 ## UGC NET CS 2016 July- paper-2

 Question 1
How many different equivalence relations with exactly three different equivalence classes are there on a set with five elements?
 A 10 B 15 C 25 D 30
Question 1 Explanation:
Step-1: Given number of equivalence classes with 5 elements with three elements in each class will be 1,2,2 (or) 2,1,2 (or) 2,2,1 and 3,1,1.
Step-2: The number of combinations for three equivalence classes are
2,2,1 chosen in (​ 5​ C​ 2​ *​ 3​ C​ 2​ *​ 1​ C​ 1​ )/2! = 15
3,1,1 chosen in(​ 5​ C​ 2​ *​ 3​ C​ 2​ *​ 1​ C​ 1​ )/2! = 10
Step-3: Total differential classes are 15+10
=25.
 Question 2
The number of different spanning trees in complete graph, K​ 4​ and bipartite graph, K​ 2,2​ have ______ and _______ respectively.
 A 14, 14 B 16, 14 C 16, 4 D 14, 4
Question 2 Explanation:
Step-1: Given complete graph K​ 4​ .To find total number of spanning tree in complete graph using standard formula is n​ (n-2) Here, n=4
=n​ (n-2)
= 4​ 2
=16
Step-2: Given Bipartite graph K​ 2,2​ . To find number of spanning tree in a bipartite graph K​ m,n​ having standard formula is m​ (n-1)​ * n​ (m-1)​ .
m=2 and n=2
= 2​ (2-1)​ * 2​ (2-1)
= 2 * 2
= 4
 Question 3
Suppose that R​ 1​ and R​ 2​ are reflexive relations on a set A. Which of the following statements is correct ?
 A R​ 1​ ∩ R​ 2​ is reflexive and R​ 1​ ∪ R​ 2​ is irreflexive. B R​ 1​ ∩ R​ 2​ is irreflexive and R​ 1​ ∪ R​ 2​ is reflexive. C Both R​ 1​ ∩ R​ 2​ and R​ 1​ ∪ R​ 2​ are reflexive. D Both R​ 1​ ∩ R​ 2​ and R​ 1​ ∪ R​ 2​ are irreflexive.
Question 3 Explanation:
A binary relation R over a set X is reflexive if every element of X is related to itself. Formally, this may be written ​ ∀ ​ x ∈X : xRx.
Ex: Let set A={0,1}
R​ 1​ ={(0,0),(1,1)} all diagonal elements we are considering for reflexive relation.
R​ 2​ ={(0,0),(1,1)} all diagonal elements we are considering for reflexive relation.
R​ 1​ ∩ R​ 2​ must have {(0,0),(1,1)} is reflexive.
R​ 1​ ∪ R​ 2​ must have {(0,0),(1,1)} is reflexive.
 Question 4
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?
 A 3/4 B 2/3 C 1/2 D 1⁄3
Question 4 Explanation:
Given data,
-- 3 cards in a box
-- 1​ st​ card: Both sides of one card is black. The card having 2 sides. We can write it as BB.
-- 2​ nd​ card: Both sides of one card is red. The card having 2 sides. We can write it as RR.
-- 3rd card: one black side and one red side.​ ​ We can write it as BR.
Step-1: The probability that the opposite side is the same colour as the one side we observed is 2⁄3 because total number of cards are 3
 Question 5
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? A 2 B 4 C 5 D 6
Question 5 Explanation:
Definition of clique is already given in question.
Definition: A clique in a simple undirected graph is a complete subgraph that is not contained in any larger complete subgraph.
Step-1: b,c,e,f is complete graph. Step-2: ‘a’ is not connected to ‘e’ and ‘b’ is not connected to ‘d’. So, it is not complete graph. Question 6
Which of the following logic expressions is incorrect?
 A 1 ⊕ 0 = 1 B 1 ⊕ 1 ⊕ 1 = 1 C 1 ⊕ 1 ⊕ 0 = 1 D 1 ⊕ 1 = 0
Question 6 Explanation:
Here, ⊕ is nothing but Ex-OR operator. The truth table for Ex-OR is According to truth table,
Option-A is TRUE
Option-B is a 1 ⊕ 1 is 0.
0 ⊕ 1 is 1(TRUE)
Option-C is 1 ⊕ 1 is 0.
0 ⊕ 0 = 0 but given 1. So, FALSE
Option-D is TRUE.
 Question 7
The IEEE-754 double-precision format to represent floating point numbers, has a length of _____ bits.
 A 16 B 32 C 48 D 64
Question 7 Explanation:
→ The IEEE-754 double-precision format to represent floating point numbers has a length of 64 bits
→ In the IEEE 754-2008 standard, the 64-bit base-2 format is officially referred to as binary64 called double in IEEE 754-1985.
→ IEEE 754 specifies additional floating-point formats, including 32-bit base-2 single precision and, more recently, base-10 representations.
 Question 8
Simplified Boolean equation for the following truth table is: A F = yz’ + y’z B F = xy’ + x’y C F = x’z + xz’ D F = x’z + xz’ + xyz
Question 8 Explanation:
Method-1: Using K-Map Method-2: Using boolean simplification
= x’y’z+x’yz+xy’z’+xyz’
= x'z(y'+y)+ xz'(y'+y)
= x'z+xz' (Since y'+y=1)
 Question 9
The simplified form of a Boolean equation (AB’ + AB’C + AC) (A’C’ + B’) is :
 A AB’ B AB’C C A’B D ABC
Question 9 Explanation:
(AB’ + AB’C + AC) (A’C’ + B’)
= (AB'+AC) (A'C'+B')
= AB'A'C' + AB'B' + ACA'C' + ACB'
= AB'B' + ACB'
= AB'(C+1)
= AB'
 Question 10
In a positive-edge-triggered JK flip-flop, if J and K both are high then the output will be _____ on the rising edge of the clock.
 A No change B Set C Reset D Toggle
Question 10 Explanation:
Positive-edge-triggered JK flip-flop is The Truth Table for the JK Function When J = 1 and K = 1 , The output continuously Toggles from 1 to 0 and 0 to 1. At the end Output is indeterminate. This condition is called as Race around Condition. This happens when Propagation Delay is less than the Pulse width.
 Question 11
Given i = 0, j = 1, k = –1 x = 0.5, y = 0.0 What is the output of the following expression in C language ?
x * y < i + j || k
 A -1 B 0 C 1 D 2
Question 11 Explanation:
x * y < i + j || k
Step-1: Evaluate x * y because multiplication has more priority than remaining operators
x * y→ 0
Step-2: i + j is 1
Step-3: (x*y) < (i+j) is 1. Because relational operators only return 1(TRUE) or 0(FALSE).
Step-4: ((x*y) < (i+j)) || k is logical OR operator.
1 || -1 will returns 1.
Note: The precedence is ((x*y) < (i+j)) || k
 Question 12
The following statement in ‘C’
int (*f())[ ];
declares
 A a function returning a pointer to an array of integers. B a function returning an array of pointers to integers. C array of functions returning pointers to integers. D an illegal statement.
Question 12 Explanation:
int (*f())[ ] declare a function returning a pointer to an array of integers.
There are 12 questions to complete.

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