KERALA UNIVERSITY- SECOND SEMESTER BCA- MATHEMATICS

KERALA UNIVERSITY PREVIOUS YEAR QUESTION PAPER

SECONG SEMESTER BCA JUNE 2012

MATHEMATICS

                                                                                          Weight:30

All the first 16 questions are compulsory. Four consecutive questions beginning with the first form a bunch. Each bunch carries 1 weightage.

1.      Show that (P ^ (P--> Q))àP is a tautology

2.      If the truth value of (P^Q) --> (R V S) is false then what will be the truth values of its components.

3.      Write the inverse of the statement “if there is forests then there is rain”

4.      Show that (P’ ^ (P V Q)) --> Q

5.      State the distributive laws in set theory.

6.      Draw the venn diagram of A’ U (B intersection C’)

7.      Find a set that satisfies the following four properties

A intersection {2,3,5,6} = {2,5}

A intersection {4,5,6,7}={1,2,4,5,6,7}

{2,4} subset of A

A subset of {1,2,4,5,6,8}

8.      Give the infix notation for a+b*c

9.      Define a digraph.

10.  Why it is not possible to draw a  3 regular graph on 5 vertices?

11.  Define an abelian group

12.  Draw the diagram of a complete graph with atleast 4 vertices

13.  Define adjacency matrix

14.  Give a necessary and sufficient condition for a function to be invertible

15.  Define a partially ordered set.

16.  Let R be an equivalence relation on a set A= {4.5.6.7} defined by R={(4,4) (5,5) (6,6) (7,7) (4,6) (6,4)}. Determine its equivalence classes.

Answer any 8 questions from among the questions 17 to 28. They carry one weight each

17.  Write an equivalent formula for P V (S < == > T) which does not involve biconditional connective and conditional connectives.

18.  Symbolize and negate the propositions:

a.      All boys run faster than all girls

b.      Some girls are more intelligent than all boys

19.  Prove or disprove

P à Q

20.  Draw the Hasse diagram of (D₄₅ , /)

21.  Determine whether the relation R={(a,b):a>=b} on the set of all real numbers is an equivalence relation.

22.  Show that the identity element of a group is unique

23.  Give an example of a finite abelian group.

24.  Determine a minimum spanning tree for the graph

25.  Determine whether the relation R={(a,b):a>=b} on the set of all real numbers is an equivalence relation.

26.  Let f and g be two functions on integers defined by f(n)=n² and g(n)=n+1. Find fog and gof

27.  Let A= {4,6,8,10} and R={(4,4) (4,10) (6,6) (6,8) (8,10)} be a relation on A. determine the symmetric closure of A.

28.  In MATLAB what is the difference between matrix operations “* “and “.*”

Answer any 5 questions from among the questions 29 to 36. They carry 2 weight each.

29.   Test the validity of the argument

“if 6 is even then 2 does not divide 7. Either 5 is not prime or 2 divides 7. But 5 is prime therefore 6 is odd.”

30.  Show that sum of squares of natural numbers is given by

[n(n+1)(2n+1)] /6

31.  Find the principal disjunctive normal form for (P à Q) à R

32.  State and prove De Morgans laws

33.  Let a relation R on the set of all integers is defines as follows:

For every a,b in z aRb if and only if a+b is divisible by 5. Check whether R is an equivalence relation or not.

34.  Let G be the set of all non zero real numbers and let a*b =(ab)/2. Show that (G,*) is an abelian group.

35.  Show that every tree of n vertices contains exactly n-1 edges.

36.  Give a depth first traversal that starts at vertex f for the graph.

Answer any 2 questions from among the questions 37 to 39. They carry 4 weight each.

37.  Check the validity of

“ All the living beings are either an animal or a plant. Johns Gold fish is alive but not a plant. All animals are cruel. Therefore there are cruel beings in the world”

38.  Let R be a relation on the set A={a,b,c,d} defined by

R={(a,b) (b,c) (d,c) (d,a) (a,d) (d,d)}. Determine

a.      Reflexive closure of R

b.      Transitive closure of R

39.  Explain breadth first algorithm with an example.