JNTU
JNTU MCA 2019 MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE Previous Question Paper (April/May - 2019 - R15)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks and may have a, b, c as sub questions.
Part A
Construct the truth table (P˄Q) ˅ (Q˄R) ˅ (P˄~R).
Answer
| P | Q | R | ~R | P˄Q | Q˄R | P˄~R | (P˄Q)˅(Q˄R)˅(P˄~R) |
|---|---|---|---|---|---|---|---|
| T | T | T | F | T | T | F | T |
| T | T | F | T | T | F | T | T |
| T | F | T | F | F | F | F | F |
| T | F | F | T | F | F | T | T |
| F | T | T | F | F | T | F | T |
| F | T | F | T | F | F | F | F |
| F | F | T | F | F | F | F | F |
| F | F | F | T | F | F | F | F |
Let G be the set of real numbers not equal to -1 and be defined by a*b=a+b+ab. Prove that < G,* > is an abelian group.
Answer
- Closure: For a, b ≠ -1, a*b = a+b+ab = (a+1)(b+1)-1. Since a, b ≠ -1, (a+1) and (b+1) are non-zero, so (a+1)(b+1) ≠ 0, thus a*b ≠ -1.
- Associativity: (a*b)*c = (a+b+ab)*c = (a+b+ab)+c+(a+b+ab)c = a+b+c+ab+bc+ca+abc. Similarly, a*(b*c) = a+(b+c+bc)+a(b+c+bc) = a+b+c+bc+ab+ac+abc. Both are equal.
- Identity: Let e be the identity: a*e = a+e+ae = a. e(1+a) = 0. Since a ≠ -1, e = 0.
- Inverse: Let a' be the inverse of a: a*a' = a+a'+aa' = 0. a'(1+a) = -a. a' = -a/(1+a). Since a ≠ -1, a' exists and a' ≠ -1.
- Commutativity: a*b = a+b+ab = b+a+ba = b*a. Thus, it is an abelian group.
How many integers between 1 and 100 have a sum of digits of integer numbers equal to 10?
Answer
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Find the particular solution for the following difference equation an+5an-1+6an-2=42.4n
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What do you mean by Isomorphic graph? When will you say that two graphs are isomorphic?
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Part B
Obtain the Principal Disjunctive normal form of P→[(P→Q)˄~(~Q˅~P)]
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Show that the following argument is valid. If today is Tuesday, I have a test in Mathematics or Economics. If my Economics professor is sick, I will not have a test in Economics. Today is Tuesday and my Economics professor is sick. Therefore I have a test in Mathematics.
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Consider the group G = {1,2,4,7,8,11,13,14} under multiplication modulo 15. Construct the multiplication table of G and verify whether G is cycle or not.
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Let f: R→R and g: R→R, where R is the set of real numbers. Find fog and gof, where f(x) = x2-2 and g(x) = x+4. State whether these functions are injective, surjective, and bijective.
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Define Lattice and write various properties of Lattice.
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Using binomial identities evaluate the sum 1.2.3 + 2.3.4 + …… + (n-2)(n-1)n.
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Suppose there are 15 red balls and 5 white balls. Assume that the balls are distinguishable and that a sample of 5 balls is to be selected.
How many samples of 5 balls are there?
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How many samples contain all red balls?
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How many samples contain 3 red balls and 2 white balls?
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How many samples contain at least 4 red balls?
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Solve the recurrence relation an-7an-1+10an-2=0 for n≥2 where a0=10 and a1=41.
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Explain and illustrate various ways of solving the recurrence relation.
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State and explain the 4-color problem for planar graphs.
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What is planar graph? Show that following graph is planar.
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