## UGC NET CS 2016 Aug- paper-2

Question 1 |

The Boolean function [~(~p ∧ q) ∧ ~( ~p ∧ ~q)] ∨ (p ∧ r) is equal to the Boolean function:

q | |

p ∧ r | |

p ∨ q | |

p |

Question 1 Explanation:

Question 2 |

Let us assume that you construct ordered tree to represent the compound proposition (~ (p ∧ q)) ↔ (~ p ∨ ~ q) Then, the prefix expression and post-fix expression determined using this ordered tree are given as ____ and _____ respectively.

↔~∧pq∨ ~ ~ pq, pq∧~p~q~∨↔ | |

↔~∧pq∨ ~ p~q, pq∧~p~q~∨↔ | |

↔~∧pq∨ ~ ~ pq, pq∧~p~~q∨↔ | |

↔~∧pq∨ ~ p~ q, pq∧~p~ ~q∨↔ |

Question 2 Explanation:

Step-1: Given compound proposition is

(~(p ∧ q))↔(~ p ∨ ~ q)

It is clearly specifying that

↔ is a root

(~(p ∧ q)) is left subtree

(~p ∨ ~q) is right subtree.

Step-2: Finally the tree is looks like

Step-3: Prefix operation traverse through Root,left and Right ↔~∧pq∨ ~ p~q Step-4: Postfix operation traverse through Left,Right and Root.

pq∧~p~q~∨↔

(~(p ∧ q))↔(~ p ∨ ~ q)

It is clearly specifying that

↔ is a root

(~(p ∧ q)) is left subtree

(~p ∨ ~q) is right subtree.

Step-2: Finally the tree is looks like

Step-3: Prefix operation traverse through Root,left and Right ↔~∧pq∨ ~ p~q Step-4: Postfix operation traverse through Left,Right and Root.

pq∧~p~q~∨↔

Question 3 |

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 ?

|A – B| ≤ |A ⊕ B| ≤ |A| + |B| ≤ |A ∪ B| | |

|A ⊕ B| ≤ |A – B| ≤ |A ∪ B| ≤ |A| + |B| | |

|A ⊕ B| ≤ |A| + |B| ≤ |A – B| ≤ |A ∪ B| | |

|A – B| ≤ |A ⊕ B| ≤ |A ∪ B| ≤ |A| + |B| |

Question 3 Explanation:

Step-1: Let A and B be sets in a finite universal set U.

Step-2: |A–B| we can also write into |A|-|A ∩ B| and equivalent Venn diagram is

Step-3: |A⊕B| We can also write into |A|+|B|- 2|A∩B| and equivalent Venn diagram is

Step-4: |A| + |B| We can represented into |A| + |B| and equivalent Venn diagram is

Step-5: |A∪B| We can also write into |A|+|B|- |A∩B| and equivalent Venn diagram is

Step-6: |A – B| ≤ |A ⊕ B| ≤ |A ∪ B| ≤ |A| + |B| is correct order.

Step-2: |A–B| we can also write into |A|-|A ∩ B| and equivalent Venn diagram is

Step-3: |A⊕B| We can also write into |A|+|B|- 2|A∩B| and equivalent Venn diagram is

Step-4: |A| + |B| We can represented into |A| + |B| and equivalent Venn diagram is

Step-5: |A∪B| We can also write into |A|+|B|- |A∩B| and equivalent Venn diagram is

Step-6: |A – B| ≤ |A ⊕ B| ≤ |A ∪ B| ≤ |A| + |B| is correct order.

Question 4 |

What is the probability that a randomly selected bit string of length 10 is a palindrome?

1/64 | |

1/32 | |

1/8 | |

1⁄4 |

Question 4 Explanation:

Palindrome is a number that remains the same when its digits are reversed.

Ex: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121...,

Step-1: Here, string length is 10. If we consider first 5 numbers as random choices like 0 or 1.

Remaining 5 numbers are fixed. Total number of possibilities are 2

Step-2: The probability 2

= 1/2

= 1/32

Ex: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121...,

Step-1: Here, string length is 10. If we consider first 5 numbers as random choices like 0 or 1.

Remaining 5 numbers are fixed. Total number of possibilities are 2

^{ 10} . But we are only considering first 5 choices. The probability is 2^{5} .Step-2: The probability 2

^{5} /2^{ 10}= 1/2

^{5}= 1/32

Question 5 |

Given the following graphs:

Which of the following is correct?

Which of the following is correct?

G _{1} contains Euler circuit and G _{2} does not contain Euler circuit. | |

G _{1} does not contain Euler circuit and G _{2} contains Euler circuit. | |

Both G _{1} and G_{ 2} do not contain Euler circuit. | |

Both G _{ 1} and G_{ 2} contain Euler circuit. |

Question 5 Explanation:

Step-1: G1 have odd number of vertices. So, it is not euler circuit.

Step-2: G2 also have odd number of vertices. So, it not euler circuit.

Question 6 |

The octal number 326.4 is equivalent to

(214.2) _{10} and (D6.8))_{ 16} | |

(212.5) _{10} and (D6.8)) _{ 16} | |

(214.5) _{10} and (D6.8)) _{ 16} | |

(214.5) _{10} and (D6.4)) _{ 16} |

Question 6 Explanation:

(326.4)

Step1: First convert given octal no. to binary number because it will be easier to solve this way.

Step 2: Now convert above binary no. into decimal .

(011010110.100)

= ( 1 * 2

= (214.5)

(326.4)

Step 2: Now convert above binary no. into decimal .

(011010110.100)

= ( 1 * 2

= (214.5)

(326.4)

_{8}= ( ?)Step1: First convert given octal no. to binary number because it will be easier to solve this way.

Step 2: Now convert above binary no. into decimal .

(011010110.100)

_{2}= ( 1 * 2

^{7}) + ( 1 × 2^{ 6}) + ( 1 × 2^{ 4}) + ( 1 × 2^{2}) + ( 1 × 2^{ 1}) · [ 1 × ( 1/2) ]= (214.5)

_{10}(326.4)

_{8}= (?)_{ 16}Step 1: Convert given octal no. into binary no.Step 2: Now convert above binary no. into decimal .

(011010110.100)

_{2}= ( 1 * 2

^{7}) + ( 1 × 2^{ 6}) + ( 1 × 2^{ 4}) + ( 1 × 2^{ 2}) + ( 1 × 2^{ 1}) · [ 1 × ( 1/2) ]= (214.5)

_{10}(326.4)

_{ 8}= (?)_{ 16}Question 7 |

Which of the following is the most efficient to perform arithmetic operations on the numbers?

Sign-magnitude | |

1’s complement | |

2’s complement | |

9’s complement |

Question 7 Explanation:

2’s complement has single representation for zero , but Sign-magnitude, 1’s complement and 9’s complement have two representations for 0 (i.e., both positive zero and negative zero).
While doing arithmetic operations like addition or subtraction using 1's complement(or 9's complement), we have to add an extra carry bit, i.e 1 to the result to get the correct answer. 2's complement doesn't require such extra calculation.

Question 8 |

The Karnaugh map for a Boolean function is given as

The simplified Boolean equation for the above Karnaugh Map is

The simplified Boolean equation for the above Karnaugh Map is

AB + CD + AB’ + AD | |

AB + AC + AD + BCD | |

AB + AD + BC + ACD | |

AB + AC + BC + BCD |

Question 8 Explanation:

Question 9 |

Which of the following logic operations is performed by the following given combinational circuit?

EXCLUSIVE-OR | |

EXCLUSIVE-NOR | |

NAND | |

NOR |

Question 9 Explanation:

Question 10 |

Match the following:

a-iii, b-ii, c-iv, d-i | |

a-ii, b-iv, c-i, d-iii | |

a-ii, b-i, c-iv, d-iii | |

a-iii, b-i, c-iv, d-ii |

Question 10 Explanation:

a. Controlled Inverter : A circuit that transmits a binary word or its1’s complement.

b. Full adder : A circuit that can add 3 bits. It adds two data bits(ai,bi) and also carry bit(ci).

c. Half adder : A logic circuit that adds 2 bits(ai,bi).

d. Binary adder : A circuit that can add two binary numbers(A and B). Collection of Full adders form a binary adder.

b. Full adder : A circuit that can add 3 bits. It adds two data bits(ai,bi) and also carry bit(ci).

c. Half adder : A logic circuit that adds 2 bits(ai,bi).

d. Binary adder : A circuit that can add two binary numbers(A and B). Collection of Full adders form a binary adder.

Question 11 |

Given i= 0, j = 1, k = –1, x = 0.5, y = 0.0 What is the output of given ‘C’ expression ?

x * 3 && 3 || j | k

x * 3 && 3 || j | k

-1 | |

0 | |

1 | |

2 |

Question 11 Explanation:

Step-1: Evaluate x * 3 because multiplication has more priority than remaining operators x * 3→ 1.5

Step-2: && is logical AND. Both the statements are TRUE, then it returns 1 otherwise 0.

1.5 && 3 is TRUE. So, it returns 1.

Step-3: j | k is bitwise OR operator. It returns -1.

Step-4: ((x * 3) && 3) || (j | k) islogical OR operator. It returns 1

Note: The precedence is ((x * 3) && 3) || (j | k)

Step-2: && is logical AND. Both the statements are TRUE, then it returns 1 otherwise 0.

1.5 && 3 is TRUE. So, it returns 1.

Step-3: j | k is bitwise OR operator. It returns -1.

Step-4: ((x * 3) && 3) || (j | k) islogical OR operator. It returns 1

Note: The precedence is ((x * 3) && 3) || (j | k)

Question 12 |

The following ‘C’ statement :
int *f [ ] ( );
declares:

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

Array of functions returning pointers to integers. | |

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

An illegal statement. |

Question 12 Explanation:

int *f [ ] ( ); It declares array of functions returning pointers to integers.

There are 12 questions to complete.