A.’NN’cannot be written as a denumerable union of
denumerable disjoint sets
B.’NN’ is well ordered

C.’NN’ is a closed subset of ‘RR’

D.None of these

Ans:A

62.E={p’in’’QQ’:2<’p^(2)’<3}.Then which is
false?

A.E is open in ‘QQ’

B.E is closed in ‘QQ’
C.E is a bounded subset of ‘QQ’

D.E is a compact subset of ‘QQ’

Ans:D

63.Let function n(x)={(n-1)x raise to n, if x is
rational(first row),1-(n-1)x raise to n,if x is irrational(second row).Then
which of the following function is discontinuous everywhere on [0,1]?

A.’g_(1)’

B.’g_(2)’

C.’g_(3)’

D.All of these

Ans:A

64.Let f(x)={ x+x raise to 2 sin 1/x,if x not equal
to 0(first row),0 if x=0(second row)

The f’(0) is

A.0

B.1

C.2

D.Does not exists

Ans:B

65.Which of the following is not uniformly
continuous on ‘RR’?

A.x

B.sin x

C.x sin x

D.’(1)/(1+x^(2))’

Ans:C

66.Let ‘s_(n)’=’1/1^(2)+1/2^(2)+….+1/(n^(2))’.Then
which is false?

A.{‘s_(n)’} is a sequence in ‘QQ’

B.{‘s_(n)’} is a Cauchy sequency in ‘QQ’

C.{‘s_(n)’} is convergent in ‘QQ’

D.None of these

Ans:C

67.Let ‘f_(n)’:’RR’’->’’RR’ defined by fn(x)={0
if x elements of (- infinity,n)(first row),n,if x elements of [n,infinity)

Then choose the most correct answer

A.’{f_(n)}’ does not convergent on ‘RR’

B.’{f_(n)}’ converges to zero function on ‘RR’

C.’{‘f_(n)’}converges to zero function on ‘RR’
uniformly

D.None of these

Ans:B

68.Which is false?

A.There exists a function f:’RR’’->’’RR’ which is
continuous exactly at one point

B.There exists a function f:’RR’’->RR’ which is
differentiable exactly at one point

C.There exists a function f:’RR’’->’’RR’ which is
discontinuous everywhere

D.None of these

Ans:D

69.Which of the following subset of ‘RR^(3)’has
positive Lebesgue measure?

70.If ‘int_(1)^(00)’f(x) dx=L where
0<L<’oo’;then ‘lim_(x->oo)’f(x)is

A.0

B.’oo’’’

C.Always exists

D.Need not exists

Ans:D

71.Which is false?

A.cos z=5 has a solution in ‘CC’

B.sin z is a polynomial in ‘CC’

C.’sin ^(z)’+’cos z^(2)’=1 for all z ‘in’’CC’

D.Every non constant polynomial in ‘CC’[x] has a
zero in ‘CC’

Ans:B

72.The image of the line y=1 under the mapping w=sin
z is

A.Parabola

B.Ellipse

C.Rectangular Hyperbola

D.None of these

Ans:B

73.The value of ‘’’int_(C)’’(1)/(z^(3)(z+1)’ dz
where C is |z|=3;is

A.2’pi’i

B.-4’pi’i

C.4’pi’i

D.0

Ans:D

74.Consider the metric space (‘NN’,d)where d is
given by d(m,n)=|’1/m’-‘1/n’|.Then which is false?

A.d is a bounded metric on ‘NN’

B.d induces the discrete topology on ‘NN’

C.’{1/n}’ converges to 0

D.This space is Hausdorff

Ans:C

75.Which is not a productive property?

A.Connectedness

B.Compactness

C.Locally connectedness

D.Path connectedness

Ans:C

76.Let X=(0,1) and Y=’RR’.Then which is true?

A.X and Y are the same as metric spaces

B.X and Y are the same as topological spaces

C.Both (a) and (b)

D.Neither (a) nor(b)

Ans:B

77.Which of the following topological property is
not preserved under a continuous function?

A.Connectedness

B.Compactness

C.First countability

D.None of these

Ans:C

78.Which is true?

A.On’RR’co-finite topology is weaker than usual
topology

B.On’RR’usual topology is weaker than co-finite
topology

C.On’RR’ co-finite topology and usual topology are
not comparable

D.On’RR’co-finite topology and usual topology are
the same

Ans:A

79.If every closed interval[a,b] with a<b is open
with respect to some topology on’RR’;then with respect to this topology,closure
of [27,37]is

A.[27,37]

B.(-‘oo’,37]

C.[27,’oo’)

D.’RR’

Ans:A

80.For the space ‘RR’with co-countable
topology,which is false?

A.Uniqueness of limits exists in this space for the
convergence of sequences

B.This space is not Hausdorff

C.{‘1/n’} is divergent in this space’’

D.None of these

Ans:D

81.If f:’RR’’->’’RR’ is a twice differentiable
function with ‘lim_(x->oo)’(2f(x)+3f’(x))=0 with
‘lim_(x->oo)’f(x)=’lim_(x->oo)’f’(x)=1;then ‘lim_(x->oo)’f’’(x)is

A.0

B.1

C.10

D.11

Ans:D

82.The third approximation ‘y_(3)’(x) for the
I.V.P.,y’=2x(1+y);y(0)=0 by Pickard’s method is

A.’x’+’x^(2)/2’+’x^(3)/6’

B.’x^(2)’+’x^(4)/2’+’x^(6)/6’

C.’x’+’x^(2)/2+x^(8)/6’

D.None of these

Ans:B

83.Consider the vector space V={f:’RR’’->’’RR’
such that f’’-2f’+f=0 over ‘RR’.Then which of the following is a basis for V?

A.{‘e^(t)’}

B.{‘e^(t),te^(t)’}

C.{e^(-t)’}

D.None of these

Ans:B

84.If F(a,b,c,x) is the hyper geometric series,then
‘lim_(b->oo)’F(a,b,a,’x/b’)is

A.’(1+x)^(a)’

B.log(1+x)

C.’e^(x)’

D.sin x

Ans:C

85.If ‘P_(n)(x)’ is the Legendre Polynomial ,then
‘P_(n)(-1)’is

A.1

B.-1

C.’(-1)^(n)’

D.0

Ans:C

86.The equation ‘U_(xxxx_)+x^(2)U_(yy)’=0 is

A.Elliptic

B.Parabolic

C.Hyperbolic

D.None of these

Ans:A

87.If a,b,c are three constants such that
‘U_(x)’=a,’U_(y)’=b and ‘U_(z)’=c with u(x,y,z)=0 at (0,0,0); then U(1,0,0)is

A.a

B.b

C.c

D.None of these

Ans:A

88.If 0<s<1,then ‘int_0^oo(x^(s-1))/(1+x)dx’is

A.’beta’(1,s)

B.’beta’(1-s,1)

C.’beta’(1-s,s)

D.Divergent

Ans:C

89.The Laplace transform L{‘(sin omegat)/(t)’} is

A.’cot^(-1)((s)/(w))’

B.’(omega)/(s^(2)+omega^(2))’

C.’(s)/(s^(2)+omega^(2))’

D.None of these

Ans:A

90.Which of the following is a periodic function on
‘RR’?

A.x-[x]

B.x

C.[x]

D.none of these

Ans:A

91.A velocity field is given by q( raist to
bar)=xi+(y+t)j.Then the stream line for this field at t=2 is

A.Circle

B.Ellipse

C.Hyperbola

D.Rectangular hyperbola

Ans:D

92.Consider the fuzzy sets A and B on X=[0,’(pi)/(2)’]where
A(x)=sin x and B(x)=cos x.Then o.8-cut of A’nn’B is

A.’Phi’

B.X

C.An uncountable subset of X

D.A denumerable subset of X

Ans:A

93.Which is false about the Cantor set?
A.Cantor set is perfect

B.Cantor set is compact

C.Cantor set is fractal

D.Cantor set is an uncountable set with positive
Lebesgue measure

Ans:D

94.The Voltra integral equation corresponding to the
differential equation y”+xy=1,y(0)=y’(0)=0 is

A.’y(x)’=’(x^(2))/(2)’-‘int_0^xty(t)dt’

B.’y(x)=x^(2)/2’-‘int_0^x(x-t)y(t)dt’

C.’y(x)=(x^(2))/(2)-‘int _0^(x)(x-t)ty(t)dt’

D.’y(x)=(x)/(2)’-‘int_0^x(x-t)ty(t)dt’

Ans:C

95.The derivative of the Weierstrass function is

A.Elliptic

B.Even

C.Pole of order 2

D.All the above

Ans:A

96.The function ‘zeta’ (z)is

A.Weierstrass (z)

B.Weierstrass raise to 1(z)

C.-Weierstrass raise to 1(z)

D.Weierstrass (z)

Ans:D

97.For the Euler’s gamma function symbol(z)symbol(1-z)is

A.’(pi)/(cospiz)’

B.’(pi)/(sinpiz)’

C.’(pi)/(cotpiz)’

D.None of these

Ans:B

98.Which of the following is not a maximal geodesic
for the cylinder ‘x_(1)^(2)’+’x_(2)^(2)’=1 in ‘RR^(3)’?

A.Vertical line

B.Horizontal circle

C.Helix

D.None of these

Ans:D

99.Which of the following linear transformation is a
possible Weingarten map for the unit n-sphere ‘x_(1)^(2)+’’x_(2)^(2)’+’x_(3)^(2)’=1
in ‘RR^(3)’ oriented by its inward unit normal?

A.Traise to
x bar=0

B.T raise to x bar=x bar

C.T raise to x bar=2 x bar

D.T raise to x bar=3 x bar

Ans:B

100.Which of the following is reducible over ‘RR’?

A.’x^(2)’+1

B.’x^(2)’+2

C.’x^(5)’+’x^(3)+’’x^(2)’+x+1

D.None of these

Ans:C

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