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Section A

Q1. a) Let G be a group and a, b, ª G. Show that the orders of ‘ab’ and ‘ba’ are equal. 5 b) Let G be a group. Suppose G has only one element of order 2. Show that ax = xa for all x ª G. 5

Q2. a) Let (G, .) be a group. Then a non empty subset H of G is a subgroup if and only if for a, b ª H, the

-1

element abª H. 5
b) Let H be a subgroup of a group G and a ª G. If (Ha)-1 (Ha)-1 = a -1 H = {(ha)-1, h ª H}, then show that: 5
Q3. a) The set An of all Permutations in Sn form a subgroup of Sn. 5
b) Is the Permutation given below is even or odd?
(1 4 7) (3 4 5) (8 7) (8 3 4 5) 5
Section B
Q4. a) If the matrices A, B and C are conformable for the indicated products then prove that: 5
A (BC) = (AB) C

-1 2[; 44-2

21

Q5. a) Solve the system of equations by Gauss Elimination method. 5

x1-x2+2x3= 0 4x1+x2+2x3 = 1 x1+x2+ x3 =-1

b) Prove that: 5

-3

]

0

b) Find inverse of the matrix over R. A =

5

xaa axa

3

= (x-a)aax aaa

Q6. a) Solve for x

a

a (x +3a) a

x

1 2+x 2

=0 3 2+x b) Find the rank of the matrix:

3

1 3+x

1

-3 5[;4-8 3

720

Also write an echelon matrix row equivalent to A.

Q7. a) Show that the transformation T : R3 ? R3 given below is linear T(x1, x2, x3) = (x1 – 3x2 – 2x3, x2 – 4x3, x3). 5

33

b) Find the matrix for the given linear transformation T : R? Rwith respect to standard

3

basis for R. T(x1, x2, x3) = (x1+x2, -x1-x2, x3). 5

3

Q8.a) Finda basisforthe subspaceWofR.W={(x,y,z) |x–2y+5z =0}. 5 b) Let V be the vector space of all real valued functions defined on R. Show that “The Set of all Even Functions” is sub space of V. 5

*** B.A/B.Sc -I (09/A) vi ***

12

]

-4

A=

16

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last updated on 28-04-2019