Engineering Mathematics: GATE CS previous year questions, 2011 to 2026

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All Engineering Mathematics questions

Hide the 4 questions not in the current syllabus

GATE CS 2026 Set 1MCQ1 markEasy (estimate)
GATE CS 2026 Set 1, Q.11
An urn contains one red ball and one blue ball. At each step, a ball is picked uniformly at random from the urn, and this ball together with another ball of the same color is pu...
Probability and Statistics
GATE CS 2026 Set 1MCQ1 markMedium (estimate)
GATE CS 2026 Set 1, Q.12
Consider $4 \times 4$ matrices with their elements from ${0, 1}$ . The number of such matrices with even number of $1$ s in every row and every column is
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2026 Set 1MCQ1 markEasy (estimate)
GATE CS 2026 Set 1, Q.13
For $n > 1$ , the maximum multiplicity of any eigenvalue of an $n \times n$ matrix with elements from $R$ is
Linear Algebra
GATE CS 2026 Set 1MSQ1 markEasy (estimate)
GATE CS 2026 Set 1, Q.20
Let $n > 1$ . Consider an $n \times n$ matrix $M$ with its elements from $R$ . Let the vector $(0, 1, 0, 0, \dots, 0) \in R^{n}$ be in the null space of $M$ . Which...
Linear Algebra
GATE CS 2026 Set 1NAT1 markEasy (estimate)
GATE CS 2026 Set 1, Q.32
Consider the function $f : R \to R$ defined as follows…
Calculus
GATE CS 2026 Set 1MSQ2 marksMedium (estimate)
GATE CS 2026 Set 1, Q.46
Let $f : R \to R$ be defined as follows: $f (x) = (\frac{∣ x ∣}{2} - x) (x - \frac{∣ x ∣}{2})$ Which of the following statements is/ar...
Calculus
GATE CS 2026 Set 1MSQ2 marksMedium (estimate)
GATE CS 2026 Set 1, Q.47
Let $G (V, E)$ be a simple, undirected graph. A vertex cover of $G$ is a subset $V^{'} \subseteq V$ such that for every $(u, v) \in E$ , $u \in V^{'}$ or $v \in V^{'}$ . Let the size of the...
Graph Theory
GATE CS 2026 Set 1NAT2 marksMedium (estimate)
GATE CS 2026 Set 1, Q.57
Let $G$ be an undirected graph, which is a path on 8 vertices. The number of matchings in $G$ is ______. (answer in integer)
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2026 Set 1NAT2 marksEasy (estimate)
GATE CS 2026 Set 1, Q.58
Let $X$ be a random variable which takes values in the set ${1, 2, 3, 4, 5, 6, 7, 8}$ . Further, $Pr (X = 1) = Pr (X = 2) = Pr (X = 5) = Pr (X = 7) = \frac{1}{6}$ and…
Probability and Statistics
GATE CS 2026 Set 2MCQ1 markEasy (estimate)
GATE CS 2026 Set 2, Q.11
For two different persons $x$ and $y$ , the predicate $M (x, y)$ denotes that $x$ knows $y$ . Consider the following statement. There is a person who does not know anyone else, but...
Propositional and First-Order Logic
GATE CS 2026 Set 2MCQ1 markEasy (estimate)
GATE CS 2026 Set 2, Q.14
The probability density function $f (x)$ of a random variable $X$ which takes real values is…
Probability and Statistics
GATE CS 2026 Set 2MSQ1 markEasy (estimate)
GATE CS 2026 Set 2, Q.26
Let $R$ be a binary relation on the set ${1, 2, \dots, 10}$ , where $(x, y) \in R$ if the product of $x$ and $y$ is square of an integer. Which of the following properties is/a...
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2026 Set 2MSQ1 markEasy (estimate)
GATE CS 2026 Set 2, Q.27
For a real number $a$ , let $I (a) = \int_{- 1}^{1} (3 x^{2} - a x + 1) d x$ . Which of the following statements is/are true?
Calculus
GATE CS 2026 Set 2NAT1 markEasy (estimate)
GATE CS 2026 Set 2, Q.31
Consider the system of linear equations given below. $a x + y = b$ $16 x + a y = 24$ Suppose the values of $a$ and $b$ are chosen such that the system of linear equations produ...
Linear Algebra
GATE CS 2026 Set 2MCQ2 marksMedium (estimate)
GATE CS 2026 Set 2, Q.36
Consider a complete graph $K_{n}$ with $n$ vertices ( $n > 4$ ). Note that multiple spanning trees can be constructed over $K_{n}$ . Each of these spanning trees is represented as a se...
Graph Theory
GATE CS 2026 Set 2NAT2 marksEasy (estimate)
GATE CS 2026 Set 2, Q.62
The determinant of a $4 \times 4$ matrix $A$ is $3$ . The value of the determinant of $2 A$ is ____________. (answer in integer)
Linear Algebra
GATE CS 2026 Set 2NAT2 marksMedium (estimate)
GATE CS 2026 Set 2, Q.63
Suppose an unbiased coin is tossed 6 times. Each coin toss is independent of all previous coin tosses. Let $E_{1}$ be the event that among the second, fourth, and sixth coin tosse...
Probability and Statistics
GATE CS 2026 Set 2NAT2 marksMedium (estimate)
GATE CS 2026 Set 2, Q.64
Consider a function $f : (0, 1) \to {0, 1}$ defined as follows. For a real number $r \in (0, 1)$ , $f (r) = 1$ if the second digit after the decimal point in $r$ is one of...
Calculus
GATE CS 2025 Set 1MCQ1 markEasy (estimate)
GATE CS 2025 Set 1, Q.17
$g (.)$ is a function from $A$ to $B$ , $f (.)$ is a function from $B$ to $C$ , and their composition defined as $f (g (.))$ is a mapping from $A$ to $C$ . If $f (.)$ and $f (g (.))$ are...
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2025 Set 1MSQ1 markEasy (estimate)
GATE CS 2025 Set 1, Q.23
Consider the given system of linear equations for variables $x$ and $y$ , where $k$ is a real-valued constant. Which of the following option(s) is/are CORRECT? $x + k y = 1$ …
Linear Algebra
GATE CS 2025 Set 1NAT1 markEasy (estimate)
GATE CS 2025 Set 1, Q.30
Let $S$ be the set of all ternary strings defined over the alphabet ${a, b, c}$ . Consider all strings in $S$ that contain at least one occurrence of two consecutive symbols, t...
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2025 Set 1NAT1 markEasy (estimate)
GATE CS 2025 Set 1, Q.31
Consider the given function $f (x)$ . $f (x) = {a x + b x^{3} + x^{2} + 1 for x < 1 for x \geq 1$ If the function is differentiable ever...
Calculus
GATE CS 2025 Set 1NAT1 markEasy (estimate)
GATE CS 2025 Set 1, Q.32
A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability $P (head) = 0.5$ and for a fake coin, $P (head) = 1$ . You pic...
Probability and Statistics
GATE CS 2025 Set 1MCQ2 marksEasy (estimate)
GATE CS 2025 Set 1, Q.41
Let $A$ be a $2 \times 2$ matrix as given. $A = [11 1 - 1]$ What are the eigenvalues of the matrix $A^{13}$ ?
Linear Algebra
GATE CS 2025 Set 1MSQ2 marksMedium (estimate)
GATE CS 2025 Set 1, Q.48
Which of the following predicate logic formulae/formula is/are CORRECT representation(s) of the statement: "Everyone has exactly one mother"? The meanings of the predicates used...
Propositional and First-Order Logic
GATE CS 2025 Set 1MSQ2 marksEasy (estimate)
GATE CS 2025 Set 1, Q.49
$A = {0, 1, 2, 3, \dots}$ is the set of non-negative integers. Let $F$ be the set of functions from $A$ to itself. For any two functions, $f_{1}, f_{2} \in F$ , we define…
Monoids and Groups
GATE CS 2025 Set 1NAT2 marksEasy (estimate)
GATE CS 2025 Set 1, Q.58
Consider a probability distribution given by the density function $P (x)$ .…
Probability and Statistics
GATE CS 2025 Set 2MCQ1 markEasy (estimate)
GATE CS 2025 Set 2, Q.11
If $A = (12 2 - 1)$ , then which ONE of the following is $A^{8}$ ?
Linear Algebra
GATE CS 2025 Set 2MCQ1 markEasy (estimate)
GATE CS 2025 Set 2, Q.12
The value of $x$ such that $x > 1$ , satisfying the equation $\int_{1}^{x} t ln t d t = \frac{1}{4}$ is
Calculus
GATE CS 2025 Set 2MCQ1 markEasy (estimate)
GATE CS 2025 Set 2, Q.14
Let $L, M,$ and $N$ be non-singular matrices of order 3 satisfying the equations $L^{2} = L^{- 1}, M = L^{8}$ and $N = L^{2}$ . Which ONE of the following is the value of the determi...
Linear Algebra
GATE CS 2025 Set 2MCQ1 markEasy (estimate)
GATE CS 2025 Set 2, Q.15
Let $P (x)$ be an arbitrary predicate over the domain of natural numbers. Which ONE of the following statements is TRUE?
Propositional and First-Order Logic · Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2025 Set 2MSQ2 marksEasy (estimate)
GATE CS 2025 Set 2, Q.42
Let $F$ be the set of all functions from ${1, \dots, n}$ to ${0, 1}$ . Define the binary relation $≼$ on $F$ as follows…
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2025 Set 2MSQ2 marksEasy (estimate)
GATE CS 2025 Set 2, Q.44
Consider a system of linear equations $P X = Q$ where $P \in R^{3 \times 3}$ and $Q \in R^{3 \times 1}$ . Suppose $P$ has an LU decomposition, $P = LU$ , where…
Linear Algebra
GATE CS 2025 Set 2NAT2 marksMedium (estimate)
GATE CS 2025 Set 2, Q.64
A quadratic polynomial $(x - α) (x - β)$ over complex numbers is said to be square invariant if $(x - α) (x - β) = (x - α^{2}) (x - β^{2})$ . Suppose from the...
Probability and Statistics · Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2025 Set 2NAT2 marksMedium (estimate)
GATE CS 2025 Set 2, Q.65
The unit interval $(0, 1)$ is divided at a point chosen uniformly distributed over $(0, 1)$ in $R$ into two disjoint subintervals. The expected length of the subinterva...
Probability and Statistics · Calculus
GATE CS 2024 Set 1MCQ1 markEasy (estimate)
GATE CS 2024 Set 1, Q.11
Let $f : R \to R$ be a function such that $f (x) = max {x, x^{3}}, x \in R$ , where $R$ is the set of all real numbers. The set of al...
Calculus
GATE CS 2024 Set 1MCQ1 markEasy (estimate)
GATE CS 2024 Set 1, Q.12
The product of all eigenvalues of the matrix $147258369$ is
Linear Algebra
GATE CS 2024 Set 1MCQ1 markEasy (estimate)
GATE CS 2024 Set 1, Q.14
Consider a permutation sampled uniformly at random from the set of all permutations of ${1, 2, 3, \dots, n}$ for some $n \geq 4$ . Let $X$ be the event that $1$ occurs before…
Probability and Statistics · Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2024 Set 1MSQ1 markEasy (estimate)
GATE CS 2024 Set 1, Q.27
Let $A$ and $B$ be two events in a probability space with $P (A) = 0.3, P (B) = 0.5$ , and $P (A \cap B) = 0.1$ . Which of the following statements is/are TRUE?
Probability and Statistics
GATE CS 2024 Set 1NAT1 markMedium (estimate)
GATE CS 2024 Set 1, Q.32
Let $A$ and $B$ be non-empty finite sets such that there exist one-to-one and onto functions (i) from $A$ to $B$ and (ii) from $A \times A$ to $A \cup B$ . The number of possible...
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2024 Set 1NAT1 markEasy (estimate)
GATE CS 2024 Set 1, Q.34
The number of spanning trees in a complete graph of 4 vertices labelled A, B, C, and D is _________
Graph Theory
GATE CS 2024 Set 1MSQ2 marksHard (estimate)
GATE CS 2024 Set 1, Q.49
Let $A$ be any $n \times m$ matrix, where $m > n$ . Which of the following statements is/are TRUE about the system of linear equations $A x = 0$ ?
Linear Algebra
GATE CS 2024 Set 1MSQ2 marksMedium (estimate)
GATE CS 2024 Set 1, Q.51
The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. Let $G$ be any graph with $n$ vertices and chromatic number $k$ . Whic...
Graph Theory
GATE CS 2024 Set 1MSQ2 marksEasy (estimate)
GATE CS 2024 Set 1, Q.52
Consider the operators $⋄$ and $□$ defined by $a ⋄ b = a + 2 b, a □ b = ab$ , for positive integers. Which of the following statements is/are TRUE?
Monoids and Groups
GATE CS 2024 Set 1NAT2 marksEasy (estimate)
GATE CS 2024 Set 1, Q.63
A bag contains 10 red balls and 15 blue balls. Two balls are drawn randomly without replacement. Given that the first ball drawn is red, the probability (rounded off to 3 decima...
Probability and Statistics
GATE CS 2024 Set 2MCQ1 markEasy (estimate)
GATE CS 2024 Set 2, Q.12
Let $p$ and $q$ be the following propositions: $p$ : Fail grade can be given. $q$ : Student scores more than 50% marks. Consider the statement: "Fail grade cannot be given when st...
Propositional and First-Order Logic
GATE CS 2024 Set 2MCQ1 markEasy (estimate)
GATE CS 2024 Set 2, Q.16
Let $f (x)$ be a continuous function from $R$ to $R$ such that $f (x) = 1 - f (2 - x)$ Which one of the following options is the CORRECT value of…
Calculus
GATE CS 2024 Set 2MCQ1 markMedium (estimate)
GATE CS 2024 Set 2, Q.17
Let $A$ be the adjacency matrix of a simple undirected graph $G$ . Suppose $A$ is its own inverse. Which one of the following statements is always TRUE?
Graph Theory · Linear Algebra
GATE CS 2024 Set 2MCQ1 markEasy (estimate)
GATE CS 2024 Set 2, Q.18
When six unbiased dice are rolled simultaneously, the probability of getting all distinct numbers (i.e., 1, 2, 3, 4, 5, and 6) is
Probability and Statistics · Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2024 Set 2NAT1 markEasy (estimate)
GATE CS 2024 Set 2, Q.34
Let $P$ be the partial order defined on the set ${1, 2, 3, 4}$ as follows $P = {(x, x) ∣ x \in {1, 2, 3, 4}} \cup {(1, 2), (3, 2), (3, 4)}$ The number of total orders on…
Sets, Relations, Functions, Partial Orders and Lattices · Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2024 Set 2MCQ2 marksMedium (estimate)
GATE CS 2024 Set 2, Q.44
Let $x$ and $y$ be random variables, not necessarily independent, that take real values in the interval $[0, 1]$ . Let $z = x y$ and let the mean values of $x, y, z$ be…
Probability and Statistics
GATE CS 2024 Set 2MSQ2 marksEasy (estimate)
GATE CS 2024 Set 2, Q.47
Let $A$ be an $n \times n$ matrix over the set of all real numbers $R$ . Let $B$ be a matrix obtained from $A$ by swapping two rows. Which of the following statements is...
Linear Algebra
GATE CS 2024 Set 2MSQ2 marksHard (estimate)
GATE CS 2024 Set 2, Q.51
Let $G$ be an undirected connected graph in which every edge has a positive integer weight. Suppose that every spanning tree in $G$ has even weight. Which of the following state...
Graph Theory · Minimum Spanning Trees
GATE CS 2024 Set 2NAT2 marksMedium (estimate)
GATE CS 2024 Set 2, Q.60
The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. The chromatic number of the following graph is _________ An 8-vertex 4-...
Graph Theory
GATE CS 2024 Set 2NAT2 marksEasy (estimate)
GATE CS 2024 Set 2, Q.63
Let $Z_{n}$ be the group of integers ${0, 1, 2, \dots, n - 1}$ with addition modulo $n$ as the group operation. The number of elements in the group $Z_{2} \times Z_{3} \times Z_{4}$ t...
Monoids and Groups
GATE CS 2023MCQ1 markEasy (estimate)
GATE CS 2023, Q.15
The Lucas sequence $L_{n}$ is defined by the recurrence relation: $L_{n} = L_{n - 1} + L_{n - 2}, for n \geq 3,$ with $L_{1} = 1$ and $L_{2} = 3$ . Which one of the options gi...
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2023MCQ1 markEasy (estimate)
GATE CS 2023, Q.18
Let $A = 1432214332144321$ and…
Linear Algebra
GATE CS 2023MSQ1 markMedium (estimate)
GATE CS 2023, Q.26
Geetha has a conjecture about integers, which is of the form $\forall x (P (x) ⟹ \exists y Q (x, y)),$ where $P$ is a statement about integers, and $Q$ is a statement abo...
Propositional and First-Order Logic
GATE CS 2023MSQ1 markEasy (estimate)
GATE CS 2023, Q.28
Let $f (x) = x^{3} + 15 x^{2} - 33 x - 36$ be a real-valued function. Which of the following statements is/are TRUE?
Calculus
GATE CS 2023NAT1 markMedium (estimate)
GATE CS 2023, Q.30
Let $A$ be the adjacency matrix of the graph with vertices ${1, 2, 3, 4, 5}$ . Undirected graph with 5 vertices {1, 2, 3, 4, 5}, edges (1,2), (2,3), (2,4), (3,5), (4,5), and se...
Linear Algebra · Graph Theory
GATE CS 2023NAT1 markEasy (estimate)
GATE CS 2023, Q.31
The value of the definite integral $\int_{- 3}^{3} \int_{- 2}^{2} \int_{- 1}^{1} (4 x^{2} y - z^{3}) d z d y d x$ is _______. (Rounded off to the nearest integer)
Calculus
GATE CS 2023MCQ2 marksMedium (estimate)
GATE CS 2023, Q.48
Let $U = {1, 2, \dots, n}$ , where $n$ is a large positive integer greater than 1000. Let $k$ be a positive integer less than $n$ . Let $A, B$ be subsets of $U$ with…
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2023MSQ2 marksMedium (estimate)
GATE CS 2023, Q.49
Let $f : A \to B$ be an onto (or surjective) function, where $A$ and $B$ are nonempty sets. Define an equivalence relation $\sim$ on the set $A$ as…
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2023MSQ2 marksMedium (estimate)
GATE CS 2023, Q.51
Let $X$ be a set and $2^{X}$ denote the powerset of $X$ . Define a binary operation $Δ$ on $2^{X}$ as follows: $A Δ B = (A - B) \cup (B - A) .$ Let $H = (2^{X}, Δ)$ . Wh...
Monoids and Groups · Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2023MSQ2 marksEasy (estimate)
GATE CS 2023, Q.53
Consider a random experiment where two fair coins are tossed. Let $A$ be the event that denotes HEAD on both the throws, $B$ be the event that denotes HEAD on the first throw, a...
Probability and Statistics
GATE CS 2023MSQ2 marksMedium (estimate)
GATE CS 2023, Q.55
Let $G$ be a simple, finite, undirected graph with vertex set ${v_{1}, \dots, v_{n}}$ . Let $Δ (G)$ denote the maximum degree of $G$ and let $N = {1, 2, \dots}$ deno...
Graph Theory · Greedy Algorithms
GATE CS 2023NAT2 marksMedium (estimate)
GATE CS 2023, Q.56
Let $U = {1, 2, 3}$ . Let $2^{U}$ denote the powerset of $U$ . Consider an undirected graph $G$ whose vertex set is $2^{U}$ . For any $A, B \in 2^{U}$ , $(A, B)$ is an edge in $G$ if an...
Graph Theory · Graph Traversals, Topological Sort and Connectivity
GATE CS 2022MCQ1 markEasy (estimate)
GATE CS 2022, Q.20
Consider the following two statements with respect to the matrices $A_{m \times n}$ , $B_{n \times m}$ , $C_{n \times n}$ and $D_{n \times n}$ . Statement 1…
Linear Algebra
GATE CS 2022MSQ1 markMedium (estimate)
GATE CS 2022, Q.27
Which of the following statements is/are TRUE for a group G ?
Monoids and Groups
GATE CS 2022NAT1 markEasy (estimate)
GATE CS 2022, Q.30
Consider a simple undirected graph of 10 vertices. If the graph is disconnected, then the maximum number of edges it can have is ____________.
Graph Theory
GATE CS 2022NAT1 markEasy (estimate)
GATE CS 2022, Q.32
The number of arrangements of six identical balls in three identical bins is______.
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2022NAT1 markEasy (estimate)
GATE CS 2022, Q.34
The value of the following limit is _____________. $lim_{x \to 0^{+}} \frac{x}{1 - e ^{2 x}}$
Calculus
GATE CS 2022MCQ2 marksMedium (estimate)
GATE CS 2022, Q.36
Which one of the following is the closed form for the generating function of the sequence ${a_{n}}_{n \geq 0}$ defined below?…
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2022MCQ2 marksEasy (estimate)
GATE CS 2022, Q.37
Consider a simple undirected unweighted graph with at least three vertices. If A is the adjacency matrix of the graph, then the number of 3-cycles in the graph is given by the t...
Graph Theory
GATE CS 2022MCQ2 marksMedium (estimate)
GATE CS 2022, Q.45
Consider solving the following system of simultaneous equations using LU decomposition. $x_{1} + x_{2} - 2 x_{3} = 4$ $x_{1} + 3 x_{2} - x_{3} = 7$ $2 x_{1} + x_{2} - 5 x_{3} = 7$ where L and U...
Linear Algebra
GATE CS 2022MSQ2 marksHard (estimate)
GATE CS 2022, Q.50
The following simple undirected graph is referred to as the Peterson graph. The Petersen graph drawn with an outer 5-cycle, an inner 5-pointed star, and 5 radial spokes connecti...
Graph Theory
GATE CS 2022MSQ2 marksMedium (estimate)
GATE CS 2022, Q.51
Consider the following recurrence: $f (1) = 1;$ $f (2 n) = 2 f (n) - 1, for n \geq 1;$ $f (2 n + 1) = 2 f (n) + 1, for n \geq 1.$ Then, which of the follo...
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2022MSQ2 marksMedium (estimate)
GATE CS 2022, Q.52
Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?
Graph Theory
GATE CS 2022MSQ2 marksMedium (estimate)
GATE CS 2022, Q.53
Which of the following is/are the eigenvector(s) for the matrix given below?…
Linear Algebra
GATE CS 2022NAT2 marksMedium (estimate)
GATE CS 2022, Q.58
Let $G (V, E)$ be a directed graph, where $V = {1, 2, 3, 4, 5}$ is the set of vertices and E is the set of directed edges, as defined by the following adjacency matrix A.…
Graph Theory
GATE CS 2021 Set 1MCQ1 markEasy (estimate)
GATE CS 2021 Set 1, Q.17 (CS Q.7)
Let $p$ and $q$ be two propositions. Consider the following two formulae in propositional logic. $S_{1} : (\neg p \land (p \lor q)) \to q$ …
Propositional and First-Order Logic
GATE CS 2021 Set 1NAT1 markEasy (estimate)
GATE CS 2021 Set 1, Q.26 (CS Q.16)
In an undirected connected planar graph $G$ , there are eight vertices and five faces. The number of edges in $G$ is __________
Graph Theory
GATE CS 2021 Set 1NAT1 markEasy (estimate)
GATE CS 2021 Set 1, Q.28 (CS Q.18)
The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter $λ$ . For a randomly picke...
Probability and Statistics
GATE CS 2021 Set 1NAT1 markMedium (estimate)
GATE CS 2021 Set 1, Q.29 (CS Q.19)
There are 6 jobs with distinct difficulty levels, and 3 computers with distinct processing speeds. Each job is assigned to a computer such that: - The fastest computer gets the...
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2021 Set 1NAT1 markEasy (estimate)
GATE CS 2021 Set 1, Q.30 (CS Q.20)
Consider the following expression. $lim_{x \to - 3} \frac{2 x + 22 - 4}{x + 3}$ The value of the above expression (rounded to 2 decimal places) is __________
Calculus
GATE CS 2021 Set 1MCQ2 marksMedium (estimate)
GATE CS 2021 Set 1, Q.44 (CS Q.34)
Let $G$ 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?
Monoids and Groups
GATE CS 2021 Set 1MCQ2 marksMedium (estimate)
GATE CS 2021 Set 1, Q.45 (CS Q.35)
Consider the two statements. $S_{1} : There exist random variables X and Y such that (E [(X - E [X]) (Y - E [Y]))^{2} > Var [X] Var [Y]$ ...
Probability and Statistics
GATE CS 2021 Set 1MCQ2 marksMedium (estimate)
GATE CS 2021 Set 1, Q.46 (CS Q.36)
Let $G = (V, E)$ be an undirected unweighted connected graph. The diameter of $G$ is defined as…
Graph Theory
GATE CS 2021 Set 1MSQ2 marksMedium (estimate)
GATE CS 2021 Set 1, Q.53 (CS Q.43)
A relation $R$ is said to be circular if $a R b$ and $b R c$ together imply $c R a$ . Which of the following options is/are correct?
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2021 Set 1NAT2 marksEasy (estimate)
GATE CS 2021 Set 1, Q.62 (CS Q.52)
Consider the following matrix. $0111101111011110$ The largest eigenvalue of the above matrix is _...
Linear Algebra
GATE CS 2021 Set 1NAT2 marksMedium (estimate)
GATE CS 2021 Set 1, Q.64 (CS Q.54)
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 edg...
Probability and Statistics
GATE CS 2021 Set 2MSQ1 markEasy (estimate)
GATE CS 2021 Set 2, Q.21 (CS Q.11)
Consider the following sets, where $n \geq 2$ : $S_{1}$ : Set of all $n \times n$ matrices with entries from the set ${a, b, c}$ $S_{2}$ : Set of all functions from the set…
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2021 Set 2MSQ1 markEasy (estimate)
GATE CS 2021 Set 2, Q.25 (CS Q.15)
Choose the correct choice(s) regarding the following propositional logic assertion $S$ : $S : ((P \land Q) \to R) \to ((P \land Q) \to (Q \to R))$
Propositional and First-Order Logic
GATE CS 2021 Set 2NAT1 markEasy (estimate)
GATE CS 2021 Set 2, Q.32 (CS Q.22)
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 numb...
Probability and Statistics
GATE CS 2021 Set 2NAT1 markEasy (estimate)
GATE CS 2021 Set 2, Q.34 (CS Q.24)
Suppose that $P$ is a $4 \times 5$ matrix such that every solution of the equation $P x = 0$ is a scalar multiple of $[25431]^{T}$ . The rank of $P$ is ______.
Linear Algebra
GATE CS 2021 Set 2NAT1 markEasy (estimate)
GATE CS 2021 Set 2, Q.35 (CS Q.25)
Suppose that $f : R \to R$ is a continuous function on the interval $[- 3, 3]$ and a differentiable function in the interval $(- 3, 3)$ such that for every $x$ in...
Calculus
GATE CS 2021 Set 2MCQ2 marksEasy (estimate)
GATE CS 2021 Set 2, Q.39 (CS Q.29)
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 zer...
Probability and Statistics
GATE CS 2021 Set 2MCQ2 marksMedium (estimate)
GATE CS 2021 Set 2, Q.43 (CS Q.33)
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 bal...
Probability and Statistics
GATE CS 2021 Set 2MCQ2 marksEasy (estimate)
GATE CS 2021 Set 2, Q.47 (CS Q.37)
For two $n$ -dimensional real vectors $P$ and $Q$ , the operation $s (P, Q)$ is defined as follows: $s (P, Q) = \sum_{i = 1}^{n} (P_{i} \cdot Q_{i})$ Let $L$ be a set of 10-dime...
Linear Algebra
GATE CS 2021 Set 2NAT2 marksEasy (estimate)
GATE CS 2021 Set 2, Q.60 (CS Q.50)
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 $A \subseteq B$ is ______.
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2020MCQ1 markEasy (estimate)
GATE CS 2020, Q.11 (CS Q.1)
Consider the functions I. $e^{- x}$ II. $x^{2} - sin x$ III. $x^{3} + 1$ Which of the above functions is/are increasing everywhere in $[0, 1]$ ?
Calculus
GATE CS 2020NAT1 markEasy (estimate)
GATE CS 2020, Q.27 (CS Q.17)
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 (roun...
Sets, Relations, Functions, Partial Orders and Lattices · Probability and Statistics
GATE CS 2020NAT1 markEasy (estimate)
GATE CS 2020, Q.28 (CS Q.18)
Let $G$ be a group of 35 elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
Monoids and Groups
GATE CS 2020MCQ2 marksEasy (estimate)
GATE CS 2020, Q.37 (CS Q.27)
Let $A$ and $B$ be two $n \times n$ matrices over real numbers. Let $rank (M)$ and $det (M)$ denote the rank and determinant of a matrix $M$ , respectively. Consider the fo...
Linear Algebra
GATE CS 2020MCQ2 marksMedium (estimate)
GATE CS 2020, Q.49 (CS Q.39)
Which one of the following predicate formulae is NOT logically valid? Note that $W$ is a predicate formula without any free occurrence of $x$ .
Propositional and First-Order Logic
GATE CS 2020NAT2 marksMedium (estimate)
GATE CS 2020, Q.52 (CS Q.42)
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 _______.
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2020NAT2 marksEasy (estimate)
GATE CS 2020, Q.55 (CS Q.45)
For $n \geq 2$ , let $a \in {0, 1}^{n}$ be a non-zero vector. Suppose that $x$ is chosen uniformly at random from ${0, 1}^{n}$ . Then, the probability that $\sum_{i = 1}^{n} a_{i} x_{i}$ ...
Probability and Statistics
GATE CS 2020NAT2 marksHard (estimate)
GATE CS 2020, Q.62 (CS Q.52)
Graph $G$ 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 _______.
Graph Theory
GATE CS 2019MCQ1 markMedium (estimate)
GATE CS 2019, Q.15 (CS Q.5)
Let $U = {1, 2, \dots, n}$ . Let $A = {(x, X) ∣ x \in X, X \subseteq U}$ . Consider the following two statements on $∣ A ∣$ . I. $∣ A ∣ = n 2^{n - 1}$ II.…
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2019MCQ1 markEasy (estimate)
GATE CS 2019, Q.19 (CS Q.9)
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?
Linear Algebra
GATE CS 2019MCQ1 markEasy (estimate)
GATE CS 2019, Q.20 (CS Q.10)
Let $G$ be an arbitrary group. Consider the following relations on $G$ : $R_{1}$ : $\forall a, b \in G$ , $a R_{1} b$ if and only if $\exists g \in G$ such that $a = g^{- 1} b g$ $R_{2}$ …
Monoids and Groups
GATE CS 2019MCQ1 markEasy (estimate)
GATE CS 2019, Q.22 (CS Q.12)
Let $G$ be an undirected complete graph on $n$ vertices, where $n > 2$ . Then, the number of different Hamiltonian cycles in $G$ is equal to
Graph Theory
GATE CS 2019MCQ1 markEasy (estimate)
GATE CS 2019, Q.23 (CS Q.13)
Compute $lim_{x \to 3} \frac{x ^{4} - 81}{2 x ^{2} - 5 x - 3}$
Calculus
GATE CS 2019NAT1 markEasy (estimate)
GATE CS 2019, Q.31 (CS Q.21)
The value of $3^{51} (mod 5)$ is ________.
Monoids and Groups
GATE CS 2019NAT1 markMedium (estimate)
GATE CS 2019, Q.32 (CS Q.22)
Two numbers are chosen independently and uniformly at random from the set ${1, 2, \dots, 13}$ . The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) bi...
Probability and Statistics
GATE CS 2019MCQ2 marksMedium (estimate)
GATE CS 2019, Q.45 (CS Q.35)
Consider the first order predicate formula $φ$ …
Propositional and First-Order Logic
GATE CS 2019NAT2 marksMedium (estimate)
GATE CS 2019, Q.54 (CS Q.44)
Consider the following matrix: $R = 1111234549162582764125$ The absolute value of the product of...
Linear Algebra
GATE CS 2019NAT2 marksMedium (estimate)
GATE CS 2019, Q.57 (CS Q.47)
Suppose $Y$ is distributed uniformly in the open interval $(1, 6)$ . The probability that the polynomial $3 x^{2} + 6 x Y + 3 Y + 6$ has only real roots is (rounded off to 1 decimal pl...
Probability and Statistics
GATE CS 2018MCQ1 markEasy (estimate)
GATE CS 2018, Q.11 (CS Q.1)
Which one of the following is a closed form expression for the generating function of the sequence ${a_{n}}$ , where $a_{n} = 2 n + 3$ for all $n = 0, 1, 2, \dots$ ?
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2018NAT1 markEasy (estimate)
GATE CS 2018, Q.25 (CS Q.15)
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...
Probability and Statistics
GATE CS 2018NAT1 markEasy (estimate)
GATE CS 2018, Q.26 (CS Q.16)
The value of $\int_{0}^{π /4} x cos (x^{2}) d x$ correct to three decimal places (assuming that $π = 3.14$ ) is _____.
Calculus
GATE CS 2018NAT1 markEasy (estimate)
GATE CS 2018, Q.27 (CS Q.17)
Consider a matrix $A = u v^{T}$ where $u = (12), v = (11)$ . Note that $v^{T}$ denotes the transpose of $v$ . The large...
Linear Algebra
GATE CS 2018NAT1 markEasy (estimate)
GATE CS 2018, Q.28 (CS Q.18)
The chromatic number of the following graph is _______. 6-vertex undirected graph consisting of an outer 4-cycle and two inner vertices with diagonal cross-connections to find c...
Graph Theory
GATE CS 2018NAT1 markEasy (estimate)
GATE CS 2018, Q.29 (CS Q.19)
Let $G$ be a finite group on 84 elements. The size of a largest possible proper subgroup of $G$ is ________.
Monoids and Groups
GATE CS 2018MCQ2 marksMedium (estimate)
GATE CS 2018, Q.36 (CS Q.26)
Consider a matrix $P$ whose only eigenvectors are the multiples of $[14]$ . Consider the following statements: (I) $P$ does not have an inverse (I...
Linear Algebra
GATE CS 2018MCQ2 marksMedium (estimate)
GATE CS 2018, Q.37 (CS Q.27)
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...
Monoids and Groups
GATE CS 2018MCQ2 marksMedium (estimate)
GATE CS 2018, Q.38 (CS Q.28)
Consider the first-order logic sentence $φ \equiv \exists s \exists t \exists u \forall v \forall w \forall x \forall y ψ (s, t, u, v, w, x, y)$ where…
Propositional and First-Order Logic
GATE CS 2018NAT2 marksMedium (estimate)
GATE CS 2018, Q.53 (CS Q.43)
Let $G$ be a graph with $100!$ vertices, with each vertex labelled by a distinct permutation of the numbers $1, 2, \dots, 100$ . There is an edge between vertices $u$ and $v$ if...
Graph Theory
GATE CS 2018NAT2 marksMedium (estimate)
GATE CS 2018, Q.54 (CS Q.44)
Consider 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 te...
Probability and Statistics
GATE CS 2017 Set 1MCQ1 markEasy (estimate)
GATE CS 2017 Set 1, Q.1 (CS Q.1)
The statement $(\neg p) ⟹ (\neg q)$ is logically equivalent to which of the statements below? I. $p ⟹ q$ II. $q ⟹ p$ III. $(\neg q) \lor p$ IV.…
Propositional and First-Order Logic
GATE CS 2017 Set 1MCQ1 markMedium (estimate)
GATE CS 2017 Set 1, Q.2 (CS Q.2)
Consider the first-order logic sentence $F : \forall x (\exists y R (x, y))$ . Assuming non-empty logical domains, which of the sentences below are implied by $F$ ? I.…
Propositional and First-Order Logic
GATE CS 2017 Set 1MCQ1 markEasy (estimate)
GATE CS 2017 Set 1, Q.3 (CS Q.3)
Let $c_{1}, \dots, c_{n}$ be scalars, not all zero, such that $\sum_{i = 1}^{n} c_{i} a_{i} = 0$ where $a_{i}$ are column vectors in $R^{n}$ . Consider the set of linear equations…
Linear Algebra
GATE CS 2017 Set 1NAT1 markEasy (estimate)
GATE CS 2017 Set 1, Q.19 (CS Q.19)
Let $X$ be a Gaussian random variable with mean 0 and variance $σ^{2}$ . Let $Y = max (X, 0)$ where $max (a, b)$ is the maximum of $a$ and $b$ . The median of $Y$ is ______.
Probability and Statistics
GATE CS 2017 Set 1NAT1 markEasy (estimate)
GATE CS 2017 Set 1, Q.20 (CS Q.20)
Let $T$ be a tree with 10 vertices. The sum of the degrees of all the vertices in $T$ is ______.
Graph Theory
GATE CS 2017 Set 1MCQ2 marksEasy (estimate)
GATE CS 2017 Set 1, Q.28 (CS Q.28)
The value of $lim_{x \to 1} \frac{x ^{7} - 2 x ^{5} + 1}{x ^{3} - 3 x ^{2} + 2}$
Calculus
GATE CS 2017 Set 1MCQ2 marksMedium (estimate)
GATE CS 2017 Set 1, Q.29 (CS Q.29)
Let $p, q$ , and $r$ be propositions and the expression $(p \to q) \to r$ be a contradiction. Then, the expression $(r \lor p) \to q$ is
Propositional and First-Order Logic
GATE CS 2017 Set 1MCQ2 marksEasy (estimate)
GATE CS 2017 Set 1, Q.30 (CS Q.30)
Let $u$ and $v$ be two vectors in $R^{2}$ whose Euclidean norms satisfy $∥ u ∥ = 2∥ v ∥$ . What is the value of $α$ such that $w = u + α v$ bisects the angle be...
Linear Algebra
GATE CS 2017 Set 1MCQ2 marksMedium (estimate)
GATE CS 2017 Set 1, Q.31 (CS Q.31)
Let $A$ be an $n \times n$ real valued square symmetric matrix of rank 2 with $\sum_{i = 1}^{n} \sum_{j = 1}^{n} A_{ij}^{2} = 50$ . Consider the following statements: (I) One eigenvalue mu...
Linear Algebra
GATE CS 2017 Set 1NAT2 marksEasy (estimate)
GATE CS 2017 Set 1, Q.47 (CS Q.47)
The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is ______.
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2017 Set 2MCQ1 markMedium (estimate)
GATE CS 2017 Set 2, Q.10 (CS Q.10)
If $f (x) = R sin (\frac{π x}{2}) + S$ , $f^{'} (\frac{1}{2}) = 2$ and $\int_{0}^{1} f (x) d x = \frac{2 R}{π}$ , then the constants $R$ and $S$ are, respe...
Calculus
GATE CS 2017 Set 2MCQ1 markEasy (estimate)
GATE CS 2017 Set 2, Q.11 (CS Q.11)
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 pl...
Propositional and First-Order Logic
GATE CS 2017 Set 2NAT1 markMedium (estimate)
GATE CS 2017 Set 2, Q.21 (CS Q.21)
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 Has...
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2017 Set 2NAT1 markEasy (estimate)
GATE CS 2017 Set 2, Q.22 (CS Q.22)
Let $P = 123 1 - 3 - 2 - 1 43$ and $Q = - 1 65 - 2 1210 - 1 65$ be two matrices....
Linear Algebra
GATE CS 2017 Set 2NAT1 markEasy (estimate)
GATE CS 2017 Set 2, Q.23 (CS Q.23)
$G$ 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
Graph Theory
GATE CS 2017 Set 2MCQ2 marksMedium (estimate)
GATE CS 2017 Set 2, Q.26 (CS Q.26)
P and Q are considering to apply for a job. The probability that P applies for the job is $\frac{1}{4}$ , the probability that P applies for the job given that Q applies for the...
Probability and Statistics
GATE CS 2017 Set 2MCQ2 marksMedium (estimate)
GATE CS 2017 Set 2, Q.31 (CS Q.31)
For any discrete random variable $X$ , with probability mass function…
Probability and Statistics
GATE CS 2017 Set 2NAT2 marksMedium (estimate)
GATE CS 2017 Set 2, Q.47 (CS Q.47)
If the ordinary generating function of a sequence ${a_{n}}_{n = 0}^{\infty}$ is $\frac{1 + z}{( 1 - z ) ^{3}}$ , then $a_{3} - a_{0}$ is equal to
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2017 Set 2NAT2 marksEasy (estimate)
GATE CS 2017 Set 2, Q.48 (CS Q.48)
If a random variable $X$ has a Poisson distribution with mean 5, then the expectation $E [(X + 2)^{2}]$ equals
Probability and Statistics
GATE CS 2017 Set 2NAT2 marksEasy (estimate)
GATE CS 2017 Set 2, Q.52 (CS Q.52)
If the characteristic polynomial of a $3 \times 3$ matrix $M$ over $R$ (the set of real numbers) is $λ^{3} - 4 λ^{2} + aλ + 30$ , $a \in R$ , and on...
Linear Algebra
GATE CS 2016 Set 1NAT1 markMedium (estimate)
GATE CS 2016 Set 1, Q.11 (CS Q.1)
Let $p, q, r, s$ represent the following propositions. $p : x \in {8, 9, 10, 11, 12}$ $q : x$ is a composite number $r : x$ is a perfect square $s : x$ is a prime number The integ...
Propositional and First-Order Logic
GATE CS 2016 Set 1MCQ1 markEasy (estimate)
GATE CS 2016 Set 1, Q.12 (CS Q.2)
Let $a_{n}$ be the number of $n$ -bit strings that do NOT contain two consecutive 1s. Which one of the following is the recurrence relation for $a_{n}$ ?
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2016 Set 1NAT1 markEasy (estimate)
GATE CS 2016 Set 1, Q.13 (CS Q.3)
The value of $lim_{x \to 4} \frac{s i n ( x - 4 )}{x - 4}$ is _____.
Calculus
GATE CS 2016 Set 1NAT1 markEasy (estimate)
GATE CS 2016 Set 1, Q.14 (CS Q.4)
A 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 _____.
Probability and Statistics
GATE CS 2016 Set 1NAT1 markEasy (estimate)
GATE CS 2016 Set 1, Q.15 (CS Q.5)
Two eigenvalues of a $3 \times 3$ real matrix $P$ are $(2 + - 1)$ and $3$ . The determinant of $P$ is _____.
Linear Algebra
GATE CS 2016 Set 1NAT2 marksMedium (estimate)
GATE CS 2016 Set 1, Q.36 (CS Q.26)
The coefficient of $x^{12}$ in $(x^{3} + x^{4} + x^{5} + x^{6} + \dots)^{3}$ is _____.
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2016 Set 1NAT2 marksMedium (estimate)
GATE CS 2016 Set 1, Q.37 (CS Q.27)
Consider the recurrence relation $a_{1} = 8$ , $a_{n} = 6 n^{2} + 2 n + a_{n - 1}$ . Let $a_{99} = K \times 1 0^{4}$ . The value of $K$ is _____.
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2016 Set 1NAT2 marksHard (estimate)
GATE CS 2016 Set 1, Q.38 (CS Q.28)
A function $f : N^{+} \to N^{+}$ , defined on the set of positive integers $N^{+}$ , satisfies the following properties…
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2016 Set 1NAT2 marksEasy (estimate)
GATE CS 2016 Set 1, Q.39 (CS Q.29)
Consider the following experiment. Step 1. Flip a fair coin twice. Step 2. If the outcomes are (TAILS, HEADS) then output Y and stop. Step 3. If the outcomes are either (HEADS,...
Probability and Statistics
GATE CS 2016 Set 2NAT1 markMedium (estimate)
GATE CS 2016 Set 2, Q.11 (CS Q.1)
Consider the following expressions: (i) $false$ (ii) $Q$ (iii) $true$ (iv) $P \lor Q$ (v) $\neg Q \lor P$ The number of expressions given above that are logically...
Propositional and First-Order Logic
GATE CS 2016 Set 2NAT1 markMedium (estimate)
GATE CS 2016 Set 2, Q.12 (CS Q.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 ________.
Calculus
GATE CS 2016 Set 2NAT1 markEasy (estimate)
GATE CS 2016 Set 2, Q.13 (CS Q.3)
The minimum number of colours that is sufficient to vertex-colour any planar graph is ________.
Graph Theory
GATE CS 2016 Set 2MCQ1 markEasy (estimate)
GATE CS 2016 Set 2, Q.14 (CS Q.4)
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 h...
Linear Algebra
GATE CS 2016 Set 2NAT1 markEasy (estimate)
GATE CS 2016 Set 2, Q.15 (CS Q.5)
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 gi...
Probability and Statistics
GATE CS 2016 Set 2NAT1 markEasy (estimate)
GATE CS 2016 Set 2, Q.16 (CS Q.6)
Suppose that the eigenvalues of matrix $A$ are 1, 2, 4. The determinant of $(A^{- 1})^{T}$ is ________.
Linear Algebra
GATE CS 2016 Set 2MCQ2 marksMedium (estimate)
GATE CS 2016 Set 2, Q.36 (CS Q.26)
A binary relation $R$ on $N \times N$ is defined as follows: $(a, b) R (c, d)$ if $a \leq c$ or $b \leq d$ . Consider the following propositions: $P$ : $R$ is reflex...
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2016 Set 2MCQ2 marksMedium (estimate)
GATE CS 2016 Set 2, Q.37 (CS Q.27)
Which one of the following well-formed formulae in predicate calculus is NOT valid?
Propositional and First-Order Logic
GATE CS 2016 Set 2MCQ2 marksMedium (estimate)
GATE CS 2016 Set 2, Q.38 (CS Q.28)
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...
Graph Theory
GATE CS 2016 Set 2NAT2 marksEasy (estimate)
GATE CS 2016 Set 2, Q.39 (CS Q.29)
The value of the expression $1 3^{99} (mod 17)$ , in the range 0 to 16, is ________.
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2015 Set 1MCQ1 markEasy (estimate)
GATE CS 2015 Set 1, Q.14 (CS Q.4)
$lim_{x \to \infty} x^{1/ x}$ is:
Calculus
GATE CS 2015 Set 1MCQ1 markMedium (estimate)
GATE CS 2015 Set 1, Q.15 (CS Q.5)
If $g (x) = 1 - x$ and $h (x) = \frac{x}{x - 1}$ , then $\frac{g ( h ( x ))}{h ( g ( x ))}$ is:
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2015 Set 1MCQ1 markEasy (estimate)
GATE CS 2015 Set 1, Q.24 (CS Q.14)
Which one of the following is NOT equivalent to $p \leftrightarrow q$ ?
Propositional and First-Order Logic
GATE CS 2015 Set 1MCQ1 markMedium (estimate)
GATE CS 2015 Set 1, Q.26 (CS Q.16)
For a set $A$ , the power set of $A$ is denoted by $2^{A}$ . If $A = {5, {6}, {7}}$ , which of the following options are TRUE? I. $\emptyset \in 2^{A}$ II.…
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2015 Set 1NAT1 markEasy (estimate)
GATE CS 2015 Set 1, Q.28 (CS Q.18)
In the LU decomposition of the matrix $[2429]$ , if the diagonal elements of $U$ are both $1$ , then the lower diagonal entry $l_{22}$ of $L$ ...
Linear Algebra
GATE CS 2015 Set 1NAT2 marksEasy (estimate)
GATE CS 2015 Set 1, Q.36 (CS Q.26)
$\sum_{x = 1}^{99} \frac{1}{x ( x + 1 )}$ The value of the summation is _____.
Calculus
GATE CS 2015 Set 1MCQ2 marksHard (estimate)
GATE CS 2015 Set 1, Q.44 (CS Q.34)
Suppose $L = {p, q, r, s, t}$ is a lattice represented by the following Hasse diagram: For any $x, y \in L$ , not necessarily distinct, $x \lor y$ and $x \land y$ are join and...
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2015 Set 1MCQ2 marksEasy (estimate)
GATE CS 2015 Set 1, Q.46 (CS Q.36)
Consider the following $2 \times 2$ 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...
Linear Algebra
GATE CS 2015 Set 1NAT2 marksEasy (estimate)
GATE CS 2015 Set 1, Q.54 (CS Q.44)
$\int_{1/ π}^{2/ π} \frac{c o s ( 1/ x )}{x ^{2}} d x$ The value of the integral is _____.
Calculus
GATE CS 2015 Set 1MCQ2 marksMedium (estimate)
GATE CS 2015 Set 1, Q.59 (CS Q.49)
Let $a_{n}$ represent the number of bit strings of length $n$ containing two consecutive 1s. What is the recurrence relation for $a_{n}$ ?
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2015 Set 1NAT2 marksEasy (estimate)
GATE CS 2015 Set 1, Q.64 (CS Q.54)
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 _____.
Graph Theory
GATE CS 2015 Set 2MCQ1 markEasy (estimate)
GATE CS 2015 Set 2, Q.13 (CS Q.3)
Consider the following two 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...
Propositional and First-Order Logic
GATE CS 2015 Set 2NAT1 markEasy (estimate)
GATE CS 2015 Set 2, Q.15 (CS Q.5)
The larger of the two eigenvalues of the matrix $[4251]$ is _____.
Linear Algebra
GATE CS 2015 Set 2NAT1 markEasy (estimate)
GATE CS 2015 Set 2, Q.19 (CS Q.9)
The number of divisors of 2100 is _____.
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2015 Set 2MCQ1 markEasy (estimate)Not in current syllabus
GATE CS 2015 Set 2, Q.22 (CS Q.12)
Consider the basic COCOMO model where $E$ is the effort applied in person-months, $D$ is the development time in chronological months, KLOC is the estimated number of delivered...
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2015 Set 2MCQ1 markEasy (estimate)
GATE CS 2015 Set 2, Q.26 (CS Q.16)
Let $R$ be the relation on the set of positive integers such that $a R b$ if and only if $a$ and $b$ are distinct and have a common divisor other than 1. Which one of the followin...
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2015 Set 2NAT1 markEasy (estimate)
GATE CS 2015 Set 2, Q.28 (CS Q.18)
The cardinality of the power set of ${0, 1, 2, \dots, 10}$ is _____.
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2015 Set 2MCQ2 marksMedium (estimate)
GATE CS 2015 Set 2, Q.36 (CS Q.26)
Let $f (x) = x^{- (1/3)}$ and $A$ denote the area of the region bounded by $f (x)$ and the X-axis, when $x$ varies from $- 1$ to 1. Which of the following statements is/are TRUE? I)...
Calculus
GATE CS 2015 Set 2NAT2 marksEasy (estimate)
GATE CS 2015 Set 2, Q.37 (CS Q.27)
Perform the following operations on the matrix $371349245105195$ : (i) Add the third row to the second row. (ii) Subtract...
Linear Algebra
GATE CS 2015 Set 2MCQ2 marksEasy (estimate)
GATE CS 2015 Set 2, Q.38 (CS Q.28)
A graph is self-complementary if it is isomorphic to its complement. For all self-complementary graphs on $n$ vertices, $n$ is
Graph Theory
GATE CS 2015 Set 2MCQ2 marksEasy (estimate)
GATE CS 2015 Set 2, Q.49 (CS Q.39)
The secant method is used to find the root of an equation $f (x) = 0$ . It is started from two distinct estimates $x_{a}$ and $x_{b}$ for the root. It is an iterative procedure involv...
Calculus
GATE CS 2015 Set 2NAT2 marksMedium (estimate)
GATE CS 2015 Set 2, Q.50 (CS Q.40)
The number of onto functions (surjective functions) from set $X = {1, 2, 3, 4}$ to set $Y = {a, b, c}$ is _____.
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2015 Set 2MCQ2 marksEasy (estimate)
GATE CS 2015 Set 2, Q.60 (CS Q.50)
In a connected graph, a bridge is an edge whose removal disconnects a graph. Which one of the following statements is true?
Graph Theory
GATE CS 2015 Set 2NAT2 marksEasy (estimate)
GATE CS 2015 Set 2, Q.64 (CS Q.54)
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 c...
Probability and Statistics
GATE CS 2015 Set 2MCQ2 marksEasy (estimate)
GATE CS 2015 Set 2, Q.65 (CS Q.55)
Which one of the following well formed formulae is a tautology?
Propositional and First-Order Logic
GATE CS 2015 Set 3NAT1 markMedium (estimate)
GATE CS 2015 Set 3, Q.15 (CS Q.5)
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 _____.
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2015 Set 3MCQ1 markEasy (estimate)
GATE CS 2015 Set 3, Q.19 (CS Q.9)
The value of $lim_{x \to \infty} (1 + x^{2})^{e^{- x}}$ is
Calculus
GATE CS 2015 Set 3NAT1 markMedium (estimate)Not in current syllabus
GATE CS 2015 Set 3, Q.21 (CS Q.11)
Consider a software program that is artificially seeded with 100 faults. While testing this program, 159 faults are detected, out of which 75 faults are from those artificially...
Probability and Statistics
GATE CS 2015 Set 3MCQ1 markEasy (estimate)
GATE CS 2015 Set 3, Q.25 (CS Q.15)
In the given matrix $101 - 1 12 201$ one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are
Linear Algebra
GATE CS 2015 Set 3NAT1 markMedium (estimate)Not in current syllabus
GATE CS 2015 Set 3, Q.31 (CS Q.21)
Consider a software project with the following information domain characteristics for calculation of function point metric.…
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2015 Set 3MCQ1 markEasy (estimate)
GATE CS 2015 Set 3, Q.33 (CS Q.23)
Suppose $U$ is the power set of the set $S = {1, 2, 3, 4, 5, 6}$ . For any $T \in U$ , let $∣ T ∣$ denote the number of elements in $T$ and $T^{'}$ denote the complement of $T$ . For...
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2015 Set 3MCQ1 markMedium (estimate)
GATE CS 2015 Set 3, Q.34 (CS Q.24)
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 tha...
Propositional and First-Order Logic
GATE CS 2015 Set 3MCQ2 marksMedium (estimate)
GATE CS 2015 Set 3, Q.43 (CS Q.33)
If the following system has non-trivial solution, $p x + q y + r z q x + r y + p z r x + p y + q z = 0 = 0 = 0$ then which one of the...
Linear Algebra
GATE CS 2015 Set 3NAT2 marksEasy (estimate)
GATE CS 2015 Set 3, Q.47 (CS Q.37)
Suppose $X_{i}$ 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…
Probability and Statistics
GATE CS 2015 Set 3MCQ2 marksMedium (estimate)
GATE CS 2015 Set 3, Q.51 (CS Q.41)
Let $R$ be a relation on the set of ordered pairs of positive integers such that $((p, q), (r, s)) \in R$ if and only if $p - s = q - r$ . Which one of the following is true abou...
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2015 Set 3MCQ2 marksHard (estimate)
GATE CS 2015 Set 3, Q.55 (CS Q.45)
If for non-zero $x$ , $a f (x) + b f (\frac{1}{x}) = \frac{1}{x} - 25$ where $a \neq = b$ then $\int_{1}^{2} f (x) d x$ is
Calculus
GATE CS 2015 Set 3NAT2 marksEasy (estimate)
GATE CS 2015 Set 3, Q.59 (CS Q.49)
Suppose $c = (c [0], \dots, c [k - 1])$ is an array of length $k$ , where all the entries are from the set ${0, 1}$ . For any positive integers $a$ and $n$ , consider the following...
Combinatorics, Counting, Recurrence Relations and Generating Functions · Asymptotic Analysis and Time/Space Complexity
GATE CS 2015 Set 3NAT2 marksMedium (estimate)
GATE CS 2015 Set 3, Q.60 (CS Q.50)
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: t 2 4 6 8 10 12 14 16 18 20 v 10 1...
Calculus
GATE CS 2015 Set 3MCQ2 marksMedium (estimate)Not in current syllabus
GATE CS 2015 Set 3, Q.65 (CS Q.55)
Consider three software items: Program-X, Control Flow Diagram of Program-Y and Control Flow Diagram of Program-Z as shown below. Program-X: sumcal(int maxint, int value) {...
Graph Theory
GATE CS 2014 Set 1MCQ1 markEasy (estimate)
GATE CS 2014 Set 1, Q.11 (CS Q.1)
Consider the statement $“Not all that glitters is gold”$ Predicate $g l i tt er s (x)$ is true if $x$ glitters and predicate $g o l d (x)$ is true if $x$ is gold. Which one of th...
Propositional and First-Order Logic
GATE CS 2014 Set 1NAT1 markMedium (estimate)
GATE CS 2014 Set 1, Q.12 (CS Q.2)
Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .
Probability and Statistics
GATE CS 2014 Set 1NAT1 markMedium (estimate)
GATE CS 2014 Set 1, Q.14 (CS Q.4)
Consider the following system of equations: $3 x + 2 y = 1$ $4 x + 7 z = 1$ $x + y + z = 3$ $x - 2 y + 7 z = 0$ The number of solutions for this system is __________________
Linear Algebra
GATE CS 2014 Set 1NAT1 markEasy (estimate)
GATE CS 2014 Set 1, Q.15 (CS Q.5)
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 _____________________.
Linear Algebra
GATE CS 2014 Set 1MCQ1 markMedium (estimate)
GATE CS 2014 Set 1, Q.16 (CS Q.6)
Let the function…
Calculus
GATE CS 2014 Set 1NAT2 marksEasy (estimate)
GATE CS 2014 Set 1, Q.47 (CS Q.37)
There are 5 bags labeled 1 to 5. All the coins in a given bag have the same weight. Some bags have coins of weight 10 gm, others have coins of weight 11 gm. I pick 1, 2, 4, 8, 1...
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2014 Set 1NAT2 marksEasy (estimate)
GATE CS 2014 Set 1, Q.56 (CS Q.46)
The function $f (x) = x sin x$ satisfies the following equation: $f^{''} (x) + f (x) + t cos x = 0$ . The value of $t$ is ______.
Calculus
GATE CS 2014 Set 1MCQ2 marksMedium (estimate)
GATE CS 2014 Set 1, Q.57 (CS Q.47)
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?
Calculus
GATE CS 2014 Set 1NAT2 marksMedium (estimate)
GATE CS 2014 Set 1, Q.58 (CS Q.48)
Four fair six-sided dice are rolled. The probability that the sum of the results being 22 is $X /1296$ . The value of $X$ is ______________.
Probability and Statistics
GATE CS 2014 Set 1NAT2 marksMedium (estimate)
GATE CS 2014 Set 1, Q.59 (CS Q.49)
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 poss...
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2014 Set 1NAT2 marksEasy (estimate)
GATE CS 2014 Set 1, Q.60 (CS Q.50)
Let $S$ denote the set of all functions $f : {0, 1}^{4} \to {0, 1}$ . Denote by $N$ the number of functions from $S$ to the set ${0, 1}$ . The value of $lo g_{2} lo g_{2} N$ is ______.
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2014 Set 1NAT2 marksMedium (estimate)
GATE CS 2014 Set 1, Q.61 (CS Q.51)
Consider an undirected graph $G$ where self-loops are not allowed. The vertex set of $G$ is ${(i, j) : 1 \leq i \leq 12, 1 \leq j \leq 12}$ . There is an edge between $(a, b)$ and…
Graph Theory
GATE CS 2014 Set 1MCQ2 marksMedium (estimate)
GATE CS 2014 Set 1, Q.62 (CS Q.52)
An ordered $n$ -tuple $(d_{1}, d_{2}, \dots, d_{n})$ with $d_{1} \geq d_{2} \geq \dots \geq d_{n}$ is called graphic if there exists a simple undirected graph with vertices having degrees…
Graph Theory
GATE CS 2014 Set 1MCQ2 marksEasy (estimate)
GATE CS 2014 Set 1, Q.63 (CS Q.53)
Which one of the following propositional logic formulas is TRUE when exactly two of $p$ , $q$ , and $r$ are TRUE?
Propositional and First-Order Logic
GATE CS 2014 Set 2NAT1 markMedium (estimate)
GATE CS 2014 Set 2, Q.11 (CS Q.1)
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 c...
Probability and Statistics
GATE CS 2014 Set 2NAT1 markEasy (estimate)
GATE CS 2014 Set 2, Q.12 (CS Q.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 o...
Probability and Statistics
GATE CS 2014 Set 2NAT1 markEasy (estimate)
GATE CS 2014 Set 2, Q.13 (CS Q.3)
The maximum number of edges in a bipartite graph on 12 vertices is _____.
Graph Theory
GATE CS 2014 Set 2NAT1 markEasy (estimate)
GATE CS 2014 Set 2, Q.14 (CS Q.4)
If the matrix $A$ is such that $A = 2 - 4 7 [195]$ then the determinant of $A$ is equal to _____.
Linear Algebra
GATE CS 2014 Set 2MCQ1 markEasy (estimate)
GATE CS 2014 Set 2, Q.15 (CS Q.5)
A non-zero polynomial $f (x)$ of degree 3 has roots at $x = 1, x = 2$ and $x = 3$ . Which one of the following must be TRUE?
Calculus
GATE CS 2014 Set 2MCQ2 marksMedium (estimate)
GATE CS 2014 Set 2, Q.56 (CS Q.46)
In the Newton-Raphson method, an initial guess of $x_{0} = 2$ is made and the sequence $x_{0}, x_{1}, x_{2} \dots$ is obtained for the function $0.75 x^{3} - 2 x^{2} - 2 x + 4 = 0$ Consider...
Calculus
GATE CS 2014 Set 2NAT2 marksMedium (estimate)
GATE CS 2014 Set 2, Q.57 (CS Q.47)
The product of the non-zero eigenvalues of the matrix…
Linear Algebra
GATE CS 2014 Set 2NAT2 marksEasy (estimate)
GATE CS 2014 Set 2, Q.58 (CS Q.48)
The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is _____.
Probability and Statistics
GATE CS 2014 Set 2NAT2 marksEasy (estimate)
GATE CS 2014 Set 2, Q.59 (CS Q.49)
The number of distinct positive integral factors of 2014 is _____.
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2014 Set 2MCQ2 marksHard (estimate)
GATE CS 2014 Set 2, Q.60 (CS Q.50)
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 th...
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2014 Set 2NAT2 marksEasy (estimate)
GATE CS 2014 Set 2, Q.61 (CS Q.51)
A cycle on $n$ vertices is isomorphic to its complement. The value of $n$ is _____.
Graph Theory
GATE CS 2014 Set 2MCQ2 marksEasy (estimate)
GATE CS 2014 Set 2, Q.63 (CS Q.53)
Which one of the following Boolean expressions is NOT a tautology?
Propositional and First-Order Logic
GATE CS 2014 Set 3MCQ1 markEasy (estimate)
GATE CS 2014 Set 3, Q.11 (CS Q.1)
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...
Propositional and First-Order Logic
GATE CS 2014 Set 3MCQ1 markMedium (estimate)
GATE CS 2014 Set 3, Q.12 (CS Q.2)
Let $X$ and $Y$ be finite sets and $f : X \to Y$ be a function. Which one of the following statements is TRUE?
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2014 Set 3NAT1 markEasy (estimate)
GATE CS 2014 Set 3, Q.13 (CS Q.3)
Let $G$ be a group with 15 elements. Let $L$ be a subgroup of $G$ . It is known that $L \neq = G$ and that the size of $L$ is at least 4. The size of $L$ is _____.
Monoids and Groups
GATE CS 2014 Set 3MCQ1 markEasy (estimate)
GATE CS 2014 Set 3, Q.14 (CS Q.4)
Which one of the following statements is TRUE about every $n \times n$ matrix with only real eigenvalues?
Linear Algebra
GATE CS 2014 Set 3NAT1 markEasy (estimate)
GATE CS 2014 Set 3, Q.15 (CS Q.5)
If $V_{1}$ and $V_{2}$ are 4-dimensional subspaces of a 6-dimensional vector space $V$ , then the smallest possible dimension of $V_{1} \cap V_{2}$ is _____.
Linear Algebra
GATE CS 2014 Set 3NAT1 markMedium (estimate)
GATE CS 2014 Set 3, Q.16 (CS Q.6)
If $\int_{0}^{2 π} ∣ x sin x ∣ d x = k π$ , then the value of $k$ is equal to _____.
Calculus
GATE CS 2014 Set 3MCQ2 marksMedium (estimate)
GATE CS 2014 Set 3, Q.56 (CS Q.46)
With respect to the numerical evaluation of the definite integral, $K = \int_{a}^{b} x^{2} d x$ , where $a$ and $b$ are given, which of the following statements is/are TRUE? I) The v...
Calculus
GATE CS 2014 Set 3MCQ2 marksMedium (estimate)
GATE CS 2014 Set 3, Q.57 (CS Q.47)
The value of the integral given below is $\int_{0}^{π} x^{2} cos x d x$
Calculus
GATE CS 2014 Set 3NAT2 marksEasy (estimate)
GATE CS 2014 Set 3, Q.58 (CS Q.48)
Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$ . If $P (\cdot)$ denotes the probability of the event, the maximum value of…
Probability and Statistics
GATE CS 2014 Set 3MCQ2 marksMedium (estimate)
GATE CS 2014 Set 3, Q.59 (CS Q.49)
Consider the set of all functions $f : {0, 1, \dots, 2014} \to {0, 1, \dots, 2014}$ such that $f (f (i)) = i$ , for all $0 \leq i \leq 2014$ . Consider the following statements: P....
Sets, Relations, Functions, Partial Orders and Lattices
GATE CS 2014 Set 3NAT2 marksMedium (estimate)
GATE CS 2014 Set 3, Q.60 (CS Q.50)
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 t...
Monoids and Groups
GATE CS 2014 Set 3MCQ2 marksEasy (estimate)
GATE CS 2014 Set 3, Q.61 (CS Q.51)
If $G$ is a forest with $n$ vertices and $k$ connected components, how many edges does $G$ have?
Graph Theory
GATE CS 2014 Set 3MCQ2 marksMedium (estimate)
GATE CS 2014 Set 3, Q.62 (CS Q.52)
Let $δ$ denote the minimum degree of a vertex in a graph. For all planar graphs on $n$ vertices with $δ \geq 3$ , which one of the following is TRUE?
Graph Theory
GATE CS 2014 Set 3MCQ2 marksEasy (estimate)
GATE CS 2014 Set 3, Q.63 (CS Q.53)
The CORRECT formula for the sentence, "not all rainy days are cold" is
Propositional and First-Order Logic
GATE CS 2013MCQ1 markEasy (estimate)
GATE CS 2013, Q.1 (CS Q.1)
A binary operation $\oplus$ on a set of integers is defined as $x \oplus y = x^{2} + y^{2}$ . Which one of the following statements is TRUE about $\oplus$ ?
Monoids and Groups
GATE CS 2013MCQ1 markEasy (estimate)
GATE CS 2013, Q.2 (CS Q.2)
Suppose $p$ is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and $p$ has a Poisson distribution with mean 3. What is the probabili...
Probability and Statistics
GATE CS 2013MCQ1 markMedium (estimate)
GATE CS 2013, Q.3 (CS Q.3)
Which one of the following does NOT equal $111 x y z x^{2} y^{2} z^{2}$ ?
Linear Algebra
GATE CS 2013MCQ1 markEasy (estimate)
GATE CS 2013, Q.22 (CS Q.22)
Which one of the following functions is continuous at $x = 3$ ?
Calculus
GATE CS 2013MCQ1 markEasy (estimate)
GATE CS 2013, Q.23 (CS Q.23)
Function $f$ is known at the following points…
Calculus
GATE CS 2013MCQ1 markEasy (estimate)
GATE CS 2013, Q.24 (CS Q.24)
Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is $1/2$ . What is the expected number of unordered cycles...
Probability and Statistics
GATE CS 2013MCQ1 markEasy (estimate)
GATE CS 2013, Q.25 (CS Q.25)
Which of the following statements is/are TRUE for undirected graphs? P: Number of odd degree vertices is even. Q: Sum of degrees of all vertices is even.
Graph Theory
GATE CS 2013MCQ2 marksMedium (estimate)
GATE CS 2013, Q.26 (CS Q.26)
The line graph $L (G)$ of a simple graph $G$ is defined as follows: - There is exactly one vertex $v (e)$ in $L (G)$ for each edge $e$ in $G$ . - For any two edges $e$ and $e^{'}$ in…
Graph Theory
GATE CS 2013MCQ2 marksEasy (estimate)
GATE CS 2013, Q.27 (CS Q.27)
What is the logical translation of the following statement? "None of my friends are perfect."
Propositional and First-Order Logic
GATE CS 2013MCQ2 marksMedium (estimate)
GATE CS 2013, Q.47 (CS Q.47)
Which one of the following is NOT logically equivalent to $\neg\exists x (\forall y (α) \land \forall z (β))$ ?
Propositional and First-Order Logic
GATE CS 2012MCQ1 markEasy (estimate)
GATE CS 2012, Q.1 (CS Q.1)
Consider the following logical inferences. $I_{1}$ : If it rains then the cricket match will not be played. The cricket match was played. Inference: There was no rain. $I_{2}$ : If it...
Propositional and First-Order Logic
GATE CS 2012MCQ1 markMedium (estimate)
GATE CS 2012, Q.9 (CS Q.9)
Consider the function $f (x) = sin (x)$ in the interval $x \in [π /4, 7 π /4]$ . The number and location(s) of the local minima of this function are
Calculus
GATE CS 2012MCQ1 markMedium (estimate)
GATE CS 2012, Q.11 (CS Q.11)
Let $A$ be the $2 \times 2$ matrix with elements $a_{11} = a_{12} = a_{21} = + 1$ and $a_{22} = - 1$ . Then the eigenvalues of the matrix $A^{19}$ are
Linear Algebra
GATE CS 2012MCQ1 markEasy (estimate)
GATE CS 2012, Q.13 (CS Q.13)
What is the correct translation of the following statement into mathematical logic? "Some real numbers are rational"
Propositional and First-Order Logic
GATE CS 2012MCQ1 markEasy (estimate)
GATE CS 2012, Q.17 (CS Q.17)
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 e...
Graph Theory
GATE CS 2012MCQ1 markEasy (estimate)
GATE CS 2012, Q.21 (CS Q.21)
Consider a random variable $X$ that takes values $+ 1$ and $- 1$ with probability $0.5$ each. The values of the cumulative distribution function $F (x)$ at $x = - 1$ and $+ 1$ are
Probability and Statistics
GATE CS 2012MCQ2 marksMedium (estimate)
GATE CS 2012, Q.26 (CS Q.26)
Which of the following graphs is isomorphic to the given graph? Given graph with 8 vertices and candidate graphs for isomorphism in options A, B, C, and D Stem graph and options...
Graph Theory
GATE CS 2012MCQ2 marksEasy (estimate)
GATE CS 2012, Q.28 (CS Q.28)
The bisection method is applied to compute a zero of the function $f (x) = x^{4} - x^{3} - x^{2} - 4$ in the interval $[1, 9]$ . The method converges to a solution after ----- iterations.
Calculus
GATE CS 2012MCQ2 marksEasy (estimate)
GATE CS 2012, Q.33 (CS Q.33)
Suppose a fair six-sided die is rolled once. If the value on the die is 1, 2, or 3, the die is rolled a second time. What is the probability that the sum total of values that tu...
Probability and Statistics
GATE CS 2012MCQ2 marksEasy (estimate)
GATE CS 2012, Q.37 (CS Q.37)
How many onto (or surjective) functions are there from an $n$ -element ( $n \geq 2$ ) set to a 2-element set?
Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2012MCQ2 marksMedium (estimate)
GATE CS 2012, Q.38 (CS Q.38)
Let $G$ be a complete undirected graph on 6 vertices. If vertices of $G$ are labeled, then the number of distinct cycles of length 4 in $G$ is equal to
Graph Theory · Combinatorics, Counting, Recurrence Relations and Generating Functions
GATE CS 2011MCQ1 markEasy (estimate)
GATE CS 2011, Q.3 (CS Q.3)
If two fair coins are flipped and at least one of the outcomes is known to be a head, what is the probability that both outcomes are heads?
Probability and Statistics
GATE CS 2011MCQ1 markEasy (estimate)
GATE CS 2011, Q.17 (CS Q.17)
$K_{4}$ and $Q_{3}$ are graphs with the following structures. Graphs K4 (complete graph on 4 vertices) and Q3 (3-dimensional hypercube graph with 8 vertices) Which one of the follow...
Graph Theory
GATE CS 2011MCQ1 markEasy (estimate)
GATE CS 2011, Q.18 (CS Q.18)
If the difference between the expectation of the square of a random variable ( $E [X^{2}]$ ) and the square of the expectation of the random variable ( $(E [X])^{2}$ ) is denoted by $R$ ,...
Probability and Statistics
GATE CS 2011MCQ2 marksEasy (estimate)
GATE CS 2011, Q.30 (CS Q.30)
Which one of the following options is CORRECT given three positive integers $x$ , $y$ and $z$ , and a predicate…
Propositional and First-Order Logic
GATE CS 2011MCQ2 marksEasy (estimate)
GATE CS 2011, Q.31 (CS Q.31)
Given $i = - 1$ , what will be the evaluation of the definite integral $\int_{0}^{π /2} \frac{c o s x + i s i n x}{c o s x - i s i n x} d x ?$
Calculus
GATE CS 2011MCQ2 marksEasy (estimate)
GATE CS 2011, Q.33 (CS Q.33)
Consider a finite sequence of random values $X = [x_{1}, x_{2}, \dots, x_{n}]$ . Let $μ_{x}$ be the mean and $σ_{x}$ be the standard deviation of $X$ . Let another finite sequence…
Probability and Statistics
GATE CS 2011MCQ2 marksEasy (estimate)
GATE CS 2011, Q.34 (CS Q.34)
A deck of 5 cards (each carrying a distinct number from 1 to 5) is shuffled thoroughly. Two cards are then removed one at a time from the deck. What is the probability that the...
Probability and Statistics
GATE CS 2011MCQ2 marksEasy (estimate)
GATE CS 2011, Q.40 (CS Q.40)
Consider the matrix as given below. $100240373$ Which one of the following options provides the CORRECT values of the e...
Linear Algebra