FBQ1: Let G = {1, -1, i, -i}. Then G is a group under usual multiplication of complex numbers, in this group, the order of i is _____.
Answer: 4
FBQ2:
Answer: (4,1)
FBQ3:
Answer: N
FBQ4: The order of (12) in is ___________________.Â
Answer: 2
FBQ5: In a permutation, any cycle of length two is called __________________.Â
Answer: Transposition
FBQ6: A field K is called _____________ of F if F is a subfield of K, thus Q is a subfield of R and R is a field extension of Q
Answer: Field extension
FBQ7:
Answer: Proper subfield
FBQ8:
Answer: Primitive
FBQ9: We call an integral domain R a _______________ if every non â€“ zero element of R which is not a unit in R can be uniquely expressed as a product of a finite number of irreducible elements of R
Answer: Unique factorization domain
FBQ10:
Answer: Greatest Common divisor
FBQ11: Given two elements a and b in a ring R, we say that c is a ______________ of a and b if c|a and c|b.Â
Answer: Common divisor
FBQ12: We call an integral domain R a ________________ if every ideal in R is a principal ideal.
Answer: Principal ideal
FBQ13: ______________.Â
Answer: 2
FBQ14: Let R be an integral domain, an element a R is called a unit or an ____________ in R if we can find bR such that ab = 1 i.e if a has a multiplicative inverse
Answer: Invertible element
FBQ15: A domain on which we can define a Euclidean valuation is called ________.
Answer: Euclidean domain
FBQ16:
Answer: Euclidean Evaluation
FBQ17:
Answer: Root of multiplicity m
FBQ18: Let F be a field and f(x) Fx we say that an element a F is a ____________ (or zero) of f(x) if f(a) = 0
Answer: Factor
FBQ19: If S is set, an object â€˜aâ€™ in the collection S is called an_________________ of S
Answer: Element
FBQ20: A set with _____________element in S is called an empty set
Answer: No
FBQ21: ____________ method is sometimes used to list the element of a large set
Answer: Roster
FBQ22: The set of rational numbers and the set of real numbers are respectively represented by the symbol_______________ and __________________.
Answer: Q and R
FBQ23: The symboldenotes __________________.
Answer: There exist
FBQ24: If A and B are two subsets of a set S, we can collect the element that are common to both A and B, we call this set the _______________of A and B.Â
Answer: Intersection
FBQ25: A relation R defined on a set S is said to be ________________ if we have
Answer: Reflexive
FBQ26: A relation R defined on a set S is said to be ________________if
Answer: Symmetric
FBQ27: A relation R defined on a set S is said to be ________________ if a R b and
Answer: Transitive
FBQ28: A relation R defined on a set S that is reflexive, symmetric and transitive is called ________________ relation
Answer: Equivalence
FBQ29: A ________________ f from a non â€“ empty set A to a non â€“ empty set B is a rule which associates with every element of A exactly on element of B
Answer: Function
FBQ30: A function is called ______________ if associates different elements of A with different element of B
Answer: Injective
FBQ31: A function is called ____________ if the range of f is B.Â
Answer: Onto
FBQ32:
Answer: Projection
FBQ33: A function that is both one to one and onto is called ________________
Answer: Bijective
FBQ34: __________________ set.
Answer: Finite
FBQ35: A set that is not ______________ is called infinite set
Answer: Finite
FBQ36: __________________Â
Answer: Bijective
FBQ37: 1 and p
Answer: Prime
FBQ38: _____________ number
Answer: Composite
FBQ39: _______________ on A.Â
Answer: Identity function
FBQ40: ______________ on S.Â
Answer: Binary operation
FBQ41:
Answer: Closed
FBQ42:
Answer: Associative
FBQ43:
Answer: Commutative
FBQ44: __________.Â
Answer: Distributive over
FBQ45:
Answer: Identity element
FBQ46: The Cayley table is named after the famous mathemathecian
Answer: Arthur Cayley
FBQ47: ____________ system consists of a set with a binary operation which satisfies certain properties is called a group
Answer: Algebraic
FBQ48:
Answer: The integral power
FBQ49: is an equivalence relation, and hence partition Z into disjoint equivalence classes called ____________ modulo n.Â
Answer: Congruence class
FBQ50: If the set X is finite, say X = (1,2,3, â€¦, n) then we denote S(x) by and each of is called a _______________ on n symbols
Answer: Permutation
MCQ1: In a principle ideal Domain an element is prime if and only if it is
Answer: Reducible
MCQ2:
Answer: I only
MCQ3:
Answer: 3x+1
MCQ4:
Answer: II only
MCQ5:
Answer: II only
MCQ6:
Answer:
MCQ7:
Answer: 1
MCQ8:
Answer: f(a) = 1
MCQ9: Express x4+Â x3+5x2-xÂ as (x2Â +x+1)+rx in Q[x]
Answer: None of the options
MCQ10: Let F be a field. Let f(x) and g(x) be two polynomials in F[x] with g(x) â‰ 0. Then I There exist two polynomial q(x) and r(x) in F[x] such that f(x) = q(x)g(x) + r(x), where degr(x) < degg(x).IIThe polynomial q(x) and r(x) are unique, which of the following is a properties of Division AlgorithmÂ
Answer: I only
MCQ11: Which of the following polynomial ring is free from zero divisor
Answer: Z6Â
MCQ12: . Let R be a ring and f(x) and g(x) be two non â€“ zero element of R[x]. Then deg(f(x)g(x)) â‰¤ degf(x) + degg(x) with equality if
Answer: R does not have a zero divisor
MCQ13: If p(x), q(x) âˆˆ Z[x] then the deg(p(x).q(x)) is
Answer: Max (deg p(x), deg q(x))
MCQ14: If f(x) = a0+a1x+â€¦+anxn and g(x) = b0+b1x+â€¦+bmxm are two polynomial in R[x], we define their product f(x).g(x) = c0+c1x+â€¦+cm+nxm+1 where ci is Â
Answer: aiÂ bi âˆ€ i = 0,1, â€¦, m+n
MCQ15: Consider the two polynomials p(x), q(x) in Z[x] by p(x) = 1+2x+3x2, q(x) = 4+5x+7x3. Then p(x) + q(x) is
Answer: 5+7x+3x2+7x3
MCQ16: Determine the degree and the leading coefficient of the polynomial Â Â Â Â Â Â Â 1+x3+x4+0.x5 is
Answer: (3,1)
MCQ17: The Degree of a polynomial written in this form deg(âˆ‘i=0naixi) if anÂ â‰ 0 is
Answer: 0
MCQ18: Let R be a domain and x âˆˆ R be nilpotent then xn = 0 for some n âˆˆ N. Since R has no zero divisors this implies that
Answer: x = 1
MCQ19: An ideal m Z of Z is maximal if and only if m is
Answer: An even number
MCQ20: Â Every maximal ideal of a ring with identity is
Answer: A field
MCQ21: Â Let R be a ring with identity. An ideal M in R is Maximal if and only if R/M is
Answer: A field
MCQ22: An ideal p of a ring R with identity is a prime ideal of R if and only if the quotient ring is
Answer: An integral domain
MCQ23: Â The characteristics of a field is either
Answer: None of the options
MCQ24: Zn is a field if and only if
Answer: n is an even number
MCQ25: Â Which of the following is an axioms of a field
Answer: Is commutative
MCQ26: Â Let R be a ring, the least positive integer n such that nx = 0 âˆ€ xÂ âˆˆ R is called
Answer: The order of R
MCQ27: Which of the following is not a property of an integral domain
Answer: Is a commutative ring
MCQ28: A non â€“ zero element in a ring R is called zero divisor in R if there exist a non â€“ zero element b in R such that
Answer: ab = 0
MCQ29: If H is a subgroup of a group G and a, b âˆˆ G then which of the following statement is true
Answer: Ha = H Iff aâˆˆ H
MCQ30: Let G be a group and aâˆˆG such that O(G) = t, then an=Â am, if and only if
Answer: None of the options
MCQ31: Which of these does not hold for â€˜Ã—â€™ distributive over , and â€˜ â€“
Answer: AÃ— (BC) = AÃ—B AÃ—C
MCQ32: The symmetric difference of two given sets A and B, denoted by A âˆ† B is defined by
Answer: A âˆ† B = ( A â€“ B) or (B â€“ A)
MCQ33: The (relative) complement (or difference) of a set A with respect to a set B denoted by B â€“ A (or B\A) is the set
Answer: B â€“ A = {x B :xâˆˆA}
MCQ34: Which of the following is of the operations and
Answer: Associative A(BC) = (AB) C and A(BC) = (AB)C for three sets A,B,C
MCQ35: The intersection of two sets A and B written as AB is
Answer: The set AB = {x:xâˆˆA and xâˆˆB}
MCQ36: A set X of n elements has
Answer: 2n subsets
MCQ37: If G is a finite group such that O(G) is neither I nor a prime, then G has
Answer: Non â€“ trivial proper subgroup
MCQ38: Which of the following is not the definition of Euler Phi â€“ function MCQ39: Every group of prime order is
Answer: Non â€“ abelian
MCQ40: An element is of infinite order if and only if all its power are
Answer: Real
MCQ41: Consider the following set of 8 2 Â´ 2 matrices over Â¢. Q8 = {Â±I, Â±A, Â±B, Â±C} where I = , A = , B = , C = and i = -1. If H = <A> is a subgroup, how many distinct right cosets does it have in Q8Â
Answer: 8
MCQ42: Let H = 4Z. How many distinct right coset of H in Z do we have?
Answer: 2
MCQ43: A function f : A B is called one â€“ one if and only if different element of B. some time is called
Answer: Bijective
MCQ44: Let G be a group, g âˆˆ G and m, n âˆˆ Z. which of the following does not hold
Answer: (gm)n = gmn
MCQ45: Let G be a group. If there exist g âˆˆ G has the form x = gn for some n âˆˆ Z then G is
Answer: A cyclic group
MCQ46: Let H = {I, (1, 2)} be a subgroup of S3. The distinct left cosets of H in S3are
Answer: H, (123)H, (12)H
MCQ47: The order of in Q8 is
Answer: 4
MCQ48: The order of (12) in S3 isÂ
Answer: 1
MCQ49: A group generated by g is given by <g> = {e, g, g2, â€¦,gm-1} the order of g is
Answer: 0
MCQ50: Let H be a subgroup of a finite group G. We call the number of distinct of H in G _____.
Answer: index