BSc Mathematics SEM I 2019 20 Nov 2019-20 MATHS ALGEBRA Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Choose correct alternative in each of the following. I. a) Every subset of positive integers has the least element 20 marks
   • b) Every non-empty subset of positive integers has the largest element
   • c) Every non- empty subset of N, has the least element
   • d)1 is the least element of every non-empty subset of positive integers Il. integers r, s and n, n> o the value of (r+s)" +(r
   • a) Is always even. b) Is always odd
   • c) Depends on n. d) Depends onr & s Il. and a/c then be is?
   • c)Valid only when d) Valid only when a= 1 IV. The integers 15239 and 15240 are?
   • a)Co-prime b) Not co-prime Their g.c.d. is O d) Their g.c.d. is a prime V. Ifevery pair of integers satisfy a = b (modn ) then n is? VI. unit digit of function from A to B, is b is bijective, is.
   • a)One-one b)Onto c)One-one and onto d)Depends on A and B A relation on z, given by a and b ifa + bis.
   • a) Reflexive b) Symmetric c) Transitive d) An equivalence relation IX. polynomial is the one whose.
   • a)Degree is 1 b)Leading coefficient is 1 c) Constant term is 1 d)All the coefficients are X. wis root of X" = 1 such that, 1 + w + + + w‘ then the smallest positive value of n is.
2. Q2 A) Attempt only ONE from the following I. and prove division algorithm in z, Il. Prove that every integer n > 1, can be expressed as a product of positive primes also prove that expression is unique except for the order in which the prime factors occur Attempt any TWO questions from the following. (12) I. Using method of finite induction prove that the following example is true for all N lI. in usual notations that for III. and prove Euclid’s Lemma IV. two integers such that (m,n) = 1 then prove that (mn) =
3. Q3 A) Attempt any ONE question from the following, & B are non-empty subsets of and Y respectively. Then prove that 8 marks
   • a) AC (f(A)) then equality holds if and only if f is injective,
   • b) B The equality holds if and only if f is surjective is equivalence relation on a non-empty set X, then prove the following
   • a) Each element of X belongs to some equivalence class of X
   • b) Any two equivalence classes of X are either disjoint or identical Union of these equivalence classes is X
4. Q3 B) Attempt any TWO questions from the following, I. given by Prove the above function with the given domain and co-domain is a bijection. Also find y is a function are any two non-empty subsets of y, prove that Is the converse true? Justify your answer A binary operation ‘o’ Is defined on — {0} as follows aob = = fora,b {0} Show that ‘o’ satisfies all the properties stated in binary operation, IV. For the following relation, defined on the given set x, find whether it is (i) reflexive 12 marks
   • (ii) symmetric (iii) transitive if the relation is equivalence state its equivalence classes X : the set of integers Z if and only if 2a + b is divisible by 3
5. Q4 A) Attempt any ONE question from the following. Prove that a polynomial in C[ x ] of degree n has exactly n complex roots, counted with II. State and prove rational root theorem 8 marks
6. Q4 B) Attempt any TWO questions from the following. I. State and prove Remainder Theorem II. dividing f(x) by g(x) find the quotient and remainder in R[x] Ill. If p(x) | a(x) b(x), a(x), b(x) in R[x] then prove that p(x)/ a(x) or p(x)/ b(x) in all rational roots and their multiplicity of 16x? + 1 12 marks
7. Q5 Attempt any FOUR questions from the following. I. the ged of the given pair of numbers and express it as a linear combination of the II. Wilson’s Theorem for p = 13 Ill. Define the following terms 20 marks
   • a) Cartesian product of two sets
   • b) Functions [ mappings ] IV. Y Z are two functions such that, gof is bijective and f is subjective, then prove that, g is injective Let f(x) R[x] let C, be a root of f(x) = 0 then prove that conjugate of is VI. all roots of = — 4x — 12 if sum of its two roots is zero

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