CS and IT GATE 2014 Set-3 Questions with Answer

Ques 40 Digital Logic

Let ⊕ denote the Exclusive OR (XOR) operation. Let '1' and '0' denote the binary constants. Consider the following Boolean expression for F over two variables P and Q:
F(P, Q) = ((1 ⊕ P) ⊕ (P ⊕ Q)) ⊕ ((P ⊕ Q) ⊕ (Q ⊕ 0))
The equivalent expression for F is

P + Q

P ⊕ Q

Ques 41 Digital Logic

Consider the following statements:
P: Good mobile phones are not cheap
Q: Cheap mobile phones are not good
L: P implies Q
M: Q implies P
N: P is equivalent to Q
Which one of the following about L, M, and N is CORRECT?

Only L is TRUE.

Only M is TRUE.

Only N is TRUE.

L, M and N are TRUE.

Ques 42 Discrete Mathematics

Consider the set of all functions f: {0, 1, ..., 2014} → {0, 1, ..., 2014} such that f(f(i)) = i, for all 0 ≤ i ≤ 2014. Consider the following statements:
P. For each such function it must be the case that for every i, f(i) = i.
Q. For each such function it must be the case that for some i, f(i) = i.
R. Each such function must be onto.
Which one of the following is CORRECT?

P, Q and R are true

Only Q and R are true

Only P and Q are true

Only R is true

Ques 43 Discrete Mathematics

There are two elements x, y in a group (G, *) such that every element in the group can be written as a product of some number of x's and y's in some order. It is known that
x * x = y * y = x * y * x * y = y * x * y * x = e
where e is the identity element. The maximum number of elements in such a group is _________.

4 to 4 is the correct answer.

Ques 44 Discrete Mathematics

If G is a forest with n vertices and k connected components, how many edges does G have?

⌊n/k⌋

⌈n/k⌉

n - k

n - k + 1

Ques 45 Discrete Mathematics

Let δ denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with δ ≥ 3, which one of the following is TRUE?

In any planar embedding, the number of faces is at least n/2 + 2.

In any planar embedding, the number of faces is less than n/2 + 2.

There is a planar embedding in which the number of faces is less than n/2 + 2.

There is a planar embedding in which the number of faces is at most n/(δ+1).

Ques 46 Discrete Mathematics

The CORRECT formula for the sentence, “not all rainy days are cold” is

∀d (Rainy(d) ∧ ¬Cold(d))

∀d (¬Rainy(d) → Cold(d))

∃d (¬Rainy(d) → Cold(d))

∃d (Rainy(d) ∧ ¬Cold(d))

Ques 47 Discrete Mathematics

Which one of the following is the most appropriate logical formula to represent the statement? “Gold and silver ornaments are precious”. The following notations are used:
G(x): x is a gold ornament
S(x): x is a silver ornament
P(x): x is precious

∀x (P(x) → (G(x) ∧ S(x)))

∀x ((G(x) ∧ S(x)) → P(x))

∃x ((G(x) ∧ S(x)) → P(x))

∀x ((G(x) ∨ S(x)) → P(x))

Ques 48 Discrete Mathematics

Let X and Y be finite sets and f: X → Y be a function. Which one of the following statements is TRUE?

For any subsets A and B of X, |f(A ∪ B)| = |f(A)| + |f(B)|

For any subsets A and B of X, f(A ∩ B) = f(A) ∩ f(B)

For any subsets A and B of X, |f(A ∩ B)| = min{|f(A)|, |f(B)|}

For any subsets S and T of Y, f⁻¹(S ∩ T) = f⁻¹(S) ∩ f⁻¹(T)

Ques 49 Discrete Mathematics

Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L ≠ G and that the size of L is at least 4. The size of L is _________.

5 is the correct answer.

Ques 50 Engineering Mathematics

If ∫₀^2π |x sin x| dx = kπ, then the value of k is equal to _________.

4 is the correct answer.

Ques 51 Engineering Mathematics

The value of the integral given below is
∫₀^πx²cos x dx

-2π

-π

2π

Ques 52 Engineering Mathematics

With respect to the numerical evaluation of the definite integral, K = ∫_a^bx²dx, where a and b are given, which of the following statements is/are TRUE?
I) The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
II) The value of K obtained using the Simpson's rule is always equal to the exact value of the definite integral.

I only

II only

Both I and II

Neither I nor II

1 2 3 4 5

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CS and IT GATE 2014 Set-3 Questions with Answer