## 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?

10 | |

15 | |

25 | |

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 (

3,1,1 chosen in(

Step-3: Total differential classes are 15+10

=25.

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! = 153,1,1 chosen in(

^{5}C_{ 2} *^{ 3} C_{2} *^{ 1} C_{ 1} )/2! = 10Step-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.14, 14 | |

16, 14 | |

16, 4 | |

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

= 4

=16

Step-2: Given Bipartite graph K

m=2 and n=2

= 2

= 2 * 2

= 4

^{(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 ?R _{1} ∩ R_{ 2} is reflexive and R_{ 1} ∪ R_{ 2} is irreflexive. | |

R _{1} ∩ R_{ 2} is irreflexive and R_{ 1} ∪ R_{ 2} is reflexive. | |

Both R _{1} ∩ R_{ 2} and R_{ 1} ∪ R_{ 2} are reflexive. | |

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

R

R

R

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?

3/4 | |

2/3 | |

1/2 | |

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

-- 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?

2 | |

4 | |

5 | |

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.

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?

1 ⊕ 0 = 1 | |

1 ⊕ 1 ⊕ 1 = 1 | |

1 ⊕ 1 ⊕ 0 = 1 | |

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.

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.

16 | |

32 | |

48 | |

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.

→ 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:

F = yz’ + y’z | |

F = xy’ + x’y | |

F = x’z + xz’ | |

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)

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 :

AB’ | |

AB’C | |

A’B | |

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'

= (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.

No change | |

Set | |

Reset | |

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

x * y < i + j || k

-1 | |

0 | |

1 | |

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

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

int (*f())[ ];

declares

a function returning a pointer to an array of integers. | |

a function returning an array of pointers to integers. | |

array of functions returning pointers to integers. | |

an illegal statement. |

Question 12 Explanation:

int (*f())[ ] declare a function returning a pointer to an array of integers.