# Kerala PSC HSST Mathematics Solved paper part 3

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
Department Higher Secondary Education

51:-int_0^ooe^(-x^(2))dx=
A:-1
B:-`(Pi)/(2)`
C:-`(sqrt(Pi))/(2)`
D:-`pi`

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

53:-`lim_(x->0)((1-cosx)sinx)/(x^(2)+x^(3))=`
A:-–1
B:-0
C:-`1/2`
D:-1

54:-`i^(321)+(1)/(i^(123))=`
A:-0
B:-2
C:-2i
D:-1 – i

55:-|z+3i| + |z–3i| = 8 represents
A:-a straight line
B:-a circle
C:-a hyperbola
D:-an ellipse

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`

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

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

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`

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

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

62:-
A:-`(3+i)/(6)`
B:-`(3-i)/(6)`
C:-`(1+3i)/(9)`
D:-`(1-3i)/(9)`

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

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

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

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)`

67:-Which of the following is not a topological property ?
A:-length and area
B:-connectedness
C:-continuity
D:-compactness

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`

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)}`

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)

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

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`

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`

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