GATE Computer Science & IT 2016 — GATE 2016 · CS · Set 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 integer $x$ ≥ 2 which satisfies $¬((p ⇒ q)∧(¬r∨ ¬s))$ is _____________ .

Let an be the number of $n$-bit strings that do NOT contain two consecutive 1s. Which one of the following is the recurrence relation for $a_n$?

$\lim_{{x \to 4}} \frac{\sin(x - 4)}{x - 4} =$ ___________.

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 .

Two eigenvalues of a 3 × 3 real matrix $P$ are $(2 + \sqrt{-1})$ and 3. The determinant of $P$ is .

Consider the Boolean operator # with the following properties:

x # 0 = x, x # 1 = x̄, x # x = 0 and x # x̄ = 1. Then x # y is equivalent to

The 16-bit 2’s complement representation of an integer is 1111 1111 1111 0101; its decimal representation is _________.

We want to design a synchronous counter that counts the sequence 0-1-0-2-0-3 and then repeats. The minimum number of J-K flip-flops required to implement this counter is _______.

A processor can support a maximum memory of 4 GB, where the memory is word-addressable (a word consists of two bytes). The size of the address bus of the processor is at least __________ bits.

A queue is implemented using an array such that ENQUEUE and DEQUEUE operations are performed efficiently. Which one of the following statements is CORRECT ($n$ refers to the number of items in the queue)?

Consider the following directed graph:

The number of different topological orderings of the vertices of the graph is ______________ .

Consider the following C program.

void f(int, short);

void main()

{

    int i = 100;

    short s = 12;

    short *p = &s;

    ____________;  // call to f()

}

Which one of the following expressions, when placed in the blank above, will NOT result in a type checking error?

The worst case running times of Insertion sort, Merge sort and Quick sort, respectively, are:

Let G be a weighted connected undirected graph with distinct positive edge weights. If every edge weight is increased by the same value, then which of the following statements is/are TRUE?

P: Minimum spanning tree of G does not change

Q: Shortest path between any pair of vertices does not change

Consider the following C program.

# include <stdio.h>

void mystery (int *ptra, int *ptrb) {  

     int *temp; 

     temp = ptrb;  

     ptrb =ptra; 

     ptra = temp; 

} 

int main () { 

    int a = 2016, b=0, c= 4, d = 42; 

    mystery (&a, &b);

    if (a < c) 

          mystery (&c, &a); 

    mystery (&a, &d); 

    printf("%d\n", a); 

}

The output of the program is __________.

Which of the following languages is generated by the given grammar?

S → aS | bS | ε

Which of the following decision problems are undecidable?

I. Given NFAs $N_1$ and $N_2$, is L(N1​) ∩ L(N2​) = ∅ ?

II. Given a CFG $G = (N,Σ,P,S)$ and a string $x ∈ Σ^∗$ , does $x ∈ L(G)$?

III. Given CFGs $G_1$ and $G_2$, is $L(G_1) = L(G_2)$?

IV. Given a TM $M$, is $L(M) = Φ$?

Which one of the following regular expressions represents the language: the set of all binary strings having two consecutive $0s$ and two consecutive $1s$?

Consider the following code segment.

x = u - t;
       y = x * v; 
       x = y + w;
       y = t - z; 
       y = x * y;

The minimum number of total variables required to convert the above code segment to static single assignment form is __________ .

Consider an arbitrary set of CPU-bound processes with unequal CPU burst lengths submitted at the same time to a computer system. Which one of the following process scheduling algorithms would minimize the average waiting time in the ready queue?

Which of the following is NOT a superkey in a relational schema with attributes V, W, X, Y, Z and primary key V Y?

Which one of the following is NOT a part of the ACID properties of database transactions?

A database of research articles in a journal uses the following schema.

(VOLUME, NUMBER, STARTPAGE, ENDPAGE, TITLE, YEAR, PRICE)

The primary key is (VOLUME, NUMBER, STARTPAGE, ENDPAGE) and the following functional dependencies exist in the schema:

(VOLUME, NUMBER, STARTPAGE, ENDPAGE) → TITLE
(VOLUME, NUMBER) → YEAR
(VOLUME, NUMBER, STARTPAGE, ENDPAGE) → PRICE

The database is redesigned to use the following schemas:

(VOLUME, NUMBER, STARTPAGE, ENDPAGE, TITLE, PRICE)
(VOLUME, NUMBER, YEAR)

Which is the weakest normal form that the new database satisfies, but the old one does not?

Which one of the following protocols is NOT used to resolve one form of address to another one?

Which of the following is/are example(s) of stateful application layer protocols?

(i) HTTP

(ii) FTP

(iii) TCP

(iv) POP3

The coefficient of $x^{12}$ in $\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}$ is ___________ .

Consider the recurrence relation $a_1 = 8, a_n = 6n^2 +2n+a_{n−1}$. Let $a_{99} = K ×10^4$ . The value of $K$ is ________ .

A function $f: \Bbb{N^+} \rightarrow \Bbb{N^+}$, defined on the set of positive integers $\Bbb{N^+}$, satisfies the following properties:

$f(n)=f(n/2)$ if $n$ is even

$f(n) = f(n+5)$ if $n$ is odd

Let $R=\{ i \mid \exists{j} : f(j)=i \}$ be the set of distinct values that $f$ takes. The maximum possible size of $R$ is __________.

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, HEADS) or (HEADS, TAILS), then output N and stop.

Step 4. If the outcomes are (TAILS, TAILS), then go to Step 1.

The probability that the output of the experiment is Y is (up to two decimal places) _____ .

Consider the two cascaded 2-to-1 multiplexers as shown in the figure

The minimal sum of products form of the output $X$ is

The size of the data count register of a DMA controller is 16 bits. The processor needs to transfer a file of 29,154 kilobytes from disk to main memory. The memory is byte addressable. The minimum number of times the DMA controller needs to get the control of the system bus from the processor to transfer the file from the disk to main memory is _________ .

The stage delays in a 4-stage pipeline are 800, 500, 400 and 300 picoseconds. The first stage (with delay 800 picoseconds) is replaced with a functionally equivalent design involving two stages with respective delays 600 and 350 picoseconds. The throughput increase of the pipeline is ________ percent.

Consider a carry lookahead adder for adding two $n$-bit integers, built using gates of fan-in at most two. The time to perform addition using this adder is

The following function computes the maximum value contained in an integer array p[ ] of size n (n >= 1).

int max (int *p,int n) {

    int a = 0, b=n-1;
    while (__________) {
        if (p[a]<= p[b]) {a = a+1;}
        else             {b = b-1;}
    }
    return p[a];
}

The missing loop condition is:

What will be the output of the following C program?

void count(int n) {
    static int d=1;
    printf("%d ", n);
    printf("%d ", d);
    d++;
    if (n>1) count(n-1);
    printf("%d ", d);
}
void main() {
    count(3);
}

What will be the output of the following pseudo-code when parameters are passed by reference and dynamic scoping is assumed?

a = 3;
void n(x) { x = x * a; print(x); }
void m(y) { a = 1; a = y - a; n(a); print(a); }
void main() { m(a); }

An operator delete(i) for a binary heap data structure is to be designed to delete the item in the ith node. Assume that the heap is implemented in an array and i refers to the i-th index of the array. If the heap tree has depth d (number of edges on the path from the root to the farthest leaf), then what is the time complexity to re-fix the heap efficiently after the removal of the element ?

Consider the weighted undirected graph with 4 vertices, where the weight of edge $\{i, j\}$ is given by the entry $W_{ij}$ in the matrix $W$.

W = $\begin{bmatrix} 0&2 &8 &5 \\ 2&0 &5 &8 \\ 8&5 &0 &x \\ 5& 8 &x &0 \end{bmatrix}$

The largest possible integer value of $x$, for which at least one shortest path between some pair of vertices will contain the edge with weight $x$ is .

Let G be a complete undirected graph on 4 vertices, having 6 edges with weights being 1, 2, 3, 4, 5, and 6. The maximum possible weight that a minimum weight spanning tree of G can have is _________.

$G = (V,E)$ is an undirected simple graph in which each edge has a distinct weight, and $e$ is a particular edge of $G$. Which of the following statements about the minimum spanning trees (MSTs) of $G$ is/are TRUE?

I. If e is the lightest edge of some cycle in $G$, then every MST of G includes $e$

II. If e is the heaviest edge of some cycle in $G$, then every MST of G excludes $e$

Let Q denote a queue containing sixteen numbers and S be an empty stack.  Head(Q) returns the element at the head of the queue Q without removing it from Q. Similarly, Top(S) returns the element at the top of S without removing it from S. Consider the algorithm given below.

The maximum possible number of iterations of the while loop in the algorithm is ___________ .

Consider the following context-free grammars:

$G_1 : S \to aS \mid B, B \to b \mid bB$

$G_2 : S \to aA \mid bB, A \to aA \mid B \mid \varepsilon,B \to bB \mid \varepsilon$

Which one of the following pairs of languages is generated by $G_1$ and $G_2$, respectively?

Consider the transition diagram of a PDA given below with input alphabet $Σ = \{a,b\}$ and stack alphabet $Γ = \{X,Z\}$. $Z$ is the initial stack symbol. Let $L$ denote the language accepted by the PDA.

Which one of the following is TRUE?

Let $X$ be a recursive language and $Y$ be a recursively enumerable but not recursive language. Let $W$ and $Z$ be two languages such that $\overline Y$ reduces to $W$, and $Z$ reduces to $\overline X$ (reduction means the standard many-one reduction). Which one of the following statements is TRUE?

The attributes of three arithmetic operators in some programming language are given below.

The value of the expression 2−5+1−7 ∗ 3 in this language is __________ .

Consider the following Syntax Directed Translation Scheme (SDTS), with non-terminals {S, A} and terminals {a, b}.

S → aA { print 1 }

S → a { print 2 }

A → Sb { print 3 }

Using the above SDTS, the output printed by a bottom-up parser, for the input aab is:

Consider a computer system with 40-bit virtual addressing and page size of sixteen kilobytes. If the computer system has a one-level page table per process and each page table entry requires 48 bits, then the size of the per-process page table is ___________ megabytes.

Consider a disk queue with requests for I/O to blocks on cylinders 47, 38, 121, 191, 87, 11, 92, 10. The C-LOOK scheduling algorithm is used. The head is initially at cylinder number 63, moving towards larger cylinder numbers on its servicing pass. The cylinders are numbered from 0 to 199. The total head movement (in number of cylinders) incurred while servicing these requests is ___________.

Consider a computer system with ten physical page frames. The system is provided with an access sequence $(a_1,a_2,\ldots,a_{20},a_1,a_2,\ldots,a_{20})$, where each ai is a distinct virtual page number. The difference in the number of page faults between the last-in-first-out page replacement policy and the optimal page replacement policy is __________ .

Consider the following proposed solution for the critical section problem. There are $n$ processes: $P_0 ...P_{n−1}$. In the code, function pmax returns an integer not smaller than any of its arguments. For all i, t[i] is initialized to zero.

Code for $P_i$ :

do {

    c[i]=1; t[i]= pmax (t[0],....,t[n-1])+1; c[i]=0;

    for every j != i in {0,....,n-1} {

        while (c[j]); 

        while (t[j] != 0 && t[j] <=t[i]);

    }

    Critical Section;

    t[i]=0;

    

    Remainder Section;

    

}  while (true);

Which one of the following is TRUE about the above solution?

Consider the following two phase locking protocol. Suppose a transaction $T$ accesses (for read or write operations), a certain set of objects $\{O_1,...,O_k\}$. This is done in the following manner:

Step 1. T acquires exclusive locks to $O_1, . . . , O_k$ in increasing order of their addresses.

Step 2. The required operations are performed.

Step 3. All locks are released.

This protocol will

Consider that B wants to send a message $m$ that is digitally signed to A. Let the pair of private and public keys for A and B be denoted by ${K_{x}}^-$ and ${K_{x}}^+$ for $x = A,B$, respectively. Let $K_x(m)$ represent the operation of encrypting $m$ with a key $K_x$ and $H(m)$ represent the message digest. Which one of the following indicates the CORRECT way of sending the message $m$ along with the digital signature to A?

An IP datagram of size 1000 bytes arrives at a router. The router has to forward this packet on a link whose MTU (maximum transmission unit) is 100 bytes. Assume that the size of the IP header is 20 bytes.

The number of fragments that the IP datagram will be divided into for transmission is ___________ .

For a host machine that uses the token bucket algorithm for congestion control, the token bucket has a capacity of 1 megabyte and the maximum output rate is 20 megabytes per second. Tokens arrive at a rate to sustain output at a rate of 10 megabytes per second. The token bucket is currently full and the machine needs to send 12 megabytes of data. The minimum time required to transmit the data is ________ seconds.

A sender uses the Stop-and-Wait ARQ protocol for reliable transmission of frames. Frames are of size 1000 bytes and the transmission rate at the sender is 80 Kbps (1Kbps = 1000 bits/second). Size of an acknowledgement is 100 bytes and the transmission rate at the receiver is 8 Kbps. The one-way propagation delay is 100 milliseconds.

Assuming no frame is lost, the sender throughput is __________ bytes/second.

Out of the following four sentences, select the most suitable sentence with respect to grammar and usage.

A rewording of something written or spoken is a ______________.

Archimedes said, “Give me a lever long enough and a fulcrum on which to place it, and I will move the world.”

The sentence above is an example of a ___________ statement.

If ‘relftaga’ means carefree, ‘otaga’ means careful and ‘fertaga’ means careless, which of the following could mean ‘aftercare’?

A cube is built using 64 cubic blocks of side one unit. After it is built, one cubic block is removed from every corner of the cube. The resulting surface area of the body (in square units) after the removal is __________.

A shaving set company sells 4 different types of razors, Elegance, Smooth, Soft and Executive. Elegance sells at Rs. 48, Smooth at Rs. 63, Soft at Rs. 78 and Executive at Rs. 173 per piece. The table below shows the numbers of each razor sold in each quarter of a year.

Which product contributes the greatest fraction to the revenue of the company in that year?

Indian currency notes show the denomination indicated in at least seventeen languages. If this is not an indication of the nation’s diversity, nothing else is.

Which of the following can be logically inferred from the above sentences?

Consider the following statements relating to the level of poker play of four players P, Q, R and S.

I. P always beats Q

II. R always beats S

III. S loses to P only sometimes

IV. R always loses to Q

Which of the following can be logically inferred from the above statements?

(i) P is likely to beat all the three other players

(ii) S is the absolute worst player in the set

If $f(x) = 2x^7 + 3x - 5$, which of the following is a factor of $f(x)$?

In a process, the number of cycles to failure decreases exponentially with an increase in load. At a load of 80 units, it takes 100 cycles for failure. When the load is halved, it takes 10000 cycles for failure. The load for which the failure will happen in 5000 cycles is ________.