Q1➡|GATE 2021 Set-1 Let p and q be two propositions. Consider the following two formulae in propositional logic. Which one of the following choices is correct?
Q4➡| GATE 2021 Set-1 Let 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 G2 on the same set of vertices with adjacency matrix N, where Which one of the following statements is true?
Q5➡| GATE 2021 Set-1 Consider 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?
Q8➡|GATE 2021 Set-1 An 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.
Q9➡|GATE 2021 Set-1 A 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 ________.
Q11➡| GATE 2021 Set-2 Let G be a connected undirected weighted graph. Consider the following two statements. S1: There exists a minimum weighted edge in G which is present in every minimum spanning tree of G. S2:If every edge in G has distinct weight, then G has a unique minimum spanning tree. Which of the following options is correct?
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 _____.
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 ______.
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.
Q18➡| GATE 2021 Set-2 A 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?
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 _______.
Q22➡|GATE 2021 Set-2 Let 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 _______.
Q25➡| GATE 2020 Let 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 ______.
Q26➡| GATE 2020 Let 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?
Q27➡| GATE 2020 The 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 ____.
Q28➡|GATE 2020 Which one of the following predicate formulae is NOT logically valid? Note that W is a predicate formula without any free occurrence of x.
Q29➡|GATE 2020 For 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 ________.
Q30➡|GATE 2020 Graph G is obtained by adding vertex s to K3,4 and making s adjacent to every vertex of K3,4. The minimum number of colours required to edge-colour G is ________.
Q33➡| GATE 2019 Let G be an arbitrary group. Consider the following relations on G: R1: ∀a,b ∈ G, aR1b if and only if ∃g ∈ G such that a = g-1bg R2: ∀a,b ∈ G, aR2b if and only if a = b-1 Which of the above is/are equivalence relation/relations?
Q34➡|GATE 2019 Let 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?
Q35➡|GATE 2019 Consider 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:
Q37➡| GATE 2019 Consider 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?
Q38➡| GATE 2019 Let 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.
Q39➡|GATE 2019 Suppose Y is distributed uniformly in the open interval (1,6). The probability that the polynomial 3x2 + 6xY + 3Y + 6 has only real roots is (rounded off to 1 decimal place) ________.
Q41➡|GATE 2019 Consider the augmented grammar given below: S’ → S S → 〈L〉 | id L → L,S | S Let I0 = CLOSURE ({[S’ → ·S]}). The number of items in the set GOTO (I0 , 〈 ) is: ___________.
Q42➡| GATE 2018 Two 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_____.
Q47➡| GATE 2018 Assume that multiplying a matrix G1 of dimension p×q with another matrix G2 of dimension q×r requires pqr scalar multiplications. Computing the product of n matrices G1G2G3…Gn can be done by parenthesizing in different ways. Define GiGi+1 as an explicitly computed pair for a given parenthesization if they are directly multiplied. For example, in the matrix multiplication chain G1G2G3G4G5G6 using parenthesization(G1(G2G3))(G4(G5G6)), G2G3 and G5G6 are the only explicitly computed pairs.
Consider a matrix multiplication chain F1F2F3F4F5, where matrices F1, F2, F3, F4 and F5 are of dimensions 2×25, 25×3, 3×16, 16×1 and 1×1000, respectively. In the parenthesization of F1F2F3F4F5 that minimizes the total number of scalar multiplications, the explicitly computed pairs is/ are
Q48➡| GATE 2018 Let G be a simple un-directed graph. Let TD be a depth first search tree of G. Let TB be a breadth first search tree of G. Consider the following statements. (I) No edge of G is a cross edge with respect to TD (A cross edge in G is between two nodes neither of which is an ancestor of the other in TD (II) For every edge (u,v) of G, if u is at depth i and v is at depth j in TB , then ∣i−j∣ = 1. Which of the statements above must necessarily be true?
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.
Q50➡| GATE 2018 Let 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
Q51➡| GATE 2018 Consider a matrix P whose only eigenvectors are the multiples of Consider the following statements. (I) P does not have an inverse (II) P has a repeated eigenvalue (III) P cannot be diagonalized Which one of the following options is correct?
Q52➡|GATE 2018 Let 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 =______.
Q53➡|GATE 2018 Consider Guwahati (G) and Delhi (D) whose temperatures can be classified as high (H), medium (M) and low (L). Let P(HG) denote the probability that Guwahati has high temperature. Similarly, P(MG) and P(LG) denotes the probability of Guwahati having medium and low temperatures respectively. Similarly, we use P(HD), P(MD) and P(LD) 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 (HG) then the probability of Delhi also having a high temperature (HD) is 0.40; i.e., P(HD ∣ HG) = 0.40. Similarly, the next two entries are P(MD ∣ HG) = 0.48 and P(LD ∣ HG) = 0.12. Similarly for the other rows. If it is known that P(HG) = 0.2, P(MG) = 0.5, and P(LG) = 0.3, then the probability (correct to two decimal places) that Guwahati has high temperature given that Delhi has high temperature is_____.
Q54➡| GATE 2017 Set-1 Let c1, cn be scalars not all zero. Such that where ai are column vectors in Rn. Consider the set of linear equations. Ax = B. where A = [a1…….an] and Set of equations has
i ➥ no solution
ii ➥ a unique solution at x = Jn where Jn denotes a n-dimensional vector of all 1
Q55➡| GATE 2017 Set-1 Consider 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))
Q56➡| GATE 2017 Set-1 The statement (¬p) ⇒ (¬q) is logically equivalent to which of the statements below? I. p ⇒ q II. q ⇒ p III. (¬q) ∨ p IV. (¬p) ∨ q
Q57➡| GATE 2017 Set-1 Let 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 ___________.
Q58➡|GATE 2017 Set-1 Let 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?
Q59➡| GATE 2017 Set-1 Let u and v be two vectors in R2 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?
Q63➡|GATE 2017 Set-2 Let 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
Q65➡| GATE 2017 Set-2 G 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 __________.
Q67➡| GATE 2017 Set-2 Consider 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 ___________.
Q68➡|GATE 2017 Set-2 For 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 gy(Z)=(1-β+βz)N. The expectation of Y is
Q69➡| GATE 2017 Set-2 P 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
Q72➡|GATE 2017 Set-2 If the characteristic polynomial of a 3 × 3 matrix M over ℝ (the set of real numbers) is λ3 – 4λ2 + aλ + 30, a ∈ ℝ, and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is __.
Q73➡| GATE 2016 Set-1 Let an be the number of n-bit strings that do NOT contain two consecutive 1s. Which one of the following is the recurrence relation for an?
Q75➡| GATE 2016 Set-1 A probability density function on the interval [a,1] is given by 1/x2 and outside this interval the value of the function is zero. The value of a is _________.
Q79➡| GATE 2016 Set-1 A function f:N+ → N+, defined on the set of positive integers N+, satisfies the following properties: f(n) = f(n/2) if n is even f(n) = f(n+5) if n is odd Let R = {i|∃j : f(j)=i} be the set of distinct values that f takes. The maximum possible size of R is ________.
Q80➡| GATE 2016 Set-1 Consider 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) __________.
Q81➡| GATE 2016 Set-2 Consider 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 _________.
Q82➡| GATE 2016 Set-2 Let 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 _______.
Q84➡| GATE 2016 Set-2 Consider 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?
Q85➡| GATE 2016 Set-2 Suppose 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 __________.
Q87➡| GATE 2016 Set-2 A 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?
Q89➡| GATE 2016 Set-2 Consider 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?
Q96➡| GATE 2015 Set-1 Suppose 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 L3 = {(x,y,z): x, y, z ∈ L} be the set of all ordered triplets of the elements of L. Let pr be the probability that an element (x,y,z) ∈ L3 chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Then
Q97➡| GATE 2015 Set-1 Consider 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?
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 __________.
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) ?
Q102➡|GATE 2015 Set-1 Consider 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?
Q103➡| GATE 2015 Set-2 Consider the following statements: S1: If a candidate is known to be corrupt, then he will not be elected. S2: If a candidate is kind, he will be elected.
Which one of the following statements follows from S1 and S2 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
Q105➡| GATE 2015 Set-2 Let 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
Q108➡|GATE 2015 Set-2 The secant method is used to find the root of an equation f(x) = 0. It is started from two distinct estimates xa and xb for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if f(xb) is very small and then xb 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? Secant
Q109➡| GATE 2015 Set-2 Let 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
Q110➡|GATE 2015 Set-2 Perform 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 ____________.
Q112➡|GATE 2015 Set-2 Let 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 ________.
Q116➡| GATE 2015 Set-3 Suppose 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 = ∅)
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 _________.
Q120➡| GATE 2015 Set-3 In 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
Q121➡| GATE 2015 Set-3 The 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 1/3rd rule is ______.
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?
Q124➡| GATE 2015 Set-3 Let 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?
Q125➡| GATE 2015 Set-3 Suppose Xi for i = 1, 2, 3 are independent and identically distributed random variables whose probability mass functions are Pr[Xi = 0] = Pr[Xi = 1]=1/2 for i = 1, 2, 3. Define another random variable Y = X1X2 ⊕ X3, where ⊕ denotes XOR. Then Pr[Y = 0|X3 = 0] =_______,
Q126➡| GATE 2014 Set-1 Consider 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?
Q127➡| GATE 2014 Set-1 Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is____________.
Q128➡| GATE 2014 Set-1 Let 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
Q129➡| GATE 2014 Set-1 Consider the following system of equations: 3x + 2y = 1 4x + 7z = 1 x + y + z = 3 x – 2y + 7z = 0 The number of solutions for this system is _.
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 ________.
Q131➡| GATE 2014 Set-1 Let 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.
Q133➡|GATE 2014 Set-1 A 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)
Q135➡|GATE 2014 Set-1 A pennant is a sequence of numbers, each number being 1 or 2. An n-pennant is a sequence of numbers with sum equal to n. For example, (1,1,2) is a 4-pennant. The set of all possible 1-pennants is {(1)}, the set of all possible 2-pennants is {(2), (1,1)}and the set of all 3-pennants is {(2,1), (1,1,1), (1,2)}. Note that the pennant (1,2) is not the same as the pennant (2,1). The number of 10- pennants is .
Q136➡| GATE 2014 Set-1 Let S denote the set of all functions f: {0,1}4 → {0,1}. Denote by N the number of functions from S to the set {0,1}. The value of log2 log2 N is .
Q137➡| GATE 2014 Set-1 Consider an un-directed graph G where self-loops are not allowed. The vertex set of G is {(i, j):1 ≤ i ≤ 12, 1 ≤ j ≤ 12}. There is an edge between (a, b) and (c, d) if |a − c| ≤ 1 and |b − d| ≤ 1. The number of edges in this graph is .
Q138➡| GATE 2014 Set-1 An ordered n-tuple (d1, d2,…, dn) with d1 ≥ d2 ≥ ⋯ ≥ dn is called graphic if there exists a simple undirected graph with n vertices having degrees d1, d2,… , dn respectively. Which of the following 6-tuples is NOT graphic?
Q140➡| GATE 2014 Set-2 The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working. Let the probability that the system is deemed functional be denoted by p. Then 100p =__________.
Q141➡| GATE 2014 Set-2 Each of the nine words in the sentence “The quick brown fox jumps over the lazy dog” is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expectedlength of the word drawn is _____________. (The answer should be rounded to one decimal place.)
Q145➡| GATE 2014 Set-2 Consider the tree arcs of a BFS traversal from a source node W in an unweighted, connected, undirected graph. The tree T formed by the tree arcs is a data structure for computing
i ➥ the shortest path between every pair of vertices.
ii ➥ the shortest path from W to every vertex in the graph.
iii ➥ the shortest paths from W to only those nodes that are leaves of T
Q146➡| GATE 2014 Set-2 In the Newton-Raphson method, an initial guess of x0 = 2 is made and the sequence x0, x1, x2 … is obtained for the function 0.75x3 – 2x2 – 2x + 4 = 0
Consider the statements (I) x3 = 0. (II) The method converges to a solution in a finite number of iterations. Which of the following is TRUE?
Q150➡| GATE 2014 Set-2 Consider the following relation on subsets of the set S of integers between 1 and 2014. For two distinct subsets U and V of S we say U < V if the minimum element in the symmetric difference of the two sets is in U. Consider the following two statements: S1: There is a subset of S that is larger than every other subset. S2: There is a subset of S that is smaller than every other subset. Which one of the following is CORRECT?
Q154➡| GATE 2014 Set-3 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 ________.
Q156➡| GATE 2014 Set-3 If V1 and V2 are 4-dimensional subspace of a 6-dimensional vector space V, then the smallest possible dimension of V1∩V2 is ______.
Q158➡| GATE 2014 Set-3 With respect to the numerical evaluation of the definite integral, K = 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.
Q159➡|GATE 2014 Set-3 Let S be a sample space and two mutually exclusive events A and B be such that A ∪ B = S. If P(∙) denotes the probability of the event, the maximum value of P(A)P(B) is .
Q160➡|GATE 2014 Set-3 Consider the set of all functions ƒ: {0,1, … ,2014} → {0,1, … ,2014} such that ƒ(ƒ(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, ƒ(i) = i. Q. For each such function it must be the case that for some i, ƒ(i) = i. R. Each such function must be onto. Which one of the following is CORRECT?
Q161➡| GATE 2014 Set-3 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_________. .
is ________.
where ai are column vectors in Rn. Consider the set of linear equations.
Set of equations has
, then the constants R and S are, respectively.
and Q= 
be two matrices.
, define the polynomial function 
, then a3 – a0 is equal to __________.
.
,if the diagonal elements of U are both 1, then the lower diagonal entry l22 of L is
is ______.
, one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are
and f'(θ) denote the derivative of f with respect to θ. Which of the following is/are TRUE?
such that f'(θ)=0.
where a and b are given, 
