# UGC NET CS 2015 Dec – II

 Question 1
How many committees of five people can be chosen from 20 men and 12 women such that each committee contains atleast three women?
 A 75240 B 52492 C 41800 D 9900
General Aptitude     UGC NET CS 2015 Dec - II
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Question 1 Explanation:
We3 can select 5 people from 20 men and 12 women having atleast 3 women: 12C3 * 20C2 + 12C4 * 20C1 + 12C5 * 20C0 = 220 * 190 + 495 * 20 + 792 = 52492 So, option (B) is correct.
 Question 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. Codes:
 A (a) only B (b) and (c) C (c) only D (d) only
Graph Theory     UGC NET CS 2015 Dec - II
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Question 2 Explanation:
• A connected multigraph has an Euler Circuit if and only if each of its vertices has even degree.Correct
• A connected multigraph has an Euler Path but not an Euler Circuit if and only if it has exactly two vertices of odd degree.Correct
• A complete graph (Kn) has a Hamilton Circuit whenever n ≥ 3.Correct
• A cycle over six vertices (C6) is not a bipartite graph but a complete graph over 3 vertices is bipartite.Incorrect
So, option (D) is cocrrect.
 Question 3
3. 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 ½ is countable.
 A (a) and (c) B (b) and (d) C (b) only D (d) only
Set Theory & Algebra     UGC NET CS 2015 Dec - II
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Question 3 Explanation:
• The set of negative integers is countable.
• The set of integers that are multiples of 7 is countable.
• The set of even integers is countable.
• The set of real numbers between 0 and ½ is countable.This is not true because we can not count set of real numbers.
For more information on countability Refer:Mathematics | Power Set and its Properties Option (D) is correct.
 Question 4
Consider the graph given below: The two distinct sets of vertices, which make the graph bipartite are:
 A (v1, v4, v6); (v2, v3, v5, v7, v8) B (v1, v7, v8); (v2, v3, v5, v6) C (v1, v4, v6, v7); (v2, v3, v5, v8) D (v1, v4, v6, v7, v8); (v2, v3, v5)
Graph Theory     UGC NET CS 2015 Dec - II
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Question 4 Explanation:
A Bipartite Graph is a graph whose vertices can be divided into two independent sets, U and V such that every edge (u, v) either connects a vertex from U to V or a vertex from V to U. In other words, for every edge (u, v), either u belongs to U and v to V, or u belongs to V and v to U. We can also say that there is no edge that connects vertices of same set. (v1, v4, v6, v7); (v2, v3, v5, v8) is a bipartite graph vertices set. So, option (C) is correct.
 Question 5
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? codes:
 A (a) and (b) B (b) and (c) C (a) and (c) D (a), (b) and (c)
Graph Theory     UGC NET CS 2015 Dec - II
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Question 5 Explanation: All trees are graceful. So, option (D) is correct.
 Question 6
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 n2>1. Suppose that n2>1, then n>1. Codes :
 A (a) and (c) B (b) and (c) C (a), (b) and (c) D (a) and (b)
Propositional and First Order Logic.     UGC NET CS 2015 Dec - II
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Question 6 Explanation:
• “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”. This is not valid argument.
• “Either Puneet is not guilty or Pankaj is telling the truth. Pankaj is not telling the truth, therefore, Puneet is not guilty”.This is valid argument.
• If n is a real number such that n>1, then n2>1. Suppose that n2>1, then n>1. This is valid argument.
So, option (B) is correct.
 Question 7
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) Codes :
 A (a) - True; (b) - True; (c) - False B (a) - True; (b) - False; (c) - False C (a) - False; (b) - False; (c) - False D (a) - True; (b) - True; (c) - True
UGC NET CS 2015 Dec - II
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Question 7 Explanation:
• ∃m ∀n P(m, n) : There exist some m which divides all n.True
• ∀n P(1, n) Every n divided by 1.True
• ∀m ∀n P(m, n) Every m divides every n False
So, option (A) is correct.
 Question 8
Match the following terms: A (1) B (2) C (3) D (4)
Propositional and First Order Logic.     UGC NET CS 2015 Dec - II
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Question 8 Explanation:
• Vacuous proof is a proof in which the implication p → q is true based on the fact that p is false.
• Trivial proof is a proof in which the implication p → q is true based on the fact that q is true.
• Direct proof is A proof in which the implication p → q is true that proceeds by showing that q must be true when p is true.
• Indirect proof a proof in which the implication p → q is true that proceeds by showing that p must be false when q is false.
So, option (A) is correct.
 Question 9
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?
 A (a) and (c) B (b) and (c) C (a) and (b) D only (a)
Propositional and First Order Logic.     UGC NET CS 2015 Dec - II
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Question 9 Explanation:
• p ∨ ~(p ∧ q) = p + (pq)` = p + p` + q` = 1 + q` = 1.This is a tautology.
• (p ∧ ~q) ∨ ~(p ∧ q) = pq` + (pq)` = pq` + p` + q` = p` + q`. This is not a tautology.
• p ∧ (q ∨ r) = pq + pr. This is not a tautology.
So, option (D) is correct
 Question 10
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 Codes:
 A (a) and (b) B (b) and (c) C (a) and (b) D (a), (b) and (c)
Set Theory & Algebra     UGC NET CS 2015 Dec - II
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There are 50 questions to complete.
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