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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 ab | ª 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}[; 4^{4}^{-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 : R^{3 }? R^{3 }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 |