Download JNTUH MCA 1st Sem R13 2019 April-May Mathematical Foundations Of Computer Science Question Paper

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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

MCA I Semester Examinations, April/May - 2019

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MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE

Time: 3hrs Max.Marks:60

Note: This question paper contains two parts A and B.

Part A is compulsory which carries 20 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 8 marks and may have a, b, c as sub questions.

PART - A

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5 x 4 Marks =20

1. a) Give the converse, contrapositive and inverse of the following statement: The hut will destroy if there is a cyclone. [4]
2. b) Define the terms: Equivalence relation, Partially ordered relation and Totally ordered relation. Give examples for each. [4]
3. c) How many integers between 1 and 1000 inclusive have the sum of the digits equal to 7. [4]
4. d) Solve the recurrence relation a_n = na_n-1 for n > 1, given that a₀ = 1. [4]

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5. e) What is a Hamiltonian graph? Discuss briefly. [4]

PART -B

5 x 8 Marks =40

1. a) Show that (P?S) can be derived from the premises ¬PvQ, ¬Qv R, R?S using CP rule.
   b) Obtain the PCNF of the (P?(Q ? R)) ? ( P?(¬Q ? ¬R)). [4+4]
   

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   OR
2. a) Show that (x) (p(x) v Q(x))= (x) p(x) v ? (x) Q(x).
   b) Use truth tables to establish whether the following statement forms a tautology or a contradiction or neither. P ?(Q? R). [4+4]
3. Define equivalence classes. Let Z be the set of integers and Let R be the relation called “congruence modulo 3” defined by R={ / X?Z ? y?Z ? (x-y) is divisible by 3}. Determine the equivalence classes generated by the elements of Z. [8]
   OR

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4. a) Draw the Hasse diagram for the Poset. <{2,4,5,10,12,20,25},/>.
   b) Let R={(b,c), (b,e), (c.e), (d,a), (c,b), (e,c)} be a relation on the set A = {a,b,c,d,e}. Find the transitive closure of the relation R. [4+4]
5. a) What is the coefficient of x²y⁵ in (2x—9y)¹⁰?
   b) How many 6 digit numbers without repetition of digits are there such that the digits are all non-zero and 1 and 2 do not appear consequently in either order? [4+4]
   OR

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6. State and explain Multinomial theorem with an example illustration. [8]
7. Solve the recurrence relation a_n-6a_n-1+9a_n-2 = 0 where a₀=1 and a₁= 6. [8]
   OR
8. Using generating function, solve the y_n+2 —4y_n+1 +3y_n=0, given y₀ =2, y₁ =4. [8]
9. Explain prim’s algorithms with suitable example. [8]
   

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   OR
10. State Graph coloring problem and describe its importance in computations. [8]