BSc Mathematics SEM II ATKT 2016-17 ATKT MATHS II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 Q.2 and Q.3 allempt any one ]
   • (a). and any three one subquestion (each 8 marks) frdm par
   • (a), 16 marks) from questions (each 4 marks) from part ) part For attempt any three.(each marks) any one, [each 8]
2. Q1 S(n, k) the stirlin & number of the se “
3. Q2 Prove that If R is an equivalence and find $(7, 3) Check’ Whether following relation is on X then R induces a partition on X and partition if exist. OF not and write corresponding
   • (b) Attempt any three, [each 4] "
4. Q1 Solve the following recurrence relation
5. Q2 Prove that if R is ane ulvaler ati any " relation on a non empty set X then or [a] N [b] = @ is true where [a] = equivalence Check whether following relation R is an equivalence or not X = the set of integer Z QR biffa — bis divisible by 4
6. Q4 Define t e following term
   • i) nt set
   • ii) Finite and infinite set
7. Q2 (a) mpt any one. [each 8]
8. Q1 State and prove Multinomial Theorem 2 )Write down all partitions of an integer n = 5&
   • (b) any three. 4]
9. Q1 Given any integer n, No, =7 Then Prove jthat ( ) =
10. Q2 Using Principle of Inclusion and Exclusion Solve following Ina class f 150 students, 70 have offered Mathematics, 80 have offered Physics and 90 have offered Chemistry. Of these, 40 students are Maths and Physics, 30 are for Maths ind Shemistry and 50 are for Physics and Chemistry. If 10 students have offered all these three subjects. Show that ,there are 20 students from this class, who have of these
11. Q3 that the number of integers from the set | to 100, which are not divisible
12. Q4 Define the signature of a permutation and find the signature of
    • (a) Attempt any one. [each tate and prove Remainder Theorem for polynomial f(x) in R[x]
13. Q2 i) Define an irreducible polynomial in R[x] and prove that x? — 1 is in li) Find g.c.d (greatest common divisor) of following pairs of Q[x]
    • b) ; Attempt any three. [each 4] Define an unit in R[x] and prove that The only’ unit polynomials in R[x] are the
14. Q2 By diving f(x) by g(x) find the quotient and remainder in R[x]
15. Q3 Prove that if P(x) is an irreducible polynomial such that p(x)| a(x)b(x) for , a(x), b(x) in R[x] then p(x)| a(x) or p(x)| b(x) in R[x]
16. Q4 Express following complex number in the polar form V3 + i
17. Q4 Attempt any three. [each 5]
18. Q1 If S(n,k) denotes number of partitions of an X into the k-parts wheren and 1 <k then prove following
19. Q2 Prove that if R is an equivalence relation on a non empty set X then Uaex[a] = X where [a] = equivalence class of a rite down (345 in cyclic form also as a product of Transposition Write down (1'3 2)(4 5) cycle of in the standard form 7
20. Q4 Foy any integer n > 2 exactly half of permutations in S,are odd and half are even
21. Q5 an associates in R[x] and prove that if f(x), g(x) are in R[x] then f(x) = c. c is suitable constant in R
22. Q6 State Demoivre’s theorm and using it prove that

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