GATE CSE & IT

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

For $n>1$, the maximum multiplicity of any eigenvalue of an $n \times n$ matrix with elements from $\mathbb{R}$ is

Let $n>1$. Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$. Let the vector ( 0,1 , $0,0, \ldots, 0) \in \mathbb{R}^n$ be in the null space of $M$. Which of...

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as follows: $$ f(x)=\left\{\begin{array}{cc} c_1 e^x-c_2 \log _e\left(\frac{1}{x}\right), & \text { if } x>0 \\...

Let $G(V, E)$ be a simple, undirected graph. A vertex cover of $G$ is a subset $V^{\prime} \subseteq V$ such that for every $(u, v) \in E, u \in V^{\prime \prime}$ or $v \in V^{\pr...

Let $G$ be an undirected graph, which is a path on 8 vertices. The number of matchings in $G$ is $\_\_\_\_$ (answer in integer)

Let $X$ be a random variable which takes values in the set $\{1,2,3,4,5,6,7,8\}$. Further, $\operatorname{Pr}(X=1)=\operatorname{Pr}(X=2)=\operatorname{Pr}(X=5)=\operatorname{Pr}(X...

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 th...

The probability density function $f(x)$ of a random variable $X$ which takes real values is $$ f(x)=\frac{1}{3 \sqrt{2 \pi}} \exp \left(-\frac{x^2}{18}\right), x \in(-\infty,+\inft...

Let $R$ be a binary relation on the set $\{1,2, \ldots, 10\}$, where $(x, y) \in, R$ if the product of $x$ and $y$ is square of an integer. Which of the following properties is/are...

For a real number $a$, let $I(a)=\int\limits_{-1}^1\left(3 x^2-a x+1\right) d x$. Which of the following statements is/are true?

Consider the system of linear equations given below. $$ \begin{aligned} a x+y & =b \\ 16 x+a y & =24 \end{aligned} $$ Suppose the values of a and b are chosen such that the system...

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 set o...

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 tosses,...

Consider a function $f:(0,1) \rightarrow\{0,1\}$ 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 the four di...

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as follows: $$ f(x)=\left(\frac{|x|}{2}-x\right)\left(x-\frac{|x|}{2}\right) $$ Which of the following statements is/are true?

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 put b...

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 ar...

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, that...

Let $A$ be a $2 \times 2$ matrix as given. $$A=\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]$$ What are the eigenvalues of the matrix $A^{13}$ ?

Consider the given system of linear equations for variables $x$ and $y$, where $k$ is a realvalued constant. Which of the following option(s) is/are CORRECT? $$\begin{aligned} & x+...

$A=\{0,1,2,3, \ldots\}$ is the set of non-negative integers. Let $F$ be the set of functions from $A$ to itself. For any two functions, $f_1, f_2 \in \mathrm{~F}$ we define $$\left...

Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. F...

Consider a probability distribution given by the density function $P(x)$. $$P(x)=\left\{\begin{array}{cc} C x^2, & \text { for } 1 \leq x \leq 4 \\ 0, & \text { for } x 4 \end{arra...

Consider the given function $f(x)$. $$f(x)=\left\{\begin{array}{cc} a x+b & \text { for } x If the function is differentiable everywhere, the value of $b$ must be _________ (Rounde...

$g(.)$ 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 onto (surjectiv...

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 pick a coin at r...

The value of $x$ such that $x>1$, satisfying the equation $\int_1^x t \ln t d t=\frac{1}{4}$ is

Consider a system of linear equations $P X=Q$ where $P \in \mathbb{R}^{3 \times 3}$ and $Q \in \mathbb{R}^{3 \times 3}$. Suppose $P$ has an $L U$ decomposition, $P=L U$, where $$L=...

A quadratic polynomial $(x-\alpha)(x-\beta)$ over complex numbers is said to be square invariant if $(x-\alpha)(x-\beta)=\left(x-\alpha^2\right)\left(x-\beta^2\right)$. Suppose fro...

If $A=\left(\begin{array}{cc}1 & 2 \\ 2 & -1\end{array}\right)$, then which ONE of the following is $A^8$ ?

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 determinant of $(M-...

Let $P(x)$ be an arbitrary predicate over the domain of natural numbers. Which ONE of the following statements is TRUE?

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 subinterval that contain...

Let $F$ be the set of all functions from $\{1, \ldots, n\}$ to $\{0,1\}$. Define the binary relation $\preccurlyeq$ on $F$ as follows: $\forall f . g \in F, f \preccurlyeq g$ if an...

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x) = \max \{x, x^3\}, x \in \mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers. The set of all po...

The product of all eigenvalues of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ is

Consider a permutation sampled uniformly at random from the set of all permutations of {1, 2, 3, ..., n } for some n ≥ 4. Let X be the event that 1 occurs before 2 in the permutati...

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?

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 va...

The number of spanning trees in a complete graph of 4 vertices labelled A, B, C, and D is __________

Let A be any n x m matrix, where m > n . Which of the following statements is/are TRUE about the system of linear equations Ax = 0 ?

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 $k$. Which of...

Consider the operators $\diamond$ and $\square$ defined by $a \diamond b=a+2 b, a \square b=a b$, for positive integers. Which of the following statements is/are TRUE?

The number of edges present in the forest generated by the DFS traversal of an undirected graph G with 100 vertices is 40. The number of connected components in G is ________

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 decimal p...

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 student sco...

Let $f(x)$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x) = 1 - f(2 - x)$ Which one of the following options is the CORRECT value of $\int_0^2 f(x) dx$?

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?

When six unbiased dice are rolled simultaneously, the probability of getting all distinct numbers (i.e., 1, 2, 3, 4, 5, and 6) is

Let $P$ be the partial order defined on the set {1,2,3,4} as follows: $P = \{(x, x) \mid x \in \{1,2,3,4\}\} \cup \{(1,2), (3,2), (3,4)\}$ The number of total orders on {1,2,3,4} t...

Let $ x $ and $ y $ be random variables, not necessarily independent, that take real values in the interval $[0,1]$. Let $ z = xy $ and let the mean values of $ x, y, z $ be $ \bar...

Let A be an n × 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/are TRUE?

Let Z n be the group of integers {0, 1, 2, ..., n − 1} with addition modulo n as the group operation. The number of elements in the group Z 2 × Z 3 × Z 4 that are their own inverse...

The Lucas sequence $$L_n$$ is defined by the recurrence relation: $${L_n} = {L_{n - 1}} + {L_{n - 2}}$$, for $$n \ge 3$$, with $${L_1} = 1$$ and $${L_2} = 3$$. Which one of the opt...

Let $$A = \left[ {\matrix{ 1 & 2 & 3 & 4 \cr 4 & 1 & 2 & 3 \cr 3 & 4 & 1 & 2 \cr 2 & 3 & 4 & 1 \cr } } \right]$$ and $$B = \left[ {\matrix{ 3 & 4 & 1 & 2 \cr 4 & 1 & 2 & 3 \cr 1 &...

Geetha has a conjecture about integers, which is of the form $$\forall x\left( {P(x) \Rightarrow \exists yQ(x,y)} \right)$$, where P is a statement about integers, and Q is a state...

Let $$f(x) = {x^3} + 15{x^2} - 33x - 36$$ be a real-valued function. Which of the following statements is/are TRUE?

The value of the definite integral $$\int\limits_{ - 3}^3 {\int\limits_{ - 2}^2 {\int\limits_{ - 1}^1 {(4{x^2}y - {z^3})dz\,dy\,dx} } } $$ is ___________. (Rounded off to the neare...

Let $$U = \{ 1,2,....,n\} $$, 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 $$|A| = |B| = k$$ and $$...

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 $${a_1} \sim {a_2}$$ if $$f({a_1}) =...

Let X be a set and 2$$^X$$ denote the powerset of X. Define a binary operation $$\Delta$$ on 2$$^X$$ as follows: $$A\Delta B=(A-B)\cup(B-A)$$. Let $$H=(2^X,\Delta)$$. Which of the...

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, and C be...

Let G be a simple, finite, undirected graph with vertex set {$$v_1,...,v_n$$}. Let $$\Delta(G)$$ denote the maximum degree of G and let N = {1, 2, ...} denote the set of all possib...

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 and on...

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 : tr(AB) = tr(BA) Statem...

Which of the following statements is/are TRUE for a group G?

Consider a simple undirected graph of 10 vertices. If the graph is disconnected, then the maximum number of edges it can have is __________.

The number of arrangements of six identical balls in three identical bins is ___________.

The value of the following limit is _____________. $$\mathop {\lim }\limits_{x \to {0^ + }} {{\sqrt x } \over {1 - {e^{2\sqrt x }}}}$$

Which one of the following is the closed form for the generating function of the sequence (a n } n $$\ge$$ 0 defined below? $${a_n} = \left\{ {\matrix{ {n + 1,} & {n\,is\,odd} \cr...

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 trac...

Consider solving the following system of simultaneous equations using LU decomposition. x 1 + x 2 $$-$$ 2x 3 = 4 x 1 + 3x 2 $$-$$ x 3 = 7 2x 1 + x 2 $$-$$ 5x 3 = 7 where L and U ar...

Consider a simple undirected weighted graph G, all of whose edge weights are distinct. Which of the following statements about the minimum spanning trees of G is/are TRUE?

Which of the properties hold for the adjacency matrix A of a simple undirected unweighted graph having n vertices?

Which of the following is/are the eigenvector(s) for the matrix given below? $$\left( {\matrix{ { - 9} & { - 6} & { - 2} & { - 4} \cr { - 8} & { - 6} & { - 3} & { - 1} \cr {20} & {...

Let p and q be two propositions. Consider the following two formulae in propositional logic. S 1 : (¬p ∧ (p ∨ q)) → q S 2 : q → (¬p ∧ (p ∨ q)) Which one of the following choices is...

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 tou...

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] S 2 : For all random variables X and Y, Cov[X, Y] = E...

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?