BSc Mathematics SEM I 2015 16 2015-16 ATKT MATHS II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 For Q.1, Q.2, Q.3 attem pt any one sub question (Each 8 Marks Subquestion (Each 4 marks) from Part (B)
2. Q3 For Q.4 attempt any three (Each 5 marks)
3. Q1 A) Attempt Any One : (Each 8 marks) ne function $(n) for positive integer n>1 and prove that if m, n Z such at ged (m,n) = 1 then $ (m, n) = $ (m) 6 (n) also find 9 (580) greatest divisor of 83120 and 750 and express it in form of 3120m + 750m, Z Are m,n unique ? Justify
   • B) Attempt Any Three : (Each 4 marks)
4. Q1 Explain Pascal's Triangle. Use it to find (a + bY’ :
5. Q2 Prove that the number of primes are infinite
6. Q3 Define least common multiple and greatest common divisor of non zero integer a and b Prove that ged (a, b) Lem [ab] = ab State first principle of finite induction and prove that 8" — 3" is divisible by 5 Vn e N
   • A) Attempt Any 8 marks)
7. Q1 Define invertible function, bejective function. Prove that composition of injective function is injective
8. Q2 Define :
   • i) Equivalence relation R on nonempty set A
   • ii) Partition of A. Prove that every partition of a nonempty set A induces equivalence
   • B) Attempt Any Three: (Each 4 marks)
9. Q1 Check whether * is binary on given
10. Q2 IR be defined by = 8x — 7. Check whether fis bijective or not. Hence find inverse if exist
11. Q3 Determine whether following relation R on set A is equivalence a not of relations among Ris defined y) R ifs is brother of y
12. Q4 Determine whether each relation from A to B is function. If it is function give its range
    • A) Attempt Any One : (Each 8 marks) State and Prove Remainder Theorem for polynomial f(x) ef(x) and Computer remainder when f(x) is divided by State and Prove factor theorem. Use it to determine whether or not factor
    • B) Attempt Any Three : (Each 4 marks) that a polynomial of degree n has at most n roots
13. Q2 Express V3 +i in polar form, also find magnitude & amplitude Find quotient and remainder when f(x) is divided by g(x)
14. Q4 Prove that a non constant polynomial f(x) F(x) can be expressed as product of and quadratic polynomial
15. Q1 State and prove Euclid's Lemma : 3) Prove =b (modn) b,n Z iffa,b leave same remainder when divided IfRis an equivalence relation on nonempty set X. Then Prove that any two equivalence classes of X are either identical or disjoint Use Demoivre's Theorem to prove that
16. Q6 State and prove Rational Root Theorem. =

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