## Loyola College M.Sc. Mathematics Nov 2008 Mathematics For Computer Applications Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 AB 31

M.C.A. DEGREE EXAMINATION – COMPUTER APPLICATION

FIRST SEMESTER – November 2008

# MT 1902 – MATHEMATICS FOR COMPUTER APPLICATIONS

Date : 11-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Part A (Answer ALL questions)                                                                              2 x 10 = 20

1. Define Lattice homomorphism between two lattices.
2. With usual notations prove that (i)(ii) .
3. Define context free grammar.
4. What is the difference between deterministic finite automata and non-deterministic finite automata?
5. Let G = (N, T, P, S), where N = {S}, T = {a}, P: {S → SS, S → a}. Check whether G is ambiguous or unambiguous.
6. Give a deterministic finite automata accepting the set of all strings over {0, 1} containing 3 consecutive 0’s.
7. If R and S be two relations defined by and , then find

RS, RR and R.

1. Let and ,. Write the matrix of

of R and sketch its graph.

1. Define ring with an example.
2. State Kuratowski’s theorem.

Part B (Answer ALL questions)                                                                              5 x 8 = 40

1. (a) Show that De Morgan’s laws given by and  hold in a

complemented, distributive lattice.

(OR)

(b) Let  be a lattice. For any  prove the following distributive inequalities:

) and .

1. (a) Show that L(G) = is accepted by the grammar G = (N, T, P, S) where N = {S,A} T = {a, b}, P consists of the following productions: S → aSA, S → aZA, Z → bZB, Z → bB, BA → AB, AB → Ab, bB → bb, bA→ ba.

(OR)

(b) Let the grammar G = ({S,A}, {a, b}, P, S) where P consists of S →aAS, S → a,               A → SbA , A → SS, A → ba. For the string aabbaa find a

(i) leftmost derivation

(ii) rightmost derivation

(iii) derivation tree.

1. (a) (i) Define deterministic finite state automata.

(ii) Draw the state diagram for the deterministic finite state automata,                            M =   where Q =, Σ ={a, b}, F =  and δ is defined as follows:

 δ a b

Check whether the string bbabab is accepted by M.                                            (3+5)

(OR)

(b) Given an non-deterministic finite automaton which accepts L. Prove that there exists a deterministic finite automaton that accepts L.

1. (a) (i) Write short on Hasse diagram.

(ii) Let  and relation  be such that  if x divides y. Draw the

Hasse diagram of .                                                                                       (4+4)

(OR)

(b) (i) Show that n3+2n is divisible by 3 using principle of mathematical induction.

(ii) If the permutations of the elements of {1,2,3,4,5} be given by

, then find

α -1-1.                                                                                        (4+4)

1. (a) Prove that there is a one- to-one correspondence between any two left cosets of H in G.

(OR)

(b) (i) If G is a graph in which the degree of every vertex is atlest two, then prove that G

contains a cycle.

(ii) Prove that the kernel of a homomorphism g from a group  to  is a subgroup

of  .                                                                                                                (4+4)

Part C (Answer ANY TWO questions)                                                                  2 x 20 = 40

16.(a)  Let G be (p,q)graph, then prove that the following statements are equivalent:

(i) G is a tree. (ii) Every two vertices of G are joined by a unique path (iii) G is connected

and  (iv) G is acyclic and p =  q+1.

(b) Let H be a subgroup of G. Then prove that any two left cosets of H in G are either

identical or have no element in common.                                                                  (14+6)

1. (a) Let be a Boolean Algebra. Define the operations + and · on the elements of B by,

. Show that  is a boolean ring with identity 1.

(b) Prove that every chain is a distributive lattice.                                                           (15+5)

1. (a) If G = (N, T, P, S) where N = {S, A,B}, T = {a,b}, and P consists of the following rules:

S → aB, S → bA, A → a, A → aS, A → bAA, B →b, B → bS, B → aBB. Then prove the following:

• S w iff w consists of an equal number of a’s and b’s
• A w iff w has one more a than it has b’s.
• B w iff w has one more b than if has a’s

(b) State and prove pumping lemma.                                                                            (10+10)

Go To Main page

## Loyola College M.C.A. Computer Application April 2008 Mathematics For Computer Applications Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 XZ 30

M.C.A. DEGREE EXAMINATION – COMPUTER APPLICATION

FIRST SEMESTER – APRIL 2008

# MT 1902 / CA 1804 – MATHEMATICS FOR COMPUTER APPLICATIONS

Date : 05/05/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION A

Answer ALL the questions.                                                                         (10 x 2 = 20)

1. Define least upper bound of a poset.
2. Define a Lattice.
3. What are the logic operators?
4. Construct a phrase structure grammar for the language .
5. Define context-sensitive language.
6. For a DFA ,

show that the string 011011 is in L(M)

1. State the Pigeon hole principle.
2. Draw the Hasse diagram for the divisors of 32.
3. Define a bipartite graph with an example.
4. Prove that every cyclic group is abelian.

# SECTION B

Answer ALL the questions.                                                                         (5 x 8 = 40)

1. (a) Prove that the complement of any element ‘a’ of a Boolean algebra is uniquely determined.           Prove also that the map  is an anti – automorphism of period  2 and  satisfies                 (a Ú b)¢ = a¢ Ù b¢, (a Ù b)¢ = a¢ Ú b¢, a¢¢ = a.

(or)

(b) Discuss ‘negation’ and explain a method of constructing the truth table for P Ú ùQ and (P Ú Q) Ú ùP

1. (a) Write a short note on principal conjunctive normal form and construct an equivalent formula             for ù.

(or)

(b) For a grammar  where P consists of the following production:

Then show that.

1. (a) Let L be a set accepted by a non-deterministic finite automaton. Then prove that there exists a deterministic finite automaton that accepts L

(or)

(b) (i) Construct an equivalent deterministic automaton for a given non-deterministic automatonwhere .

(ii) If R and S are equivalence relations on the set X, prove that R Ç S is also an equivalence relation on X.

1. (a) Prove that the equivalence relation ~ defined on the set A decomposes the set A into mutually disjoint equivalence classes.

(or)

(b) (i) A computer password consists of a letter of the alphabet followed by 3 or 4 digits. Find the total number of passwords that can be formed and the number of passwords in which no digit repeats.

(ii) Find the minimum number of students in a class to be sure that four out of them are born in the same month.

1. (a) (i) Prove that a subgroup N of a group G is a normal subgroup of G iff the product of two left cosets of N in G is again a left coset N in G.

(ii) Define ring with an example.

(or)

(b) Prove that the following statements are equivalent for a connected graph G.

• G is Eulerian
• Every point of G has even degree
• The set of edges of G can be partitioned into cycles.

SECTION C

Answer any TWO questions.                                                                       (2 x 20 = 40)

1. (a) Explain conditional and bi-conditional connectives with an example.

(b) Define a Non – Deterministic Finite automata.

(c) For the non deterministic finite automaton,

give the transition table and show that 0100110 is in L (M).

(10 + 2 + 8)

1. (a) State and prove pumping lemma for regular sets.

(b) List any four applications of pumping lemma.

(c) Prove that if  and  be one-to-one onto functions, then  is also one-to-one onto and .

(10 + 4 + 6)

1. (a) Show that in a graph G, any u – v walk contains a u – v path.

(b) Prove that a closed walk of odd length contains a cycle.

(c) State and prove Lagrange theorem.                                                            (4 + 4 + 12)

## Loyola College M.C.A. Computer Application Nov 2008 Mathematics For Computer Applications Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 AB 31

M.C.A. DEGREE EXAMINATION – COMPUTER APPLICATION

FIRST SEMESTER – November 2008

# MT 1902 – MATHEMATICS FOR COMPUTER APPLICATIONS

Date : 11-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Part A (Answer ALL questions)                                                                              2 x 10 = 20

1. Define Lattice homomorphism between two lattices.
2. With usual notations prove that (i)(ii) .
3. Define context free grammar.
4. What is the difference between deterministic finite automata and non-deterministic finite automata?
5. Let G = (N, T, P, S), where N = {S}, T = {a}, P: {S → SS, S → a}. Check whether G is ambiguous or unambiguous.
6. Give a deterministic finite automata accepting the set of all strings over {0, 1} containing 3 consecutive 0’s.
7. If R and S be two relations defined by and , then find

RS, RR and R.

1. Let and ,. Write the matrix of

of R and sketch its graph.

1. Define ring with an example.
2. State Kuratowski’s theorem.

Part B (Answer ALL questions)                                                                              5 x 8 = 40

1. (a) Show that De Morgan’s laws given by and  hold in a

complemented, distributive lattice.

(OR)

(b) Let  be a lattice. For any  prove the following distributive inequalities:

) and .

1. (a) Show that L(G) = is accepted by the grammar G = (N, T, P, S) where N = {S,A} T = {a, b}, P consists of the following productions: S → aSA, S → aZA, Z → bZB, Z → bB, BA → AB, AB → Ab, bB → bb, bA→ ba.

(OR)

(b) Let the grammar G = ({S,A}, {a, b}, P, S) where P consists of S →aAS, S → a,               A → SbA , A → SS, A → ba. For the string aabbaa find a

(i) leftmost derivation

(ii) rightmost derivation

(iii) derivation tree.

1. (a) (i) Define deterministic finite state automata.

(ii) Draw the state diagram for the deterministic finite state automata,                            M =   where Q =, Σ ={a, b}, F =  and δ is defined as follows:

 δ a b

Check whether the string bbabab is accepted by M.                                            (3+5)

(OR)

(b) Given an non-deterministic finite automaton which accepts L. Prove that there exists a deterministic finite automaton that accepts L.

1. (a) (i) Write short on Hasse diagram.

(ii) Let  and relation  be such that  if x divides y. Draw the

Hasse diagram of .                                                                                       (4+4)

(OR)

(b) (i) Show that n3+2n is divisible by 3 using principle of mathematical induction.

(ii) If the permutations of the elements of {1,2,3,4,5} be given by

, then find

α -1-1.                                                                                        (4+4)

1. (a) Prove that there is a one- to-one correspondence between any two left cosets of H in G.

(OR)

(b) (i) If G is a graph in which the degree of every vertex is atlest two, then prove that G

contains a cycle.

(ii) Prove that the kernel of a homomorphism g from a group  to  is a subgroup

of  .                                                                                                                (4+4)

Part C (Answer ANY TWO questions)                                                                  2 x 20 = 40

16.(a)  Let G be (p,q)graph, then prove that the following statements are equivalent:

(i) G is a tree. (ii) Every two vertices of G are joined by a unique path (iii) G is connected

and  (iv) G is acyclic and p =  q+1.

(b) Let H be a subgroup of G. Then prove that any two left cosets of H in G are either

identical or have no element in common.                                                                  (14+6)

1. (a) Let be a Boolean Algebra. Define the operations + and · on the elements of B by,

. Show that  is a boolean ring with identity 1.

(b) Prove that every chain is a distributive lattice.                                                           (15+5)

1. (a) If G = (N, T, P, S) where N = {S, A,B}, T = {a,b}, and P consists of the following rules:

S → aB, S → bA, A → a, A → aS, A → bAA, B →b, B → bS, B → aBB. Then prove the following:

• S w iff w consists of an equal number of a’s and b’s
• A w iff w has one more a than it has b’s.
• B w iff w has one more b than if has a’s

(b) State and prove pumping lemma.                                                                            (10+10)

Go To Main page

## Loyola College B.C.A. Computer Application April 2012 Mathematics For Computer Applications Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.C.A. DEGREE EXAMINATION – COMPUTER APPL.

SECOND SEMESTER – APRIL 2012

# MT 2101 – MATHEMATICS FOR COMPUTER APPLICATIONS

Date : 23-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Part A

Answer ALL questions:                                                                               (10 x 2 = 20)

1. Give an example of skew symmetric matrix.
2. Prove that .
3. If α and β are the roots of the equation, find α+β, αβ.
4. Find the first order partial derivatives for .
5. Evaluate .
6. Write down the Bernoulli’s formula for integration.
7. Find the complementary function for .
8. Form partial differential equation by eliminating arbitrary constants from .
9. Write the approximation formula to find the root using Regula Falsi method.
10. How many types in Simpson’s rule.

Part B

Answer any FIVE questions:                                                                       (5 x 8 = 40)

1. Find the rank of the matrix .
2. Prove that .
3. Solve the equation  whose roots are in A.P.
4. If  where ,then prove that .
5. Evaluate .
6. Solve the equation .
7. Solve .
8. Apply Simpson’s  rule to evaluate  correct to 2 decimal places by dividing the range into 8 equal parts.

Part C

Answer any TWO questions:                                                                      (2 x 20 = 40)

1. (a)Find the Eigen values and Eigen vectors of the matrix .              (12)

(b)Prove that .                                (8)

1. (a)Solve .                                                                (12)

(b)Find the radius of curvature for the curve  at .                                    (8)

1. (a)Prove that .                                                             (8)

(b)Solve the equation .                                                 (12)

1. (a)Using Newton-Raphson method find the root of the equation       (15)

(b)The velocity of a particle at distance S from a point on it’s path is given by the following table

 S(ft) 0 10 20 30 40 50 60 V(ft/s) 47 58 64 65 61 52 38

Estimate the time taken to travel 60 ft using Trapezoidal rule.                                  (5)

Go To Main page

© Copyright Entrance India - Engineering and Medical Entrance Exams in India | Website Maintained by Firewall Firm - IT Monteur