Discrete Mathematics Past Paper PDF

1 CTH EDUCATION Discrete Mathematics Unit – 01 : Logic & Proofs 1. What is proposition ? explain its type. 2. Explain different types of operators in propositional logic with truth table. 3. Prove t...

1 CTH EDUCATION Discrete Mathematics Unit – 01 : Logic & Proofs 1. What is proposition ? explain its type. 2. Explain different types of operators in propositional logic with truth table. 3. Prove that the given function (P ∪ (P→Q)) is tautology. 4. Define predicate & quantifier with example. 5. Let P(x) denote the statement “x > 20″. What are the truth values of P(21) and P(15) ? 6. Explain different rules of Inference. 7. Define the following terms : a. Proofs b. Normal forms c. Universal quantifiers d. Existential quantifiers Unit-02 : Introduction to SET THEORY: 8. Define set. Explain different types of sets. 9. If set A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}. Write the universal set for all three sets. 10. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}. Find A ∩ (B ∪ C). 11. If A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find A – B and B – A. 12. Find the power set of Z = {2, 7, 9} and a total number of elements. 13. If A = {1, 2, 4, 7, 9} and B = {2, 3, 7, 8, 9} then find A △ B ? 14. If n(A) = 20 and n(B) = 30 and n(A U B) = 40 then n(A ∩ B) is? 15. If n(A - B) = 18, n(A ∪ B) = 70 and n(A ∩ B) = 25, then find n(B). 16. In a group of 60 people, 27 like cold drinks and 42 like hot drinks and each person likes at least one of the two drinks. How many like both coffee and tea? 17. There are 35 students in art class and 57 students in dance class. Find the number of students who are either in art class or in dance class. 18. If A = { 1, 3, 5} and B = {2, 3}, then Find : (i) A × B (ii) B × A (iii) A × A (iv) (B × B) 19. If A and B are two sets, and A × B consists of 6 elements: If three elements of A × B are (2, 5) (3, 7) (4, 7) find A × B. Unit-03 : Relation Between two sets 20. What are relation ? Explain in details with example. 21. Let A={x,y,z} and B={1,2}. Find the number of relations from A to B. 22. Explain equivalence relation with suitable examples. 23. Find the inverse relation of R = {(4, 8), (-3, -6), (7.1, 2.3)} Diploma : CSE (All Paper) By : Alok Sir (Mob. No.: +91-80 84 370 470) 2 CTH EDUCATION 24. Let A = {1, 3, 5, 7} and B = {p, q, r}. Let R be a relation from A into B defined by R = {(1, p), (3, r), (5, q), (7, p), (7, q)} find the domain and range of R. 25. Define reflexive, symmetric and transitive relation with suitable example. Unit-04 : Functions 26. Define function. Explain in brief different types of function. 27. Let A = {-3, -2, -1, 0, 1, 2, 3} and B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Consider the rule f(x) = x². Find domains (f) and range (f). 28. What is difference between relation and a function? Is every relation a function? Why? 29. Let f(x) = ((4x – 3))/((5x + 1)) , x ∈ R. Find: (a) f(-1) (b) f(1) (c) f(-1) + f(1) 30. Determine the domain and range of the following relation. {(3,−4),(6,37),(10,−10),(−31,2)} Unit-05 : Recurrence Relation 31. Find the homogenious solution of given recurrence relation Fn = 5Fn−1 − 6Fn−2 where F0 = 1 and F1 = 4 32. Find the homogenious solution of given recurrence relation Fn = 10Fn−1 − 25Fn−2 where F0 = 3 and F1 = 17 33. Fn - 2Fn−1 = 3n , where F0 = 1. Solve the given reccurence relation for a. Homogenious solution b. Particular solution 34. Solve the given recurrence relation & find the total solution : Fn = 3Fn−1 + 10Fn−2 + 7.5n where F0=4 and F1=3 Unit-05 : Graph Theory 35. What is graph & sub-graph ? Explain with suitable example. 36. Solve and check whether the given graph G1 and G2 are isomorphic or not ? 37. Explain in brief the term walk, path and circuit with example. 38. Define Euler graph. Which of the following is Euler graphs. Diploma : CSE (All Paper) By : Alok Sir (Mob. No.: +91-80 84 370 470) 3 CTH EDUCATION Unit – 07 : Counting 39. Explain basic principle of counting in details. 40. Solve & find the value of n if 4 x np3 = n+1P3 ? 41. In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together? 42. Out of 6 men and 5 women in a room, in how many ways can be 3 men and 2 women chosen? 43. Explain pigeonhole principle in details. 44. Find the minimum number of students in a class to be sure that three of them are born in the same month. 45. A box contain 10 blue balls, 20 red balls, 8 green balls, 15 yellow balls and 25 white balls. How many balls must be choose to ensure that we have 12 balls of the same colour. Unit-08 : Probability Theory 46. What is probability ? Define the term event & sample space. 47. Q. If P(A) = 7/13, P(B) = 9/13 and P(A∩B) = 4/13, evaluate P(A|B). 48. If P(A) = 1/3, P(B) = 1/4 and P(A ∪ B) = 5/12, Find P(A/B) and P(B/A) ? 49. If A and B are two events and P(A) = 3/8, P(B) = 1/2, P(A∩B) = 1/4 then the value of P(Ac∩Bc) = K/8, find the value of K. [ P(Ac∩Bc) = 1 - P(A ∪ B)] 50. A coin is tossed twice at random. What is the probability of getting a. At least one head b. The same face ? 51. Three coins are tossed at a time find the probability of getting a. At most one head b. At least one tail 52. A dice is thrown, what is the probability that the number obtained is a prime number. 53. Find the Mean and Variance for a single dice when it is thrown ? x 1 2 3 4 5 6 P(x) 1/6 1/6 1/6 1/6 1/6 1/6 54. 3 dice are thrown , find the Mean and Variance for the some of the numbers on them. 55. Find the probability of getting 9 exactly 2 in 3 times with a pair of dice. 56. If a coin is tossed 5 times, find the probability of : a. Exactly 2 heads b. At least 4 heads. Telegram Link : https://t.me/ctheducation19 App Link : http://on-app.in/app/home?orgCode=swffh For : Video Lecture & PDF Notes Download CTH Education app from Play Store. Diploma : CSE (All Paper) By : Alok Sir (Mob. No.: +91-80 84 370 470)