1.

If the lines == and t = = 2t are parallel, then the value of

k is

(A) (B) (c) (D)

2. Let Aand B be the conics with equations 222 + 3y + 5æ +6=0 and 22 +3y2 +
a+y+l=0respectively. Then

(A) A and B are ellipses

(B) Ais an ellipse and B is a circle
(C) A is a parabola and B is a hyperbola
(D) A is a parabola and B is an ellipse

3. The equation of the plane passing through the points (1, 1,-1), (4, 1, -4), (4, 4,2) is
(A) z-2y +z+2=0 (C) 2* -ytz+2=0
(B) -2y +2z +2 = 0 (D) 2 -y+2z -2= 0

4. The smallest value of the polynomial in a- 3a + 3z+6 the interval 0, 2]is

(A) 6 (B) 0 (C) 7 (D) 8

5. The function f(a) = - log(1 +t a) for z >0

1+

(A) Always increases (C) Increases in some finite interval

(B) Always decreases (D) None of these

6. The value of T+0

lim (²) is

(A) O (B) e () (D) 1

1
 7. There are 10 eggs in a refrigerator, of which 2 are rotten. If 3 eggs are taken for
cooking, what is the probability that at least one of the eggs are rotten?

488
(A) 6 (B) (O) 00 (D)

8. How many distinct 4 digit even number can be formed using the digits 0,1, 2, 3, 4,
5 without repetition of digits?

(A) 144 (B) 156 (C) 180 (D) 216

The series E-1"sin() is

T=1

(A) absolutely convergent

(B) convergent but not absolutely convergent
(C) absolutely convergent but not convergent
(D) diverges
sin na
10. Let {f(æ)} be a sequence of real functions defined by fn(e) =
Vn
Then {fn(z)}
(A) converges pointwise but not uniformly

(B) converges uniformly but {f()} does not converge for each z
(C) converges uniformly and {f()} converges
(D) not pointwise convergent
sin z
if z > 0
11. Consider the function f: (0, o)’R defined by f(a) =
if z= 0

(A) f is Riemann integrable but not Lebesgue integrable.

(B) is Lebesgue integrable but not Riemann integrable.
(C) f is both Riemann integrable and Lebesgue integrable.
(D) f is neither Riemann integrable nor Lebesgue integrable.

2
 2:+1
12. If the imaginary part of iz +1
is -2, then the locus of the point representing z is

(A) Circle (B) Straight line (C) Parabola (D) Hyperbola

13. The value of the integral [tan z dz, where C is the circle |2| = 2 is

(A) 4ni (B) -4Ti (C) 2ri (D) -2mi

14. Find the bi-linear transformation which mnaps the points z = 1, i, 1 in to the
points w=i, 0, -i
1+iz
(A) 1-i7 (B)
1- iz (D) !-:
1+ iz 1-z 1+z

15. Which of the following is true

(A) (Z4, +) is isomorphic to (Z;, x) where Z; is the non zero elements of Zs

(B) (Z4,+) has an element of order 3
(C) An infinite cyclic group may have three generators
(D) If G is an infinite cyclic group, then there exists no isomorphism from G to G
other than identity isomorphism.

16. Number of subgroups of a cyclic group of order 2024

(A) 16 (B) 20 (C) 200 (D) 1760

17. Number of cyclic subgroups of order 20 in Z10o0 x Z2s

(A) 16 (B) 40 (C)) 48 (D) 100

18. Which of the following is always true ?

(A) If R is a sub ring of a ring R with unity then R contains unity.

(B) IF R is a ring with a.a = a for every a E R, then a +a=0
(C) If (R, +, .) is a ring and a, bE R, then (a+b) = a² + 2a.b + b², where a? = a.a
(D) Every commutative ring has a multiplicative identity.
19. The degree of the splitting feld of -3 over Q is

(A) 1 (B) 3 (C) 4 (D) 6

20. Number of elements in the field Za<el/(=² + 1)

(A) 3 (B) 8 (C) 9 (D) infinite

21. Which of the following is irreducible in Za[z]

(A) z + +1 (C) a+z +a+1

(B) 22 + 2r + 2a -1 (D) z+a+z++1

22. The eigen values of a 3 x 3 matrix A are given by 1, -2, 3, then

(A) A- =(5/-24 - A) (C) 4=6I -24 +4)

(B) A- =(5/+ 2A- A?) (D) 4-(5/+ 2A +A2)
23. A 2 x 2 real matrix A is diagonalizable if and only if

(A) (trace A)? < 4det A (C) (trace A)² > 4det A
(B) (trace A) = 4 det A (D) (trace A)² = det A

24. Consider the three equations -2r +y+z=a, a - 2y +z =b, z +y-2: = c. The
system has no solutions, unless a +b+c=

(A) 1 (C) 0
(B) -1 (D) any odd integer

25. What is the condition on a, b, csothat v= (a,b, c) in R belongs to W= span(uj, ug, ua)
where uj = (1,2,0), ug = (-1,1, 2) and ug (3,0, -4) ?
(A) 40- 26 + 3c =0 (C) 4a - 26 +c= 0
(B) 3a 26 +3c=0 (D) 4a --b+3c =0

4
26. What is the rank of the matrix A whose rows are (1, 1, -1), (2, 3, -1), and (3, 1, -5)?

(A) 1 (C) 3
(B) 2 (D) rank not defined

27. Consider the vector space V = P(t) of polynomials over the field of real numbers.
Let H:V’V be the third derivative operator, then the nullity of His

(A) 1 (B) 2 (C) 3 (D) 4

28. Let L be the linear transformation on R that reflects each point P across the line
y= kz, where k > 0. Which of the following are the eigen vectors of T,

(A) (&, 1) (B) (-k, 1) (C) (2k, 5) (D) (k, -1)

29. If the determinant of a3x3 matrix is 11, then the value of the square of determinant
formed by the co-factors will be

(A) 11 (B) 121 (C) 1331l (D) 14641

30. The remainder obtained on dividing 2l00 by 77 is

(A) 24 (B) 75 (C) 26 (D) 23

31. The differential equation of all parabolas having their axis along z - azis and focus
at origin is
dy
(A) 2a t y ) -y=0
dy dy
(B) 2z -y=0

dy
(C) 2r -y=0

dy
(D) 2rda +y)+y=0

5
32. Which of the following is an integrating factor of the differential equation (ry +
v)dz + 2('y + z+y')dy = 0
1
(A) y² (B) () (D) y

dy
33. The particular integral of the differential equation+25y = cos 5z is
T COs 5x -T Sin 5a T COs 5T cos 5z+ sin 5
(A) 10
(B) 10
(C) 10
(D) 10

34. The Partial Differential equation of the family of curves given by z = ary+b, where
a and b are arbitrary constants is

(A) pr+ qy = 0 (B) pa? + qy² = 0 (C) pr - qy =0 (D) py + qz = 0

35. The solution of the linear Partial Differential equation

6 8=2rty
+ is
(A) z= f(y + 3z) + faly + 2r) +ety
(B) = fi(y -4z) + sly-2r) +ety
() z= f(y + 4z) + faly + 2z) + ety
(D) = f(y 4r) + faly + 2r) +ety
36. Let S = fn+:n¬N} be asubset of R. Then
(A) S has a limit point and is not compact

(B) S has no limit point and is compact

(C) S has no limit point and is not compact
(D) S has a limit point and is compact

37. Which of the following sequence of sets in R satisfy the hypothesis of Cantor's
intersection theorem?

(A) A, := (0, n]; n e N (C) A, := (0, );n e N

(B) A, := [0, );neN (D) A, := (0,1 - :n eN
38. Let X =:neNU{O} be endowed with discrete topology. Which of the
following is false?
(A) X is closed (C) X is compact

(B) X is not connected (D) X = {:neN} is not dense in X

39. Let S = (R, T1) and S = (R, ) be such that TË is the usual topology and 2 is
the discrete topology on R. If f: S S, and g: S, ’ S, are two functions, then
necessarily

(A) f is continuous (C) f is not continuous

(B) g is continuous (D) g is not continuous

40. Which among the following pairs are homeomorphic?

(A) [0, 1] and {z¬C: |=|=1} (C) (0,1) and (V2, oo)
(B) [0, 1) and (1,2) U{3} (D) R and R\Q

41. Which among the following is a norm that does not arise from an innerproduct?

(A) R? with |(21, z2)|l= /aft

(B) 2 with lI(..| =(} )
(C) C0,
(D) C[0, 1] with |f| = ze[0,1]
sup |f(*)|

42. Which of the following is not an orthonormal basis for R?

(A) {( 0). (*-9), (0,0, 1)}

(B) {(1,0, 0), (0, -1,0), (0,0, 1)}
(C) {( )-0).(* -a)}
(D) {(%-)(0, ). (0,1,0)}

7
43. What will be the range of the function f() = 2- |x- 5| is
A) -oo, 1] B) (-00, 2] C) (-o, 2) D) (-0, 1)

44. If points (a,0), (0,b) and (1,1) are collinear then what is the value of a+b
A) -ab B) ab C) 1/ab D) -1/ab

45. Find the minimum value of the function f(x) = x2 - x+ 2 is

A) 1/2 B) 3/4 C) 7/4 D) 1/4

46. The value of the integral cr sin xdx dy is equal to

A) 1 B) 2 C) 3 D) -1

47. The number of urjective maps from a set of 4 elements to a set of 3 elements is
A) 36 B) 64 C) 69 D) 81

43. A man is known to speak truth 2 out of 3 times. He throws a die and reports that it is
a 6. Then the probability that it is actually a 6 is
A) 2/3 B) 3/4 C) 3/8 D) 2/7

49. lim {Vn+i-/nj

A) 0 B) 1/2 C) 1 D) o

50. ,Ixl dx is
A) -1 B) 1 C) 0 D) 2

51. From the following function, pick the function which is1-cos
uniformly
x
continuous on (0,1)
A) f«) =B) f«) =c) fa) =n D) f(*) =

52. The harmonic conjugate of x - y' is

A) *²+y² B) 4xy C) 2xy D) y² x²

53. Let f(z) and f2) be analytic a domain D. Then

A) f(z)is zero for all z
B) f(2)is a constant function
C) f(z)is a real valued function but not constant
D)f(2)is imaginary valued function but not constant

54. Jc T -1 dz where C is the circle z = 2

A) 4ni B) ri C)0 D) 2ri
55 The function f(z) =has
el/+1 at z=0 is
A) a removable singularity C) a pole
B) an isolated essential singularity D) a non-isolated essential singularity

56. Let Z,(a) be an extension of the field Z2, where a is a zero of x² +x +1¬
Z,(x). Then a + a+1 =
A) a? B) a³ C) a D) a5

57. Which of the following is not a class of cyclic group

A)all groups of order 4 C) all groups of order 33
B) all groups of order D) all goups of order 15

58. Which of the following is a generator of the group Z X Z2

A) (2,6) B) (2,3) C) (3,4) D) (3,5)

59. Let f(x) = x³+ 2x2 + 1 and g(*) = 2x2 +x+2. Then over Z
A) f(x)and g (x) are irreducible
B) f(x)is irreducible and g(*)is not
C) g(r)is irreducible and f(*) is not
D) Neither f(x)nor g()is irreducible
60. Which of the following is a subspace of M,(R), the vector space of nxn matrices
A) The set of all non-invertible real matrices
B) The set of all matrices A with det A=0
C) The set of all matrices A with trace A=0
D) None of the above

61. Let T: R2 ’ R2by T(x,y)=(2x+y,3x+2y),then inverse of T is

A) T-(x,y) = (2x -y, 3x - 2y) c) T-(x,y) = (2x -y, -3x + 2y)
B) T-(z,y) = (-2x + y, 3x - 2y) D) T-(«,y) = (-2x + y, -3x + 2y)

-1 1 0 0
0 -1 0
62. The minimal polynomial of 0 0 2 0|s
0 0 2J
A) (* +1)(x-2) C) (* + 1)'(x - 2)²
B) (*+ 1)(x- 2)2 D) (x +1)2(x-2)
63. For a positive integer n, let p, denote the vector space of polynomials in one variable
x with real coefficients and with degree < n. Consider the map T:P,+P, defined
by T(p(*)) = p(*?). Then
A) T is a linear transformation and rank (T)=5
B) T is a linear transformation and rank (T)=3
C) T is a linear transformation and rank (T)=2
D) T is not a linear transformation
1+x 1+x+x*
64. The determinant|1 1+y 1+y+ y²| is equal to
LË 1+z 1+z+z?J
A)(2-x)(2- y)(y- x) c) (*- y)(«- z)(-z)
B) (*-y)?(y- z)(2 - x)? D) (*2-yy²-z²)(2?- x)
65. If A is a 5 XX 5 real matrix with trace 15 and if 2 and 3 are eigenvalues of A, each with
algebraic multiplicity 2, then the determinant of A is equal to
A)0 B) 24 C) 120 D) 180

66. The rank of the linear transformation T: R ’ R+ given by

T(%, y,z, w) = (x- y, x - 2y, x -3y, x - 4y)is
A) 2 B) 1 C) 4 D) 3

67. Which of the following is not a linear transformation T: R³’R3

A)T(x,y, z) = (y, x,0) c) T(*, y,z) = (0,0,0)
B)T(x, y, z) = (*y, yZ, xz) D) T(X, y, z) = (* + y,y + z,x + z)

68. The number of positive divisors of 50000 is

A) 20 B)30 C) 40 D) 50

69. The period of the function y = Isinx|+ |cos x|31is

A) B) TI C) 2
D) 2

70. The degree of the equation ydy= x(dy+ x is

A) 0 B) 1 C) 2 D) 3

71. Solution of (1 + y²) dx = (tan- y - r) dy is

A) x= tany 1+ Ce- tany C) y= tanx - 1+ Ce-tan- x
B) x = tan-y + Ce-tanx D) y= tanx+ Ce-tanx

72. A particular solution of the equation y +y = secx is

A) y= x sin x + cos log sin x C) y=x cos x + sin x log cos x
B) y = x sin x + cos x log cos x D) y = x cosx + sin x log sin x

10
73. The general solution of x(y -)+ y(z -x)=z(*-y) is
A) f(x+y +z, xyz) =0 C)f(xy yz, xyz) = 0
B) f++;.xyz) =0 D) f(*²-y², z) = 0
74 The partial differential equation representing the family of curves
Z = (x- a)? + (y- b) ²is
A) Z =p²+q? ) 4Z =p² -q?
B) 22 = p² + q² D) 4Z =p + q²
75 Which of the following is false
A) Product of TË space is a T, space
B) Product of completely regular space is completely regular
C) Product of first countable space is first countable
D) Product of two second countable space is second countable
76 Let R be the set of real numbers and t be the semi-open interval topology on R. Then
which of the following is true for (R,r)
A) (R,r) is a second countable space C) (R,r) is a metrizable space
B) (R,r) is a separable space D) (R,r) is a compact space
77. Let t be the topology on R consisting of R,$,and all open intervals of the form (a,o0)
where aER. Then the closure of the interval A=[0,1] is
A) (0,1] B) (-00, 1 ) C) (0, oo) D) R
78. Consider the norms || l,,I 2. ll Ilo on R". Thenfor all xE R2, which one of the
following is not true.
A) lx, s Vl| xll2
B) |xllo s llxll, s lxll, D) l<|l, s Ilxl2 s llxllo

79. Let H be a Hilbert space. If x,y E H are such that ||xl| = 6, ||x + yl| = 16 and
x- yll =4. Then |lyll is
A)2 B) 8 C) 10 D) 12

80. Let R² be the usual inner product space andu= (1,1). Define f: R' ’R by
fu: (x) = (x, u). Then G
A) 1 B) 2 c) v2 D)

11