Q51➡|NTA UGC NET November 2017 Paper 3 Consider the following two well-formed formulas in propositional logic. F1 : P ⇒ ¬ P F2 : (P ⇒ ¬ P) ∨ (¬ P ⇒ P) Which of the following statements is correct?
Q53➡| NTA UGC NET November 2017 Paper 3 Find the equation of the circle x2+y2=1 in terms of x’y’ coordinates, assuming that the xy coordinate system results from a scaling of 3 units in the x’ direction and 4 units in the y’ direction.
Q54➡| NTA UGC NET January 2017 Paper 3 Suppose there are n stations in a slotted LAN. Each station attempts to transmit with a probability P in each time slot. The probability that only one station transmits in a given slot is_____.
Q55➡| NTA UGC NET January 2017 Paper 2 Consider a sequence F00 defined as : F00(0) = 1, F00(1) = 1 F00(n) = ((10 ∗ F00(n – 1) + 100)/ F00(n – 2)) for n ≥ 2 Then what shall be the set of values of the sequence F00 ?
Q56➡| NTA UGC NET January 2017 Paper 2 The functions mapping R into R are defined as : f(x) = x3 – 4x, g(x) = 1/(x 2 + 1) and h(x) = x4 . Then find the value of the following composite functions : hog(x) and hogof(x)
Q59➡| NTA UGC NET January 2017 Paper 2 Consider a Hamiltonian Graph G with no loops or parallel edges and with |V(G)| = n ≥ 3. Then which of the following is true ?
i ➥ deg(v) ≥n/2 for each vertex v.
ii ➥ |E(G)| ≥1/2(n – 1) (n – 2) + 2
iii ➥ deg (v) + deg(w) ≥ n whenever v and w are not connected by an edge
Q60➡| NTA UGC NET July 2016 Paper 2 How many different equivalence relations with exactly three different equivalence classes are there on a set with five elements?
Q61➡| NTA UGC NET July 2016 Paper 2 The number of different spanning trees in complete graph, K4 and bipartite graph, K2,2 have_________and______respectively.
Q63➡| NTA UGC NET July 2016 Paper 2 There are three cards in a box. Both sides of one card are black, both sides of one card are red, and the third card has one black side and one red side. We pick a card at random and observe only one side. What is the probability that the opposite side is the same colour as the one side we observed?
Q64➡| NTA UGC NET July 2016 Paper 2 A clique in a simple undirected graph is a complete subgraph that is not contained in any larger complete subgraph. How many cliques are there in the graph shown below?
Q66➡| NTA UGC NET July 2016 Paper 3 Let ν(x) mean x is a vegetarian, m(y) for y is meat, and e(x, y) for x eats y. Based on these, consider the following sentences : I. ∀x ν(x ) ⇔ (∀y e(x, y) ⇒ ¬m(y)) II. ∀x ν(x ) ⇔ (¬(∃ym(y) ∧e(x, y))) III. ∀x (∃y m(y) ∧e(x, y)) ⇔ ¬ν(x) One can determine that.
Q67➡| NTA UGC NET July 2016 Paper 3 Consider the following logical inferences : I1 : If it is Sunday then school will not open. The school was open. Inference : It was not Sunday. I2 : If it is Sunday then school will not open. It was not Sunday. Inference : The school was open. Which of the following is correct ?
i ➥ Both I1 and I2 are correct inferences.
ii ➥ I1 is correct but I2 is not a correct inference.
iii ➥ I1 is not correct but I2 is a correct inference.
Q68➡| NTA UGC NET August 2016 Paper 2 Let A and B be sets in a finite universal set U. Given the following: |A – B|, |A ⊕ B|, |A| + |B| and |A ∪ B| Which of the following is in order of increasing size ?
Q71➡| NTA UGC NET August 2016 Paper 3 The symmetric difference of two sets S1 and S2 is defined as S1⊖S2 = {x|x ∈ S1 or x ∈ S2, but x is not in both S1 and S2} The nor of two languages is defined as nor (L1, L2) = {w|w |∈L1 and w |∈ L1}. Which of the following is correct?
i ➥ The family of regular languages is closed under symmetric difference but not closed under nor.
ii ➥ The family of regular languages is closed under nor but not closed under symmetric difference.
iii ➥ The family of regular languages are closed under both symmetric difference and nor.
iv ➥ The family of regular languages are not closed under both symmetric difference and nor.
Q72➡| NTA UGC NET December 2015 Paper 2 How many committees of five people can be chosen from 20 men and 12 women such that each committee contains at least three women?
Q73➡| NTA UGC NET December 2015 Paper 2 Which of the following statement(s) is/are false ? (a) A connected multigraph has an Euler Circuit if and only if each of its vertices has even degree. (b) A connected multigraph has an Euler Path but not an Euler Circuit if and only if it has exactly two vertices of odd degree. (c) A complete graph (Kn) has a Hamilton Circuit whenever n ≥ 3. (d) A cycle over six vertices (C6) is not a bipartite graph but a complete graph over 3 vertices is bipartite.
Q74➡| NTA UGC NET December 2015 Paper 2 Which of the following is/are not true? (a) The set of negative integers is countable. (b) The set of integers that are multiples of 7 is countable. (c)The set of even integers is countable. (d)The set of real numbers between 0 and 1/2 is countable.
Q76➡| NTA UGC NET December 2015 Paper 2 A tree with n vertices is called graceful, if its vertices can be labelled with integers 1, 2,….n such that the absolute value of the difference of the labels of adjacent vertices are all different. Which of the following trees are graceful? A-
Q77➡| NTA UGC NET December 2015 Paper 2 Which of the following arguments are not valid ? (a) “If Gora gets the job and works hard, then he will be promoted. If Gora gets promotion, then he will be happy. He will not be happy, therefore, either he will not get the job or he will not work hard”. (b) “Either Puneet is not guilty or Pankaj is telling the truth. Pankaj is not telling the truth, therefore, Puneet is not guilty”. (c) If n is a real number such that n >1, then n 2 >1. Suppose that n 2 >1, then n >1.
Q78➡| NTA UGC NET December 2015 Paper 2 Let P(m, n) be the statement “m divides n” where the Universe of discourse for both the variables is the set of positive integers. Determine the truth values of the following propositions. (a)∃m ∀n P(m, n) (b)∀n P(1, n) (c) ∀m ∀n P(m, n)
Q80➡| NTA UGC NET December 2015 Paper 2 Consider the compound propositions given below as: (a)p ∨ ~(p ∧ q) (b)(p ∧ ~q) ∨ ~(p ∧ q) (c)p ∧ (q ∨ r) Which of the above propositions are tautologies?
Q81➡| NTA UGC NET December 2015 Paper 2 Which of the following property/ies a Group G must hold, in order to be an Abelian group? (a)The distributive property (b)The commutative property (c)The symmetric property
Q82➡| NTA UGC NET December 2015 Paper 2 Suppose that from given statistics, it is known that meningitis causes stiff neck 50% of the time, that the proportion of persons having meningitis is 1 / 50000 , and that the proportion of people having stiff neck is 1/20. Then the percentage of people who had meningitis and complain about stiff neck is:
Q83➡| NTA UGC NET December 2015 Paper 2 How many solutions are there for the equation x + y + z + u = 29 subject to the constraints that x ≥ 1, y ≥ 2, z ≥ 3 and u ≥ 0?
Q86➡| NTA UGC NET December 2015 Paper 3 Consider the conditional entropy and mutual information for the binary symmetric channel. The input source has alphabet X={0,1} and associated probabilities {1/2, 1/2}. The channel matrix is where p is the transition probability. Then the conditional entropy is given by:
Q88➡| NTA UGC NET December 2015 Paper 3 Consider an experiment of tossing two fair dice, one black and one red. What is the probability that the number on the black die divides the number on red die?
Q89➡| NTA UGC NET December 2015 Paper 3 In how many ways can 15 indistinguishable fish be placed into 5 different ponds, so that each pond contains at least one fish ?
Q90➡| NTA UGC NET December 2015 Paper 2 Consider a Hamiltonian Graph (G) with no loops and parallel edges. Which of the following is true with respect to this Graph (G) ? (a) deg (v) ≥ n / 2 for each vertex of G (b) |E(G)| ≥ 1 / 2 (n – 1) (n – 2) + 2 edges (c) deg (v) + deg (w) ≥ n for every n and v not connected by an edge.
Q91➡| NTA UGC NET December 2015 Paper 2 “If my computations are correct and I pay the electric bill, then I will run out of money. If I don’t pay the electric bill, the power will be turned off. Therefore, if I don’t run out of money and the power is still on, then my computations are incorrect.” Convert this argument into logical notations using the variables c, b, r, p for propositions of computations, electric bills, out of money and the power respectively. (Where ¬ means NOT)
Q93➡| NTA UGC NET December 2015 Paper 2 Consider a proposition given as : x ≥ 6, if x 2 ≥ 5 and its proof as: If x ≥ 6, then x 2 = x.x ≥ 6.6 = 36 ≥ 25 Which of the following is correct w.r.to the given proposition and its proof? (a)The proof shows the converse of what is to be proved. (b)The proof starts by assuming what is to be shown. (c)The proof is correct and there is nothing wrong.
Q97➡| NTA UGC NET December 2014 Paper 2 Consider a set A = {1, 2, 3, …….., 1000}. How many members of A shall be divisible by 3 or by 5 or by both 3 and 5 ?
Q98➡| NTA UGC NET December 2014 Paper 2 A certain tree has two vertices of degree 4, one vertex of degree 3 and one vertex of degree 2. If the other vertices have degree 1, how many vertices are there in the graph ?
Q100➡| NTA UGC NET December 2014 Paper 2 A computer program selects an integer in the set {k : 1 ≤ k ≤ 10,00,000} at random and prints out the result. This process is repeated 1 million times. What is the probability that the value k=1 appears in the printout at least once ?
Q101➡| NTA UGC NET December 2014 Paper 2 If we define the functions f, g and h that map R into R by : f(x) = x 4 , g(x) = √ x2 + 1 , h(x) = x2 + 72, then the value of the composite functions ho(gof) and (hog)of are given as.
Q104➡| NTA UGC NET June 2014 Paper 2 The notation ∃!xP(x) denotes the proposition “there exists a unique x such that P(x) is true”. Give the truth values of the following statements : I. ∃!xP(x) → ∃xP(x). II. ∃!x ¬ P(x) → ¬∀xP(x)
Q105➡| NTA UGC NET June 2014 Paper 2 Give a compound proposition involving propositions p, q and r that is true when exactly two of p, q and r are true and is false otherwise.
Q107➡| NTA UGC NET June 2014 Paper 2 Consider a complete bipartite graph km,n. For which values of m and n does this, complete graph have a Hamilton circuit
Q108➡| NTA UGC NET June 2014 Paper 2 How many cards must be chosen from a deck to guarantee that at least I. two aces of two kinds are chosen. II. two aces are chosen. III. two cards of the same kind are chosen. IV. two cards of two different kinds are chosen.
Q110➡| NTA UGC NET June 2014 Paper 3 ________predicate calculus allows quantified variables to refer to objects in the domain of discourse and not to predicates or functions.
Q111➡| NTA UGC NET December 2013 Paper 2 Let P(m, n) be the statement “m divides n” where the universe of discourse for both the variables is the set of positive integers. Determine the truth values of each of the following propositions : I. ∀m ∀n P(m, n), II. ∃m ∀n P(m, n)
Q112➡| NTA UGC NET December 2013 Paper 2 Let f and g be the functions from the set of integers to the set integers defined by f(x) = 2x + 3 and g(x) = 3x + 2 Then the composition of f and g and g and f is given as.
