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ALGEBRAALGEBRAALGEBRAALGEBRA

TEST :

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1) Let C be the set of ordered pairs (, ) of real numbers and multiplication in C are

defined by the equations

(, ) + (, ) ( + , + ) (, ) ! (, ) ( " , + ) Then the

multiplicati#e identity is

$) (%, %) &) (1, 1) C) (%, 1) ') (1, %)

) hich one of the follo*in is a in *ithout unity-

$) The set of all inteers *ith respect to addition and multiplication!

&) The set of all e#en inteers *ith respect to addition and multiplication!

C) The set of all rational *ith respect to addition and multiplication!

') The set of all real *ith respect to addition and multiplication!

.) hich one of the follo*in is not an inteeral domain-

$) The set of all inteers *ith respect to addition and multiplication

&) The set of all rational *ith respect to addition and multiplication!

C) The set of numbers of the form + √ *ith a and b as inteers *ith respect

to addition and multiplication!

') The set of all ordered pairs (, ) of real numbers *ith respect to addition and

multiplication defined by the equations!

(, ) + (, ) ( + , + )

(, ) + (, ) (, )

/) hich one of the follo*in set of #ectors in (.) is linearly dependent-

$) 0(1,%,%), (%,1,%), (%,%,1) &) 0(1,1,%), (.,1,.), (2,.,.)

C) 0("1,,1), (.,1, ") ') 0(,.,2), (/,3,2)!

2) Let 4 be any field and 5 be a subfield of 4! be the field of real numbers and C the

field of comple6 numbers! Then *hich one of the follo*in statements is not correct-

$) 4 is a #ector space o#er 4 &) 4 is a #ector space o#er 5

C) is a #ector space o#er C ') C is a #ector space o#er

7) The #alue of a for *hich the set of #ectors

0(1 + , 1 , ), ( + , + , ), (. + , . + , . + ) is not a basis for . is

$) % &) 1 C) ') .!

K. Sellavel, M.Sc., M.Phil., B.Ed., - www.Padasalai.Net


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8) hich one of the follo*in statements is *ron-

$) E#ery roup 9 *ith identity e such that is $belian

&) ;f 9 is an $belian roup then for all a, b 9 ()

C) ;f e#ery element of a roup 9 is its o*n in#erse then 9 is $belian!') ;f 9 is an $belian roup then e#ery element of 9 is its o*n in#erse!

of the roup 9 is a subroup of 9 if and only if

, ? ? "1 !

&) $ necessary and sufficient condition for a non=empty subset > of a roup 9 to

be a subroup is that , ? "1 !

C) The intersection of t*o subroups of a roup is also a subroup!

') union of t*o subroups of a roup is also a subroup!

11) ;f 9 is roup of order


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C) .. + + . + / ') .. + + / + /

12) The basis for the #ector space B 0( % ) , , is

$) 0@1 %

% %A , @

% 1

% %A , @

% %

% 1A &) 0@

1 %

% %A , @

% 1

% %A , @

% %

1 %A

C) 0@% %% %

A , @1 %% %

A , @% 1% %

A ') 0@% %% %

A , @1 11 1

A , @% %% 1

A

17) $ny ordered interal domain is characteristic of

$) 1 &) % C) D ') rime!

18) Let 9 0@ A ,

F! Go* (, +) is

$) $ roup *ith identity element @1 %

% 1A &) Got a roup

C) $n $belian roup ') $ non= $belian roup

1


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/) The Set 0@ "

A , is a rin under matri6 addition and multiplication!

The in#erse of @ "

A is

$) @" " A &) @

"" A C) @

+" "A ') @

" " "A

2) ;f a non=empty subset of a #ector space B o#er a field 4 is a subspace of B, then Q,

R , ?

$) Q + R &) Q + R C) Q + R ') Q

7) hich one of the follo*in sets of #ectors is linearly dependent-

$) 0(1,/, "), (",1,.), ("/,11,2) &) 0(1,,1), (,1,%), (1, "1,)

C) 0(1,%,%), (%,1,%), (1,1,%) ') 0(%,%,%), (,2,.), ("1,%,7)

8) hich one of the follo*in sets of #ectors is not a basis for B3 ()-

$) 0(1,%,%), (1,1,%) &) 0(1,%,%), (%,1,%), (%,%,1)

C) 0(1,%,%), (%,1,%), (1,1,1) ') 0(1,1,%), (%,1,1), (1,%,1)

is a subroup of 9 and G is a normal subroup of 9, then

$) >G is a subroup of 9 &) > is a normal subroup of 9

C) $ M & is a normal subroup of 9 ') > N G is a normal subroup of 9.1) hich one of the follo*in is a #ector space- *ith usual addition and scalar

multiplication defined by

$) , %, &) , ,

C) , , ') , | |, | | |

.) ;f $ and & are t*o subspaces of a #ector space B o#er a field 4, then

$) $ N & is a subspace of B &) $ & is a subspace of B

C) $ M & is a subspace of B ') $& is a subspace of B

K. Sellavel, M.Sc., M.Phil., B.Ed., - www.Padasalai.Net

33-maths-material-algebra.pdf0_;+

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