GATE DM PYQ [2010-2025]

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Question 1

Suppose the predicate F(x, y, t) is used to represent the statement that person x can fool person y at time t. Which one of the statements below expresses best the

meaning of the formula ∀x∃y∃t(¬F(x, y,t))?[ MCQ || GATE 2010 || 2 Marks ]

• Everyone can fool some person at some time

• No one can fool everyone all the time

• Everyone cannot fool some person all the time

• No one can fool some person at some time

Question 2

What is the possible number of reflexive relations on a set of 5 elements? [ MCQ || GATE 2010 || 1 Marks ]

• 2¹⁰

• 2¹⁵

• 2²⁰

• 2²⁵

Question 3

Consider the set S = {1, ω, ω ² }, where ω and ω ²  are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms. [ MCQ || GATE 2010 || 1 Marks ]

• A group

• A ring

• An integral domain

• A field

Question 4

K4 and Q3 are graphs with the following structures.

.

Which one of the following statements is TRUE in relation to these graphs? [ MCQ || GATE 2011 || 1 Marks ]

• K4 is planar while Q3 is not

• Both K4 and Q3 are planar

• Q3 is planar while K4 is not

• Neither K4 nor Q3 is planar

Question 5

Which one of the following options is CORRECT given three positive integers x,y and z, and a predicate

P(x) = ¬(x=1) ∧ ∀y(∃z(x=y*z) ⇒ (y=x) ∨ (y=1) [ MCQ || GATE 2011 || 2 Marks ]

• P(x) being true means that x is a prime number

• P(x) being true means that x is a number other than 1

• P(x) is always true irrespective of the value of x

• P(x) being true means that x has exactly two factors other than 1 and x

Question 6

Let G be a simple undirected planar graph on 10 vertices with 15 edges. If G is a connected graph, then the number of bounded faces in any embedding of G on the plane is equal to ____. [ MCQ || GATE 2012 || 1 Marks ]

• 3

• 4

• 5

• 6

Question 7

Which of the following graphs is isomorphic

to [ MCQ || GATE 2012 || 2 Marks ]

• 

• 

• 

• 

Question 8

Consider the following logical inferences.

I₁ : If it rains then the cricket match will not be played. The cricket match was played.

Inference: There was no rain.

I₂ : If it rains then the cricket match will not be played. It did not rain.

Inference: The cricket match was played.

Which of the following is TRUE? [ MCQ || GATE 2012 || 1 Marks ]

• Both I₁ and I₂ are correct inferences

• I₁ is correct but I₂ is not a correct inference

• I₁ is not correct but I₂ is a correct inference

• Both I₁ and I₂ are not correct inferences.

Question 9

What is the correct translation of the following statement into mathematical logic?

“Some real numbers are rational” [ MCQ || GATE 2012 || 1 Marks ]

• ∃x (real(x) ∨ rational(x))

• ∀x (real(x) → rational(x))

• ∃x (real(x) ∧ rational(x))

• ∃x (rational(x) → real(x))

Question 10

How many onto (or surjective) functions are there from an n-element (n ≥ 2) set to a 2-element set? [ MCQ || GATE 2012 || 2 Marks ]

• 2ⁿ

• 2ⁿ -1

• 2ⁿ -2

• 2(2ⁿ -2)

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