**Kerala PSC HSST Mathematics previous question paper part 3**

**Paper Code: 44/2017/OL**

**Category Code: 315/2015**

**Exam: HSST(Jr) Mathematics NCA**

**Date of Test 02-06-2017**

51:-int_0^ooe^(-x^(2))dx=

A:-1

B:-`(Pi)/(2)`

C:-`(sqrt(Pi))/(2)`

D:-`pi`

Answer:- Option-C

52:-Bolzano-Weierstrass theorem

A:-Every convergent sequence of real numbers is bounded

B:-A bounded sequence of real numbers has a convergent subsequence

C:-Every sequence of real numbers has a convergent subsequence

D:-A sequence of non-negative real numbers is bounded if and only if it is convergent

Answer:- Option-B

53:-`lim_(x->0)((1-cosx)sinx)/(x^(2)+x^(3))=`

A:-–1

B:-0

C:-`1/2`

D:-1

Answer:- Option-C

54:-`i^(321)+(1)/(i^(123))=`

A:-0

B:-2

C:-2i

D:-1 – i

Answer:- Option-C

55:-|z+3i| + |z–3i| = 8 represents

A:-a straight line

B:-a circle

C:-a hyperbola

D:-an ellipse

Answer:- Option-D

56:-Harmonic conjugate of `u(x,y)=x^(2)-y^(2)` is

A:-`v(x, y)=x^(2)+y^(2)`

B:-`v(x, y)=(x+y)^(2)`

C:-`v(x, y)=(x-y)^(2)`D:-`v(x, y)=2 xy`

Answer:- Option-D

57:-Let C be the positively oriented circle |z| = 4. Then `oint_(C)(z^(2)dz)/(z-1)+oint_(C)(z^(2)dz)/((z-1)^(2))=`

A:-`6pii`

B:-`2pii`

C:-`pii`

D:-0

Answer:- Option-A

58:-If f(z) is continuous in a simply connected domain D and if `oint_(C)f(z)dz=0` for every closed path in D, then

f(z) is analytic in D

A:-Liouville's theorem

B:-Morera's theorem

C:-Cauchy's integral theorem

D:-Cauchy's integral formula

Answer:- Option-B

59:-The radius of convergence of the power series `sum_(n=0)^oo``((2n)!)/((n!)^(2))(z-2)^(n)` is

A:-0

B:-`1/4`

C:-`1/2`

D:-`oo`

Answer:- Option-B

60:-At z = 0, the function `f(z)=e^((1)/(z))` has

A:-a removable singularity

B:-a simple pole

C:-an essential singularity

D:-no singular point

Answer:- Option-C

61:-Let `f(z)=(1-cosz)/(z^(5))` . Then f(z) has

A:-a pole of order 3 and residue `(-1)/(24)` at z = 0

B:-a pole of order 5 and residue `(-1)/(24)` at z = 0

C:-a pole of order 3 and residue `(1)/(5)` at z = 0

D:-a pole of order 5 and residue `(1)/(5)` at z = 0

Answer:- Option-A

62:-

A:-`(3+i)/(6)`

B:-`(3-i)/(6)`

C:-`(1+3i)/(9)`

D:-`(1-3i)/(9)`

Answer:- Option-B

63:-Which of the following is false ?

A:-Every order topology is Hausdorff

B:-Subspace of a Hausdorff space is Hausdorff

C:-Every Hausdorff space is normal

D:-Product of two Hausdorff space is Hausdorff

Answer:- Option-C

64:-Deleted comb space is

A:-connected and path connected

B:-connected but not path connected

C:-not connected but path connected

D:-neither connected nor path connected

Answer:- Option-B

65:-Which of the following need not be a normal space ?

A:-product of two normal spaces

B:-a metrizable space

C:-a compact Hausdorff space

D:-a regular space with a countable basis

Answer:- Option-A

66:-Which of the following is false ?

A:-the one point compactification of the real line `RR` is homeomorphic to an ellipse

B:-the one point compactification of the open interval (0, 1) is homeomorphic to closed interval [0, 1]

C:-the one point compactification of the open interval (0, 1) is homeomorphic to the circle `S^(1)`

D:-the one point compactification of `RR^(2)` is homeomorphic to the sphere `S^(2)`

Answer:- Option-B

67:-Which of the following is not a topological property ?

A:-length and area

B:-connectedness

C:-continuity

D:-compactness

Answer:- Option-A

68:-Let d be a metric defined on `RR` by

Then

A:-d is a pseudo metric on `RR`

B:-d is the usual metric on `RR`

C:-d is the Euclidean metric on `RR`

D:-d is the trivial metric on `RR`

Answer:- Option-D

69:-Which of the following is not a basis for `RR^(3)` ?

A:-`{(1, 1, 1), (1, 1, 0), (1, 0, 0)}`

B:-`{(1, 1, 1), (0, 1, 1), (1, 0, 0)}`

C:-`{(1, 1, 1), (0, 1, 1), (0, 0, 1)}`

D:-`{(1, 0, 0), (0, 1, 0), (0, 0, 1)}`

Answer:- Option-B

70:-Let `T : RR^(3)->RR^(3)` be a map defined on `RR^(3)` . Then which of the following is not a linear transformation ?

A:-T (x, y, z) = (y, x, 0)

B:-T (x, y, z) = (x + y, y + z, z + x)

C:-T (x, y, z) = (xy, yz, xz)

D:-T (x, y, z) = (0, 0, 0)

Answer:- Option-C

71:-Let `P_(5)(x)` be the set of all real polynomials of degree `<= ` 5.Then dimension of the vector space

`P_(5)(x)` over `RR` is``

A:-0

B:-1

C:-5

D:-6

Answer:- Option-D

72:-Let `T:RR^(4)->RR^(5)` be defined by `T(x_(1),x_(2),x_(3),x_(4))=(x_(1),x_(2),x_(3),x_(4),0)` Then the

dimension of the null space is

A:-`n(T)=0`` `

B:-` ``n(T)=1`

C:-` ``n(T)=4`

D:-` ``n(T)=5`

Answer:- Option-A

73:-Characteristic polynomial of `[[1, -1, 0],[0, 1, -1],[-1, 0, 1]]` is

A:-`lambda^(3)+3lambda^(2)+3Lambda+1=0`

B:-`lambda^(3)-3lambda^(2)+3Lambda-1=0`

C:-`lambda^(3)-3lambda^(2)+3Lambda-2=0`

D:-`lambda^(3)-3lambda^(2)+3Lambda=0`

Answer:- Option-D

74:-Let `2x+y-z=4`

`x+3y+2z=1`

`3x+4y+z=5`

The above system of equation is

A:-homogeneous and consistentB:-nonhomogeneous and inconsistent

C:-consistant and has unique solution

D:-consistant and has infinite solution

Answer:- Option-D

75:-Which of the following map `T:RR^(3)->RR` is a linear functional ?

A:-`T(x,y,z)=5`

B:-`T(x, y, z) = x^(2)`

C:-`T(x,y,z)=-2x+y`

D:-`T(x,y, z)=xy+6`

Answer:- Option-C

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