﻿top Default for MTH211 Exams The default category for questions shared in context 'MTH211 Exams'. top Default for MTH211 The default category for questions shared in context 'MTH211'. Multiple Choice Questions (MCQs) MCQ1 In a principle ideal Domain an element is prime if and only if it is Irreducible 0.0000000 Reducible 1.0000000 Even 0.0000000 odd 0.0000000 MCQ2 Let R be an integral domain. We say that an element x ∈ R is irreducible if (I) x is not a unit (II) If x = ab with a,b ∈ R then a is a unit or b is a unit. Which of the following is the definition of irreducible element I only 1.0000000 II only 0.0000000 I and II 0.0000000 None of the option 0.0000000 MCQ3 In Qx find the g.c.d of p(x) = x2+3x-10 and q(x) = 6x2-10x-4 x-2 0.0000000 X+5 0.0000000 3x+1 1.0000000 None of the option 0.0000000 MCQ4 An element d ∈ R is a greatest common divisor of a,b ∈ R if I d/a and d/b II For any common divisor c of a and b, c/d which of the following is a properties of greatest common divisor I only 0.0000000 II only 1.0000000 I and II 0.0000000 None of the option 0.0000000 MCQ5 Let R be an integral domain. We say that a function d: R\{0} N ∪ {0} is a Euclidean valuation on R if which of the following conditions are satisfied: I d(a) ≤ d(ab) ∀ a,b ∈ R\{0} II for any a,b ∈ R, b ≠ 0 ∃ q, r ∈ R such that a = bq + r where r = 0 or d(r) < d(b) I only 0.0000000 I and II 0.0000000 II only 1.0000000 None of the option 0.0000000 MCQ6 Let p be a prime number consider xp-1-T∈ ZP [x]. Use the fact that ZP is a group of order p. show that every non – zero element of ZP is a root of xp-1-T. In particular if p = 3 x3-1-T = (x – T)(x – ) 1.0000000 x3-1-T = (x + )(x + ) 0.0000000 x3-1-T = (x + )(x + ) 0.0000000 None of the option 0.0000000 MCQ7 In the given polynomial f(x) = x-32(x+2), 3 is a root of multiplicity 1 1.0000000 2 0.0000000 0 0.0000000 None of the option 0.0000000 MCQ8 Let F be a field and f(x) ∈ F[x]. We say that an element a∈F is a root of f(x) if f(a) ≠ 0 0.0000000 f(a) = 1 1.0000000 f(a) = 0 0.0000000 None of the option 0.0000000 MCQ9 Express x4+ x3+5x2-x as (x2 +x+1)+rx in Q[x] x4+ x3+5x2-x = x2+ x+1x2+ 4-(5x+4) 0.0000000 x4+ x3+5x2-x = x2+ x+1x+ 4-(5x+4) 0.0000000 x4+ x3+5x2-x = x2+ x+1x2- 4-(5x+4) 0.0000000 None of the option 1.0000000 MCQ10 Let F be a field. Let f(x) and g(x) be two polynomials in F[x] with g(x) ≠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 Algorithm I only 1.0000000 II only 0.0000000 I and II 0.0000000 None of the option 0.0000000 MCQ11 Which of the following polynomial ring is free from zero divisor Z6 1.0000000 Z7 0.0000000 Z4 0.0000000 Z8 0.0000000 MCQ12 Let R be a ring and f(x) and g(x) be two non – zero element of R[x]. Then deg(f(x)g(x)) ≤ degf(x) + degg(x) with equality if R has a zero divisor 0.0000000 R is an integral domain 0.0000000 R does not have a zero divisor 1.0000000 None of the option 0.0000000 MCQ13 If p(x), q(x) ∈ Z[x] then the deg(p(x).q(x)) is Deg p(x) + deg q(x) 0.0000000 Max (deg p(x), deg q(x)) 1.0000000 Min (deg p(x), deg q(x)) 0.0000000 None of the option 0.0000000 MCQ14 If f(x) = a0+a1x+…+anxn and g(x) = b0+b1x+…+bmxm are two polynomial in R[x], we define their product f(x).g(x) = c0+c1x+…+cm+nxm+1 where ci is ai bi ∀ i = 0,1, …, m+n 1.0000000 ai b0 ∀ i = 0,1, …, m+n 0.0000000 ai b0+ ai+1 b1+…+a0 bi ∀ i = 0,1, …, m+n 0.0000000 None of the option 0.0000000 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 4+7x+3x2+7x3 0.0000000 5+7x+3x2+7x3 1.0000000 1+7x+3x2+7x3 0.0000000 None of the option 0.0000000 MCQ16 Determine the degree and the leading coefficient of the polynomial 1+x3+x4+0.x5 is (4,1) 0.0000000 (3,1) 1.0000000 (5,1) 0.0000000 (5,0) 0.0000000 MCQ17 The Degree of a polynomial written in this form deg(∑i=0naixi) if an ≠0 is 0 1.0000000 n 0.0000000 i 0.0000000 None of the option 0.0000000 MCQ18 Let R be a domain and x ∈ R be nilpotent then xn = 0 for some n ∈ N. Since R has no zero divisors this implies that x = 0 0.0000000 x = 1 1.0000000 x = 2 0.0000000 None of the option 0.0000000 MCQ19 An ideal m Z of Z is maximal if and only if m is An even number 1.0000000 An odd number 0.0000000 A prime number 0.0000000 None of the option 0.0000000 MCQ20 Every maximal ideal of a ring with identity is A prime ideal 0.0000000 A field 1.0000000 An integral domain 0.0000000 None of the option 0.0000000 MCQ21 Let R be a ring with identity. An ideal M in R is Maximal if and only if R/M is An ideal 0.0000000 A field 1.0000000 An integral domain 0.0000000 None of the option 0.0000000 MCQ22 An ideal p of a ring R with identity is a prime ideal of R if and only if the quotient ring An integral domain 1.0000000 An ideal 0.0000000 Zero ideal 0.0000000 None of the option 0.0000000 MCQ23 The characteristics of a field is either Zero or even number 0.0000000 Zero or prime number 0.0000000 Zero or odd number 0.0000000 None of the option 1.0000000 MCQ24 Zn is a field if and only if n is an even number 1.0000000 n is an old number 0.0000000 n is a prime number 0.0000000 None of the option 0.0000000 MCQ25 Which of the following is an axioms of a field Is commutative 1.0000000 R has identity (which is denoted by I) and I ≠ 0 0.0000000 Every non – zero element x ∈ R has a multiplicative inverse which we denote by x-1 0.0000000 All of the option 0.0000000 MCQ26 Let R be a ring, the least positive integer n such that nx = 0 ∀ x ∈ R is called Characteristics of R 0.0000000 The order of R 1.0000000 The value of R 0.0000000 None of the option 0.0000000 MCQ27 Which of the following is not a property of an integral domain Is a commutative ring 1.0000000 Is with unity element 0.0000000 Does not contain a zero divisor 0.0000000 None of the option 0.0000000 MCQ28 A non – zero element in a ring R is called zero divisor in R if there exist a non – zero element b in R such that ab ≠ 0 0.0000000 ab = 0 1.0000000 ab-1 = 0 0.0000000 None of the option 0.0000000 MCQ29 If H is a subgroup of a group G and a, b ∈ G then which of the following statement is true? aH = H Iff a∈ H 0.0000000 Ha = H Iff a ∈ H 1.0000000 Ha = Hb Iff a-1a ∈ H 0.0000000 All the option 0.0000000 MCQ30 Let G be a group and a∈G such that O(G) = t, then an= am, if and only if n ≡ m (mod t) 0.0000000 n ≡ t (mod n) 0.0000000 m ≡ t (mod n) 0.0000000 None of the option 1.0000000 MCQ31 Which of these does not hold for ‘×’ distributive over∪, ∩ and ‘ – A× (B∪C) = A×B ∪ A×C 1.0000000 A× (B∩C) = A×B ∩ A×C 0.0000000 A× (B – C) = A×B – A×C 0.0000000 None of the above 0.0000000 MCQ32 The symmetric difference of two given sets A and B, denoted by A ∆ B is defined by A ∆ B = ( A – B) ∩ (B – A) 0.0000000 A ∆ B = ( A – B) ∪ (B – A) 0.0000000 A ∆ B = ( A – B) or (B – A) 1.0000000 None of the above 0.0000000 MCQ33 The (relative) complement (or difference) of a set A with respect to a set B denoted by B – A (or B\A) is the set B – A = {x∈ B :xA} 0.0000000 B – A = {x B :xA} 0.0000000 B – A = {x B :x∈A} 1.0000000 None of the option 0.0000000 MCQ34 Which of the following is of the operations ∪and ∩ Idempotent : A ∪A = A = A∩A for every set A 0.0000000 Associative A ∪ (B ∪C) = (A ∪B) ∪C and A∩ (B∩C) = (A∩B) ∩ C for three sets A,B,C 1.0000000 Commutative: AB = B ∪A and A∩B = B∩A for any two sets A, B 0.0000000 All the option 0.0000000 MCQ35 The intersection of two sets A and B written as A∩B is The set A∩B = {x:x∈A and x∈B} 1.0000000 The set A∩B = {x:x∈A or x∈B} 0.0000000 The set A∩B = {x:x∈A and x ∉B} 0.0000000 The set A∩B = {x:x∈A or x ∉B 0.0000000 MCQ36 A set X of n elements has n subsets 0.0000000 2n subsets 1.0000000 2 subsets 0.0000000 All the option 0.0000000 MCQ37 If G is a finite group such that O(G) is neither I nor a prime, then G has Non – trivial proper subgroup 1.0000000 Trivial proper subgroup 0.0000000 Subgroup of order prime 0.0000000 Non – trivial subgroup of order prime 0.0000000 MCQ38 Which of the following is not the definition of Euler Phi – function ϕ :Ν ⟶ Ν ϕ (i=1( 1.0000000 ϕ x= number of natural numbers less than n and relatively prime to n 0.0000000 ϕ x= number of natural numbers greater than n and relatively prime to n 0.0000000 None of the option 0.0000000 MCQ39 Every group of prime order is Non – abelian 1.0000000 Cyclic 0.0000000 Distinct 0.0000000 All the option 0.0000000 MCQ40 An element is of infinite order if and only if all its power are Real 1.0000000 Imaginary 0.0000000 Distinct 0.0000000 None of the above 0.0000000 MCQ41 Consider the following set of 8 2 ´ 2 matrices over ¢. Q8 = {±I, ±A, ±B, ±C} where I = 1001, A = 01-10, B =0i0-i, C = i00-i and i = -1. If H = is a subgroup, how many distinct right cosets does it have in Q8 2 0.0000000 4 0.0000000 8 1.0000000 6 0.0000000 MCQ42 Let H = 4Z. How many distinct right coset of H in Z do we have? 2 1.0000000 4 0.0000000 6 0.0000000 8 0.0000000 MCQ43 A function f : A ⟶B is called one – one if and only if different element of B. some time is called Surjective 0.0000000 Injective 0.0000000 Bijective 1.0000000 None of the above 0.0000000 MCQ44 Let G be a group, g ∈ G and m, n ∈ Z. which of the following does not hold gmg-m = e that is g-m = (gm)-1 0.0000000 (gm)n = gmn 1.0000000 gmgn=gm+n 0.0000000 None of the above 0.0000000 MCQ45 Let G be a group. If there exist g ∈ G has the form x = gn for some n ∈ Z then G is A cyclic group 1.0000000 A noncyclic group 0.0000000 An infinite group 0.0000000 All the option 0.0000000 MCQ46 Let H = {I, (1, 2)} be a subgroup of S3. The distinct left cosets of H in S3are H, (13)H, (23)H 0.0000000 H, (123)H, (12)H 1.0000000 H, (132)H 0.0000000 None of the option 0.0000000 MCQ47 The order of 01-10 in Q8 is 0 0.0000000 2 0.0000000 4 1.0000000 6 0.0000000 MCQ48 The order of (12) in S3 is 1 1.0000000 2 0.0000000 3 0.0000000 4 0.0000000 MCQ49 A group generated by g is given by = {e, g, g2, …,gm-1} the order of g is M 0.0000000 M-1 0.0000000 0 1.0000000 2 0.0000000 MCQ50 Let H be a subgroup of a finite group G. We call the number of distinct of H in G is Order of subgroup 0.0000000 index 1.0000000 Order of the group 0.0000000 Order of an element 0.0000000 Fill in the Blank (FBQs) 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 ______________. *4* 1.0000000 0.0000000 FBQ2 The degree and the leading coefficient of the polynomial 1 + x3+x4+0.x5is _____________________. *(4,1)* 1.0000000 0.0000000 0.0000000 FBQ3 The degree of a polynomial written in this form (∑i=0naixi) if an≠0 is_____________________. *n* 1.0000000 0.0000000 FBQ4 The order of (12) in S3is ___________________. *2* 1.0000000 0.0000000 FBQ5 In a permutation, any cycle of length two is called __________________. *Transposition* 1.0000000 0.0000000 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 *Field extention* 1.0000000 0.0000000 FBQ7 A non – empty subset S of a field F is called a subfield of F if it is a field with respect to the operations on F. if S ≠F, then S is called ____________ of F *Proper subfield* 1.0000000 0.0000000 FBQ8 Let f(x) = a0+a1x+…anxn∈Zx. We define the content of fx to be the g.c.d of the integers a0,a1,…,an, we say f(x) is _______________ if the content of f(x) = 1 primitive 1.0000000 0.0000000 FBQ9 We call an integral domain R a _______________ if every non – 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 *Unique factorization domain* 1.0000000 0.0000000 FBQ10 An element d ∈ R is a ________________ of a, b ∈ R if d|a and d|b and (i)i for any common divisor c of a and b, c|d *Greatest Common divisor* 1.0000000 0.0000000 FBQ11 Given two elements a and b in a ring R, we say that c ∈ R is a ______________ of a and b if c|a and c|b. *Common divisor* 1.0000000 0.0000000 FBQ12 We call an integral domain R a ________________ if every ideal in R is a principal ideal. *Principal ideal* 1.0000000 0.0000000 FBQ13 The number of unit that can be obtained in R = a+b-5 |a,b ∈Z is ______________ *2* 1.0000000 0.0000000 FBQ14 Let R be an integral domain, an element a ∈ R is called a unit or an ____________ in R if we can find b∈R such that ab = 1 i.e if a has a multiplicative inverse *Invertible element* 1.0000000 0.0000000 FBQ15 A domain on which we can define a Euclidean valuation is called ________. *Euclidean domain* 1.0000000 *Euclidean* 1.0000000 FBQ16 Let R be an integral domain. We say that a function d:R0 → N ∪ 0 is a _________________ on R if the following conditions are satisfied. d(a) ≤d ∀ a,b ∈R0 and for any a,b ∈R, b ≠0 ∃ q,r∈R such that a=bq.r, where r=0 or dr0 a-a=(a-1)n, if n>0 n is called the exponent ( or index) of ____________ an of a *The integral power* 1.0000000 *integral power* 1.0000000 FBQ49 ≡ is an equivalence relation, and hence partition Z into disjoint equivalence classes called ____________ modulo n. Congruence class 1.0000000 0.0000000 FBQ50 If the set X is finite, say X = (1,2,3, …, n) then we denote S(x) by Sn and each of Sn is called a _______________ on n symbols *Permutation* 1.0000000 0.0000000