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(x

^{2}+ 1) + 3x^{2}is given as
(A) O(xlogx) (B)
O(x

^{2})
(C) O(x

^{3}) (D) O(x^{2}logx)
Answer: B

**Explanation:**

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

^{2}+1) + 3x^{2}
Note (x+1) is O(x) and x

^{2}+1≤2x^{2}when x>1
So,
log x

^{2}+1≤log(2x^{2})=log 2+ log x^{2}=log 2+ 2 log x
≤
3 log x if x >2

Thus, log x

^{2}+1 is O(log x)
The first part of f(x) is O(x log x)

Also, 3x

^{2}is O(x^{2})
So, f(x) is O(max(x log x, x

^{2}))=O(x^{2}) as x log x ≤ x^{2}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:**

f

*o*g(x)=f(g(x))=f(3x+2)=2(3x+2)+3=6x+7
g

*o*f(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 K_{5}(B) K_{3,3}and K_{6}
(C) K

_{3,3}or K_{5}(D) K_{2,3}and K_{5}
Answer: C

**Explanation:**

**A graph is non-planar if and only if it contains a subgraph that is homeomorphic to either K**

*Kuratowski’s Theorem:*_{5}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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