Define the rank of the matrix. 1. 2. Find sum and product of the Eigen values of 3 Define eigen value and eigen vector of a matrix. 4 . The product of two eigen values of the matrix is 16, find the third eigen value. How many different ways we can arrange the 7 letters of the word “SUCCESS”? 5 If f and g are two functions defined by f(x) = 2x+3 , g(x) = 3x+2 find 6 7 8 9 10 11 12 Is Find the symmetric difference of {1, 3, 5} and {1, 2, 3}. Define Partition of sets. What is symbol for the statement ‘some rational numbers are powers of 2’? Define Proposition. Define Tautology with an example. Find the truth table of where denotes the conditional. What is the Language generated by the grammar? 13 14 15 ) Define context free language. Define a regular grammar. Define the derivation tree for the grammar with productions for the string 16 “ abbaaa”. 17 18 19 20 Define Finite state automata. Define DFA. Define Finite State Automata. What are regular expressions? PART B UNIT 1 1 Identify and by using Cayley – Hamilton theorem and also verify the theorem , where 2 Verify Cayley Hamilton Theorem and hence identify the inverse of the matrix 1 A= 3 Test for consistency of the following system of equations and solve if it is consistent x+y+z = 6 , x+2y-2z = -3, 2x+3y+z = 11 Identify the values of ‘a’ and ’ b’ for which the system of equations x+y+2z = 2, 2x-y+3z = 2, 5x-y+az = b has (i) no solution (ii) unique solution (iii) infinitely many solution. 4 UNIT 2 . 5 Test for consistency of the following system of equations and solve if it is consistent. x+2y+z = 3 , 2x+3y+2z = 5 , 3x-5y+5z = 2, 3x+9y-z = 4 6 Identify the values of ‘a’ and ’ b’ for which the system of equations x+y+z = 3 x+2y+2z=6 x+ay+3z=b has (i) no solution (ii) unique solution (iii) infinitely many solution. 1 If f(x) = x+2, g(x) = x-2 and h(x) = 3x then identify ( . 2 Identify the relation congruence modulo m is an equivalence relation over the set of positive integers. 3 There are 2504 computer science students at a school. Of these, 1876 have taken a course in Pascal, 999 have taken a course in Fortran, and 345 have taken a course in C. further, 876 have taken course in both Pascal and Fortran, 231 have taken course in both Fortran and C, and 290 have taken course in both Pascal and C. If 189 of these students have taken courses in Fortran, Pascal and C, Identify the number of 2504 students have not taken a course in any of these three programming languages. 4 Identify the number of positive integers ‘n’ where 1≤n≤1000 and n is not divisible by 3,5,7,and 9. 5 Let A= { 1,2,3,4} and R = { (1,1), (1,3), (2,3),(3,4), (4,1) (4,2) } be a relation on A. Identify Reflexive, Symmetric, Transitive closure of A. 6 A survey of 100 students, it was found that 30 studied mathematics, 54 studied Statistics, 25 studied Operations research, 1 studied all the three subjects. 20 studied mathematics and Statistics, 3 studied Mathematics and Operations research and 15 studied Statistics and Operations research.i) Identify the number of students studied none of the subjects. ii) Identify the number of studied only mathematics. iii) Identify the number of studied Mathematics and Operations research 2 but not Statistics . UNIT 3 1 Construct using truth table, show that (P(QR)) ((QR) (QR)) R 2 Solve the premises “Every living thing is a plant or an animal”. “John’s gold fish is alive and it is not a plant”. “All animals have heart”. Hence “John’s gold fish has a heart 3 Construct that 4 Construct the principal disjunctive normal form and principal is irrational by giving a proof using contradiction. conjunction form of the statement 5 Identify the PCNF and PDNF of UNIT 4 1 2 UNIT 5 Identify the types of grammar. Construct the transition diagram and transition table for machine modulo counter 3. 3 Identify the Language generated by the context free grammar whose production rules are S→aSa,,S→bSb,S→λ. 4 Construct pumping lemma for regular languages. 1 For the following NFA, construct DFA.(* denotes final state) State/input 0 1 , 2 State/input 0 1 Construct the following NFA to its equivalent DFA {p,q} {p} {r} {r} {s} {s} 3 {s} 3 Construct NFA that recognizes the regular expression 4 Construct the regular expression €-transitions . to an 5 For the following NFA, construct DFA.(* denotes final State/input 0 1 state) , 6 State/input 0 , For the following NFA, construct DFA. ***** 4 1