PPSC Lecturer Mathematics Solved Paper-2011

30. Number of non-lsomorphic groups of order 8 is.

(A) 4

(B) 2

(D) 5

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Q.31 Center of the group of quatemions $Q_8$ is of order (A) 1

(B) 2

(D) 4

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Q.32 a. (b x c) is not equal to

(A) a . (c x b)

(B) (a x b) . c

(D) a . (a x b)

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Q.33 Let G be a group. Then the derived group G ‘ is subgroup of G.

(A) cyclic

(B) abelian

(D) none of these

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Q.34 Let G be a group. Then the factor group G/G is

(A) abelian

(B) cyclic

(D) none of these

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Q.35 Finite simple abelian groups are of order

(A) 4

(B) prime power

(D) prime number

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Q.36 Set of integers Z is

(A) Field

(B) group under multiplication

(D) division ring

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Q.37 Set of integers Z is__________ of the set Q of rationals

(A) prime ideal

(B) subring

(D) none of these

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Q.38 Solution set of the equation 1+ cos x = 0 is

(A) $\left \{\pi+n\pi:n \in z \right \}$

(B) $\left \{2n\pi:n \in z \right \}$

(D) $\left \{\pi+2n\pi:n \in z \right \}$

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39. Non-zero elements of a fleld form a group under

(A) addition

(B) multiplication

(D) division

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Q.40 Let Q be the set of rational numbers. Then $Q({\sqrt{3}})=\left \{ a + b\sqrt{3}: a,b \in Q \right \}$ is a vector space over g with dimension

(A) 1

(B) 2

(D) 4

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Q.41 Let W be a subspace of the space R³. If dim W = 0 then W is a

(A) line through the origin 0

(B) plane through the origin 0

(D) a point

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Q.42 Let $P_n$ (t) be a vector space of all polynomials of degree $\leq$ n: Then

(A) dim $P_n$ (t) = n-1

(B) dim $P_n$ (t) = n

(D) 2

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Q.43 A one to one linear transformation preserves,

(A) basis but not dimension

(B) basis and dimension

(D) None of these

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Q.44 In the group (Z, °) of all integers where a ° b = a+b+1 for a, b ɛ Z, the inverse of-3 is

(A) -3

(B) 0

(D) 1

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Q.45 The set Z of all integers is not a vector space over the field R of real numbers under ordinary addition ‘+’ multiplication ‘ × ‘ of real numbers, because

(A) (Z, +) is a ring

(B) (Z, +, x) is not a field

(D) ordinary multiplication of real numbers does not define a scalar multiplication of Z by R.

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Q.46 Let G be an abelian group. Then $\Phi: G\rightarrow G$ given by______________ is an automorphism

(A) $\Phi$ (x) = x³

(B) $\Phi$ (x) = e

(D) $\Phi$ (x) = x

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Q.47 Let G be a group in which g² = 1 for all g is G. Then G is

(A) abelian

(B) cyclic

(D) Non abelian

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Q.48 Let G = (a, b: b² = 1 = a³ , $ab=ba^{-1}$ ). Then the number of distinct left cosets of H = <b> in G is non-abelian

(A) 1

(B) 2

(D) 3

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Q.49 A linear transformation T: U $\rightarrow$ V is one-to-one if and only if kernel of T is equal to

(A) U

(B) V

(D) Im(T)

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Q.50 For a scalar point function ☆(x; y, z), div grad ☆ is

(A) A scalar point function

(B) vector point function

(D) neither

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Q.51 A particle moves along a curve F = ( $e^{-1}$ , 2cos3t, 2 sin3t) where t time is. The velocity at t= 0 is

(A) (-1, 0, 6)

(B) (-1,-6, 0)

(D) (-1, 2, 2)

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Q.52 The coordinate surfaces for the cylindrical coordinates x =rcos $\theta$ , y = r sin $\theta$ , z =z are given by

(A) $r=c,\theta=c$

(B) $r=c_1,\theta=c,z=c_3$

(D) $\theta=c_2,z=c_3$

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Q.53 The metric coefficients in cylindrical coordinates are

(A) (1, 1, 1)

(B) (1,0, 1)

(D) neither

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Q.54 The value of the quantity $\delta_i x_ix_j$ is

(A) $x_i$

(B) zero

(D) $x_i x_j$

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Q.55 A tensor of rank 5 in a space of 4 dimensions has components

(A) 5

(B) 4

(D) 1024

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Q.56 A vector is said to be irrotational if

(A) $\nabla \overline{F} = 1$

(B) $\nabla \overline{F} = 0$

(D) none

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Q.57 The moment of inertia of a rigid hemisphere of mass M and radius a about a diameter of a base is

(A) Ma²/5

(B) Ma²/2

(D) more information needed

Q.58 Radius of gyration of a rigid body of mass 4 gm having moment of inertia 32 gm(cm)² is:

(A) 8 (cm)²

(B) 2 $\sqrt{2}$ cm

(D) 2 $\sqrt{2}$ gm

(D) more information needed

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Q.59 Equation for the ellipsoid of inertia for a rigid body having moments and products of inertia $1_{xx}$ = 18 units $1_{yy}$ = 18 units, $1_{zz}$ = 36 units, $1_{xy}$ = -13.5 units, $1_{xz}$ = 0, $1_{yz}$ = 0.