Download PTU MCA 2020 March 2nd Sem 26052 Mathematical Foundations Of Computer Science Question Paper

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RollNo. | [ [ [ I [ [ [ 1] [ ] Total No. of Pages : 02

Total No. of Questions : 09

MCA (2014 Batch) (Sem.-2)

MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE

Subject Code : MCA-201

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M.Code : 26052

Time : 3 Hrs. Max. Marks : 100

INSTRUCTIONS TO CANDIDATES :

1. SECTIONS-A, B, C & D contains TWO questions each carrying TWENTY marks each and students has to attempt any ONE question from each SECTION.
2. SECTION-E is COMPULSORY consisting of TEN questions carrying TWENTY marks in all.

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SECTION-A

1. a) Define a Hamiltonian circuit in a graph. Give an example of a graph which has a Hamiltonian circuit and an example of a graph which does not have a Hamiltonian circuit.
   b) State and prove five-color problem.
2. A connected multigraph has an Euler circuit. Prove that each of its vertices has even degree.

SECTION-B

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1. In a survey of 25 students, it was found that 15 had taken Mathematics, 12 had taken Physics and 11 had taken Chemistry, 5 had taken Mathematics and Chemistry, 9 had taken Mathematics and Physics, 4 had taken Physics and Chemistry and 3 had taken all the three subjects. Find the number of students studying (a) only Physics; (b) Physics and Chemistry but not Mathematics; (c) Atleast one of the three subjects.
2. a) Partition the set A = {1, 2, 3, ...., 10} using the minsets generated by B1 = {1, 7, 8}, B2 = {1, 6,9,10}, B3 = {1, 9, 10}. Also represent the minsets thus generated through a Venn diagram.
   b) Define a Relation. Discuss the properties of relations.

SECTION-C

1. Prove by the principle of mathematical induction that for all n € N:
   

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   1²+2²+3²+ ... +n²= 1/6 n(n+1)(2n+1)
2. Show that (? x) (P(x) v Q(x)) => (? x) P(x) v (?x) Q(x).

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SECTION-D

1. a) Find the inverse of the matrix :
   1 1 0
   1 0 1
   

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   1 2 2
   b) Discuss matrix addition, scalar multiplication and multiplication of matrices by taking suitable example.
2. Solve the following system using Gauss-Jordan elimination :
   3x1+x2+ x3+ 2x4=13
   2x1 —4x2 + 14x3 — x4 =-10
   

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   5x1 + 11x2 — 7x3 + 8x4 =59
   2x1 + 5x2 — 4x3 — 3x4 =39

SECTION-E

1. Write briefly :
   a) What is a bipartite graph and a complete bipartite graph?
   

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   b) Differentiate between directed and undirected graph.
   c) What is chromatic number?
   d) What is a universal set? Give an example.
   e) State the DeMorgan’s laws.
   f) What are the different types of quantifiers? What is the purpose of each?
   

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   g) Define Proposition.
   h) What is the difference between Equivalence and Implication?
   i) What is transpose of a matrix? Give an example.
   j) What is an Identity matrix? Give an example.

NOTE : Disclosure of Identity by writing Mobile No. or Marking of passing request on any paper of Answer Sheet will lead to UMC against the Student.

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