSIR Dec. 2014]

2 d2 f df d2 f df
(a) x 2
x  x2  1 f 0 (b) x
2
2
 2x  x2 1 f 0
dx dx dx dx

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2 d2 f df d2 f df
(c) x 2
x  x2  1 f 0 (d) x
2
2
x  x2  1 f 0
dx dx dx dx

d 2x dx
12. Consider the differential equation 2
3  2x 0 . If x 0 at t 0 and x 1 at
dt dt
t 1 , the value of x at t 2 , will be : [CSIR June 2015]

(a) e2  1 (b) e2  e (c) e  2 (d) 2e

dx
I3. The solution of the differential equation 2 1  x 2 , with initial condition x = 0 at t =
dt
0 is : [CSIR Dec. 2015]

­ S ­ S
°° sin 2t , 0 d t d 4 °°sin 2t , 0 d t  4
x ® x ®
(a) °sinh 2t , S (b) ° 1, S
tt tt
°̄ 4 °̄ 2

­ S
°°sin 2t , 0 d t  4
x ® (d) x 1  cos 2t , t t 0
(c) ° 1, S
tt
°̄ 4

14. The Hermite polynomial H n x satisfies the differential equation :

d 2H n dH n
 2x  2nH n x 0 [CSIR Dec. 2015]
dx 2 dx

f
1
The corresponding generating function G t , x ¦ n! H n x t n satisfies the equation :
n 0

w 2G wG wG w 2G wG wG
(a)  2x  2t 0 (b)  2x  2t 2 0
wx 2 wx wt wx 2 wx wt

w 2G wG wG w 2G wG w 2G
(c)  2x 2 0 (d)  2x 2 0
wx 2 wx wt wx 2 wx wxwt

15. A ball of mass m is dropped from a tall building with zero initial velocity. In addition to
gravity, the ball experiences a damping force of the form Jv , where v is its instantaneous
velocity and J is a constant. Given the values m 10kg / s and g 10 m / s 2 , the distance
travelled (in metres) in time t in seconds, is [CSIR Dec. 2016]
t t 2 t
(a) 10 t  1  e (b) 10 t  1  e (c) 5t  1  e (d) 5t 2

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 36

dy
16. Consider the linear differential equation xy . If y 2 at x 0 , then the value of y at
dx
x 2 is : [GATE 2016]

(a) e2 (b) 2e2 (c) e2 (d) 2e2

1 m2
17. A function y z satisfies the ordinary differential equation y  y 2 y 0 , where m
2 z
= 0,1,2,3......
Consider the four statements P, Q, R, S as given below: [GATE 2015
P : z m and z  m are linearly independent solutions for all values of m.

Q : z m and z  m are linearly.independent solutions for all values of m > 0.

R : In z and 1 are linearly independent solutions for rn = 0.
S : z m and ln z are linearly independent solutions for all values of m.
The correct option for the combination of valid statements is
(a) P, R and S only (b) P and R only (c) Q and R only (d) R and S only

d2y
18. The solution of the differential equation y 0
dt 2

subject to the boundary conditions y 0 1 and y f 0 is : [GATE 2014]

(a) cos t  sin t (b) cosh t  sinh t (c) cos t  sin t (d) cosh t  sinh t

dy x
19. The solutions to the differential equation dx 
y  1 are a family of : [GATE 2011]

(a) Circles with different radii (b) Circles with different centres
(c) Straight tines with different slopes
(d) Straight lines with different intercepts on the y-axis

d2y
20. The solution of the differential equation for y t y 2 cosh t , subject to the
dt 2
dy
initial conditions y 0 0 and 0 is : [GATE 2010]
dt t 0

1
(a) cosh t  t sinh t (b)  sinh t  t cosh t
2

(c) t cosh t (d) t sinh t

21. Given the recurrence relation for the Legendre polynomials :

2n  1 xPn x n  1 Pn 1 x  nPn 1 x

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 37

Which of the following integrals has a non zero value?

1 1
³x ³ xPn
2
(a) Pn x Pn 1 x dx (b) x Pn  2 x dx
1 1

1 1
2
³ ³x
2
(c) x ª¬ Pn x º¼ dx (d) Pn x Pn  2 x dx
1 1

22. Which one of the following curves gives the solution of the differential equation
dx
k1  k2 x k3 , where k1 , k2 and k3 are positive real constants with initial conditions
dt
x 0 at t 0 ? [GATE 2009]

(a) (b)

(c) (d)

23. Consider the bessel equation:

d2y 1 dy
2
 y 0 v 0
dz z dz

Which of the following statements is correct? [GATE 2008]

(a) Equation has regular singular points at z = 0 and z f
(b) Equation has 2 linearly independent solutions that are entire
(c) Equation has an entire solution and a second linearly independent solution is singular at
z=0
(d) Limit z o f , taken along X- axis, exists for both the linearly independent solutions
2
24. The points, where the series solution of the Legendre differential equation 1  x

d2y dy 3 § 3 ·
2
 2x  ¨  1¸ y 0 will diverge, are located at : [GATE 2007]
dx dx 2 © 2 ¹

3 5
(a) 0 and 1 (b) 0 and –1 (c) –1 and 1 (d) and
2 2

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dy
25. Solution of the differential equation x y x 4 , with the boundary condition that y 1 at
dx
x 1 , is : [GATE 2005]

x4 4x 4x4 1 x4 4
(a) y 4
5x  4 (b) y  (c) y  (d) y 
5 5 5 5x 5 5x
Statement for Linked Answer Q. 26 and Q. 27 :

d2y dy
For the differential equaiton 2
2 y 0 [GATE 2005]
dx dx

26. One of the solutions is :

(c) x  x
2 2
(a) e x (b) ln x (d) x x

27. The second linearly independent solution is :

(a) x x (b) xe x (c) x2e x (d) x 2e x

28. Consider the differential eqtuation d 2 x / dt 2  2dx / dt  x 0 . At time t 0 , it is given

that x 1 and dx / dt 0 . At t 1 , the value of x is given by : [GATE 2003]
(a) 1 / e (b) 2 / e (c) 1 (d) 3 / e
29. The solution of the differential equation,

d2y x dy x
1 x 2
x y x 0 is : [GATE 2003]
dx dx

(a) Ax 2  B (b) Ax  Be  x (c) Ax  Be x (d) Ax  Bx 2

where A and B are constants.
30. The solution of the system ofdifferential equations,

dy dz
y  z and 4 y  z
dx dx
is given by (for Aand B are arbitrary constants) [GATE 2001]

(a) y x Ae3 x  Be x ; z x 2 Ae3 x  2 Be  x

(b) y x Ae3 x  Be x ; z x 2 Ae3 x  2 Be  x

(c) y x Ae3 x  Be x ; z x 2 Ae3 x  2 Be  x

(d) y x Ae3 x  Be x ; z x 2 Ae3 x  2 Be x

d2y dy
31. The differential equation 2
2 y 0 has the complete solution, in terms of arbitrary
dx dx
constants A and B as : [TIFR 2013]

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(a) A exp x  Bx exp x (b) A exp x  B exp  x

(c) A exp x  Bx exp  x (d) x A exp x  B exp  x

32. The solution of the integral equation f x x  ³ dt f t has the graphical form :
0

[TIFR 2014]

(a) (b)

(c) (d)

33. Consider the differential equation :

d2y § dy ·
2
4 ¨ y  ¸
dx © dx ¹

1
with the boundary conditions that y x 0 at x . When plotted as a function of x, for
5
x t 0 , we can say with certainity that the value of y : [TIFR 2015]
(a) first increases, then decreases to zero
(b) first decreases, then increases to zero
(c) has an extremum in the range 0  x  1
(d) oscillates from positive to neagtive with amplitude decreasing to zero
34. The generating function for a set of polynomials in x is given by :
1
f x, t 1  2 xt  t 2

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The third polynomial (order x 2 ) in this set is [TIFR 2015]

(a) 2 x 2  1 (b) 4 x 2  1 (c) 2x 2  x (d) 4 x 2  1

dy
35. The function y x satisfies the differential equation x y ln y  ln x  1 with the
dx
initial condition y 1 3 . What will he the value of y 3 ? [TIFR 2015]

36. Write down x t , where x t is the solution of the following differential equation :

§d ·§ d ·
¨ dt  2 ¸¨ dt  1¸ x 1
© ¹© ¹

dx 1
with the boundary conditions 0, x t  [TIFR 2017]
dt t t 0 2
0

37. Consider the differential equation :

dG x
 kG x G x
dx
where k is a constant. Which of the following statement is true? [JEST 2013]
(a) Both G(x) and G'(x) are continuous at x = 0
(b) G(x) is continuous at x = 0 but G'(x) is not
(c) G(x) is discontinuous at x = 0
(d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k.

39. What are the solutions to f " x  2 f ' x  f x 0? [JEST 2015]

(a) c1e x / x (b) c1x  c2 / x (c) c1 xe x  c2 (d) c1e x  c2 xe x

39. Given the equations H '  aG and G ' bH , G can have oscillatory solutions :
(a) a, b (b) for no choice of a, b
(c) If a  0 and b ! 0 (d) a  0 and b  0
40. Consider a forced harmonic oscilleator which obeys the differential equation :

d2y
y sin t .
dt 2
Which one of the following is thesolution of the differential equationl with initial condition
y (0) = 0?

t
(a) y t 6sin t (b) y t 16sin t  cos t
2

t t
(c) y t 12sin t  cos t (d) y t 6cos t  cos t
2 2

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41. The solution of the differential equation

dy 2 y
 0
dx 3 x
with the condition y = 2 when x = 3 is given by :

(a) x 2 y 3 72 (b) x 2 y 3 108 (c) x 3 y 2 108 (d) x3 y 2 72

dy 1
42. From the solution of the differential equation :
dx 1  x2
what can you regarding the following series ?

1 1 1 1
1      ...
3 5 7 9
(a) The series is divergent (b) The series is absolutely convergent

S
(c) The series converges to (d) The series converges to 0
4
43. Consider the system of differential equations

dx dy
 y, x
dt dt
Plotting the various solutions in the x, y-plane one obtains :
(a) Hyperbola (b) Parabola (c) Circles (d) Straight lines

f
44. If f x x4  2 ¦ an Pn x , then a is :
3
n 0

1
(a) (b) 1 (c) –1 (d) 0
2

J1 x
2
45. The value of J x is :
1
2

(a) 0 (b) tan x (c) cot x (d) tanh x

46. Legendre’s differential equation :

d2y dy
1  x2 2
 2x  n n 1 y 0
dx dx

has an ordinary point at x =

(a) –1 (b) 0 (c) 1 (d) f

2 d2y dy
47. For the differential equation: z 2
z y 0
dz dz

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(a) z f is an ordinary point

(b) z 0 is a regular singular and z f is an irregular singular point
(c) z 0 and z f both g-e regular singular points
(d) z 0 is an irregular singular point

48. The Legendre polynomial Pn x , where 1 d x d 1

1
³ dxPn
2
(a) Is singular at x r1 (b) Satisfies x 1
1

1
³ dxPn
2
(c) Satisfies x 0 for n t 1 (d) Is always an even function of x
1

49. In order to obtain the solution of the initial value problem of the equation ofmotion,

d 2x dx
2
5  16 x 0 . How many initial conditions are required?
dt dt

(a) 0 (b) 1 (c) 2 (d) 3

2 d 2x dy
50. For the Bessel equation, x 2
x  x2  n2 y 0 , where n is an integer, the
dy dx
maximum number of linearly independent solutions, well-defined at x = 0 is :
(a) zero (b) one (c) two (d) three

d4y
51. The general solution of the linear differential equation, 0 is, u x equal to :
dx 4

(a) 0 (b) c3t  c4

(c) c2t 2  c3t  c4 (d) c2t 3  c2 t 2  c3t  c4

52. The regular singular points of the associated Legendre differential equationjn the finite
domain are
(a) 1, –1 (b) 0 (c) f (d) 2

d2y
53. The Wronskian of the linear independent solutions of the differential equation  4y 0
dx 2
is :
(a) Constant (b) 0 (c) f (d) Undetermined

3 x2  1
54. Given the Legendre polynormals P0 x 1, P1 x x and P2 x then the
2
2
polynomial 3 x  x  1 can be expressed as :

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(a) P2 x  P1 x (b) 2P2 x  P1 x

(c) P2 x  P1 x (d) 2P2 x  P1 x  P0 x

dy
55. The solution ofthe differential equation x  y 1 , given the condition y = 0 at x = 1 is:
dx

x2  1  x x 1
(a) y 1  x 2 An x (b) y e (c) y (d) y sinh x
x x

1.(d) 2.(d) 3.(d) 4.(d) 5.(c) 6.(a) 7.(a)

8.(a) 9.(b) 10.(d) 11.(c) 12.(b) 13.(c) 14.(a)
15.(b) 16.(d) 17.(c) 18.(d) 19.(a) 20.(d) 21.(d)
22.(a) 23.(c,d) 24.(c) 25.(d) 26.(a) 27.(b) 28.(b)
29.(b) 30.(a) 31.(a) 32.(b) 33.(c) 34.(d) 35.(81)
36.() 37.(c) 38.(d) 39.(d) 40.(c) 41.(a) 42.(c)
43.(c) 44.(d) 45.(b) 46.(b) 47.(a) 48.(c) 49.(c)
50.(b) 51.(d) 52.(a) 53.(a) 54.(b) 55.(c)

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SECOND : FOURIER SERIES, FOURIER TRANSFORM, LAPLACE TRANSFORM

1. Consider the periodic function f t with time period T as shown hi the fugure below :

[CSIR June 2015]

1
The spikes are located at t 2n  1 , where n = 0, +1, +2.... are Dirac delta functions
2
f
of strength r1 . The amplitudes an in the fourier expansion f t ¦ an ei 2 Sn /T are
n f
given by :

n 1 nS nS
(a) 1 (b) sin (c) i sin (d) nS
nS 2 2

2. Fourier Transform of the derivative of the dirac G function namely G x is proportional

to : [CSIR Dec. 2013]
(a) 0 (b) 1 (c) sin k (d) ik

3. The graph of a real periodic function f x for range f, f shown below :

[CSIR June 2014]

Which of the following graphs represents the real part of its Fourier transform?

(a) (b)

(c) (d)

f
Dx
4. The Fourier transform of f x is : f k ³ dxe f x . If f x DG x  EG ' x
f
JG ' x where G x is the Dirac Delta function (and prime denotes derivative), what is

f k ? [CSIR Dec. 2015]

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(a) D  iE k  i Jk 2 (b) D  E k  Jk 2 (c) D  iEk  Jk 2 (d) iD  E k  i Jk 2

5. A solution y x satisfies the following differential equation :

d2y
2
 Z2 y G x  a
dx

f
³ dxe
 ikx
where Z is positive. The Fourier transform f k f x of f and the solution
f
of the equation are, respectively [CSIR Dec. 2015]

eik D1 ª Z xa Z xa º eik D 1 Z x a

(a) 2 2 and 2Z ¬«
e e (b) and e
k Z ¼» k Z2 2 2Z

eik D 1 ª iZ x a iZ x a º eik D 1 ª iZ x a iZ x a º

(c) and e e (d) and e e
k Z2 2 «
2Z ¬ »¼ k Z2 2 «
2Z ¬ »¼

6. What is the Fourier Transform ³ dxeikx f x of

f
dn
f x G x ¦ n
G x
n 1 dx

where G x is the Dirac delta function? [CSIR June 2016]

1 1 1 1
(a) (b) (c) (d)
1  ik 1  ik k i k i

­1 for 2n  x d 2n  1 ½
7. The graph of the function f x ® ¾ (where n = 0,1,2 .......). Its
¯0 for 2n  1  x d 2n  2 ¿
laplae transform f s is : [CSIR Dec. 2011]

1 1
1  e s 1  e s
(a) (b) (c) s 1  e s (d) s 1  e s
s s

1
8. The inverse Laplace tranform of s 2 s  1 is : [CSIR June 2013]

1 2 1 1 2 1 2
(a) t e (b) t  1  e 1 (c) t  1  e 1 (d) t 1  e 1
2 2 2

9. The Laplace transform of 6t 3  3sin 4t is : [CSIR June 2015]

36 12 36 12 18 12 36 12
(a) 4
 2 (b) 4
 2 (c) 4
 2 (d) 3
 2
s s  16 s s  16 s s  16 s s  16

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f
1
10. The Fourier transform ³ dx f x eikx of the function f x
2
x 2
is :
f

[CSIR Dec. 2016]

S  2k S  2k
(a)  2k (b)  2k (c) e (d) e
2Se 2Se 2 2

­t
° , 0t T
11. The Laplace transform of f t ®T is : [CSIR Dec. 2016]
°¯ 1 t !T

1  e  sT 1  e sT 1  e  sT 1  e sT
(a)  (b) (c) (d)
s 2T s 2T s 2T s 2T

12. A periodic function f x x for S  x  S has the Fourier series representation

f
§ 2·
¦ ¨©  n ¹¸
n
f x 1 sin nx
n 1

f
1
Using this, one finds the sum ¦ 2 to be : [GATE 2004]
n n 1

S2 S2
(a) 2ln 2 (b) (c) (d) S ln 2
3 6

13. f x symmetric periodic function of x i.e. f x f  x . Then in general, the Fourier

series of the function f x will be of the form : [GATE 2013]

f f
(a) f x ¦ an cos nkx  b sin nkx (b) f x a0  ¦ an cos nkx
n 1 n 1

f f
(c) f x ¦ bn sin nkx (d) f x a0  ¦ bn sin nkx
n 1 n 1

14. A periodic function f x of period 2S is defined in the interval S  x  S as :

­ 1 for  S  x  0
f x ® [GATE 2016]
¯ 1 for 0  x  S

The approximate Fourier series expansion for f x is :

4ª 1 1 º
(a) f x «sin x  sin 2 x  sin 5 x  .....»
S¬ 3 5 ¼

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4ª 1 1 º
(b) f x « sin x  sin 3x  sin 5 x  .....»
S¬ 3 5 ¼

4ª 1 1 º
(c) f x « cos x  cos3 x  cos 5 x  .....»
S¬ 3 5 ¼

4ª 1 1 º
(d) f x « cos x  cos3x  cos5 x  .....»
S¬ 3 5 ¼
15. The Heaviside function is defined as :

­ 1 for t ! 0
H t ®
¯ 1 for t  0
and its fourier transform is given by 2i / Z . The fourier transform of
1ª § 1· § 1 ·º
« H ¨ t  ¸  H ¨ t  ¸ » is : [GATE 2015]
2¬ © 2¹ © 2 ¹¼

Z Z
sin cos
2 2 Z
(a) Z (b) Z (c) sin (d) 0
2
2 2

16. If the fourier transform F ªG

¬ x  a ¼º exp i 2Sva , then F 1 cos 2Sva will
correspond to : [GATE 2008]

(a) G x  a  G x  a (b) a constant

1 1
ªG x  a  iG x  a ¼º ªG x  a  G x  a ¼º
2¬ 2¬
(c) (d)

17. The ken Fourier component of f x G x is :

1/2 3/2
(a) 1 (b) zero (c) 2S (d) 2S

18. The Fourier transforin F k of a function f x is defined as :

f
F k ³f dxf x exp ikx

f
then F k for f x exp  x 2 is [Given : ³f exp  x 2 dx S] [GATE 2008]

§ k 2 · S § k 2 ·
S exp ¨
¨ 4 ¸¸ ¨¨ ¸¸ (d) S exp  k 2
(a) S exp  k (b) (c) 2 exp
© ¹ © 2 ¹

³e
ikx
19. The Fourier transform of the function f x is F k f x dx . The fourier transform

of df x / dx is : [GATE 2003]

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(a) dF k / dk (b) ³F k / dk (c) ikF k (d) ikF k

20. Fourier transform of which of the following functions does not exist? [GATE 2002]

(a) e x (b) xe x (d) e x

2 2 2
(c) e x

21. Which of the following pairs of thegiven function F t and its Laplace transform f s is
not correct? [GATE 2013]

1
(a) F t G t ,f s 1 (singularity at 0) (b) F t 1, f s s!0
s

1 1
s!0 F t tekt , f s s ! k, s ! 0
(c) F t sin kt , f s (d) sk
2
s2  k 2

­ 0 for x ! 3
22. If f x ® then, the Laplace transform of f x is : [GATE 2010]
¯ x  3 for x t 3

(a) s 2 e3x (b) s 2e 3x (c) s 2 (d) s 2 e3x

23. If f s is the Laplace transform of f t the Laplace transform of f at , where a is a

constnat, is [GATE 20051

1  1 
(a) f s (b) f s/a (c) f s (d) f s / a
a a

S
F x ,s ! 0
24. The Laplace transform of f t sin St is s  S2
2 . Therefore, the Laplace

transform of t sin St is :

S 2S 2Ss 2S
(a) s 2 s 2  S2 (b) s 2 s 2  S 2 2 (c) s 2  S 2 2 (d) s 2  S 2 2

25. A function f x is defined in the range 1 d x d 1 by

­1  x for x t 0
f x ®
¯1  x for x  0
The first few terms in the Fonrier series approximating this function are [TIFR 2010]

1 4 4 1 4 4
(a)  2 cos Sx  2 cos3Sx  .. (b)  2 sin Sx  2 sin 3Sx  ..
2 S 9S 2 S 9S

4 4 1 4 4
(c) 2
cos Sx  2
cos3Sx  .. (d)  2 cos Sx  2 cos 3Sx  ..
S 9S 2 S 9S

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26. The student is asked to find a series approximation for the function f x in the domain
1 d x d 1 , as indicated by the thick line in the figure below..

A student represents the function by a sum of three terms

Sx Sx
f x | a0  a1 cos  a2 sin [TIFR 2013]
2 2
Which of the following would be the best choices for the coefficients a0 , a1 and a2 ?

1 2 2
(a) a0 1, a1  , a2 0 (b) a0 , a1  , a2 0
3 3 3

2 2 1
(c) a0 , a1 0, a2  (d) a0  , a1 0, a2 1
3 3 3

27. Consider the waveform x t shown in the diagram below..

The Fourier series for x t which gives the closest approximation to this waveform is

2ª St 1 4St 1 3St º
(a) x t « cos  cos  cos  ....» [TIFR 2017]
S¬ T 2 T 3 T ¼

2ª St 1 2 St 1 3St º
(b) x t «  sin  sin  sin  ....»
S¬ T 2 T 3 T ¼

2 ª St 1 2St 1 3St º
(c) x t « sin  sin  sin  ....»
S¬ T 2 T 3 T ¼

2ª St 1 4St 1 6St º
(d) x t «  cos  cos  cos  ....»
S¬ T 2 T 3 T ¼
28. The output intensity I of radiation from a single mode of resonant cavity obeys

d Z0
I  I
dt Q

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where Q is the quality factor of the cavity and Z0 is the resonant frequency. The form of
the frequency spectrum of the output is : [JEST 2016]
(a) Delta function (b) Gaussian (c) Lorentzian (d) Exponential

1.(c) 2.(d) 3.(b) 4.(c) 5.(b) 6.(b) 7.(c)

8.(c) 9.(a) 10.(d) 11.(b) 12.(c) 13.(b) 14.(a)
15.(a) 16.(d) 17.(c) 18.(b) 19.(c) 20.(c) 21.(c)
22.(d) 23.(b) 24.(c) 25.(a) 26.(b) 27.(c) 28.(c)

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SECTION: PROBABILITY
1. An unbiased dice is thrown three times successively. The probability that the number
ofdots on the uppermost surface add up to 16 is [CSIR Dec. 2011]
(a) 1/16 (b) 1/36 (c) 1/108 (d) 1/216
2. A ball is picked at random from one of the two boxes that contain 2 black and 3 white and
3 black and 4 white balls respectively. What is the probability that it is white? [CSIR
(a) 34/70 (b) 41/70 (c) 36/70 (d) 29/70 June 2012]
3. In a series of five cricket matches, one of the captains calls ‘heads’ every time when the
toss is iken. The probability that he will win 3 times and lose 2 times is. [CSIR Dec 2012]
(a) 1/8 (b) 5/8 (c) 3/16 (d) 5/16
4. Consider three particles A, B and C, each with attribute S that can take two values ±1.
Let S A 1, S B 1 and SC 1 at a given instant. In the next instant, each S value can
change to S with probability 1/3. The probability that S A  S B  SC remains unchanged
is : [CSIR June 2013]
(a) 2/3 (b) 1/3 (c) 2/9 (d) 4/9

1 2 3 4 5 6
5. A loaded dice has the probabilities , , , , , of turning up 1,2,3,4,5 and 6
21 21 21 21 21 21
respectively. If it is thrown twice, what is the probability that the sum of the numbers that
turn up is even? [CSIR Dec. 2013]
(a) 144/441 (b) 225/441 (c) 221/441 (d) 220/441
6. A child makes a random walk on a square lattice of lattice constant 'a', taking a step in the
north, east, south or west directions with probabilities 0.255, 0.255, 0.245, 0.245 respectively.
After a large number of steps N, the expected position of the child w.r.t the strating point
is at a distance : [CSIR Dec. 2013]

(a) 2 u 102 Na in the north-east direction

(b) 2 N u 102 a in the north-east direction

(c) 2 2 u102 Na in the south-east direction (d) 0

7. In one dimension, a random walker takes a step with equal probability to the left or right.
What is the probability that the walker returns to the starting point after 4 steps?
[CSIR June 2014]
(a) 3/8 (b) 5/16 (c) 1/4 (d) 1/16

1
8. Let, y x1  x2  P , where x1 and x2 are independent and identically distributed
2

y4
Gaussian random variables of mean P and standard deviation V . Then is :
V4
[CSIR June 2014]

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(a) 1 (b) 3/4 (c) 1/2 (d) 1/4

9. Two independent random variables m and n, which can take the integer values 0, 1, 2, .....
f , follow the Poisson distribution, with distinct mean values P and v respectively. Then
[CSIR Dec. 2014]
(a) the probability distribution of the random variable l m  n is a binomial distribution
(b) the probability distribution of the random variable r m  n is also a Poisson distribution
(c) the variance of the random variable l m  n is equal to P  v
(d) the mean value of the random variable r m  n is equal to 0
10. A random walker takes a step ofunit length in the positive direction with probabitity 2/3
and a step of unit length in the negative direction with probability 1/3. The mean displacement
of the walker after n steps is : [CSIR Dec. 2014]
(a) n/3 (b) n/8 (c) 2n/3 (d) 0
11. Three real variables a, b and c are each randomly chosen from a uniform probability
distribution in the interval [0, 1]. The probability that a  b ! 2c is [CSIR June 2015]
(a) 3/4 (b) 2/3 (c) 1/2 (d) 1/4
12. Consider a random walker on a square lattice. At each step the walker moves to a
nearest neighbour site with equal probability for each of the four sites.The walker starts
at the origin and 3 steps. The probability that during this walk no site is visited more than
once is : [CSIR Dec. 2015]
(a) 12/27 (b) 27/64 (c) 3/8 (d) 9/16
13. Let X and Y be two independent random variables, each of which follow a normal distribution
with same standard deviation V , but with means P and P reslaectively. Then the sum
X  Y follows a : [CSIR June 2016]
(a) distribution with two peaks at P and mean 0 and standard deviations V 2
(b) normal distribution with mean 0 and standard deviations 2V
(c) distribution with two peaks at P and mean 0 and standard deviation V 2

(d) normal distribution with mean 0 and standard deviation V 2

14. A box of volume V containing N moleculesofan ideal gas is divided by wall with a hole
into two conpartrnents, the volume of the smaller compartment V/3, the variance of the
number of particles in it, is [CSIR June 2016]

N 2N N
(a) (b) (c) N (d)
3 9 3
15. Consider two radioactive atoms, each of which has a decay rate of l per year. The
probability that at least one of them decays in the first two years is : [CSIR Dec. 2016]

1 3 2
(a) (b) (c) 1  e4 (d) 1  e2
4 4

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16. Consider a random walk on an infinite two-dimensional triangular lattice, a part of which
is shown in the figure below.

If the probabilities ofmoving to any of the nearest neighbour sites are equal, what is the
probability that the walker returns to the starting position at the end ofexactly three steps?
[CSIR Dee. 2016]

1 1 1 1
(a) (b) (c) (d)
36 216 18 12
17. A 100 page book is known to have 200 printing errors distributed randomly through the
pages. The probability that one of the pages will be found to be completely free oferrors
is closest to : [TIFR 2011]
(a) 67% (b) 50% (c) 25% (d) 13%
18. The probability function for a variable x which assumes only positive values is

§ x·
f x x exp ¨  ¸ [TIFR 2014]
© O¹

where O ! 0 . The ratio x / x where x̂ is the most probable value and x is the mean
value of the variable x, is

1 O 1
(a) 2 (b) (c) (d) 1
1 O O
19. A random number generator outputs +1 or –1 with equal probability every time it is run.
After it is run 6 times, what is the probability that sum of the answers generated is zero?
Assume that the individual runs are independent of each other. [TIFR 2015]
(a) 1/2 (b) 5/6 (c) 5/16 (d) 15/32
20. In a triangular lattice a particle moves from a lattice point to any of its 6 neighbouring
points with equal probability, as shown in the figure on the right. [TIFR 2016]

The probability that the particle is back at its starting point after 3 moves is :
(a) 5/18 (b) 1/6 (c) 1/18 (d) 1/36

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21. An unbiased die is cast twice. The probability that the positive difference (bigger-smaller)
between the two numbers is 2 is : [JEST 2012]
(a) 1/9 (b) 2/9 (c) 1/6 (d) 1/3
22. A box contains 100 coins out of which 99 fair coins and 1 is a double-headed coin.
Suppose you choose a coin at random and toss it 3 times. It turns out that the results of all
3 tosses are heads. What is the probability that the coin you have drawn is the double-
headed one? [JEST 2013]
(a) 0.99 (b) 0.925 (c) 0.075 (d) 0.01
23. There are on average 20 buses per hour at a point, but at random times. The probability
that there are no buses in five minutes is closest to. [JEST 2013]
(a) 0.07 (b) 0.60 (c) 0.36 (d) 0.19
24. Two drunks start out together at the origin, each having equal probability of making a step
simultaneously to the left or right along the x-axis. The probability that they meet after n
steps is [JEST 2013]

1 2n! 1 2n! 1 1
(a) n 2 (b) n 2 (c) n
2n ! (d) n!
4 n! 2 n! 2 4n

25. If the distribution fitriction of x is f x xe  x / O over the interval 0  x  f , then mean

value of x is : [JEST 2013]
(a) O (b) 2O (c) O / 2 (d) 0
26. If two ideal dice are rolled once, what is the probability of getting atleast one 6?
[JEST 2015]
(a) 11/36 (b) 1/36 (c) 10/36 (d) 5/36
27. You receive on average emairs per day during a 365 days. The number of days on average
on which you do not receive any emails in that year are : [JEST 2016]
(a) More than 5 (b) More than 2 (c) 1 (d) None of the above

1
28. The mean value of random variable x with probability density p x exp
V 2S

ª  x 2  Px / 2V2 º is : [JEST 2016]

«¬ »¼

(a) 0 (b) P / 2 (c) P / 2 (d) V

1.(b) 2.(b) 3.(d) 4.(d) 5.(b) 6.(a) 7.(a)

8.(d) 9.(c) 10.(a) 11.(c) 12.(d) 13.(d) 14.(b)
15.(c) 16.(c) 17.(d) 18.(a) 19.(c) 20.(c) 21.(b)
22.(c) 23.(d) 24.(a) 25.(b) 26.(a) 27.(b) 28.(c)

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SECTION : MISCELLANEOUS TOPICS

1. Which of the following matrices is an element of the group SU 2 ? [CS1R June 2011]

§1 i 1 · § 1 3·
¨ ¸ ¨ ¸
§ 1 1· ¨ 3 3¸ §2i i · ¨ 2 2 ¸
(a) ¨ ¸ (b) ¨ 1 1 i ¸ (c) ¨ ¸ (d) ¨ 3 1 ¸
© 0 1¹ ¨ ¸ © 3 1 i ¹ ¨ ¸
© 3 3¹ © 2 2 ¹

2. The character table of C3 , the group of symmetries of an equilateral triangle is given

below :

0 1 2
F F F
1 f1 1 1 b
2 f2 1 a c
3 f3 1 1 d

In the above C1 , C2 , C3 denotes the three classes of C3v containing 1, 3 and 2 elements

respectively, and F 0 1 2
,F and F are the characters of the free representations
0 1 2
* ,* and * of C3v ,

(A) The entries a, b, c and d in this table are, respectively.

(a) 2, 1, –1, 0 (b) –1, 2, 0, –1 (c) –1, 1, 0, –1 (d) –1, 1, 1, –1

(B) The reducible representation * of C3v with character F 4, 0,1 decomposses into

its irreducible representations * 0 1 2 as : [CSIR June 2011]

,* ,*

(a) 2* 0 1 (b) * 0 1 (c) * 0 1 2 (d) 2* 1

 2*  3* * *
3. The radioactive decay of a certain material satisfies Poisson statistics with a mean rate of
O per second, what should be the minimum duration of counting (in seconds) so that the
relative error is less than 1% ? [CSIR June 2912]

(a) 100 / O (b) 104 / O 2 (c) 104 / O (d) 1 / O

4. A bag contains many balls, each with a number painted on it. There are exactly n balls
which have tie number n (namely one ball with 1, two balls with 2, and so on until N balls
with N on them). An experiment consists of choosing a ball at random, noting the number
on it and returning it to the bag. If the experiment is repeated a large number of times, the
average value of the number will tend to : [CSIR June 2012]

2N 1 N N 1 N N 1
(a) (b) (c) (d)
3 2 2 2

5. The approximation cos T 1 is valid up to 3 decimal places as long as T is less than:

(take 180q / S 57.29q ) [CSIR June 2013]

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(a) 1.28º (a) 1.81º (b) 3.28º (d) 4.01º

6. The solution of the partial differential equation

w2 w2
2
u x, t  u x, t 0
wt wt 2

satisfying the boundary conditions u 0, t 0 u L, t and initial conditions u x,0 sin

w
Sx / L and u x, t sin 2Sx / L is [CSIR June 2013]
wx t 0

L
(a) sin Sx / L cos Sx / L  sin 2Sx / L cos 2Sx / L
2S

(b) 2sin Sx / L cos Sx / L  sin Sx / L cos 2Sx / L

L
(c) sin Sx / L cos 2Sx / L  sin 2Sx / L sin Sx / L
S

L
(d) sin Sx / L cos Sx / L  sin 2Sx / L sin 2Sx / L
2S
7. The expression : [CSIR Dec. 2013]

§ w2 w2 w2 w2 · 1
¨ 2  2 2 2¸
¨ wx ¸ 2 2 is proportional to :
© 1 wx2 wx3 wx4 ¹ x1  x2  x3  x4
2 2

(a) G x1  x2  x3  x4 (b) G x1 G x2 G x3 G x4

3/2 2
(c) x12  x22  x32  x42 (d) x12  x22  x32  x42

8. Let A. and B be two vectors in three dimensional Euclidean space. Under rotation, the
tensor product Tij Ai B j . [CSIR Dec. 2013]

(a) reduces to a direct sum of three 3 dimesional representations

(b) is an irreducible 9 dimensiona representation
(c) reduces to a direct sum of a 1-dimensional a 3-dimensional and a 5-dimensional
irreducible representations
(d) reduces to a direct sum of a 1-dimensional and an 5-dimensional irrducible representation
9. 69 three sets of data A, B.and C from an experiment, represented by ×, †, and O, are
plotted on a log-log scale. Each of these are fitted with straight lines as shown in the
figure. [CS1R Dec. 2013]

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The functional dependence y x for the sets A, B and C are, respectively..

x 1 1
(a) x , x and x 2 (b)  , x and 2x (c) 2
, x and x 2 (d) , x and x 2
2 x x
10. The following data is obtained in an expriment that measures the viscosity K as a function
ofmolecular weight M for a set of polymers. [CSIR June 2014]

M Da K kPa  s

990 0.28 r 0.03

5032 30 r 2
10191 250 r 10
19825 2000 r 200
The relation that best describes the dependence of K on M is :

(a) K  M 4/9 (b) K  M 3/2 (c) K  M 2 (d) K  M 3

1
11. The integral - ³ xdx is to be evaluated up to 3 decimal places using Simpson’s 3-point
0
rule. If the interval [0, 1] is divided into 4 equal parts, the correct result is :
[CSIR June 2014]
(a) 0.683 (b) 0.667 (c) 0.657 (d) 0.638

12. The function ( ) x, y, z, t cos z  vt  Re sin x  iy satisfies the equation :

[CSIR June 2014]

1 w 2) § w 2 w2 w2 · § 1 w2 w2 · § w2 w2 ·
(a) 2 2 ¨ 2 ¨   ¸) ¨
(b) ¨ 2 2  ¸) ¨¨ 2  2 ¸¸ )
v wt © wt wy 2 wz 2 ¹¸ © v wt wz 2 ¹¸ © wt wy ¹

§ 1 w2 w2 · § w2 w2 · § w2 1 w2 · § w2 w2 ·
(c) ¨¨ 2 2  2 ¸¸ ) ¨¨ 2  ¸)
2¸ (d) ¨¨ 2  2
¸)
2¸
¨¨ 2  ¸)
2¸
© v wt wz ¹ © wt wy ¹ © wz v wt ¹ © wt wy ¹

13. Let D and E be complex numbers. Which of the following sets of matrices forms a
group under matrix inultiplication? [CSIR Dec. 2014]

§D E· §1 D·
(a) ¨ ¸ (b) ¨ ¸ , where DE z 1
© 0 0¹ ©E 1 ¹

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§D D· § D E ·
(c) ¨ ¸ . where DE is real (d) ¨¨ * * ¸¸ , where D  E
2 2
1
©E E¹ © E D ¹

14. A particle moves in two dimensions on the ellipse x 2  4 y 2 8 . At a particular instant it

is at the point (x, y) = (2, 1) and the x-component of its velocity is 6 (in suitable units).
Then the y-component of its velocity is. [CSIR June 2015]
(a) –3 (b) –2 (c) 1 (d) 4

15. The rank-2 tensor xi , x j where xi are the Cartesian coordinates of the position vector in
three dimensions, has 6 independent elements. Under rotation, these 6 elements decompose
into irreducible sets (that is the elements o f each set transform only into linear combinations
of elements in that set) containing : [CSIR June 2015]
(a) 4 and 2 elements (b) 5 and 1 elements
(c) 3, 2 and 1 elements (d) 4, 1 and 1 elements

dy
16. Consider the differential equation x 2  y with the initial conditiony y 2 at x 0.
dx
Let y 1 and y 1/2 be the solutions at x 1 obtained using Euler’s forward algorithm

with step size 1 and 1/2 respectively. The value of y 1  y 1/2 / y 1/2 is :

[CSIR June 2015]

1 1
(a)  (b) –1 (c) (d) 1
2 2

w2 f w2 f
17. Let f x, t be a solution of the wave equation v2 in 1-dimension. If at
wt 2 wx 2
wf
e x and
2
t 0, f x,0 x,0 0 for all x, then f x, t for all future times t > 0 is
wt
described by : [CSIR June 2015]
(a) e x 2  v2 t 2
2
(c) e xvt

1  xvt
2
3  x vt
2
1ª  x vt
2
 x  vt
2
º
(c) e  e (d) « e e
4 4 2¬ ¼»
18. In the scattering of some elementary particles, the scattering cross-section a is found to
depend on the total energy E and the fundamental constants h (Planck’s constant) and c
(the speed of light in vacuum). Using dimensional analysis, the dependence of a on these
quantities is given by : [CSIR Dec. 2015]
2
hc hc § hc · hc
(a) (b) 3/ 2 (c) ¨ ¸ (d)
E E ©E¹ E

1
19. If y tanh x , then x is : [CSIR Dec. 2015]

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§ y 1· § y 1 · y 1 y 1
(a) ln ¨ ¸ (b) ln ¨ ¸ (c) ln (d) ln
© y 1 ¹ © y 1¹ y 1 y 1

8 1 1
20. The value of the integral ³0 x 2  5 dx , evaluated using Simpson's 3 rule with h 2 , is

[CSIR Dec. 2015]

(a) 0.565 (b) 0.620 (c) 0.698 (d) 0.736

21. The Gauss hypergeometric function F a, b, c; z , defined by the factor series series
expansion around z 0 as
f a a  1 ... a  n  1 b b  1 ... b  n  1
F a, b, c; z ¦ c c  1 .. c  n  1 n !
zn
h 0

satisfies the equation relation : [CSIR June 2016]

d c
(a) F a, b, c; z F a  1, b  1, c  1; z
dz ab

d c
(b) F a, b, c; z F a  1, b  1, c  1; z
dz ab

d ab
(c) F a, b, c; z F a  1, b  1, c  1; z
dz c

d ab
(d) F a, b, c; z F a  1, b  1, c  1; z
dz c
22. Using dimensional analysis, Planck defined a characteristic temperature TP from powers
ofthe gravitiorial constant G, Planck’s constant h, Bolti.mann constant k B , and the speed
of light c in vacuum. The expression for TP is proportional to : [CSIR June 2016]

hc  hc3 G hk B2
(a) (b) (c) (d)
k B2 G k B2 G hc 4 k B2 Gc3

23. The integral equation

I x, t O ³ dx ' dt '

is equivalent to the diffe

Q.23 to Q. 42 Page Missing
f
1
43. The sum ¦ m 1  m
is equal to : [JEST 2015]
m 1

1
(a) 9 (b) 99  1 (c) 99  1 (d) 11

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1 1 1
44. The sun of the infinite series 1     ... is : [JEST 2016]
3 5 7

S S
(a) 2S (b) S (c) (d)
2 4
G G
45. Given the condition ’2 ) 0 , the solution of the equation ’2 < k’).’) is given by:
[JEST 2016]

(a) < k) 2 / 2 (b) < k)2 (c) < k ) An) (d) < k )An) / 2

1.(b) 2.(c) 3.(c) 4.(a) 5.(b) 6.(d) 7.(b)

8.(c) 9.(d) 10.(d) 11.(c) 12.(a) 13.(d) 14.(a)
15.(b) 16.(b) 17.(d) 18.(c) 19.(d) 20.(a) 21.(d)
22.(a) 23.(d) 24.(d) 25.(b) 26.(c) 27.(d) 28.(c)
29.(c) 30.(c) 31.(b) 32.(d) 33.(b) 34.(d) 35.(b)
36.(c) 37.(343) 38.(An) 39.(a) 40.(c) 41.(d) 42.(d)
43.(a) 44.(d) 45.(a)

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