BSc Mathematics SEM I ATKT MATHEMATICS PAPER II Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2. Figures to the right indicate full marks

1. Q1 Choose correct alternative in each of the following: 20 marks
   • i. What is the GCD of 187 and 1797?
   • (c) 1797 (d) None of these
   • ii. The sum of all binomial coefficients in the expansion of (a + is
   • (c) 1024 (d) None of these ili. For positive integers a and b, gcd(a, b) (a, b) is
   • (c) ab (d) None of these
   • iv. Let X and Y be two non-empty sets and Y be a function
   • (a) EX (b)
   • v. IfA = = {1,2,3,4}. Which of the following is not a fz = {(a, 1), Cb, — None of these
   • vi. Arelation R = {(1,1), (2,2), (1,2), (2,1)} in set X =
   • (c) Ris symmetric (d). None of these Let A = {1,2,3,4}. Which of the following is a partition of A?
   • (d) None of these If 1— 4iis root of polynomial f(x) of degree 2 then.is also a
   • (c) 1+ 4i (d) None of these
   • ix. Degree of constant polynomial is
   • (c) 2 (d) None of these
   • x. of the roots of the quadratic polynomial (k —8 is 2, then the value of k is
   • (c) 38 (d) None of these
2. Q2 a) Attempt any ONE question from the following: 8 marks
   • i. Show that two integers a and b, not both zero have a unique positive g.c.d. which can be expressed in the form of ma + nb, where m and n are integers
   • ii. Let the integer n > 1, has the prime factorization n= . then show that where denotes Euler’s phi function
   • b) Attempt any TWO questions from the following: 12
3. Q1 For integers a,b if (a,4) = 2, (b,4) = 2 then prove that State and prove Euclid’s Lemma
   • ii. Find gcd 879 andb=1216 and express it in form Prove = using principle of
4. Q3 a) Attempt any ONE question from the following: 8 marks
   • i. B— C be bijective functions. Prove that gof : C is also a bijective function. Is the converse true? Prove that an equivalence relation in a non-empty set gives a partition of that set
   • b) Attempt any TWO questions from the following: 12
   • i. Define composition of two functions. If f: R and g :R— Rare bijective functions given by f(x) = 4x + 3 and g(x) = 2x3 then verify that = Let f: X > Y be any function and be two non-empty subsets of X. Show that f(Ai U Ag) = U ui. Define a binary operation on Z as ach = ab, for a,b Check whether satisfies commutative and associative properties Also find identity element and inverse element of a if they
   • iv. Define following with suitable examples, a) Identity function
5. Q4 a) Attempt any ONE question from the following: 8 marks
   • i. a) Prove that a polynomial of degree n has atmost n roots
   • b) If f(x) R[x] anda Cisa root of f(x), then prove that its conjugate @ is also a root of f(x) ie Tf = ag t+ be a polynomial in R[x] with integer coefficients and a rational number Z,(p,q) =1,q #0, is a root of f(x), prove that play and Hence prove that if f(x) is a monic polynomial then it’s rational root is an integer
   • b) Attempt any TWO questions from the following: 12
   • i. Find G.C.D. of = and
   • ii. in f(x) = x? — 4x? — 4x + k if sum of two of its roots is equal to the third root
   • iii. Solve 23x + given that two of its roots are
   • iv. Find the fourth roots of unity and show that their sum is
6. Q5 Attempt any FOUR questions from the following: 20 marks
   • a) Prove that for any arbitrary integer a
   • b) State Wilson’s theorem and verify it for prime 11 such that f(x) , show that f is bijective and hence find the inverse of f
   • d) A relation Rin Zis defined as iff 5x — y is divisible by 4” Show that Ris a equivalence relation
   • e) State Factor theorem and hence check whether x+lisa
   • f) Find the multiplicity of the root 2 of f(x) = x

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