GATE Computer Science & IT 2014 — GATE 2014 · CS · Set 3

Consider the following statements:

P: Good mobile phones are not cheap

Q: Cheap mobile phones are not good

L: P implies Q

M: Q implies P

N: P is equivalent to Q

Which one of the following about L, M, and N is CORRECT?

Let $X$ and $Y$ be finite sets and $f: X \to Y$ be a function. Which one of the following statements is TRUE?

Let $G$  be a group with 15 elements. Let $L$ be a subgroup of $G$. It is known that $L \neq G$ and that the size of $L$ is at least 4. The size of $L$  is __________.

Which one of the following statements is TRUE about every ݊$n \times n$ matrix with only real eigenvalues?

If $V_1$ and $V_2$ are 4-dimensional subspaces of a 6-dimensional vector space $V$, then the smallest possible dimension of $V_1 ∩ V_2$ is ______.

If $\int \limits_0^{2 \pi} |x \: \sin x| dx=k\pi$ then the value of $k$ is equal to ______ .

Consider the following minterm expression for

$F(P,Q,R,S) = \sum 0,2,5,7,8,10,13,15$

The minterms 2, 7, 8 and 13 are ‘do not care’ terms. The minimal sum-of-products form for $F$ is

Consider the following combinational function block involving four Boolean variables $x,\:y,\:a,\:b$ where $x,\:a,\:b$ are inputs and $y$ is the output.

f (x, y, a, b)
{
    if (x is 1) y = a;
    else y = b;
}

Which one of the following digital logic blocks is the most suitable for implementing this function?

Consider the following processors (ns stands for nanoseconds). Assume that the pipeline registers have zero latency

P1: Four-stage pipeline with stage latencies 1 ns, 2 ns, 2 ns, 1 ns.

P2: Four-stage pipeline with stage latencies 1 ns, 1.5 ns, 1.5 ns, 1.5 ns.

P3: Five-stage pipeline with stage latencies 0.5 ns, 1 ns, 1 ns, 0.6 ns, 1 ns.

P4: Five-stage pipeline with stage latencies 0.5 ns, 0.5 ns, 1 ns, 1 ns, 1.1 ns.

Which processor has the highest peak clock frequency?

Let $A$ be the square matrix of size $n \times n$. Consider the following pseudocode. What is the expected output?

C = 100;
for i = 1 to n do
    for j = 1 to n do
    {
        Temp = A[i][j] + C;
        A[i][j] = A[j][i];
        A[j][i] = Temp - C;
    }
for i = 1 to n do
    for j = 1 to n do
        output (A[i][j]);

The minimum number of arithmetic operations required to evaluate the polynomial $P(X) = X^5+4X^3+6X+5$ for a given value of $X$, using only one temporary variable is ______.

Consider the following rooted tree with the vertex labeled P as the root:

The order in which the nodes are visited during an in-order traversal of the tree is

Suppose depth first search is executed on the graph below starting at some unknown vertex. Assume that a recursive call to visit a vertex is made only after first checking that the vertex has not been visited earlier. Then the maximum possible recursion depth (including the initial call) is _________.

You have an array of $n$ elements. Suppose you implement quicksort by always choosing the central element of the array as the pivot. Then the tightest upper bound for the worst case performance is

The length of the shortest string NOT in the language (over $Σ = \{ a,b \}$) of the following regular expression is ______________.

$a^*b^*(ba)^*a^*$

Let $Σ$ be a finite non-empty alphabet and let $2^{Σ^*}$ be the power set of $Σ^*$. Which one of the following is TRUE?

One of the purposes of using intermediate code in compilers is to

Which of the following statements are CORRECT?

1) Static allocation of all data areas by a compiler makes it impossible to implement recursion.

2) Automatic garbage collection is essential to implement recursion.

3) Dynamic allocation of activation records is essential to implement recursion.

4) Both heap and stack are essential to implement recursion.

In the context of modular software design, which one of the following combinations is desirable?

A system uses 3 page frames for storing process pages in main memory. It uses the Least Recently Used (LRU) page replacement policy. Assume that all the page frames are initially empty. What is the total number of page faults that will occur while processing the page reference string given below?

4, 7, 6, 1, 7, 6, 1, 2, 7, 2

What is the optimized version of the relation algebra expression $\pi_{A1}(\pi_{A2}(\sigma_{F1}(\sigma_{F2}(r))))$ where $A1, A2$ are sets of attributes in $r$ with $A1 \subset A2$ and $F1,F2$ are Boolean expressions based on the attributes in $r$ ?

A prime attribute of a relation scheme $R$ is an attribute that appears

In the following pairs of OSI protocol layer/sub-layer and its functionality, the INCORRECT pair is

A bit-stuffing based framing protocol uses an 8-bit delimiter pattern of 01111110. If the output bit-string after stuffing is 01111100101, then the input bit-string is

Host A (on TCP/IP v4 network A) sends an IP datagram D to host B (also on TCP/IP v4 network B). Assume that no error occurred during the transmission of D. When D reaches B, which of the following IP header field(s) may be different from that of the original datagram D?

(i) TTL     (ii) Checksum     (iii) Fragment Offset

An IP router implementing Classless Inter-domain Routing (CIDR) receives a packet with address 131.23.151.76. The router’s routing table has the following entries:

$\begin{array}{|c|c|} \hline \textbf {Prefix} & \textbf {Output Interface Identifier} \\\hline \text {131.16.0.0/12} & \text{3 } \\\hline \text{131.28.0.0/14} & \text{5} \\\hline \text{131.19.0.0/16} & \text{2} \\\hline \text{131.22.0.0/15} & \text{1} \\\hline \end{array}$

The identifier of the output interface on which this packet will be forwarded is ______.

Every host in an IPv4 network has a 1-second resolution real-time clock with battery backup. Each host needs to generate up to 1000 unique identifiers per second. Assume that each host has a globally unique IPv4 address. Design a 50-bit globally unique ID for this purpose. After what period (in seconds) will the identifiers generated by a host wrap around?

An IP router with a Maximum Transmission Unit (MTU) of 1500 bytes has received an IP packet of size 4404 bytes with an IP header of length 20 bytes. The values of the relevant fields in the header of the third IP fragment generated by the router for this packet are

Consider the transactions T1, T2, and T3 and the schedules S1 and S2 given below

T1: r1(X); r1(Z); w1(X); w1(Z)

T2: r2(Y); r2(Z); w2(Z) T3: r3(Y); r3(X); w3(Y)

S1: r1(X); r3(Y); r3(X); r2(Y); r2(Z); w3(Y); w2(Z); r1(Z); w1(X); w1(Z)

S2: r1(X); r3(Y); r2(Y); r3(X); r1(Z); r2(Z); w3(Y); w1(X); w2(Z); w1(Z)

Which one of the following statements about the schedules is TRUE?

Consider the relational schema given below, where eId of the relation dependent is a foreign key referring to empId of the relation employee. Assume that every employee has at least one associated dependent in the dependent relation.

employee (empId, empName, empAge)

dependent(depId, eId, depName, depAge)

Consider the following relational algebra query:

$\Pi_{empId}\:(employee) - \Pi_{empId}\:(employee \bowtie_{(empId=eID) \wedge (empAge \leq depAge)} dependent)$

The above query evaluates to the set of empIds of employees whose age is greater than that of

A system contains three programs and each requires three tape units for its operation. The minimum number of tape units which the system must have such that deadlocks never arise is _________.

An operating system uses shortest remaining time first scheduling algorithm for pre-emptive scheduling of processes. Consider the following set of processes with their arrival times and CPU burst times (in milliseconds):

$\small \begin{array}{|c|c|c|} \hline \textbf{Process} & \textbf{Arrival Time} & \textbf{Burst Time}\\\hline \text{P1} & 0 & 12 \\ \text{P2} & 2 & 4 \\ \text{P3} & 3 & 6 \\ \text{P4} & 8 & 5 \\\hline \end{array}$

The average waiting time (in milliseconds) of the processes is _________.

Consider a paging hardware with a TLB. Assume that the entire page table and all the pages are in the physical memory. It takes 10 milliseconds to search the TLB and 80 milliseconds to access the physical memory. If the TLB hit ratio is 0.6, the effective memory access time (in milliseconds) is _________.

Consider the basic block given below.

a = b + c
c = a + d
d = b + c
e = d - b
a = e + b

The minimum number of nodes and edges present in the DAG representation of the above basic block respectively are

Which one of the following problems is undecidable?

Consider the following languages over the alphabet $Σ = \{0,1,c\}$:

$L_1 = \left\{0^n1^n\mid n \geq 0\right\}$

$L_2 = \left\{wcw^r \mid w \in \{0,1\}^*\right\}$

$L_3 = \left\{ww^r \mid w \in \{0,1\}^*\right\}$

Here, $w^r$ is the reverse of the string $w$. Which of these languages are deterministic Context-free languages?

Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre-specified pair of integers $i,j$ with $i < j$. Given a shortcut $(i,j)$, if you are at position $i$ on the number line, you may directly move to $j$. Suppose $T(k)$ denotes the smallest number of steps needed to move from $k$ to 100. Suppose further that there is at most 1 shortcut involving any number, and in particular, from 9 there is a shortcut to 15. Let $y$ and $z$ be such that $T(9) = 1 + \min(T(y),T(z))$. Then the value of the product $yz$ is _____.

Consider the decision problem $2CNFSAT$ defined as follows:

$\left\{ \phi \mid \phi \text{ is a satisfiable propositional formula in CNF with at most two literals per clause}\right\}$

For example, $\phi = (x1 \vee x2) \wedge (x1 \vee \bar{x3}) \wedge (x2 \vee x4)$ is a Boolean formula and it is in $2CNFSAT$.

The decision problem $2CNFSAT$ is

Suppose we have a balanced binary search tree $T$ holding $n$ numbers. We are given two numbers $L$ and $H$ and wish to sum up all the numbers in $T$ that lie between $L$ and $H$. Suppose there are $m$ such numbers in $T$. If the tightest upper bound on the time to compute the sum is $O(n^a\log^bn+m^c\log^dn)$, the value of $a+10b+100c+1000d$ is ______.

Consider a hash table with 100 slots. Collisions are resolved using chaining. Assuming simple uniform hashing, what is the probability that the first 3 slots are unfilled after the first 3 insertions?

Consider the pseudocode given below. The function DoSomething() takes as argument a pointer to the root of an arbitrary tree represented by the leftMostChild-rightSibling representation. Each node of the tree is of type treeNode.

typedef struct treeNode* treeptr;

struct treeNode
{
    treeptr leftMostChild, rightSibling;
};

int DoSomething (treeptr tree)
{
    int value=0;
    if (tree != NULL) {
        if (tree->leftMostChild == NULL)
            value = 1;
        else
            value = DoSomething(tree->leftMostChild);
        value = value + DoSomething(tree->rightSibling);
    }
    return(value);
}

When the pointer to the root of a tree is passed as the argument to DoSomething, the value returned by the function corresponds to the 

Consider the C function given below. Assume that the array listA contains n (> 0) elements, sorted in ascending order.

int ProcessArray(int *listA, int x, int n)
{
    int i, j, k;
    i = 0;
    j = n-1;
    do {
        k = (i+j)/2;
        if (x <= listA[k])
            j = k-1;
        if (listA[k] <= x)
            i = k+1;
    } while (i <= j);
    if (listA[k] == x)
        return(k);
    else
        return -1;
}

Which one of the following statements about the function ProcessArray is CORRECT?

An instruction pipeline has five stages, namely, instruction fetch (IF), instruction decode and register fetch (ID/RF), instruction execution (EX), memory access (MEM), and register writeback (WB) with stage latencies 1 ns, 2.2 ns, 2 ns, 1 ns, and 0.75 ns, respectively (ns stands for nanoseconds). To gain in terms of frequency, the designers have decided to split the ID/RF stage into three stages (ID, RF1, RF2) each of latency 2.2/3 ns. Also, the EX stage is split into two stages (EX1, EX2) each of latency 1 ns. The new design has a total of eight pipeline stages. A program has 20% branch instructions which execute in the EX stage and produce the next instruction pointer at the end of the EX stage in the old design and at the end of the EX2 stage in the new design. The IF stage stalls after fetching a branch instruction until the next instruction pointer is computed. All instructions other than the branch instruction have an average CPI of one in both the designs. The execution times of this program on the old and the new design are $P$ and $Q$ nanoseconds, respectively. The value of $P/Q$ is __________.

The memory access time is 1 nanosecond for a read operation with a hit in cache, 5 nanoseconds for a read operation with a miss in cache, 2 nanoseconds for a write operation with a hit in cache and 10 nanoseconds for a write operation with a miss in cache. Execution of a sequence of instructions involves 100 instruction fetch operations, 60 memory operand read operations and 40 memory operand write operations. The cache hit-ratio is 0.9. The average memory access time (in nanoseconds) in executing the sequence of instructions is __________.

The above synchronous sequential circuit built using JK flip-flops is initialized with $Q_2Q_1Q_0 = 000$. The state sequence for this circuit for the next 3 clock cycles is

With respect to the numerical evaluation of the definite integral, $K = \int \limits_a^b \:x^2 \:dx$, where $a$ and $b$ are given, which of the following statements is/are TRUE?

The value of $K$ obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.

The value of $K$ obtained using the Simpson's rule is always equal to the exact value of the definite integral.

The value of the integral given below is 

$\int \limits_0^{\pi} \: x^2 \: \cos x\:dx$

Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$. If $P(.)$ denotes the probability of the event, the maximum value of $P(A)P(B)$ is_____.

Consider the set of all functions $f:\{0,1, \dots,2014\} \to \{0,1,\dots, 2014\}$ such that $f\left(f\left(i\right)\right)=i$, for all  $0 \leq i \leq 2014$. Consider the following statements:

$P$. For each such function it must be the case that for every $i,f(i) = i$.

$Q$. For each such function it must be the case that for some $i,f(i)=i$.

$R$. Each function must be onto.

Which one of the following is CORRECT?

There are two elements $x,y$ in a group ($G$,∗) such that every element in the group can be written as a product of some number of x's and $y$'s in some order. It is known that

$x*x=y*y=x*y*x*y=y*x*y*x=e$

​where $e$  is the identity element. The maximum number of elements in such a group is ____.

​

If $G$ is the forest with $n$ vertices and $k$ connected components, how many edges does $G$ have?

Let $\delta$ denote the minimum degree of a vertex in a graph. For all planar graphs on $n$ vertices with $\delta \geq 3$, which one of the following is TRUE?

The CORRECT formula for the sentence, "not all Rainy days are Cold" is

Consider the following relational schema:

employee(empId,empName,empDept)
customer(custId,custName,salesRepId,rating)

salesRepId is a foreign key referring to empId of the employee relation. Assume that each employee makes a sale to at least one customer. What does the following query return?

SELECT empName
FROM employee E
WHERE NOT EXISTS (SELECT custId
                  FROM customer C
                  WHERE C.salesRepId = E.empId
                  AND C.rating <> 'GOOD');

Let ⊕ denote the exclusive OR (XOR) operation. Let '1' and '0' denote the binary constants. Consider the following Boolean expression for $F$ over two variables $P$ and $Q$:

$F(P,Q)=\left( \left(1 \oplus P \right) \oplus \left( P \oplus Q \right )\right ) \oplus \left(\left(P \oplus Q\right) \oplus \left(Q \oplus 0\right)\right)$

The equivalent expression for $F$ is

$\underset{\text{I}}{\underline{\text{While trying to collect}}}$ an envelope $\underset{\text{II}}{\underline{\text{from under the table,}}}$$\underset{\text{III}}{\underline{\text{Mr. X fell down}}}$  and  $\underset{\text{IV}}{\underline{\text{was losing consciousness.}}}$

Which one of the above underlined parts of the sentence is NOT appropriate?

If she _______________ how to calibrate the instrument, she _______________ done the experiment.

Choose the word that is opposite in meaning to the word “coherent”.

Which number does not belong in the series below?

2, 5, 10, 17, 26, 37, 50, 64

The table below has question-wise data on the performance of students in an examination. The marks for each question are also listed. There is no negative or partial marking in the examination.

$\begin{array}{|c|c|c|c|c|} \hline \textbf{Q No.} & \textbf{Marks} &\textbf{Answered}&\textbf{Answered}&\textbf{Not}\\&&\textbf{ Correctly} &\textbf{Wrongly} & \textbf{Attempted} \\\hline \text{1} & \text{2} &\text{21} &\text{17} &\text{6}\\\hline \text{2} & \text{3} &\text{15} & \text{27}& \text{2}\\\hline \text{3} & \text{2} &\text{23} & \text{18} & \text{3}\\\hline \end{array}$

What is the average of the marks obtained by the class in the examination?

A dance programme is scheduled for 10.00 a.m. Some students are participating in the programme and they need to come an hour earlier than the start of the event. These students should be accompanied by a parent. Other students and parents should come in time for the programme. The instruction you think that is appropriate for this is

By the beginning of the 20th century, several hypotheses were being proposed, suggesting a paradigm shift in our understanding of the universe. However, the clinching evidence was provided by experimental measurements of the position of a star which was directly behind our sun.

Which of the following inference(s) may be drawn from the above passage?

(i) Our understanding of the universe changes based on the positions of stars

(ii) Paradigm shifts usually occur at the beginning of centuries

(iii)Stars are important objects in the universe

(iv) Experimental evidence was important in confirming this paradigm shift

The Gross Domestic Product (GDP) in Rupees grew at 7% during 2012-2013. For international comparison, the GDP is compared in US Dollars (USD) after conversion based on the market exchange rate. During the period 2012-2013 the exchange rate for the USD increased from Rs. 50/ USD to Rs. 60/ USD. India’s GDP in USD during the period 2012-2013

The ratio of male to female students in a college for five years is plotted in the following line graph. If the number of female students in 2011 and 2012 is equal, what is the ratio of male students in 2012 to male students in 2011?

Consider the equation: $(7526)_8 − (Y)_8 = (4364)_8$ , where $(X)_N$ stands for $X$ to the base $N$. Find $Y$.