GATE Computer Science & IT 2017 — GATE 2017 · CS · Set 1

The statement $(¬p)⇒(¬q)$ is logically equivalent to which of the statements below?

I.    $p \Rightarrow q$

II.    $q \Rightarrow p$

III.    $\left ( ¬q \right ) \vee p$

IV. $\left ( ¬p \right ) \vee q$

Consider the first-order logic sentence $F:\forall x(\exists yR(x,y))$. Assuming non-empty logical domains, which of the sentences below are implied by $F$ ?

I.    $\exists y(\exists xR(x,y))$

II    $\exists y(\forall xR(x,y))$

III.    $\forall y(\exists xR(x,y))$

IV.    $¬\exists x(\forall y¬R(x,y))$

Let $c_{1},\ldots,c_{n}$ be scalars, not all zero, such that $\sum_{i=1}^{n}c_{i}a_{i}=0$, where $a_{i}$ are column vectors in $\mathbb{R}^{n}$.

Consider the set of linear equations

$Ax = b$

where $A=\left [ a_{1}.....a_{n} \right ]$ and $b=\sum_{i=1}^{n}a_{i}$. The set of equations has

Consider the following functions from positive integers to real numbers:

$10, \sqrt n, \ n ,\ log_2 \ n, \frac {100} {n}$

The CORRECT arrangement of the above functions in increasing order of asymptotic complexity is:

Consider the following table:

Match the algorithms to the design paradigms they are based on.

Let $T$ be a binary search tree with $15$ nodes. The minimum and maximum possible heights of $T$ are:

Note: The height of a tree with a single node is $0$.

The n-bit fixed-point representation of an unsigned real number $X$uses $f$ bits for the fraction part. Let $i = n-f$. The range of decimal values for $X$ in this representation is

Consider the C code fragment given below.

typedef struct node {
int data;
node* next;
} node;
void join(node* m, node* n) {
node* p = n;
while(p->next != NULL) {
p = p->next;
}
p->next = m;
}
Assuming that m and n point to valid NULL-terminated linked lists, invocation of join will

When two 8-bit numbers $A_{7}\cdots A_{0}$ and $B_{7}\cdots B_{0}$ in 2's complement representation (with $A_{0}$ and $B_{0}$ as the least significant bits) are added using a ripple-carry adder, the sum bits obtained are $S_{7}\cdots S_{0}$ and the carry bits are $C_{7}\cdots C_{0}$. An overflow is said to have occurred if

Consider the following context-free grammar over the alphabet $\Sigma = \{a,b,c\}$ with $S$ as the start symbol:

$S \rightarrow abScT \mid abcT$

$T \rightarrow bT \mid b$

Which one of the following represents the language generated by the above grammar?

Consider the C  struct defined below:

The base address of student is available in register R1. The field student.grade can be accessed efficiently using:

Consider the following intermediate program in three address code
p = a - b
q = p * c
p = u * v
q = p + q

Which one of the following corresponds to a static single assignment form of the above code?

Consider the following C code:

The code suffers from which one of the following problems:

Consider a TCP client and a TCP server running on two different machines. After completing data transfer, the TCP client calls close to terminate the connection and a FIN segment is sent to the TCP server. Server-side TCP responds by sending an ACK, which is received by the client-side TCP. As per the TCP connection state diagram (RFC 793), in which state does the client-side TCP connection wait for the FIN from the server-side TCP?

A sender $S$ sends a message $m$ to receiver $R$, which is digitally signed by $S$ with its private key. In this scenario, one or more of the following security violations can take place.

(I) $S$ can launch a birthday attack to replace $m$ with a fraudulent message.

(II) $A$ third party attacker can launch a birthday attack to replace $m$ with a fraudulent message.

(III) $R$ can launch a birthday attack to replace $m$ with a fraudulent message.

Which of the following are possible security violations?

The following functional dependencies hold true for the relational schema {V, W, X, Y, Z}:
V → W
VW → X
Y → VX
Y → Z
Which of the following is irreducible equivalent for this set of functional dependencies?

Consider the following grammar:

p → xQRS

Q → yz | z

R → w | ε

S → y

What is FOLLOW($Q$)?

Threads of a process share

Let $X$ be a Gaussian random variable with mean 0 and variance $\sigma ^{2}$. Let $Y$ = max($X$,0) where max($a,b$) is the maximum of $a$ and $b$. The median of $Y$ is ______________ .

Let $T$ be a tree with 10 vertices. The sum of the degrees of all the vertices in $T$  is ________

Consider the Karnaugh map given below, where $X$ represents "don't care" and blank represents 0.

Assume for all inputs $\left ( a,b,c,d \right )$, the respective complements $\left ( \bar{a}, \bar{b}, \bar{c}, \bar{d} \right )$ are also available. The above logic is implemented using 2-input NOR gates only. The minimum number of gates required is ____________ .

Consider the language $L$ given by the regular expression $(a+b)^{*} b (a+b)$ over the alphabet $\{a,b\}$. The smallest number of states needed in a deterministic finite-state automaton (DFA) accepting $L$  is ___________ .

Consider a database that has the relation schema EMP (EmpId, EmpName, and DeptName). An instance of the schema EMP and a SQL query on it are given below:

SELECT AVG(EC.Num) FROM EC WHERE (DeptName, Num) IN (SELECT DeptName, COUNT(EmpId) AS EC(DeptName, Num) FROM EMP GROUP BY DeptName)

The output of executing the SQL query is _____________ .

Consider the following CPU processes with arrival times (in milliseconds) and length of CPU bursts (in milliseconds) as given below:

If the pre-emptive shortest remaining time first scheduling algorithm is used to schedule the processes, then the average waiting time across all processes is _______ milliseconds.

Consider a two-level cache hierarchy with L1 and L2 caches. An application incurs 1.4 memory accesses per instruction on average. For this application, the miss rate of L1 cache is 0.1; the L2 cache experiences, on average, 7 misses per 1000 instructions. The miss rate of L2 expressed correct to two decimal places is ________.

Let $G=\left ( V,E \right )$ be $any$ connected, undirected, edge-weighted graph. The weights of the edges in $E$ are positive and distinct. Consider the following statements:

(I)   Minimum Spanning Tree of $G$  is always unique.

(II)  Shortest path between any two vertices of $G$ is always unique.

Which of the above statements is/are necessarily true?

A multithreaded program $P$ executes with $x$ number of threads and uses $y$ number of locks for ensuring mutual exclusion while operating on shared memory locations. All locks in the program are non-reentrant, i.e., if a thread holds a lock $l$, then it cannot re-acquire lock $l$ without releasing it. If a thread is unable to acquire a lock, it blocks until the lock becomes available. The minimum value of $x$ and the minimum value of $y$ together for which execution of $P$ can result in a deadlock are:

The value of $\displaystyle \lim_{x\rightarrow 1} \frac{x^{7}-2x^{5}+1}{x^{3}-3x^{2}+2}$

Let $p, q$ and $r$ be propositions and the expression $\left ( p\rightarrow q \right )\rightarrow r$ be a contradiction. Then, the expression $\left ( r\rightarrow p \right )\rightarrow q$ is

Let $u$ and $v$ be two vectors in $\mathbf{R}^{2}$ whose Euclidean norms satisfy ‖$u$‖=2‖$v$‖. What is the value of $\alpha$ such that $w = u + \alpha v$ bisects the angle between $u$ and $v$?

Let $A$ be $n\times n$ real valued square symmetric matrix of rank 2 with $\sum_{i=1}^{n}\sum_{j=1}^{n}A^{2}_{ij} = 50.$. Consider the following statements.

I.    One eigenvalue must be in [−5,5]

II.    The eigenvalue with the largest magnitude must be strictly greater than 5

Which of the above statements about eigenvalues of $A$ is/are necessarily CORRECT?

A computer network uses polynomials over $GF(2)$ for error checking with 8 bits as information bits and uses $x^{3}+x+1$ as the generator polynomial to generate the check bits. In this network, the message 01011011 is transmitted as:

Consider a combination of T and D flip-flops connected as shown below. The output of the D flip-flop is connected to the input of the T flip-flop and the output of the T flip-flop is connected to the input of the D flip-flop.

Initially, both Q0 and Q1 are set to 1 (before the 1st clock cycle). The outputs

If $G$ is a grammar with productions

$S\rightarrow SaS\mid aSb\mid bSa\mid SS\mid\epsilon$

where $S$ is the start variable, then which one of the following strings is not generated by $G$ ?

Consider the following two functions.    

The output printed when fun1(5) is called is

Consider the C functions foo and bar given below:

Invocations of foo(3) and bar(3) will result in:

Consider the context-free grammars over the alphabet $\left \{ a, b, c \right \}$ given below. $S$ and $T$ are non-terminals.

$G_{1}:S\rightarrow aSb \mid T, T \rightarrow cT \mid \epsilon$

$G_{2}:S\rightarrow bSa \mid T, T \rightarrow cT \mid \epsilon$

The language $L\left ( G_{1} \right )\cap L(G_{2})$ is

Consider the following languages over the alphabet $\Sigma = \left \{ a, b, c \right \}$. Let $L_{1} = \left \{ a^{n}b^{n}c^{m}\mid m,n \geq 0 \right \}$ and $L_{2} = \left \{ a^{m}b^{n}c^{n}\mid m,n \geq 0 \right \}$.

Which of the following are context-free languages?

I.    $L_{1} \cup L_{2}$

II.    $L_{1} \cap L_{2}$

Let $A$ and $B$ be finite alphabets and let # be a symbol outside both $A$ and $B$. Let $f$ be a total function from $A^*$ to $B^*$. We say $f$ is computable if there exists a Turing machine $M$ which given an input $x$∈$A^*$, always halts with $f(x)$ on its tape. Let $L_f$ denote the language $\Bigl \{x\# f(x) \mid x\in A^{*} \Bigr \}$. Which of the following statements is true:

Recall that Belady’s anomaly is that the page-fault rate may increase as the number of allocated frames increases. Now, consider the following statements:

S1: Random page replacement algorithm (where a page chosen at random is replaced) suffers from Belady’s anomaly

S2: LRU page replacement algorithm suffers from Belady’s anomaly

Which of the following is CORRECT?

Consider a database that has the relation schemas EMP(EmpId, EmpName, DeptId), and DEPT(DeptName, DeptId). Note that the DeptId can be permitted to be NULL in the relation EMP. Consider the following queries on the database expressed in tuple relational calculus.

I.    {$t$ | ∃u ∈ EMP(t[EmpName] = u[EmpName] ∧ ∀v ∈ DEPT(t[DeptId] ≠ v[DeptId]))}

II.    {$t$ | ∃u ∈ EMP(t[EmpName] = u[EmpName] ∧ ∃v ∈ DEPT(t[DeptId] ≠ v[DeptId]))}

III.    {$t$ | ∃u ∈ EMP(t[EmpName] = u[EmpName] ∧ ∃v ∈ DEPT(t[DeptId] = v[DeptId]))}

Which of the above queries are safe?

In a database system, unique timestamps are assigned to each transaction using Lamport's logical clock. Let $TS(T_1)$ and $TS(T_2)$ be the timestamps of transactions $T_1$ and $T_2$ respectively. Besides, $T_1$ holds a lock on the resource $R$, and $T_2$ has requested a conflicting lock on the same resource $R$. The following algorithm is used to prevent deadlocks in the database system assuming that a killed transaction is restarted with the same timestamp.

                                      if  $TS(T_2)$ < $TS(T_1)$ then

$​​T_1$ is killed

                                      else $​​T_2$ waits.

Assume any transaction that is not killed terminates eventually. Which of the following is TRUE about the database system that uses the above algorithm to prevent deadlocks?

Consider the following grammar:

stmt → if expr then expr else expr; stmt | Ò

expr → term relop term | term

term → id | number

id → a | b | c

number →[0−9] 

where relop is a relational operator (e.g., <,>,…), Ò refers to the empty statement, and if, then, else are terminals. 
Consider a program $P$ following the above grammar containing ten if terminals. The number of control flow paths in $P$ is________ . For example. the program

if e1then e2else e3

has 2 control flow paths. e1→e2 and e1→e3. 

In a RSA cryptosystem, a participant A uses two prime numbers p = 13 and q = 17 to generate her public and private keys. If the public key of A is 35, then the private key of A is __________ .

The values of parameters for the Stop-and-Wait ARQ protocol are as given below:

• Bit rate of the transmission channel =1 Mbps.

• Propagation delay from sender to receiver =0.75 ms.

• Time to process a frame =0.25 ms.

• Number of bytes in the information frame =1980.

• Number of bytes in the acknowledge frame =20.

• Number of overhead bytes in the information frame =20.

Assume there are no transmission errors. Then, the transmission efficiency (expressed in percentage) of the Stop-and-Wait ARQ protocol for the above parameters is _____________ (correct to 2 decimal places).

Consider a database that has the relation schema CR(StudentName, CourseName). An instance of the schema CR is as given below.

$\begin{array}{|c|c|} \hline \textbf{StudentName} & \textbf {CourseName} \\\hline \text {SA} & \text{CA} \\\hline \text{SA} & \text{CB}\\\hline \text{SA} & \text{CC} \\\hline \text{SB} & \text{CB} \\\hline \text{SB}& \text{CC} \\\hline \text{SC} & \text{CA}\\\hline \text{SC}&\text{CB} \\\hline \text{SC} & \text{CC} \\\hline \text {SD} & \text{CA} \\\hline \text{SD} & \text{CB}\\\hline \text{SD} & \text{CC} \\\hline \text{SD} & \text{CD} \\\hline \text{SE}& \text{CD} \\\hline \text{SE} & \text{CA}\\\hline \text{SE}&\text{CB} \\\hline \text{SF}& \text{CA} \\\hline \text{SF} & \text{CB }\\\hline\text{SF} & \text{CC} \\\hline \end{array}$

The following query is made on the database.

$T1 \leftarrow \pi _{CourseName}\left ( \sigma _{StudentName=SA}\left ( CR \right ) \right )$

$T2 \leftarrow CR\div T1$

The number of rows in $T2$ is ______________ .

The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is ____________ .

Let $A$ be an array of 31 numbers consisting of a sequence of 0's followed by a sequence of 1's. The problem is to find the smallest index $i$ such that $A[i]$ is 1 by probing the minimum number of locations in $A$. The worst case number of probes performed by an optimal algorithm is ____________.

Consider a RISC machine where each instruction is exactly 4 bytes long. Conditional and unconditional branch instructions use PC-relative addressing mode with Offset specified in bytes to the target location of the branch instruction. Further the Offset is always with respect to the address of the next instruction in the program sequence. Consider the following instruction sequence

$\begin{array}{ll} \text{Instr. No.} & \text{Instruction} \\\hline \text{i:} & \text{add R2, R3, R4} \\ \text{i+1:} & \text{sub R5, R6, R7} \\ \text{i+2:} & \text{cmp R1, R9, R10} \\ \text{i+3:} & \text{beq R1, Offset} \\ \end{array}$

If the target of the branch instruction is $i$, then the decimal value of the Offset is ____________ .

Instruction execution in a processor is divided into 5 stages, Instruction Fetch (IF), Instruction Decode (ID), Operand fetch (OF), Execute (EX), and Write Back (WB). These stages take 5, 4, 20, 10 and 3 nanoseconds (ns) respectively. A pipelined implementation of the processor requires buffering between each pair of consecutive stages with a delay of 2 ns. Two pipelined implementation of the processor are contemplated:

I.  a naive pipeline implementation (NP) with 5 stages and

II. an efficient pipeline (EP) where the OF stage is divided into stages OF1 and OF2 with execution times of 12 ns and 8 ns respectively.

The speedup (correct to two decimal places) achieved by EP over NP in executing 20 independent instructions with no hazards is _________ .

Consider a 2-way set associative cache with 256 blocks and uses LRU replacement. Initially the cache is empty. Conflict misses are those misses which occur due to the contention of multiple blocks for the same cache set. Compulsory misses occur due to first time access to the block. The following sequence of access to memory blocks :

      {0,128,256,128,0,128,256,128,1,129,257,129,1,129,257,129}

is repeated 10 times. The number of conflict misses experienced by the cache is _________ .

Consider the expression $(a-1) * (((b+c)/3)+d)$. Let $X$ be the minimum number of registers required by an optimal code generation (without any register spill) algorithm for a load/store architecture, in which 

1.  only load and store instructions can have memory operands and 

2.  arithmetic instructions can have only register or immediate operands.

The value of $X$ is _____________ .

Consider the following C program.

#include<stdio.h> #include<string.h> void printlength(char *s, char *t) { unsigned int c=0; int len = ((strlen(s) - strlen(t)) > c) ? strlen(s) : strlen(t); printf("%d\n", len); } void main() { char *x = "abc"; char *y = "defgh"; printlength(x,y); }

Recall that strlen is defined in string.ℎ as returning a value of type size_t, which is an unsigned int. The output of the program is __________ .

A cache memory unit with capacity of $N$ words and block size of $B$ words is to be designed. If it is designed as a direct mapped cache, the length of the TAG field is 10 bits. If the cache unit is now designed as a 16-way set-associative cache, the length of the TAG field is ____________ bits.

The output of executing the following C program is _______________ .

#include<stdio.h>
int total(int v) {
static int count = 0;
while(v) {
count += v&1;
v >>= 1;
}
return count;
}
void main() {
static int x=0;
int i=5;
for(; i>0; i--) {
x = x + total(i);
}
printf("%d\n", x);
}

After Rajendra Chola returned from his voyage to Indonesia, he ________ to visit the temple in Thanjavur.

Research in the workplace reveals that people work for many reasons _______________ .

Rahul, Murali, Srinivas and Arul are seated around a square table. Rahul is sitting to the left of Murali. Srinivas is sitting to the right of Arul. Which of the following pairs are seated opposite each other?

Find the smallest number $y$ such that $y×162$  is a perfect cube.

The probability that a $k$-digit number does NOT contain the digits 0,5, or 9 is

"The hold of the nationalist imagination on our colonial past is such that anything inadequately or improperly nationalist is just not history."

Which of the following statements best reflects the author's opinion?

Six people are seated around a circular table. There are at least two men and two women. There are at least three right-handed persons. Every woman has a left-handed person to her immediate right. None of the women are right-handed. The number of women at the table is

The expression $\large \frac{(x+y) - |x-y|}{2}$ is equal to :

Arun, Gulab, Neel and Shweta must choose one shirt each from a pile of four shirts coloured red, pink, blue and white respectively. Arun dislikes the colour red and Shweta dislikes the colour white. Gulab and Neel like all the colours. In how many different ways can they choose the shirts so that no one has a shirt with a colour he or she dislikes?

A contour line joins locations having the same height above the mean sea level. The following is a contour plot of a geographical region. Contour lines are shown at 25 $m$ intervals in this plot. If in a flood, the water level rises to 525 $m$, which of the villages $P, Q, R, S, T$ get submerged?