**GATE CSE Engineering Mathematics PYQ**

Q1➡ | GATE 2021 Set-1Let p and q be two propositions. Consider the following two formulae in propositional logic. Which one of the following choices is correct? |

i ➥ S1 is a tautology but S2 is not a tautology. |

ii ➥ Both S1 and S2 are tautologies. |

iii ➥ Neither S1 and S2 are tautology. |

iv ➥ S1 is not a tautology but S2 is a tautology. |

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Q2➡ | GATE 2021 Set-1Consider the following expression The value of the above expression (rounded to 2 decimal places) is____ |

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Q3➡ | GATE 2021 Set-1In an undirected connected planar graph G, there are eight vertices and five faces. The number of edges in G is _____. |

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Q4➡ | GATE 2021 Set-1Let G = (V, E) be an undirected unweighted connected graph. The diameter of G is defined as: Let M be the adjacency matrix of G. Define graph G _{2} on the same set of vertices with adjacency matrix N, whereWhich one of the following statements is true? |

i ➥ |

ii ➥ |

iii ➥ |

iv ➥ |

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Q5➡ | GATE 2021 Set-1Consider the two statements. S1: There exist random variables X and Y such that (E[(X-E(X)) (Y-E(Y))]) ^{2}>Var[X] Var[Y] S2: For all random variables X and Y, Cov[X,Y]=E[|X-E[X]| |Y-E[Y]|] Which one of the following choices is correct? |

i ➥ Both S1 and S2 are false. |

ii ➥ S1 is false, but S2 is true. |

iii ➥ Both S1 and S2 are true. |

iv ➥ S1 is true, but S2 is false. |

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Q6➡ | GATE 2021 Set-1Let G be a group of order 6, and H be a subgroup of G such that 1 < |H| < 6. Which one of the following options is correct? |

i ➥ G may not be cyclic, but H is always cyclic. |

ii ➥ Both G and H may not be cyclic. |

iii ➥ Both G and H are always cyclic. |

iv ➥ G is always cyclic, but H may not be cyclic. |

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Q7➡ | GATE 2021 Set-1 A relation R is said to be circular of aRb and bRc together imply cRa. Which of the following options is/are correct? |

i ➥ If a relation S is circular and symmetric, then S is an equivalence relation. |

ii ➥ If a relation S is transitive and circular, then S is an equivalence relation. |

iii ➥ If a relation S is reflexive and symmetric, then S is an equivalence relation. |

iv ➥ If a relation S is reflexive and circular, then S is an equivalence relation. |

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Q8➡ | GATE 2021 Set-1An articulation point in a connected graph is a vertex such that removing the vertex and its incident edges disconnects the graph into two or more connected components. Let T be a DFS tree obtained by doing DFS in a connected undirected graph G. Which of the following options is/are correct? |

i ➥ A leaf of T can be an articulation point in G. |

ii ➥ If u is an articulation point in G such that x is an ancestor of u in T and y is a descendent of u in T, then all paths from x to y in G must pass through u. |

iii ➥ Root of T is an articulation point in G if and only if it has 2 or more children |

iv ➥ Root of T can never be an articulation point in G. |

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Q9➡ | GATE 2021 Set-1A sender (S) transmits a signal, which can be one of the two kinds: H and L with probabilities 0.1 and 0.9 respectively, to a receiver (R). In the graph below, the weight of edge (u, v) is the probability of receiving v when u is transmitted, where u, v ∈ {H, L}. For example, the probability that the received signal is L given the transmitted signal was H, is 0.7. If the received signal is H, the probability that the transmitted signal was H (rounded to 2 decimal places) is ________. |

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Q10➡ | GATE 2021 Set-1Consider the following matrix. The largest eigenvalue of the above matrix is___________ |

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Q11➡ | GATE 2021 Set-2Let G be a connected undirected weighted graph. Consider the following two statements. S _{1}: There exists a minimum weighted edge in G which is present in every minimum spanning tree of G.S _{2}:If every edge in G has distinct weight, then G has a unique minimum spanning tree.Which of the following options is correct? |

i ➥ S_{1} is false and S_{2} is true. |

ii ➥ Both S_{1} and S_{2} are true. |

iii ➥ Both S_{1} and S_{2} are false. |

iv ➥ S_{1} is true and S_{2} is false. |

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Q12➡ | GATE 2021 Set-2Choose the correct choice(s) regarding the following propositional logic assertion S: |

i ➥ S is a tautology. |

ii ➥ The antecedent of S is logically equivalent to the consequent of S. |

iii ➥ S is a contradiction. |

iv ➥ S is neither a tautology nor a contradiction. |

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Q13➡ | GATE 2021 Set-2 |

i ➥ There does not exist an injunction from S_{1} to S_{2}. |

ii ➥ There exists a bijection from S_{1} to S_{2}. |

iii ➥ There does not exist a bijection from S_{1} to S_{2}. |

iv ➥ There exists a surjection from S_{1} to S_{2}. |

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Q14➡ | GATE 2021 Set-2 |

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Q15➡ | GATE 2021 Set-2 Suppose that P is a 45 matrix such that every solution of the equation Px=0 is a scalar multiple of [2 5 4 3 1] ^{T}. The rank of P is _____. |

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Q16➡ | GATE 2021 Set-2 For a given biased coin, the probability that the outcome of a toss is a head is 0.4. This coin is tossed 1,000 times. Let X denote the random variable whose value is the number of times that head appeared in these 1,000 tosses. The standard deviation of X (rounded to 2 decimal places) is ______. |

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Q17➡ | GATE 2021 Set-2 In an examination, a student can choose the order in which two questions (QuesA and QuesB) must be attempted. • If the first question is answered wrong, the student gets zero marks. • If the first question is answered correctly and the second question is not answered correctly, the student gets the marks only for the first question. • If both the questions are answered correctly, the student gets the sum of the marks of the two questions. The following table shows the probability of correctly answering a question and the marks of the question respectively. Assuming that the student always wants to maximize her expected marks in the examination, in which order should she attempt the questions and what is the expected marks for that order (assume that the questions are independent)? A First QuesB and then QuesA. Expected marks 22. |

i ➥ First QuesB and then QuesA. Expected marks 22. |

ii ➥ First QuesA and then QuesB. Expected marks 14. |

iii ➥ First QuesB and then QuesA. Expected marks 14. |

iv ➥ First QuesA and then QuesB. Expected marks 16. |

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Q18➡ | GATE 2021 Set-2A bag has r red balls and b black balls. All balls are identical except for their colours. In a trial, a ball is randomly drawn from the bag, its colour is noted and the ball is placed back into the bag along with another ball of the same colour. Note that the number of balls in the bag will increase by one, after the trial. A sequence of four such trials is conducted. Which one of the following choices gives the probability of drawing a red ball in the fourth trial? |

i ➥ |

ii ➥ |

iii ➥ |

iv ➥ |

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Q19➡ | GATE 2021 Set-2 For two n-dimensional real vectors P and Q, the operation s(P, Q) is defined as follows: |

i ➥ 9 |

ii ➥ 10 |

iii ➥ 100 |

iv ➥ 11 |

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Q20➡ | GATE 2021 Set-2Consider the following directed graph: Which of the following is/are correct about the graph? |

i ➥ The graph does not have a topological order. |

ii ➥ The graph does not have a strongly connected component. |

iii ➥ A depth-first traversal staring at vertex S classifies three directed edges as back edges. |

iv ➥ For each pair of vertices u and v, there is a directed path from u to v. |

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Q21➡ | GATE 2021 Set-2 In a directed acyclic graph with a source vertex s, the quality-score of s directed path is defined to be the product of the weights of the edges on the path. Further, for a vertex v other than s, the quality-score of v is defined to be the maximum among the quality-scores of all the paths from s to v. The quality-score of s is assumed to be 1. The sum of the quality-scores of all the vertices in the graph shown above is _______. |

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Q22➡ | GATE 2021 Set-2Let S be a set consisting of 10 elements. The number of tuples of the form (A, B) such that A and B are subsets of S, and A ⊆ B is _______. |

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Q23➡ | GATE 2020Consider the functions I. e ^{-x}II. x ^{2}-sin xIII. √(x ^{3}+1)Which of the above functions is/are increasing everywhere in [0,1]? |

i ➥ I and III only |

ii ➥ II and III only |

iii ➥ III only |

iv ➥ II only |

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Q24➡ | GATE 2020Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ________. |

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Q25➡ | GATE 2020Let R be the set of all binary relations on the set {1,2,3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is ______. |

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Q26➡ | GATE 2020Let A and B be two n×n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements, I. rank(AB) = rank(A) rank(B) II. det(AB) = det(A) det(B) III. rank(A + B) ≤ rank(A) + rank(B) IV. det(A + B) ≤ det(A) + det(B) Which of the above statements are TRUE? |

i ➥ I and IV only |

ii ➥ III and IV only |

iii ➥ II and III only |

iv ➥ I and II only |

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Q27➡ | GATE 2020The number of permutations of the characters in LILAC so that no character appears in its original position, if the two L’s are indistinguishable, is ____. |

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Q28➡ | GATE 2020Which one of the following predicate formulae is NOT logically valid? Note that W is a predicate formula without any free occurrence of x. |

i ➥ ∀x(p(x) ∨ W) ≡ ∀x p(x) ∨ W |

ii ➥ ∃x(p(x) ∧ W) ≡ ∃x p(x) ∧ W |

iii ➥ ∃x(p(x) → W) ≡ ∀x p(x) → W |

iv ➥ ∀x(p(x) → W) ≡ ∀x p(x) → W |

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Q29➡ | GATE 2020For n>2, let a ∈ {0,1} ^{n} be a non-zero vector. Suppose that x is chosen uniformly at random from {0,1}^{n}. Then, the probability that is an odd number is ________. |

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Q30➡ | GATE 2020Graph G is obtained by adding vertex s to K _{3,4} and making s adjacent to every vertex of K_{3,4}. The minimum number of colours required to edge-colour G is ________. |

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Q31➡ | GATE 2019Let U = {1,2,…,n}. Let A = {(x,X)|x ∈ X, X ⊆ U}. Consider the following two statements on |A|. I. II. Which of the above statements is/are TRUE? |

i ➥ Both I and II |

ii ➥ Neither I nor II |

iii ➥ Only I |

iv ➥ Only II |

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Q32➡ | GATE 2019Let G be an undirected complete graph on n vertices, where n > 2. Then, the number of different Hamiltonian cycles in G is equal to |

i ➥ (n-1)! |

ii ➥ (n-1)! /2 |

iii ➥ n! |

iv ➥ 1 |

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Q33➡ | GATE 2019Let G be an arbitrary group. Consider the following relations on G: R _{1}: ∀a,b ∈ G, aR_{1}b if and only if ∃g ∈ G such that a = g^{-1}bgR _{2}: ∀a,b ∈ G, aR_{2}b if and only if a = b^{-1}Which of the above is/are equivalence relation/relations? |

i ➥ R_{1} only |

ii ➥ Neither R_{1} and R_{2} |

iii ➥ R_{2} only |

iv ➥ R_{1} and R_{2} |

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Q34➡ | GATE 2019Let X be a square matrix. Consider the following two statements on X. I. X is invertible. II. Determinant of X is non-zero. Which one of the following is TRUE? |

i ➥ II implies I; I does not imply II. |

ii ➥ I does not imply II; II does not imply I. |

iii ➥ I implies II; II does not imply I. |

iv ➥ I and II are equivalent statements. |

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Q35➡ | GATE 2019Consider Z = X – Y, where X, Y and Z are all in sign-magnitude form. X and Y are each represented in n bits. To avoid overflow, the representation of Z would require a minimum of: |

i ➥ n – 1 bits |

ii ➥ n + 1 bits |

iii ➥ n bits |

iv ➥ n + 2 bits |

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Q36➡ | GATE 2019 |

i ➥ 108/7 |

ii ➥ Limit does not exist |

iii ➥ 1 |

iv ➥ 53/12 |

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Q37➡ | GATE 2019Consider the first order predicate formula φ: ∀x[(∀z z|x ⇒ ((z = x) ∨ (z = 1))) ⇒ ∃w (w > x) ∧ (∀z z|w ⇒ ((w = z) ∨ (z = 1)))] Here ‘a|b’ denotes that ‘a divides b’, where a and b are integers. Consider the following sets: S1. {1, 2, 3, …, 100} S2. Set of all positive integers S3. Set of all integers Which of the above sets satisfy φ? |

i ➥ S2 and S3 |

ii ➥ S1 and S3 |

iii ➥ S1 and S2 |

iv ➥ S1, S2 and S3 |

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Q38➡ | GATE 2019Let G be any connected, weighted, undirected graph. I. G has a unique minimum spanning tree, if no two edges of G have the same weight. II. G has a unique minimum spanning tree, if, for every cut of G, there is a unique minimum-weight edge crossing the cut. Which of the above statements is/are TRUE? |

i ➥ Both I and II |

ii ➥ I only |

iii ➥ Neither I nor II |

iv ➥ II only |

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Q39➡ | GATE 2019Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial 3x ^{2} + 6xY + 3Y + 6 has only real roots is (rounded off to 1 decimal place) ________. |

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Q40➡ | GATE 2019Consider the following matrix: The absolute value of the product of Eigen values of R is ________. |

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Q41➡ | GATE 2019Consider the augmented grammar given below: S’ → S S → 〈L〉 | id L → L,S | S Let I _{0} = CLOSURE ({[S’ → ·S]}). The number of items in the set GOTO (I_{0} , 〈 ) is: ___________. |

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Q42➡ | GATE 2018Two people, P and Q, decide to independently roll two identical dice, each with 6 faces, numbered 1 to 6. The person with the lower number wins. In case of a tie, they roll the dice repeatedly until there is no tie. Define a trial as a throw of the dice by P and Q. Assume that all 6 numbers on each dice are equi-probable and that all trials are independent. The probability (rounded to 3 decimal places) that one of them wins on the third trial is_____. |

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Q43➡ | GATE 2018 |

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Q44➡ | GATE 2018 |

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Q45➡ | GATE 2018The chromatic number of the following graph is ._ |

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Q46➡ | GATE 2018Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is_________. |

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Q47➡ | GATE 2018Assume that multiplying a matrix G1 of dimension p×q with another matrix G _{2 }of dimension q×r requires pqr scalar multiplications. Computing the product of n matrices G_{1}G_{2}G_{3}…G_{n} can be done by parenthesizing in different ways. Define G_{i}G_{i}+1 as an explicitly computed pair for a given parenthesization if they are directly multiplied. For example, in the matrix multiplication chain G_{1}G_{2}G_{3}G_{4}G_{5}G6 using parenthesization(G_{1}(G_{2}G_{3}))(G_{4}(G_{5}G_{6})), G_{2}G_{3} and G_{5}G_{6} are the only explicitly computed pairs.Consider a matrix multiplication chain F _{1}F_{2}F_{3}F_{4}F_{5}, where matrices F_{1}, F_{2}, F_{3}, F_{4} and F_{5 }are of dimensions 2×25, 25×3, 3×16, 16×1 and 1×1000, respectively. In the parenthesization of F_{1}F_{2}F_{3}F_{4}F_{5} that minimizes the total number of scalar multiplications, the explicitly computed pairs is/ are |

i ➥ F_{1}F_{2} and F_{3}F_{4} only |

ii ➥ F_{2}F_{3 }only |

iii ➥ F_{3}F_{4} only |

iv ➥ F_{1}F_{2} and F_{4}F_{5 }only |

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Q48➡ | GATE 2018Let G be a simple un-directed graph. Let T _{D} be a depth first search tree of G. Let T_{B} be a breadth first search tree of G. Consider the following statements.(I) No edge of G is a cross edge with respect to T _{D} (A cross edge in G is between two nodes neither of which is an ancestor of the other in T_{D}(II) For every edge (u,v) of G, if u is at depth i and v is at depth j in T _{B} , then ∣i−j∣ = 1.Which of the statements above must necessarily be true? |

i ➥ I only |

ii ➥ II only |

iii ➥ Both I and II |

iv ➥ Neither I nor II |

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Q49➡ | GATE 2018 Consider the first-order logic sentence φ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s,t,u,v,w,x,y)where ψ(s,t,u,v,w,x,y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements. Which one of the following statements is necessarily true? |

i ➥ There exists at least one model of φ with universe of size less than or equal to 3. |

ii ➥ There exists no model of φ with universe of size less than or equal to 3 |

iii ➥ There exists no model of φ with universe of size greater than 7. |

iv ➥ Every model of φ has a universe of size equal to 7. |

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Q50➡ | GATE 2018Let N be the set of natural numbers. Consider the following sets, P: Set of Rational numbers (positive and negative) Q: Set of functions from {0, 1} to N R: Set of functions from N to {0, 1} S: Set of finite subsets of N Which of the above sets are countable? |

i ➥ Q and S only |

ii ➥ P and S only |

iii ➥ P and R only |

iv ➥ P, Q and S only |

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Q51➡ | GATE 2018Consider a matrix P whose only eigenvectors are the multiples of Consider the following statements. (I) does not have an inverseP(II) has a repeated eigenvalueP(III) cannot be diagonalizedPWhich one of the following options is correct? |

i ➥ Only I and II are necessarily true |

ii ➥ Only II and III are necessarily true |

iii ➥ Only I and III are necessarily true |

iv ➥ Only II is necessarily true |

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Q52➡ | GATE 2018Let G be a graph with 100! vertices, with each vertex labeled by a distinct permutation of the numbers 1, 2, …, 100. There is an edge between vertices u and v if and only if the label of u can be obtained by swapping two adjacent numbers in the label of v. Let y denote the degree of a vertex in G, and z denote the number of connected components in G. Then, y + 10z =______. |

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Q53➡ | GATE 2018Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (H), medium (M) and low (L). Let P(H _{G}) denote the probability that Guwahati has high temperature. Similarly, P(M_{G}) and P(L_{G}) denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use P(H_{D}), P(M_{D}) and P(L_{D}) for Delhi.The following table gives the conditional probabilities for Delhi’s temperature given Guwahati’s temperature. Consider the first row in the table above. The first entry denotes that if Guwahati has high temperature (H _{G}) then the probability of Delhi also having a high temperature (H_{D}) is 0.40; i.e., P(H_{D} ∣ H_{G}) = 0.40. Similarly, the next two entries are P(M_{D} ∣ H_{G}) = 0.48 and P(L_{D} ∣ H_{G}) = 0.12. Similarly for the other rows.If it is known that P(H _{G}) = 0.2, P(M_{G}) = 0.5, and P(L_{G}) = 0.3, then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is_____. |

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Q54➡ | GATE 2017 Set-1Let c _{1}, c_{n} be scalars not all zero. Such that where a_{i} are column vectors in R^{n}. Consider the set of linear equations.Ax = B. where A = [a _{1}…….a_{n}] and Set of equations has |

i ➥ no solution |

ii ➥ a unique solution at x = J_{n} where J_{n} denotes a n-dimensional vector of all 1 |

iii ➥ finitely many solutions |

iv ➥ infinitely many solutions |

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Q55➡ | GATE 2017 Set-1Consider the first-order logic sentence F: ∀x(∃yR(x,y)). Assuming non-empty logical domains, which of the sentences below are implied by F? I. ∃y(∃xR(x,y)) II. ∃y(∀xR(x,y)) III. ∀y(∃xR(x,y)) IV. ¬∃x(∀y¬R(x,y)) |

i ➥ IV only |

ii ➥ II and III only |

iii ➥ I and IV only |

iv ➥ II only |

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Q56➡ | GATE 2017 Set-1The statement (¬p) ⇒ (¬q) is logically equivalent to which of the statements below? I. p ⇒ q II. q ⇒ p III. (¬q) ∨ p IV. (¬p) ∨ q |

i ➥ I and IV only |

ii ➥ I only |

iii ➥ II and III only |

iv ➥ II only |

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Q57➡ | GATE 2017 Set-1Let X be a Gaussian random variable with mean 0 and variance σ2. Let Y = max(X, 0) where max(a,b) is the maximum of a and b. The median of Y is ___________. |

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Q58➡ | GATE 2017 Set-1Let A be m×n real valued square symmetric matrix of rank 2 with expression given below. Consider the following statements (i) One eigenvalue must be in [-5, 5]. (ii) The eigenvalue with the largest magnitude must be strictly greater than 5. Which of the above statements about engenvalues of A is/are necessarily CORRECT? |

i ➥ (I) only |

ii ➥ Neither (I) nor (II) |

iii ➥ (II) only |

iv ➥ Both (I) and (II) |

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Q59➡ | GATE 2017 Set-1Let u and v be two vectors in R ^{2} whose Euclidean norms satisfy ||u||=2||v||. What is the value of α such that w = u + αv bisects the angle between u and v? |

i ➥ -1/2 |

ii ➥ 2 |

iii ➥ 1/2 |

iv ➥ 1 |

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Q60➡ | GATE 2017 Set-1Let p, q and r be prepositions and the expression (p → q) → r be a contradiction. Then, the expression (r → p) → q is |

i ➥ a contradiction |

ii ➥ always TRUE when q is TRUE |

iii ➥ a tautology |

iv ➥ always TRUE when p is FALSE |

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Q61➡ | GATE 2017 Set-1The value of |

i ➥ is 1 |

ii ➥ does not exist |

iii ➥ is 0 |

iv ➥ is -1 |

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Q62➡ | GATE 2017 Set-1The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is __________. |

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Q63➡ | GATE 2017 Set-2Let p, q, r denote the statements “It is raining”, “It is cold”, and “It is pleasant”, respectively. Then the statement “It is not raining and it is pleasant, and it is not pleasant only if it is raining and it is cold” is represented by |

i ➥ (¬p ∧ r) ∨ (r → (p ∧ q)) |

ii ➥ (¬p ∧ r) ∧ ((p ∧ q) → ¬r) |

iii ➥ (¬p ∧ r) ∨ ((p ∧ q) → ¬r) |

iv ➥ (¬p ∧ r) ∧ (¬r → (p ∧ q)) |

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Q64➡ | GATE 2017 Set-2If f(x) = R sin(πx/2) + S, f'(1/2) = √2 and , then the constants R and S are, respectively. |

i ➥ |

ii ➥ |

iii ➥ |

iv ➥ |

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Q65➡ | GATE 2017 Set-2G is an undirected graph with n vertices and 25 edges such that each vertex of G has degree at least 3. Then the maximum possible value of n is __________. |

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Q66➡ | GATE 2017 Set-2Let P= and Q= be two matrices. Then the rank of P+Q is __________. |

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Q67➡ | GATE 2017 Set-2Consider the set X = {a,b,c,d e} under the partial ordering R = {(a,a),(a,b),(a,c),(a,d),(a,e),(b,b),(b,c),(b,e),(c,c),(c,e),(d,d),(d,e),(e,e)}. The Hasse diagram of the partial order (X,R) is shown below: The minimum number of ordered pairs that need to be added to R to make (X,R) a lattice is ___________. |

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Q68➡ | GATE 2017 Set-2For any discrete random variable X, with probability mass function P(X=j)=pj, pj≥0, j∈{0, …, N} and , define the polynomial function . For a certain discrete random variable Y, there exists a scalar β∈[0,1] such that g _{y}(Z)=(1-β+βz)^{N}. The expectation of Y is |

i ➥ Nβ(1 – β) |

ii ➥ Nβ |

iii ➥ N(1 – β) |

iv ➥ Not expressible in terms of N and β alone |

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Q69➡ | GATE 2017 Set-2P and Q are considering to apply for job. The probability that p applies for job is 1/4. The probability that P applies for job given that Q applies for the job 1/2 and The probability that Q applies for job given that P applies for the job 1/3.The probability that P does not apply for job given that Q does not apply for the job |

i ➥ 4/5 |

ii ➥ 5/6 |

iii ➥ 7/8 |

iv ➥ 11/12 |

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Q70➡ | GATE 2017 Set-2If a random variable X has a Poisson distribution with mean 5, then the expectation E[(X + 2) ^{2}] equals ________. |

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Q71➡ | GATE 2017 Set-2If the ordinary generating function of a sequence is , then a _{3} – a_{0} is equal to __________. |

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Q72➡ | GATE 2017 Set-2If the characteristic polynomial of a 3 × 3 matrix M over ℝ (the set of real numbers) is λ ^{3} – 4λ^{2} + aλ + 30, a ∈ ℝ, and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is .__ |

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Q73➡ | GATE 2016 Set-1Let a _{n} be the number of n-bit strings that do NOT contain two consecutive 1s. Which one of the following is the recurrence relation for a_{n}? |

i ➥ a_{n} = 2a_{n-1} + 2a_{n-2} |

ii ➥ a_{n} = 2a_{n-1} + a_{n-2} |

iii ➥ a_{n} = a_{n-1} + a_{n-2} |

iv ➥ a_{n} = a_{n-1} + 2a_{n-2} |

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Q74➡ | GATE 2016 Set-1 |

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Q75➡ | GATE 2016 Set-1A probability density function on the interval [a,1] is given by 1/x ^{2} and outside this interval the value of the function is zero. The value of a is _________. |

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Q76➡ | GATE 2016 Set-1Two eigenvalues of a 3 × 3 real matrix P are (2 + √-1) and 3. The determinant of P is __________. |

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Q77➡ | GATE 2016 Set-1The coefﬁcient of x ^{12} in (x^{3} + x^{4} + x^{5} + x^{6} + …)^{3} is ____________. |

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Q78➡ | GATE 2016 Set-1 Consider the recurrence relation a _{1} = 8, a_{n} = 6n^{2} + 2n + a_{n-1}. Let a_{99} = K × 10^{4}. The value of K is ._ |

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Q79➡ | GATE 2016 Set-1A function f:N ^{+} → N^{+}, deﬁned on the set of positive integers N^{+}, satisﬁes the following properties:f(n) = f(n/2) if n is evenf(n) = f(n+5) if n is oddLet R = {i|∃j : f(j)=i} be the set of distinct values that f takes. The maximum possible size of R is ________. |

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Q80➡ | GATE 2016 Set-1Consider the following experiment. • Step1: Flip a fair coin twice.• Step2: If the outcomes are (TAILS, HEADS) then output Y and stop.• Step3: If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output N and stop.• Step4: If the outcomes are (TAILS, TAILS), then go to Step 1.The probability that the output of the experiment is Y is (up to two decimal places) __________. |

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Q81➡ | GATE 2016 Set-2Consider the following expressions: (i) false (ii) Q (iii) true (iv) P ∨ Q (v) ¬Q ∨ P The number of expressions given above that are logically implied by P ∧ (P ⇒ Q) is _________. |

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Q82➡ | GATE 2016 Set-2Let f(x) be a polynomial and g(x) = f'(x) be its derivative. If the degree of (f(x) + f(-x)) is 10, then the degree of (g(x) – g(-x)) is _______. |

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Q83➡ | GATE 2016 Set-2The minimum number of colours that is sufﬁcient to vertex-colour any planar graph is _________. |

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Q84➡ | GATE 2016 Set-2Consider the systems, each consisting of m linear equations in n variables. I. If m < n, then all such systems have a solution II. If m > n, then none of these systems has a solution III. If m = n, then there exists a system which has a solution Which one of the following is CORRECT? |

i ➥ I, II and III are true |

ii ➥ Only II and III are true |

iii ➥ Only III is true |

iv ➥ None of them is true |

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Q85➡ | GATE 2016 Set-2Suppose that a shop has an equal number of LED bulbs of two different types. The probability of an LED bulb lasting more than 100 hours given that it is of Type 1 is 0.7, and given that it is of Type 2 is 0.4. The probability that an LED bulb chosen uniformly at random lasts more than 100 hours is __________. |

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Q86➡ | GATE 2016 Set-2Suppose that the eigenvalues of matrix A are 1, 2, 4. The determinant of (A ^{-1})^{T} is _______. |

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Q87➡ | GATE 2016 Set-2A binary relation R on ℕ × ℕ is deﬁned as follows: (a,b)R(c,d) if a≤c or b≤d. Consider the following propositions: • P: R is reﬂexive • Q: R is transitive Which one of the following statements is TRUE? |

i ➥ Both P and Q are true. |

ii ➥ P is true and Q is false. |

iii ➥ P is false and Q is true. |

iv ➥ Both P and Q are false. |

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Q88➡ | GATE 2016 Set-2 Which one of the following well-formed formulae in predicate calculus is NOT valid? |

i ➥ (∀x p(x) ⇒ ∀x q(x)) ⇒ (∃x ¬p(x) ∨ ∀x q(x)) |

ii ➥ (∃x p(x) ∨ ∃x q(x)) ⇒ ∃x (p(x) ∨ q(x)) |

iii ➥ ∃x (p(x) ∧ q(x)) ⇒ (∃x p(x) ∧ ∃x q(x)) |

iv ➥ ∀x (p(x) ∨ q(x)) ⇒ (∀x p(x) ∨ ∀x q(x)) |

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Q89➡ | GATE 2016 Set-2Consider a set U of 23 different compounds in a Chemistry lab. There is a subset S of U of 9 compounds, each of which reacts with exactly 3 compounds of U. Consider the following statements: • I. Each compound in U\S reacts with an odd number of compounds. • II. At least one compound in U\S reacts with an odd number of compounds. • III. Each compound in U\S reacts with an even number of compounds. Which one of the above statements is ALWAYS TRUE? |

i ➥ Only I |

ii ➥ Only II |

iii ➥ Only III |

iv ➥ None |

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Q90➡ | GATE 2015 Set-1If g(x) = 1 – x and h(x)= x/x-1 , then g(h(x)) / h(g(x)) is: |

i ➥ h(x)/g(x) |

ii ➥ -1/x |

iii ➥ g(x)/h(x) |

iv ➥ x/(1-x)2 |

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Q91➡ | GATE 2015 Set-1. |

i ➥ ∞ |

ii ➥ 0 |

iii ➥ 1 |

iv ➥ not defined |

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Q92➡ | GATE 2015 Set-1 Which one of the following is NOT equivalent to p↔q? |

i ➥ ( ⌉p∨q)∧(p∨ ⌉q) |

ii ➥ ( ⌉p∨q)∧(p→q) |

iii ➥ ( ⌉p ∧ q) ∨ (p ∧ ⌉q) |

iv ➥ ( ⌉p ∧ ⌉q) ∨ (p ∧ q) |

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Q93➡ | GATE 2015 Set-1 |

i ➥ I and III only |

ii ➥ II and III only |

iii ➥ I,II and III only |

iv ➥ I,II and IV only |

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Q94➡ | GATE 2015 Set-1In the LU decomposition of the matrix ,if the diagonal elements of U are both 1, then the lower diagonal entry l of _{22}L is |

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Q95➡ | GATE 2015 Set-1 |

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Q96➡ | GATE 2015 Set-1Suppose L = {p, q, r, s, t} is a lattice represented by the following Hasse diagram: For any x, y ∈ L, not necessarily distinct, x ∨ y and x ∧ y are join and meet of x, y respectively. Let L ^{3 }= {(x,y,z): x, y, z ∈ L} be the set of all ordered triplets of the elements of L. Let p_{r} be the probability that an element (x,y,z) ∈ L^{3 }chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Then |

i ➥ p_{r} = 0 |

ii ➥ p_{r} = 1 |

iii ➥ 0 < p_{r} ≤ 1/5 |

iv ➥ 1/5 < p_{r }< 1 |

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Q97➡ | GATE 2015 Set-1Consider the operations f(X, Y, Z) = X’YZ + XY’ + Y’Z’ and g(X′, Y, Z) = X′YZ + X′YZ′ + XY Which one of the following is correct? |

i ➥ Both {f} and {g} are functionally complete |

ii ➥ Only {f} is functionally complete |

iii ➥ Only {g} is functionally complete |

iv ➥ Neither {f} nor {g} is functionally complete |

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Q98➡ | GATE 2015 Set-1 Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is __________. |

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Q99➡ | GATE 2015 Set-1 Let a _{n} represent the number of bit strings of length n containing two consecutive 1s. What is the recurrence relation for a_{n}? |

i ➥ a_{n-2} + a_{n-1} + 2^{n-2} |

ii ➥ a_{n-2} + 2a_{n-1} + 2^{n-2} |

iii ➥ 2a_{n-2} + a_{n-1} + 2^{n-2} |

iv ➥ 2a_{n-2} + 2a_{n-1} + 2^{n-2} |

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Q100➡ | GATE 2015 Set-1 Let G = (V, E) be a simple undirected graph, and s be a particular vertex in it called the source. For x ∈ V, let d(x)denote the shortest distance in G from s to x. A breadth first search (BFS) is performed starting at s. Let T be the resultant BFS tree. If (u, v) is an edge of G that is not in T, then which one of the following CANNOT be the value of d(u) – d(v) ? |

i ➥ -1 |

ii ➥ 0 |

iii ➥ 1 |

iv ➥ 2 |

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Q101➡ | GATE 2015 Set-1 |

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Q102➡ | GATE 2015 Set-1Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are –1 and 7. What are the values of a and b? |

i ➥ a=6, b=4 |

ii ➥ a=4, b=6 |

iii ➥ a=3, b=5 |

iv ➥ a=5, b=3 |

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Q103➡ | GATE 2015 Set-2Consider the following statements: S _{1}: If a candidate is known to be corrupt, then he will not be elected.S _{2}: If a candidate is kind, he will be elected.Which one of the following statements follows from S _{1} and S_{2 }as per sound inference rules of logic? |

i ➥ If a person is known to corrupt, he is kind |

ii ➥ If a person is not known to be corrupt, he is not kind |

iii ➥ If a person is kind, he is not known to be corrupt |

iv ➥ If a person is not kind, he is not known to be corrupt |

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Q104➡ | GATE 2015 Set-2The cardinality of the power set of {0, 1, 2, … 10} is __________. |

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Q105➡ | GATE 2015 Set-2Let R be the relation on the set of positive integers such that aRb if and only if a and b are distinct and have a common divisor other than 1. Which one of the following statements about R is true? |

i ➥ R is symmetric and reflexive but not transitive |

ii ➥ R is reflexive but not symmetric and not transitive |

iii ➥ R is transitive but not reflexive and not symmetric |

iv ➥ R is symmetric but not reflexive and not transitive |

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Q106➡ | GATE 2015 Set-2The number of divisors of 2100 is ______. |

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Q107➡ | GATE 2015 Set-2The larger of the two eigenvalues of the matrix is ______. |

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Q108➡ | GATE 2015 Set-2The secant method is used to find the root of an equation f(x) = 0. It is started from two distinct estimates x and _{a}x for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if _{b}f(x is very small and then _{b})x is the solution. The procedure is given below. Observe that there is an expression which is missing and is marked by? Which is the suitable expression that is to be put in place of? So that it follows all steps of the secant method?_{b}Secant |

i ➥ x_{b} – (f_{b}–f(x_{a})) f_{b} /(x_{b}–x_{a}) |

ii ➥ x_{a} – (f_{a}–f(x_{a})) f_{a} /(x_{b}–x_{a}) |

iii ➥ x_{b} – (x_{b}–x_{a}) f_{b} /(f_{b}–f(x_{a})) |

iv ➥ x_{a} – (x_{b}–x_{a}) f_{a }/(f_{b}–f(x_{a})) |

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Q109➡ | GATE 2015 Set-2Let f(x) = x ^{-(1/3)} and A denote the area of the region bounded bu f(x) and the X-axis, when x varies from -1 to 1. Which of the following statements is/are TRUE? I) f is continuous in [-1,1] II) f is not bounded in [-1,1] III) A is nonzero and finite |

i ➥ II only |

ii ➥ III only |

iii ➥ II and III only |

iv ➥ I, II and III |

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Q110➡ | GATE 2015 Set-2Perform the following operations on the matrix (i) add the third row to the second row (ii) Subtract the third column from the first column The determinant of the resultant matrix is ____________. |

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Q111➡ | GATE 2015 Set-2The number of onto function (surjective function) from set X = {1, 2, 3, 4} to set Y = {a, b, c} is _______. |

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Q112➡ | GATE 2015 Set-2Let X and Y denote the sets containing 2 and 20 distinct objects respectively and F denote the set of all possible functions defined from X to Y. Let f be randomly chosen from F. The probability of f being one-to-one is ________. |

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Q113➡ | GATE 2015 Set-2 A graph is self-complementary if it is isomorphic to its complement for all self-complementary graphs on n vertices, n is |

i ➥ A multiple of 4 |

ii ➥ Even |

iii ➥ Odd |

iv ➥ Congruent to 0 mod 4, or, 1 mod 4 |

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Q114➡ | GATE 2015 Set-2In a connected graph, a bridge is an edge whose removal disconnects a graph. Which one of the following statements is true? |

i ➥ A tree has no bridges |

ii ➥ A bridge cannot be part of a simple cycle |

iii ➥ Every edge of a clique with size 3 is a bridge (A clique is any complete sub graph of a graph) |

iv ➥ A graph with bridges cannot have a cycle |

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Q115➡ | GATE 2015 Set-2Which one of the following well formed formulae is a tautology? |

i ➥ ∀x ∃y R(x,y) ↔ ∃y ∀x R(x,y) |

ii ➥ (∀x [∃y R(x,y) → S(x,y)]) → ∀x∃y S(x,y) |

iii ➥ [∀x ∃y (P(x,y) → R(x,y)] ↔ [∀x ∃y (¬ P(x,y)∨R(x,y)] |

iv ➥ ∀x ∀y P(x,y) → ∀x ∀y P(y,x) |

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Q116➡ | GATE 2015 Set-3Suppose U is the power set of the set S = {1, 2, 3, 4, 5, 6}. For any T ∈ U, let |T| denote the number of element in T and T’ denote the complement of T. For any T, R ∈ U, let TR be the set of all elements in T which are not in R. Which one of the following is true? |

i ➥ ∀X ∈ U (|X| = |X’|) |

ii ➥ ∃X ∈ U ∃Y ∈ U (|X| = 5, |Y| = 5 and X ∩ Y = ∅) |

iii ➥ ∀X ∈ U ∀Y ∈ U (|X| = 2, |Y| = 3 and X \ Y = ∅) |

iv ➥ ∀X ∈ U ∀Y ∈ U (X \ Y = Y’ \ X’) |

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Q117➡ | GATE 2015 Set-3In the given matrix , one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are |

i ➥ |

ii ➥ |

iii ➥ |

iv ➥ |

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Q118➡ | GATE 2015 Set-3The value of is |

i ➥ 0 |

ii ➥ 1/2 |

iii ➥ 1 |

iv ➥ ∞ |

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Q119➡ | GATE 2015 Set-3 The number of 4 digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set {1, 2, 3} is _________. |

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Q120➡ | GATE 2015 Set-3In a room there are only two types of people, namely Type 1 and Type 2. Type 1 people always tell the truth and Type 2 people always lie. You give a fair coin to a person in that room, without knowing which type he is from and tell him to toss it and hide the result from you till you ask for it. Upon asking, the person replies the following: “The result of the toss is head if and only if I am telling the truth.”Which of the following options is correct? |

i ➥ The result is head |

ii ➥ The result is tail |

iii ➥ If the person is of Type 2, then the result is tail |

iv ➥ If the person is of Type 1, then the result is tail |

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Q121➡ | GATE 2015 Set-3The velocity v (in kilometer/minute) of a motorbike which starts from rest, is given at fixed intervals of time t(in minutes) as follows: The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson’s 1/3 ^{rd} rule is ______. |

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Q122➡ | GATE 2015 Set-3 If the following system has non-trivial solution, px + qy + rz = 0 qx + ry + pz = 0 rx + py + qz = 0 then which one of the following options is True? |

i ➥ p – q + r = 0 or p = q = –r |

ii ➥ p + q – r = 0 or p = –q = r |

iii ➥ p + q + r = 0 or p = q = r |

iv ➥ p – q + r = 0 or p = –q = –r |

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Q123➡ | GATE 2015 Set-3If for non-zero x,A: B: C: D: |

i ➥ A |

ii ➥ B |

iii ➥ C |

iv ➥ D |

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Q124➡ | GATE 2015 Set-3Let R be a relation on the set of ordered pairs of positive integers such that ((p,q),(r,s)) ∈ R if and only if p – s = q – r. Which one of the following is true about R? |

i ➥ Both reflexive and symmetric |

ii ➥ Reflexive but not symmetric |

iii ➥ Not reflexive but symmetric |

iv ➥ Neither reflexive nor symmetric |

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Q125➡ | GATE 2015 Set-3Suppose X _{i} for i = 1, 2, 3 are independent and identically distributed random variables whose probability mass functions are Pr[X_{i} = 0] = Pr[X_{i} = 1]=1/2 for i = 1, 2, 3. Define another random variable Y = X_{1}X_{2} ⊕ X_{3}, where ⊕ denotes XOR. Then Pr[Y = 0|X _{3} = 0] =_______, |

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Q126➡ | GATE 2014 Set-1Consider the statement “Not all that glitters is gold” Predicate glitters(x) is true if x glitters and predicate gold(x) is true if x is gold. Which one of the following logical formulae represents the above statement? |

i ➥ ∀x: glitters(x) ⇒ ¬gold(x) |

ii ➥ ∀x: gold(x) ⇒ glitters() |

iii ➥ ∃x: gold(x) ∧ ¬glitters(x) |

iv ➥ ∃x: glitters(x) ∧ ¬gold(x) |

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Q127➡ | GATE 2014 Set-1Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is____________. |

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Q128➡ | GATE 2014 Set-1Let G = (V,E) be a directed graph where V is the set of vertices and E the set of edges. Then which one of the following graphs has the same strongly connected components as G? |

i ➥ G1=(V,E1) where E1={(u,v)|(u,v)∉E} |

ii ➥ G2=(V,E2 )where E2={(u,v)│(u,v)∈E} |

iii ➥ G3=(V,E3) where E3={(u,v)|there is a path of length≤2 from u to v in E} |

iv ➥ G4=(V4,E) where V4 is the set of vertices in G which are not isolated |

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Q129➡ | GATE 2014 Set-1Consider the following system of equations: 3x + 2y = 14x + 7z = 1 x + y + z = 3 x – 2y + 7z = 0 The number of solutions for this system is ._ |

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Q130➡ | GATE 2014 Set-1 The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is ______ .__ |

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Q131➡ | GATE 2014 Set-1Let the function where and f'(θ) denote the derivative of f with respect to θ. Which of the following is/are TRUE? (I) There exists θ ∈ such that f'(θ)=0. (II) There exists θ ∈ such that f'(θ)≠0. |

i ➥ I only |

ii ➥ II only |

iii ➥ Both I and II |

iv ➥ Neither I nor II |

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Q132➡ | GATE 2014 Set-1The function f(x) = x sin x satisfies the following equation: fuu(x) + f(x) +t cosx = 0. The value of t is . |

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Q133➡ | GATE 2014 Set-1A function f(x) is continuous in the interval [0,2]. It is known that f(0) = f(2) = −1 and f(1) = 1. Which one of the following statements must be true? |

i ➥ There exists a y in the interval (0,1) such that f(y) = f(y + 1) |

ii ➥ For every y in the interval (0,1), f(y) = f(2− y) |

iii ➥ The maximum value of the function in the interval (0,2) is 1 |

iv ➥ There exists a y in the interval (0,1) such that f(y) = −f(2− y) |

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Q134➡ | GATE 2014 Set-1Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is X/1296.The value of X is . |