UGC NET/JRF Computer Science December 2013 Paper II - Part 4

31.       The dual of a Boolean expression is obtained by interchanging

(A) Boolean sums and Boolean products

(B) Boolean sums and Boolean products or interchanging 0's and 1's

(C) Boolean sums and Boolean products and interchanging 0's & 1's

(D) Interchanging 0's and 1's

Answer: C

32.       Given that (292)₁₀ = (1204)_x in some number system x. The base x of that number system is

(A) 2                   (B) 8

(C) 10                 (D) None of the above

Answer: D

Explanation:

6|292

6|48| 4

6|8|   0

6|1|   2

6|1|   1

Solution: (1204)₆

So here, x=6

33.       The sum of products expansion for the function

F(x, y, z) = (x + y)z’ is given as

(A) x’y’z + xyz’ + x’yz’

(B) xyz + xyz’ + xy’z’

(C) xy’z’ + x’y’z’ + xyz’

(D) xyz’ + xy’z’ + x’yz’

Answer: D

Explanation:

Use Boolean identities to expand the product and simplify.

F(x, y, z)=(x + y)z’

                =xz’+yz’                                  Distributive law

                =x1z’+1yz’                             Identity law

                =x(y+y’)z’+(x+x’)yz’              Unit property

                =xyz’+xy’z’+xyz’+x’yz’         Distributive law

                =xyz’+xy’z’+x’yz’                  Idempotent law

34.       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)

(A) Both I and II are true

(B) Both I and II are false

(C) I - false & II - true

(D) I - true & II – false

Answer: C

35.       Big - O estimate for

f(x) = (x + 1) log(x² + 1) + 3x² is given as

(A) O(xlogx)      (B) O(x²)

(C) O(x³)            (D) O(x²logx)

Answer: B

Explanation:

Big-O notation of f(x)=(x+1) log(x²+1) + 3x²

Note (x+1) is O(x) and x²+1≤2x² when x>1

 So, log x²+1≤log(2x²)=log 2+ log x²=log 2+ 2 log x

     ≤ 3 log x if x >2

Thus, log x²+1 is O(log x)

The first part of f(x) is O(x log x)

Also, 3x² is O(x²)

So, f(x) is O(max(x log x, x²))=O(x²) as  x log x ≤ x² for x >1

36.       How many edges are there in a forest of t-trees containing a total of n vertices ?

(A) n+t                (B) n-t

(C) n*t                (D) n^t

Answer: B

37.       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

(A) 6x + 7, 6x + 11

(B) 6x + 11, 6x + 7

(C) 5x + 5, 5x + 5

(D) None of the above

Answer: A

Explanation:

fog(x)=f(g(x))=f(3x+2)=2(3x+2)+3=6x+7

gof(x)=g(f(x))=g(2x+3)=3(2x+3)+2=6x+11

38.       If n and r are non-negative integers and n≥r, then p(n + 1, r) equals to

(A) P(n,r)(n+1) / (n+1-r)

(B) P(n,r)(n+1) / (n-1+r)

(C) p(n,r)(n-1) / (n+1-r)

(D) p(n,r)(n+1) / (n+1+r)

Answer: A

Explanation:

p(n, r) = n!/(n-r)!

p( n+1, r) = (n+1)!/(n+1-r)!

= (n+1) n! /(n+1-r) (n-r)!

= P(n, r)(n+1)/(n+1-r)

39.       A graph is non-planar if and only if it contains a subgraph homeomorphic to

(A) K_3,2 or K₅     (B) K_3,3 and K₆

(C) K_3,3 or K₅     (D) K_2,3 and K₅

Answer: C

Explanation:

Kuratowski’s Theorem: A graph is non-planar if and only if it contains a subgraph that is homeomorphic to either K₅ or K_3,3.

40.    Which of the following statements are true ?

I.   A circuit that adds two bits, producing a sum bit and a carry bit is called half adder.

II.  A circuit that adds two bits, producing a sum bit and a carry bit is called full adder.

III. A circuit that adds two bits and a carry bit producing a sum bit and a carry bit is called full adder.

IV.  A device that accepts the value of a Boolean variable as input and produces its complement is called an inverter.

(A) I & II              (B) II & III

(C) I, II, III           (D) I, III & IV

Answer: D

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