GATE Computer Science & IT 2021 — GATE 2021 · CS · Set 1

Suppose that $L_1$ is a regular language and $L_2$ is a context-free language. Which one of the following languages is NOT necessarily context-free?

Let P be an array containing $n$  integers. Let $t$  be the lowest upper bound on the number of comparisons of the array elements, required to find the minimum and maximum values in an arbitrary array of $n$  elements. Which one of the following choices is correct?

Consider the following three functions.

$f_1 = 10^n$$f_2 = n^{log \ n}$$f_3 = n^{\sqrt n}$

Which one of the following options arranges the functions in the increasing order of asymptotic growth rate?

Consider the following statements.

 S1: The sequence of procedure calls corresponds to a preorder traversal of the activation tree.

S2: The sequence of procedure returns corresponds to a postorder traversal of the activation tree.

Which one of the following options is correct?

Consider the following statements.

S1: Every SLR(1) grammar is unambiguous but there are certain unambiguous grammars that are not SLR(1).

S2: For any context-free grammar, there is a parser that takes at most $O(n^3)$ time to parse a string of length $n$.

Which one of the following options is correct?

Let the representation of a number in base 3 be 210. What is the hexadecimal representation of the number?

Let p and q be two propositions. Consider the following two formulae in propositional logic.

$S1: (\neg p \land (p \lor q)) \to q$

$S2: q \to (\neg p \land (p \lor q))$

Which one of the following choices is correct?

Consider the following two statements.

S1: Destination MAC address of an ARP reply is a broadcast address.

S2: Destination MAC address of an ARP request is a broadcast address.

Which one of the following choices is correct?

Consider the following array.

Which algorithm out of the following options uses the least number of comparisons (among the array elements) to sort the above array in ascending order ?

A binary search tree $T$ contains  $n$  distinct elements. What is the time complexity of picking an element in $T$  that is smaller than the maximum element in $T$ ?

In the context of operating systems, which of the following statements is/are correct with respect to paging?

Let $⟨M⟩$ denote an encoding of an automaton $M$. Suppose that Σ={0,1}. Which of the following languages is/are NOT recursive?

Which of the following standard C library functions will always invoke a system call when executed from a single-threaded process in a UNIX/Linux operating system?

Consider a linear list based directory implementation in a file system. Each directory is a list of nodes, where each node contains the file name along with the file metadata, such as the list of pointers to the data blocks. Consider a given directory foo. Which of the following operations will necessarily require a full scan of foo for successful completion?

In an undirected connected planar graph G, there are eight vertices and five faces. The number of edges in G is _________.

Consider the following undirected graph with edge weights as shown:

 The number of minimum-weight spanning trees of the graph is ___________.

The lifetime of a component of a certain type is a random variable whose probability density function is exponentially distributed with parameter 2. For a randomly picked component of this type, the probability that its lifetime exceeds the expected lifetime (rounded to 2 decimal places) 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 toughest job and the slowest computer gets the easiest job.

        - Every computer gets at least one job.

The number of ways in which this can be done is ___________.

Consider the following expression.

$\displaystyle \lim_{x\rightarrow-3}\frac{\sqrt{2x+22}-4}{x+3}$

he value of the above expression (rounded to 2 decimal places) is ___________.

Consider the following sequence of operations on an empty stack.

$Push(54);push(52);pop();push(55);push(62);s=pop();$

Consider the following sequence of operations on an empty queue.

$enqueue(21);enqueue(24);dequeue();enqueue(28);enqueue(32);q=dequeue();$

The value of s + q is ________.

Consider a computer system with a byte-addressable primary memory of size 232 bytes. Assume the computer system has a direct-mapped cache of size 32 KB (1 KB = 210 bytes), and each cache block is of size 64 bytes.

The size of the tag field is ________ bits.

A relation r(A,B) in a relational database has 1200 tuples. The attribute A has integer values ranging from 6 to 20, and the attribute B has integer values ranging from 1 to 20. Assume that the attributes A and B are independently distributed.

The estimated number of tuples in the output of σ(A>10)∨(B=18)(r) is ____________.

Consider the following representation of a number in IEEE 754 single-precision floating point format with a bias of 127.

Here, S,E and F denote the sign, exponent, and fraction components of the floating point representation.

The decimal value corresponding to the above representation (rounded to 2 decimal places) is ____________.

Three processes arrive at time zero with CPU bursts of 16, 20 and 10 milliseconds. If the scheduler has prior knowledge about the length of the CPU bursts, the minimum achievable average waiting time for these three processes in a non-preemptive scheduler (rounded to nearest integer) is _____________ milliseconds.

Consider the following grammar (that admits a series of declarations, followed by expressions) and the associated syntax directed translation (SDT) actions, given as pseudo-code

$\begin{array}{lll} P & \rightarrow & D^* E^* \\ D & \rightarrow & \textsf{int ID} \{ \text{record that } \textsf{ID.} \text{lexeme is of type} \textsf{ int\}} \\ D & \rightarrow & \textsf{bool ID} \{ \text{record that } \textsf{ID.} \text{lexeme is of type} \textsf{ bool\}} \\ E& \rightarrow & E_1 +E_2 \{ \text{check that } E_1. \text{type}=E_2. \text{type} = \textsf{int}; \text{set } E.\text{type }:= \textsf{int} \} \\ E & \rightarrow & !E_1 \{ \text{check that } E_1. \text{type} = \textsf{bool}; \text{ set } E.\text{type} := \textsf{bool} \} \\ E & \rightarrow & \textsf{ID} \{ \text{set } E. \text{type } := \textsf{int} \} \end{array}$

With respect to the above grammar, which one of the following choices is correct?

The following relation records the age of 500 employees of a company, where empNo ( indicating the employee number) is the key: 

$empAge(\underline{empNo},age)$

Consider the following relational algebra expression: 

$\Pi_{empNo}(empAge \Join_{(age>age1)} \rho_{empNo1,age1}(empAge))$

 What does the above expression generate?

Consider a 3-bit counter, designed using T flip-flops, as shown below:

 Assuming the initial state of the counter given by PQR as 000, what are the next three states?

Assume that a 12-bit Hamming codeword consisting of 8-bit data and 4 check bits is $d_8 d_7 d_6 d_5 c_8 d_4 d_4 d_3 d_2 c_4 d_1 c_2 c_1$,where the data bits and the check bits are given in the following tables:

$\overset{\textbf{Data bits}}{\begin{array}{|c|c|c|c|} \hline d_{8} & d_{7} & d_{6} & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} \\\hline 1 & 1 & 0 & x & 0 & 1 & 0 & 1 \\\hline \end{array}}\qquad \overset{\textbf{Check bits}}{\begin{array}{|c|c|c|c|} \hline c_{8} & c_{4} & c_{2} & c_{1} \\\hline y & 0 & 1 & 0 \\\hline \end{array}}$

Which one of the following choices gives the correct values of $x$ and $Y$ ?

Consider the following recurrence relation.

$T\left ( n \right )=\left\{\begin{array} {lcl} T(n ∕ 2)+T(2n∕5)+7n & \text{if} \; n>0\\1 & \text{if}\; n=0 \end{array}\right.$

Which one of the following options is correct?

Consider the following context-free grammar where the set of terminals is {a,b,c,d,f}.

$\begin{array}{lll} \text{S} & \rightarrow & d \: a \: \text{T} \mid \text{R} \: f \\ \text{T} & \rightarrow & a \: \text{S} \: \mid \: b \: a \: \text{T} \: \mid \epsilon \\ \text{R} & \rightarrow & c \: a \: \text{T} \: \text{R} \: \mid \epsilon \end{array}$

The following is a partially-filled LL(1) parsing table.

$\begin{array} {c c c } & a & b & c & d & f & \$ \\\hline \text{S} & & & \boxed{1} & \text{S} \rightarrow da \text{T} & \boxed{2} & \\\hline \text{T} & \text{T} \rightarrow a\text{S} & \text{T} \rightarrow ba\text{T} & \boxed{3} & & \text{T} \rightarrow \varepsilon & \boxed{4}\\\hline \text{R} & & & \text{R} \rightarrow ca\text{T}\text{R} & & \text{R} \rightarrow \varepsilon & \end{array}$

Which one of the following choices represents the correct combination for the numbered cells in the parsing table (“blank” denotes that the corresponding cell is empty)?

Let $r_i(z)$ and $w_i(z)$ denote read and write operations respectively on a data item $z$ by a transaction $T_i$ . Consider the following two schedules.

$S_1: r_1(x)r_1(y)r_2(x)r_2(y)w_2(y)w_1(x)$

$S_2: r_1(x)r_2(x)r_2(y)w_2(y)r_1(y)w_1(x)$

Which one of the following options is correct?

Consider the relation $R(P,Q,S,T,X,Y,Z,W)$  with the following functional dependencies.

$PQ\rightarrow X;\quad P\rightarrow YX;\quad Q\rightarrow Y; \quad Y\rightarrow ZW$

 Consider the decomposition of the relation R into the constituent relations according to the following two decomposition schemes.

$D_1:\quad R=[(P,Q,S,T);\;(P,T,X);\;(Q,Y);\;(Y,Z,W)]$

$D_2:\quad R=[(P,Q,S);\;(T,X);\;(Q,Y);\;(Y,Z,W)]$

Which one of the following options is correct?

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?

Consider the two statements. 

$S_1 :$ There exist random variables $X$ and $Y$ such that

$\left(\mathbb E[(X-\mathbb E(X))(Y-\mathbb E(Y))]\right)^2>\textsf{Var}[X]\textsf{Var}[Y]$

$S_2 :$ For all random variables $X$ and $Y$,

$Y, \textsf{Cov}[X,Y]=\mathbb E \left[|X-\mathbb E[X]|\;|Y-\mathbb E[Y]|\right ]$

Which one of the following choices is correct?

Let G=(V,E) be an undirected, unweighted, connected graph. Its diameter is the maximum, over all vertex pairs, of the length of a shortest path between them.

Let M be the adjacency matrix of G and let P=M2. Define a simple graph G2 on the same vertex set with adjacency matrix N by setting Nii=0 and, for i≠j, Nij=1 if Mij>0 or Pij>0; otherwise Nij=0.

Which one of the following statements is true?

Consider the following C program. 

#include <stdio.h>

int main()
{
    int i, j, count;
    count = 0;
    i = 0;
    for (j = -3; j <= 3; j++)
    {
        if ((j >= 0) && (i++))
        {
            count = count + j;
        }
    }
    count = count + i;
    printf("%d", count);
    return 0;
 }

Consider the following language:

$L= \{ w \in \{0,1\}^* \mid w \text{ ends with the substring } 011 \}$

Which one of the following deterministic finite automata accepts $L$ ? 

For a Turing machine $M, ⟨M⟩$ denotes an encoding of $M$. Consider the following two languages.

$\begin{array}{ll} L_1 = \{ \langle M \rangle \mid M \text{ takes more than } 2021 \text{ steps on all inputs} \} \\ L_2 = \{ \langle M \rangle \mid M\text{ takes more than } 2021 \text{ steps on some input} \} \end{array}$

Which one of the following options is correct?

Define $R_n$ to be the maximum amount earned by cutting a rod of length n meters into one or more pieces of integer length and selling them. For $i > 0$ , let  $p[i]$ denote the selling price of a rod whose length is i meters. Consider the array of prices:

$\text{p}[1]=1,\text{p}[2]=5,\text{p}[3]=8,\text{p}[4]=9,\text{p}[5]=10,\text{p}[6]=17,\text{p}[7]=18$

Which of the following statements is/are correct about $R_7$ ?

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?

Consider the following Boolean expression.

$F=(X+Y+Z)(\overline X +Y)(\overline Y +Z)$

Which of the following Boolean expressions is/are equivalent to $\overline F$ (complement of $F$)?

A relation $R$ is said to be circular if $aRb$ and bRc together imply $cRa$. Which of the following options is/are correct?

A TCP server application is programmed to listen on port number P on host S. A TCP client is connected to the TCP server over the network. Consider that while the TCP connection was active, the server machine S crashed and rebooted. Assume that the client does not use the TCP keepalive timer. Which of the following behaviors is/are possible?

Consider two hosts P and Q connected through a router R. The maximum transfer unit (MTU) value of the link between P and R is 1500 bytes, and between R and Q is 820 bytes.

A TCP segment of size 1400 bytes was transferred from P to Q through R, with IP identification value as 0×1234. Assume that the IP header size is 20 bytes. Further, the packet is allowed to be fragmented, i.e., Don’t Fragment (DF) flag in the IP header is not set by P.

Which of the following statements is/are correct?

Consider the following pseudocode, where S is a semaphore initialized to 5 in line #2 and counter is a shared variable initialized to 0 in line #1. Assume that the increment operation in line #7 is not atomic. 

int counter =0;

Semaphore S= init(5);

void parop(void)

{

           wait(S);

           wait(S);

           counter++;

           signal(S)

           signal(S);

     }

If five threads execute the function parop concurrently, which of the following program behavior(s) is/are possible?

Consider a dynamic hashing approach for 4-bit integer keys:

1. There is a main hash table of size 4.

2. The 2 least significant bits of a key is used to index into the main hash table.

3. Initially, the main hash table entries are empty.

4. Thereafter, when more keys are hashed into it, to resolve collisions, the set of all keys corresponding to a main hash table entry is organized as a binary tree that grows on demand.

5. First, the 3rd least significant bit is used to divide the keys into left and right subtrees.

6. To resolve more collisions, each node of the binary tree is further sub-divided into left and right subtrees based on the 4th least significant bit.

7. A split is done only if it is needed, i.e., only when there is a collision.

Which of the following sequences of key insertions can cause the above state of the hash table (assume the keys are in decimal notation)?

Consider the following ANSI C function:

        int SimpleFunction(int Y[], int n, int x)

        {

            int total = Y[0], loopIndex;

                for (loopIndex=1; loopIndex<=n-1; loopIndex++)

                    total=x*total +Y[loopIndex];

            return total;

        }

Let Z be an array of 10 elements with Z[i]=1, for all i such that 0≤i≤9. The value returned by SimpleFunction(Z,10,2) is __________ .

Consider the sliding window flow-control protocol operating between a sender and a receiver over a full-duplex error-free link. Assume the following:

• The time taken for processing the data frame by the receiver is negligible.

• The time taken for processing the acknowledgement frame by the sender is negligible.

• The sender has infinite number of frames available for transmission.

• The size of the data frame is 2,000 bits and the size of the acknowledgement frame is 10 bits.

• The link data rate in each direction is 1 Mbps (= 106 bits per second).

• One way propagation delay of the link is 100 milliseconds.

The minimum value of the sender's window size in terms of the number of frames, (rounded to the nearest integer) needed to achieve a link utilization of 50% is_____________.

Consider the following C code segment:
    a = b + c;
    e = a + 1;
    d = b + c;
    f = d + 1;
    g = e + f;

In a compiler, this code segment is represented internally as a directed acyclic graph (DAG). The number of nodes in the DAG is _____________ .

In a pushdown automaton P=(Q,Σ,Γ,δ,q0,F), a transition of the form,

where $p,q \in Q$$a \in \Sigma \cup \{ \epsilon \}$, and $X,Y \in \Gamma \cup \{ \epsilon \}$ represents

$(q,Y) \in \delta(p,a,X).$

Consider the following pushdown automaton over the input alphabet $\Sigma = \{a,b\}$ and stack alphabet $\Gamma = \{ \#, A\}$. 

The number of strings of length 100 accepted by the above pushdown automaton is ___________ .

Consider the following matrix.

$\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}$

The largest eigenvalue of the above matrix is __________.

A five-stage pipeline has stage delays of 150,120,150,160 and 140 nanoseconds. The registers that are used between the pipeline stages have a delay of 5 nanoseconds each. The total time to execute 100 independent instructions on this pipeline, assuming there are no pipeline stalls, is _______ nanoseconds.

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 __________.

Consider the following instruction sequence where registers $R1,R2$ and $R3$ are general purpose and $MEMORY[X]$ denotes the content at the memory location $X$.

$\begin{array}{llc} \textbf{Instruction} & \textbf{Semantics} & \textbf{Instruction Size} \text{ (bytes)} \\ \hline \text{MOV } \text{R1}, (5000) & \text{R1} \leftarrow \text{MEMORY}[5000] & 4 \\ \hline \text{MOV } \text{R2}, (\text{R3}) & \text{R2} \leftarrow \text{MEMORY[R3]} & 4 \\ \hline \text{ADD} \text{R2}, \text{R1} & \text{R2} \leftarrow \text{R1} + \text{R2} & 2 \\ \hline \text{MOV } (\text{R3}), \text{R2} & \text{MEMORY[R3]} \leftarrow \text{R2} & 4 \\ \hline \text{INC } \text{R3} & \text{R3} \leftarrow \text{R3}+1 & 2 \\ \hline \text{DEC } \text{R1} & \text{R1} \leftarrow \text{R1}-1 & 2 \\ \hline \text{BNZ } 1004 & \text{Branch if not zero to the} & 2 \\ & \text{given absolute address}& \\ \hline \text{HALT} & \text{Stop} & 1 \\ \hline \end{array}$

Assume that the content of the memory location 5000 is 10, and the content of the register $R3$ is 3000. The content of each of the memory locations from 3000 to 3010 is 50. The instruction sequence starts from the memory location 1000. All the numbers are in decimal format. Assume that the memory is byte addressable.

After the execution of the program, the content of memory location 3010 is ____________ .

The ratio of boys to girls in a class is 7 to 3.

Among the options below, an acceptable value for the total number of students in the class is:

A polygon is convex if, for every pair of points, P and Q belonging to the polygon, the line segment PQ lies completely inside or on the polygon.

Which one of the following is NOT a convex polygon?

Consider the following sentences:

    (i) Everybody in the class is prepared for the exam.

    (ii) Babu invited Danish to his home because he enjoys playing chess.

Which of the following is the CORRECT observation about the above two sentences?

A circular sheet of paper is folded along the lines in the directions shown. The paper, after being punched in the final folded state as shown and unfolded in the reverse order of folding, will look like _______.

_____ is to surgery as writer is to ________

Which one of the following options maintains a similar logical relation in the above sentence?

We have 2 rectangular sheets of paper, M and N, of dimensions 6 cm x 1 cm each. Sheet M is rolled to form an open cylinder by bringing the short edges of the sheet together. Sheet N is cut into equal square patches and assembled to form the largest possible closed cube. Assuming the ends of the cylinder are closed, the ratio of the volume of the cylinder to that of the cube is __________

There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag.

The probability that at least two chocolates are identical is ___________

Given below are two statements 1 and 2, and two conclusions I and II.

    Statement 1: All bacteria are microorganisms.

    Statement 2: All pathogens are microorganisms.

    Conclusion I: Some pathogens are bacteria.

    Conclusion II: All pathogens are not bacteria.

Based on the above statements and conclusions, which one of the following options is logically CORRECT?

Some people suggest anti-obesity measures (AOM) such as displaying calorie information in restaurant menus. Such measures sidestep addressing the core problems that cause obesity: poverty and income inequality.

Which one of the following statements summarizes the passage?