GATE CS question library

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x)=\max \left\{x, x^3\right\}, x \in \mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers. The set of all points where $f(x)$ is NOT differentiable is

The product of all eigenvalues of the matrix $\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$ is

Consider a system that uses $5$ bits for representing signed integers in $2$ 's complement format. In this system, two integers $A$ and $B$ are represented as $A$=$01010$ and $B$=$11010$. Which one of the following operations will result in either an arithmetic overflow or an arithmetic underflow?

Consider a permutation sampled uniformly at random from the set of all permutations of $\{1,2,3, \cdots, n\}$ for some $n \geq 4$. Let $X$ be the event that $1$ occurs before $2$ in the permutation, and $Y$ the event that $3$ occurs before $4$. Which one of the following statements is TRUE?

​​​​Which one of the following statements is FALSE?

A user starts browsing a webpage hosted at a remote server. The browser opens a single TCP connection to fetch the entire webpage from the server. The webpage consists of a top-level index page with multiple embedded image objects. Assume that all caches (e.g., DNS cache, browser cache) are all initially empty. The following packets leave the user's computer …

Given an integer array of size $N$, we want to check if the array is sorted (in either ascending or descending order). An algorithm solves this problem by making a single pass through the array and comparing each element of the array only with its adjacent elements. The worst-case time complexity of this algorithm is

Consider the following $\mathrm{C}$ program: #include <stdio.h> int main() { int a=6; int b = 0; while (a<10) { a = a / 12+1 ; a += b ;} printf ("%d", a); return 0 ; } Which one of the following statements is CORRECT?

​​​Consider the following $\mathrm{C}$ program: #include <stdio.h> void fX (); int main(){ fX(); return 0 }; void fX () { char a; if ((a=g e t c h a r()) ! = '\n') fX(); if (a ! = '\n') putchar (a); } Assume that the input to the program from the command line is $1234$ followed by a newline character. Which one of the following statements is …

Let $\mathrm{S}$ be the specification: "Instructors teach courses. Students register for courses. Courses are allocated classrooms. Instructors guide students." Which one of the following $\mathrm{ER}$ diagrams CORRECTLY represents $\mathrm{S}$?

In a $\mathrm{B}^+$ tree, the requirement of at least half-full $(50 \%)$ node occupancy is relaxed for which one of the following cases?

Which of the following statements about a relation $\mathbf{R}$ in first normal form $\text{(1NF)}$ is/are TRUE?

​Let $L_1, L_2$ be two regular languages and $L_3$ a language which is not regular. Which of the following statements is/are always TRUE?

Which of the following statements about threads is/are TRUE?

Which of the following process state transitions is/are NOT possible?

Which of the following is/are Bottom-Up Parser(s)?

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?

​​Consider the circuit shown below where the gates may have propagation delays. Assume that all signal transitions occur instantaneously and that wires have no delays. Which of the following statements about the circuit is/are CORRECT?

​$\mathrm{TCP}$ client $\mathrm{P}$ successfully establishes a connection to $\text{TCP}$ server $\mathrm{Q}$. Let $\text{N}_{P}$ denote the sequence number in the $\text{SYN}$ sent from $\mathrm{P}$ to $\mathrm{Q}$. Let $\text{N}_{Q}$ denote the acknowledgement number in the SYN ACK from $\text{Q}$ to $\text{P}$. Which of the following …

Consider a $5$-stage pipelined processor with Instruction Fetch (IF), Instruction Decode (ID), Execute (EX), Memory Access (MEM), and Register Writeback (WB) stages. Which of the following statements about forwarding is/are CORRECT?