S2 BCA Model paper kerala university mathematics

S2 BCA Model Exam October 2018

Total:80

Section A(All questions are compulsory)

1.      What is meant by Boolean expression?

2.      Find the contrapositive of the following statement :If jack is a poet then he is poor

3.      How many relations are there on the set {a,b,c,d} ?

4.      If x=<1,0,0,1> and y = <0,1,0,0> then what is the hamming distance H(x,y) ?

5.      Define  equivalence relation

6.      When will you say that a function is invertible?

7.      Give an example of a finite abelian group.

8.      What is a complete graph?

9.      Define language.

10.   Give example of commutative ring.

Section B(Answer any 8 questions)

11.   Test the validity of the argument

P→Q

Q

_____

P

12.   Show that p↔q ≡ (p v q) → (p ^q)

13.   Find the roots of f(x) over Z₁₀ where f(x) = 2x² +4x+4

14.   Prove that the operation U on sets is commutative

15.   Briefly describe shortest path algorithm

16.   Write the prefix notation of (x-1)³(2x+1)

17.   Distinguish between Euler graph and Hamiltonian graph

18.   Explain the resolution principle with a suitable example

19.   What is a finite state automata?

20.   What are the functions of * .* ^ .^ in MATLAB

21.   What is a spanning tree?

22.   Obtain the DFSA that accepts strings with even numbers of x’s and y’s

Section C(Answer any 6)

23.   Prove that 1 + 3 + 5+ … + 2n-1 = n²

24.   Prove that equality modulo m is an equivalence relation

25.   Define a fuzzy set. Explain the union and intersection of two fuzzy sets

26.   Define hamming distance and state its properties

27.   Define group

28.   What is Boolean algebra?

29.   Explain DFS algorithm with a suitable example.

30.   Explain the different types of grammar.

31.   Prove that a tree with n vertices have n-1 edges

Section D(Answer any 2)

32.   Differentiate the contrapositive and contradiction method of proof. Prove by contradiction “ if 3n + 2 is odd then n is odd”. Prove that  is irrational by giving a proof by contradiction.

33.   A. State and prove the inclusion exclusion theorem in set theory

b.      Write warshall’s algorithm. Let A = {1,2,3,4} and R be the relation {(1,2) , (2,3),(3,4),(2,1)} find the transitive closure by using Warshall’s algorithm

34.   A. Define isomorphism. Let G be the group of real numbers under addition and let G’ be the group of positive real numbers under multiplication. Show that f:G→G’ defined by f(x)=e^x is an isomorphism.

B. Consider an algebraic system (Q,*) where Q is the set of all rational numbers and * is a binary relation defined by a*b = a+b-ab Ɐ a,b in Q. determine whether (Q,*) is a group.

35. a.  Prove that a hamming code can correct all combinations of k or fewer errors if and only if the minimum distance between any two codes is atleast 2k+1

              b. State and prove De Morgan’s laws in set theory.