1. GCD(a,b) = GCD(b,a mod b)
a) True
b) False
Answer
Answer: a [Reason:] The statement is true. For example, GCD(55,22) = GCD(22,55 mod 22) = GCD(22,11) = 11
Consider the Following properties Properties
G-i) Closure
G-ii) Associative
G-iii) Identity Element
G-iv) Inverse Element
G-v) Commutative
Consider the Following properties Properties
R-i) Closure under multiplication
R-ii) Associativity of multiplication
R-iii) Distributive Law
R-iv) Commutativity of multiplication
R-v) Multiplicative Identity
R-vi) No zero divisors
R-vii) Multiplicative Inverse
2. All groups satisfy properties
a) G-i to G-v
b) G-i to G-iv
c) G-i to R-v
d) R-i to R-v
Answer
Answer: b [Reason:] Group G denoted by {G,o}, is a set of elements that satisfy the properties G-i to G-iv.
3. An Abelian Group satisfies the properties
a) G-i to G-v
b) G-i to R-iv
c) G-i to R-v
d) R-i to R-v
Answer
Answer: a [Reason:] An Abelian group is a group that satisfies the Commutative property also.
4. A Ring satisfies the properties
a) R-i to R-v
b) G-i to G-iv
c) G-i to R-v
d) G-i to R-iii
Answer
Answer: d [Reason:] A ring R denoted by {R, + , x} is a set of elements with two binary operations addition and multiplication and satisfy axioms G-i to R-iii.
5. A Ring is said to be commutative if it also satisfies the property
a) R-vi
b) R-v
c) R-vii
d) R-iv
Answer
Answer: d [Reason:] A Ring is said to be commutative if it also satisfies the property R-iv: Commutativity of multiplication.
6. An ‘Integral Domain’ satisfies the properties
a) G-i to G-iii
b) G-i to R-v
c) G-i to R-vi
d) G-i to R-iii
Answer
Answer: c [Reason:]An ‘Integral Domain’ satisfies the properties G-i to R-vi.
7. A Field satisfies all the properties above from G-i to R-vi.
a) True
b) False
Answer
Answer: a [Reason:] A Field satisfies all the properties above from G-i to R-vi and is denoted by {F, +, x}.
8. In modular arithmetic : (a/b) = b(a^-1)
a) True
b) False
Answer
Answer: b [Reason:] This statement is not true. The correct version would be : (a/b) = a(b-1).
9. a.(b.c) = (a.b).c is the representation for which property?
a) G-ii
b) G-iii
c) R-ii
d) R-iii
Answer
Answer: a [Reason:] a.(b.c) = (a.b).c represents the Associative property.
10. a(b+c) = ac+bc is the representation for which property?
a) G-ii
b) G-iii
c) R-ii
d) R-iii
Answer
Answer: d [Reason:] a(b+c) = ac+bc represents the Distributive Property.
11. For the group Sn of all permutations of n distinct symbols, what is the number of elements in Sn?
a) n
b) n-1
c) 2n
d) n!
Answer
Answer: d [Reason:] There there are n distinct symbols there will be n! elements.
12. For the group Sn of all permutations of n distinct symbols, Sn is an abelian group for all values of n.
a) True
b) False
Answer
Answer: b [Reason:] For n>2 it does not form a Abelian Group.
13. Is S a ring from the following multiplication and addition tables?
+ a b x a b
a a b a a a
b b a b a b
a) Yes
b) No
c) Can’t Say
d) Insufficient Data
Answer
Answer: a [Reason:] S is a ring as it satisfies the properties G-i to R-iii.
14. Does the set of residue classes (mod 3) form a group with respect to modular addition?
a) Yes
b) No
c) Can’t Say
d) Insufficient Data
Answer
Answer: a [Reason:] Yes. The identity element is 0, and the inverses of 0, 1, 2 are respectively 0, 2, 1.
15. Does the set of residue classes (mod 3) form a group with respect to modular addition?
a) Yes
b) No
c) Can’t Say
d) Insufficient Data
Answer
Answer: b [Reason:] No. The identity element is 1, but 0 has no inverse.
