# GATE COMPUTER SCIENCE 2015 SET 3 WITH ANSWERS SOLVED

 QuestiON 1
We __________ our friend’s birthday and we ________ how to make it up to him.
 A Completely forgot — don’t just know B Forgot completely — don’t just know Completely forgot — just don’t know D Forgot completely — just don’t know

Explanation:

We Completely forgot our friend’s birthday and we just don’t know how to make it up to him.

 Question 2
Choose the statement where underlined word is used correctly.
 A The industrialist had a personnel jet B I write my experience in my personnel diary C All personnel are being given the day off D Being religious is a personnel aspect

Explanation:

personnel mean a body of persons employed in an organization or place of work.

 Question 3

Based on the given statements, select the most appropriate option to solve the given question. What will be the total weight of 10 poles each of same weight?

```Statements:
(I) One fourth of the weight of a pole is 5 kg
(II) The total weight of these poles is 160 kg more
than the total weight of two poles.```
 A Statement II alone is not sufficient B Statement II alone is not sufficient C Either I or II alone is sufficient D Both statements I and II together are not sufficient.

Explanation:

```(I) is sufficient. We can determine total weight of 1o
poles as 10 * 4 * 5 = 200 Kg

(II) is also sufficient, we can determine the weight.
Let x be the weight of 1 pole.
10*x - 2*x = 160 Kg
x = 20 Kg
10x = 200 Kg```

 Question 4
Consider a function f(x) = 1 – |x| on –1 ≤ x ≤ 1. The value of x at which the function attains a maximum and the maximum value of the function are:
 A 0, –1 B –1, 0 C 0, 1 D –1, 2

Explanation:

```f(x) = 1 – |x|
f(0) = 1
f(1) = f(-1) = 0
f(0.5) = f(-0.5) = 0.5
```

The maximum is attained at x = 0, and the maximum value is 1.

 Question 5
A generic term that includes various items of clothing such as a skirt, a pair of trousers and a shirt is
 A fabric B textile C fibre D apparel

Explanation:

apparel means clothing.

 Question 6

If the list of letters, P, R, S, T, U is an arithmetic sequence, which of the following are also in arithmetic sequence?

```1. 2P, 2R, 2S, 2T, 2U
2. P–3, R–3, S–3, T–3, U–3
3. P2, R2, S2, T2, U2
```
 A 1 only B 1 and 2 C 2 and 3 D 1 and 3

Explanation:

1 is AP with common difference as double of that of original series. 2 is AP with common difference as same as original series.

 Question 7
Four branches of a company are located at M, N, O and P. M is north of N at a distance of 4 km; P is south of O at a distance of 2 km; N is southeast of O by 1 km. What is the distance between M and P in km?
 A 5.34 B 6.74 C 28.5 D 45.49

Explanation:

 Question 8
In a triangle PQR, PS is the angle bisector of ∠ QPR and ∠ QPS = 60°. What is the length of PS?
 A a B b C c D d

Explanation:

```Area of a Δ = 1/2 * ac sinB
= 1/2 * bc sinA
= 1/2 ab sinC

Here area (ΔPQR) = area (ΔPQS) + area (ΔPRS)

1/2 rq sin120 = 1/2 PS * r sin60 + 1/2 PS * q sin60

=> PS = rq/(r+q)```

 Question 9
```If p, q, r, s are distinct integers such that:
f(p, q, r, s) = max (p, q, r, s)
g(p, q, r, s) = min (p, q, r, s)
h(p, q, r, s) = remainder of (p × q) / (r × s)  if (p × q) > (r × s) OR
remainder of (r × s) / (p × q) if (r × s) > (p × q)
Also a function fgh (p, q, r, s) =
f(p, q, r, s) × g(p, q, r, s) × h(p, q, r, s).
Also the same operation are valid with two variable functions
of the form f(p, q).

What is the value of fg(h(2, 5, 7, 3), 4, 6, 8)?```
 A 6 B 7 C 8 D 9

Explanation:

```The question asks value of fg (h(2, 5, 7, 3), 4, 6, 8)

We need to first find value of h(2, 5, 7, 3)

h is defined as
h(p, q, r, s) = remainder of (p × q) / (r × s)  if (p × q) > (r × s)
remainder of (r × s) / (r × q)  if (r × s) > (p × q)

h(2, 5, 7, 3) = remainder of (7 * 3) / (2 * 5)  since 7*3 > 2*5
= 1

fg(1, 4, 6, 8) = f(1, 4, 6, 8) * g(1, 4, 6, 8)
= max(1, 4, 6, 8) * min(1, 4, 6, 8)
= 8 * 1
= 8```

 Question 10
Out of the following four sentences, select the most suitable sentence with respect to grammar and usage:
 A Since the report lacked needed information, it was of no use to them B The report was useless to them because there were no needed information in it C Since the report did not contain the needed information, it was not real useful to them D Since the report lacked needed information, it would not had been useful to them

 Question 11

Consider the following transaction involving two bank accounts x and y.

`read(x);  x := x – 50;  write(x);  read(y);  y := y + 50;  write(y)`

The constraint that the sum of the accounts x and y should remain constant is that of

 A Atomicity B Consistency C Isolation D Durability

Explanation:

Consistency in database systems refers to the requirement that any given database transaction must only change affected data in allowed ways, that is sum of x and y must not change.

 Question 12
Consider two decision problems Q1, Q2 such that Q1 reduces in polynomial time to 3-SAT and 3-SAT reduces in polynomial time to Q2. Then which one of the following is consistent with the above statement?
 A Q1 is in NP, Q2 is NP hard B Q2 is in NP, Q1 is NP hard C Both Q1 and Q2 are in NP D Both Q1 and Q2 are in NP hard

Explanation:

```Q1 reduces in polynomial time to 3-SAT
==> Q1 is in NP

3-SAT reduces in polynomial time to Q2
==> Q2 is NP Hard.  If Q2 can be solved in P, then 3-SAT
can be solved in P, but 3-SAT is NP-Complete, that makes
Q2 NP Hard```

 Question 13

Consider the following two statements.

```S1: If a candidate is known to be corrupt, then he will not be elected.
S2: If a candidate is kind, he will be elected.```

Which one of the following statements follows from S1 and S2 as per sound inference rules of logic?

 A If a person is known to be corrupt, he is kind B If a person is not known to be corrupt, he is not kind C If a person is kind, he is not known to be corrupt D If a person is not kind, he is not known to be corrupt

Explanation:

```S1: If a candidate is known to be corrupt, then
he will not be elected.
S2: If a candidate is kind, he will be elected.

If p → q, then ¬q → ¬p
So from S1, elected → not corrupt
and S2 is,   kind → elected

Therefore, kind → not corrupt```

 Question 14
A Software Requirements Specification (SRS) document should avoid discussing which one of the following?
 A User interface issues B Non-functional requirements C Design specification D Interfaces with third party software

Explanation:

A software requirements specification (SRS) is a description of a software system to be developed, laying out functional and non-functional requirements, and may include a set of use cases that describe interactions the users will have with the software. (Source Wiki) Design Specification should not be part of SRS.

 Question 15

The larger of the two eigenvalues of the matrix

```⎡ 4   5 ⎤
⎣ 2   1 ⎦```

is ____

 A 5 B 6 C 7 D 8

Explanation:

```The character equation for given matrix is

| 4-λ    5 | = 0
| 2     1-λ|

(4-λ)*(1-λ) - 10 = 0
λ2 - 5λ - 6 = 0
(λ+1)*(λ-6) = 0
λ = -1, 6

Greater of two Eigenvalues is 6.```

 Question 16
With reference to the B+ tree index of order 1 shown below, the minimum number of nodes (including the root node) that must be fetched in order to satisfy the following query: “Get all records with a search key greater than or equal to 7 and less than 15” is ________
 A 4 B 5 C 6 D 7

Explanation:

```We can get all values in range from 7 to 59 by accessing 5 nodes.
1) First search 7 in a leaf node.
2) Once 7 is found, linearly traverse till 15 is found.

See following diagram```

 Question 17
The minimum number of JK flip-flops required to construct a synchronous counter with the count sequence (0, 0, 1, 1, 2, 2, 3, 3, 0, 0,…) is ________
 A 0 B 1 C 2 D 3

Explanation:

Ans = 3, mod 8 up counter using 3 JK flip flops. Ignore the output of LSB

 Question 18
A link has a transmission speed of 106 bits/sec. It uses data packets of size 1000 bytes each. Assume that the acknowledgment has negligible transmission delay, and that its propagation delay is the same as the data propagation delay. Also assume that the processing delays at nodes are negligible. The efficiency of the stop-and-wait protocol in this setup is exactly 25%. The value of the one-way propagation delay (in milliseconds) is ___________.
 A 4 B 8 C 12 D 16

Explanation:

```In stop and wait, protocol next packet is sent only when
acknowledgement of previous packet is received.  This

Tansmission speed = 106

Time to send a packet = (1000 * 8) bits / 106
= 8 miliseconds

Since link utilization or efficiency is 25%, total time
taken for 1 packet is 8 * 100/25 = 32 miliseconds.

Total time is twice the one way propagation delay plus
transmission delay. Propagation delay has to be considered
for packet and ack both.
Transmission delay is considered only for packet as the
question says that trans. time for ack is negligible.

Let propagation delay be x.
2x + 8 = 32.
x = 12.```

 Question 19
The number of divisors of 2100 is _________.
 A 42 B 36 C 78 D 72

Explanation:

```2100  =  22 * 31 * 52 * 71

There are 3 choices for powers of 2,  the choices are 0, 1, 2
Similarly, there are 2 choices for powers of 3,  3 choices for power of 5
and 2 choices for power of 7.

So total number of divisors is 3 * 2 * 3  * 2 = 36```

 Question 20
A binary tree T has 20 leaves. The number of nodes in T having two children is
 A 18 B 19 C 17 D Any number between 10 and 20

Explanation:

The number of nodes with two children is always one less than the number of leaves. See Handshaking Lemma and Interesting Tree Properties for proof.

 Question 21

Consider the following C function.

 `int` `fun (``int` `n)` `{` `  ``int` `x=1, k;` `  ``if` `(n==1) ``return` `x;` `  ``for` `(k=1; k

The return value of fun(5) is __________.

 A 0 B 26 C 51 D 71

Explanation:

```fun(5) = 1 + fun(1) * fun(4) + fun(2) * fun(3) +
fun(3) * fun(2) + fun(4) * fun(1)
= 1 + 2*[fun(1)*fun(4) + fun(2)*fun(3)]

Substituting fun(1) = 1
= 1 + 2*[fun(4) + fun(2)*fun(3)]

Calculating fun(2), fun(3) and fun(4)
fun(2) = 1 + fun(1)*fun(1) = 1 + 1*1 = 2
fun(3) = 1 + 2*fun(1)*fun(2) = 1 + 2*1*2 = 5
fun(4) = 1 + 2*fun(1)*fun(3) + fun(2)*fun(2)
= 1 + 2*1*5 + 2*2 = 15

Substituting values of fun(2), fun(3) and fun(4)
fun(5) = 1 + 2*[15 + 2*5] = 51```

 Question 22
Consider the basic COCOMO model where E is the effort applied in person-months, D is the development time in chronological months, KLOC is the estimated number of delivered lines of code (in thousands) and ab, bb, cb, db have their usual meanings. The basic COCOMO equations are of the form.
 A E = ab(KLOC) exp(bb), D = cb(E) exp(db) B D = ab(KLOC) exp(bb), E = cb(D) exp(db) C E = ab exp(bb), D = cb(KLOC) exp(db) D E = ab exp(db), D = cb(KLOC) exp(bb)

Explanation:

In Basic COCOMO, following are true.

Effort Applied (E) = ab(KLOC)bb [ person-months ]
Development Time (D) = cb(Effort Applied)db [months]
People required (P) = Effort Applied / Development Time [count]

 Question 23
Which one of the following statements is NOT correct about HTTP cookies?
 A A cookies is a piece of code that has the potential to compromise the security of an Internet user B A cookie gains entry to the user’s work area through an HTTP header C A cookie has an expiry date and time D Cookies can be used to track the browsing pattern of a user at a particular site

Explanation:

Cookies are not piece of code, they are just strings typically in the form of key value pairs.

 Question 24
In the context of abstract-syntax-tree (AST) and control-flow-graph (CFG), which one of the following is True?
 A In both AST and CFG, let node N2 be the successor of node N1. In the input program, the code corresponding to N2 is present after the code corresponding to N1 B For any input program, neither AST nor CFG will contain a cycle C The maximum number of successors of a node in an AST and a CFG depends on the input program D Each node in AST and CFG corresponds to at most one statement in the input program

Explanation:

```(A) is false, In CFG (Control Flow Graph), code of N2
may be present before N1 when there is a loop or goto.

(B) is false, CFG (Control Flow Graph) contains cycle
when input program has loop.

(C) is true, successors in AST and CFG depedend on
input program

(D) is false, a single statement may belong to a block
of statements.```

 Question 25

Consider the following function written in the C programming language. The output of the above function on input “ABCD EFGH” is

 `void` `foo (``char` `*a)` `{` `  ``if` `(*a && *a != ` `)` `  ``{ ` `     ``foo(a+1);` `     ``putchar``(*a);` `  ``}` `}`

 A ABCD EFGH B ABCD C HGFE DCBA D DCBA

Explanation:

The program prints all characters before ‘ ‘ or ‘\0’ (whichever comes first) in reverse order.

 Question 26
Let R be the relation on the set of positive integers such that aRb if and only if a and b are distinct and have a common divisor other than 1. Which one of the following statements about R is True?
 A R is symmetric and reflexive but not transitive B R is reflexive but not symmetric and not transitive C R is transitive but not reflexive and not symmetric D R is symmetric but not reflexive and not transitive

Explanation:

R cannot be reflexive as ‘a’ and ‘b’ have to be distinct in aRb. R is symmetric if a and b have a common divisor, then b and a also have. R is not transitive as aRb and bRc doesn’t mean aRc. For example 3 and 15 have common divisor, 15 and 5 have common divisor, but 3 and 5 don’t have.

 Question 27
Consider a complete binary tree where the left and the right subtrees of the root are max-heaps. The lower bound for the number of operations to convert the tree to a heap is
 A Ω(logn) B Ω(n) C Ω(nlogn) D Ω(n2)

Explanation:

The answer to this question is simply max-heapify function. Time complexity of max-heapify is O(Log n) as it recurses at most through height of heap.

```// A recursive method to heapify a subtree with root at given index
// This method assumes that the subtrees are already heapified
void MinHeap::MaxHeapify(int i)
{
int l = left(i);
int r = right(i);
int largest = i;
if (l < heap_size && harr[l] < harr[i])
largest = l;
if (r < heap_size && harr[r] < harr[smallest])
largest = r;
if (largest != i)
{
swap(&harr[i], &harr[largest]);
MinHeapify(largest);
}
}```

 Question 28
The cardinality of the power set of {0, 1, 2 . . ., 10} is _________.
 A 1024 B 1023 C 2048 D 2043

Explanation:

The power set has 2n elements. For n = 11, size of power set is 2048.

 Question 29

Match the following:

```      List-I                  List-II
A. Lexical analysis       1. Graph coloring
B. Parsing                2. DFA minimization
C. Register allocation    3. Post-order traversal
D. Expression evaluation  4. Production tree```
```Codes:
A B C D
(a) 2 3 1 4
(b) 2 1 4 3
(c) 2 4 1 3
(d) 2 3 4 1```
 A a B b C c D d

Explanation:

```Register allocation is a variation of Graph Coloring problem.

Lexical Analysis uses DFA.

Parsing makes production tree

Expression evaluation is done using tree traversal```

 Question 30
Identify the correct order in which a server process must invoke the function calls accept, bind, listen, and recv according to UNIX socket API.
 A listen, accept, bind recv B bind, listen, accept, recv C bind, accept, listen, recv D accept, listen, bind, recv

Explanation:

bind, listen, accept and recv are server side socket API functions.

```bind() associates a socket with a socket address structure,
i.e. a specified local port number and IP address.

listen() causes a bound TCP socket to enter listening state.

accept() accepts a received incoming attempt to create a new
TCP connection from the remote client,

recv() is used to receive data from a remote socket.```

A server must first do bind() to tell operating system the port number on which it would be listing, then it must listen to receive incoming connection requests on the bound port number. Once a connection comes, the server accepts using accept(), then starts receiving data using recv().

 Question 31

Consider the following statements:

```1. The complement of every Turning decidable
language is Turning decidable
2. There exists some language which is in NP
but is not Turing decidable
3. If L is a language in NP, L is Turing decidable```

Which of the above statements is/are True?

 A Only 2 B Only 3 C Only 1 and 2 D Only 1 and 3

Explanation:

```1 is true: Complement of Turing decidable is Turing Decidable.

3 is true. All NP problems are Turing decidable
(See http://www.geeksforgeeks.org/np-completeness-set-1/)

2 is false:  The definition of NP itself says solvable in
polynomial time using non-deterministic Turing machine.```

 Question 32
An unordered list contains n distinct elements. The number of comparisons to find an element in this list that is neither maximum nor minimum is
 A Θ(nlogn) B Θ(n) C Θ(logn) Θ(1)

Explanation:

We only need to consider any 3 elements and compare them. So the number of comparisons is constants, that makes time complexity as Θ(1) The catch here is, we need to return any element that is neither maximum not minimum. Let us take an array {10, 20, 15, 7, 90}. Output can be 10 or 15 or 20 Pick any three elements from given liar. Let the three elements be 10, 20 and 7. Using 3 comparisons, we can find that the middle element is 10.

 Question 33
A system has 6 identical resources and N processes competing for them. Each process can request atmost 2 resources. Which one of the following values of N could lead to a deadlock?
 A 1 B 2 C 3 D 4

Explanation:

It seems to be wrong question in GATE exam. Below seems to be the worst case situation. In below situation P1 and P2 can continue as they have 2 resources each

```P1 --> 2
P2 --> 2
P3 --> 1
P4 --> 1```

 Question 34
Assume that for a certain processor, a read request takes 50 nanoseconds on a cache miss and 5 nanoseconds on a cache hit. Suppose while running a program, it was observed that 80% of the processor’s read requests result in a cache hit. The average read access time in nanoseconds is____________.
 A 10 B 12 C 13 D 14

Explanation:

The average read access time in nanoseconds = 0.8 * 5 + 0.2*50 = 14

 Question 35
A computer system implements a 40 bit virtual address, page size of 8 kilobytes, and a 128-entry translation look-aside buffer (TLB) organized into 32 sets each having four ways. Assume that the TLB tag does not store any process id. The minimum length of the TLB tag in bits is _________
 A 20 B 10 C 11 D 22

Explanation:

```Total virtual address size = 40

Since there are 32 sets, set offset = 5

Since page size is 8kilobytes, word offset = 13

Minimum tag size = 40 - 5- 13 = 22```

 Question 36

Let f(x) = x –(1/3) and A denote the area of the region bounded by f(x) and the X-axis, when x varies from –1 to 1. Which of the following statements is/are True?

```1. f is continuous in [–1, 1]
2. f is not bounded in [–1, 1]
3. A is nonzero and finite```
 A 2 only B 3 only C 2 and 3 only D 1, 2 and 3

Explanation:

1 is false: function is not a Continuous function. As a change of 1 in x leads to ∞ change in f(x). For example when x is changed from -1 to 0. At x = 0, f(x) is ∞ and at x = 1, f(x) is finite. 2 is True: f(x) is not a bounded function as it becomes ∞ at x = 0. 3 is true: A denote the area of the region bounded by f(x) and the X-axis. This area is bounded, we can calculate it by doing integrating the function

 Question 37

Perform the following operations on the matrix

```⎡ 3  4   45⎤
⎢ 7  8  105⎥
⎣13  2  195⎦
1. Add the third row to the second row.
2. Subtract the third column from the first column.```

The determinant of the resultant matrix is _____________.

 A 0 B 1 C 50 D 100

Explanation:

The given matrix is singular matrix as its last column is a multiple of first column. If we multiply 15 with first column, we get the third column.

 Question 38
A graph is self-complementary if it is isomorphic to its complement. For all self-complementary graphs on n vertices, n is
 A A multiple of 4 B Even C Odd D Congruent to 0 mod 4, or 1 mod 4

Explanation:

An n-vertex self-complementary graph has exactly half number of edges of the complete graph, i.e., n(n − 1)/4 edges, and (if there is more than one vertex) it must have diameter either 2 or 3. Since n(n −1) must be divisible by 4, n must be congruent to 0 or 1 mod 4; for instance, a 6-vertex graph cannot be self-complementary. Source:http://en.wikipedia.org/wiki/Self-complementary_graph

 Question 39

Consider the intermediate code given below:

```1. i = 1
2. j = 1
3. t1 = 5 * i
4. t2 = t1 + j
5. t3 = 4 * t2
6. t4 = t3
7. a[t4] = –1
8. j = j + 1
9. if j <= 5 goto(3)
10. i = i + 1
11. if i < 5 goto(2)```

The number of nodes and edges in the control-flow-graph constructed for the above code, respectively, are

 A 5 and 7 B 6 and 7 C 5 and 5 D 7 and 8

Explanation:

Below is control flow graph of above code

.

 Question 40
Consider six memory partitions of size 200 KB, 400 KB, 600 KB, 500 KB, 300 KB, and 250 KB, where KB refers to kilobyte. These partitions need to be allotted to four processes of sizes 357 KB, 210 KB, 468 KB and 491 KB in that order. If the best fit algorithm is used, which partitions are NOT allotted to any process?
 A 200 KB and 300 KB B 200 KB and 250 KB C 250 KB and 300 KB D 300 KB and 400 KB

Explanation:

Best fit allocates the smallest block among those that are large enough for the new process. So the memory blocks are allocated in below order.

```357 ---> 400
210 ---> 250
468 ---> 500
491 ---> 600```

Sot the remaining blocks are of 200 KB and 300 KB

 Question 41

A Young tableau is a 2D array of integers increasing from left to right and from top to bottom. Any unfilled entries are marked with ∞, and hence there cannot be any entry to the right of, or below a ∞. The following Young tableau consists of unique entries.

```1     2     5      14
3     4     6      23
10    12    18     25
31    ∞     ∞       ∞```

When an element is removed from a Young tableau, other elements should be moved into its place so that the resulting table is still a Young tableau (unfilled entries may be filled in with a ∞). The minimum number of entries (other than 1) to be shifted, to remove 1 from the given Young tableau is ____________

 A 2 B 5 C 6 D 18

Explanation:

Initially

```1     2     5      14
3     4     6      23
10    12    18     25
31    ∞     ∞       ∞```

When 1 is removed, it is replaced by the smallest adjacent which is 2.

```2     2     5      14
3     4     6      23
10    12    18     25
31    ∞     ∞       ∞```

When 2 is moved in place of 1, it is replaced by smallest adjacent which is 4

```2     4     5      14
3     4     6      23
10    12    18     25
31    ∞     ∞       ∞```

When 4 is moved in place of 2, it is replaced by smallest adjacent which is 6

```2     4     5      14
3     6     6      23
10    12    18     25
31    ∞     ∞       ∞```

When 6 is moved in place of 4, it is replaced by smallest adjacent which is 18.

```2     4     5      14
3     6     18     23
10    12    18     25
31    ∞     ∞       ∞```

When 18 is moved in place of 6, it is replaced by smallest adjacent which is 25.

```2     4     5      14
3     6     18     23
10    12    25     25
31    ∞     ∞       ∞```

When 25 is moved in place of 18, it is replaced by smallest adjacent which is ∞.

```2     4     5      14
3     6     18     23
10    12    25      ∞
31    ∞     ∞       ∞```

Shifted numbers are 2, 4, 6, 18, 25

 Question 42

Consider two relations R1(A, B) with the tuples (1, 5), (3, 7) and R1(A, C) = (1, 7), (4, 9). Assume that R(A,B,C) is the full natural outer join of R1 and R2. Consider the following tuples of the form (A,B,C)

```  a = (1, 5, null),
b = (1, null, 7),
c = (3, null, 9),
d = (4, 7, null),
e = (1, 5, 7),
f = (3, 7, null),
g = (4, null, 9).```

Which one of the following statements is correct?

 A R contains a, b, e, f, g but not c, d B R contains a, b, c, d, e, f, g C R contains e, f, g but not a, b D R contains e but not f, g

Explanation:

```Below is R1
A   |   B
-----------
1   |   5
3   |   7

Below is R2
A   |   C
-----------
1   |   7
4   |   9

Full outer join of above two is

A   |   B   |   C
-------------------
1   |   5   |   7
3   |   7   |  NULL
4   |  NULL |   9```

So the full outer join contains e = (1, 5, 7), f = (3, 7, null), g = (4, null, 9).

 Question 43
Which one of the following hash functions on integers will distribute keys most uniformly over 10 buckets numbered 0 to 9 for i ranging from 0 to 2020?
 A h(i) =i2 mod 10 B h(i) =i3 mod 10 C h(i) = (11 ∗ i2) mod 10 D h(i) = (12 ∗ i) mod 10

Explanation:

Since mod 10 is used, the last digit matters. If you do cube all numbers from 0 to 9, you get following

```Number    Cube    Last Digit in Cube
0        0              0
1        1              1
2        8              8
3        27             7
4        64             4
5        125            5
6        216            6
7        343            3
8        512            2
9        729            9```

Therefore all numbers from 0 to 2020 are equally divided in 10 buckets. If we make a table for square, we don’t get equal distribution. In the following table. 1, 4, 6 and 9 are repeated, so these buckets would have more entries and buckets 2, 3, 7 and 8 would be empty.

```Number   Square     Last Digit in Cube
0        0              0
1        1              1
2        4              4
3        9              9
4        16             6
5        25             5
6        36             6
7        49             9
8        64             4
9        81             1```

 Question 44
Assume that the bandwidth for a TCP connection is 1048560 bits/sec. Let α be the value of RTT in milliseconds (rounded off to the nearest integer) after which the TCP window scale option is needed. Let β be the maximum possible window size with window scale option. Then the values of α and β are.
 A 63 milliseconds 65535 × 214 B 63 milliseconds 65535 × 216 C 500 milliseconds 65535 × 214 D 500 milliseconds 65535 × 216

Explanation:

Since sequence number in TCP header is limited to 16 bits, the maximum window size is limited. When bandwidth delay product of a link is high, scaling is required to efficiently use link. TCP allows scaling of windows when bandwidth delay product is greater than 65,535 (Refer this). The bandwidth delay product for given link is 1048560 * α. Window scaling is needed when this value is more than 65535 bytes, i.e., when α is greater than 65535 * 8 / 1048560 or 0.5 seconds. Scaling is done by specifying a one byte shift count in the header options field. The true receive window size is left shifted by the value in shift count. A maximum value of 14 may be used for the shift count value. Therefore maximum window size with scaling option is 65535 × 214.

 Question 45

Consider alphabet ∑ = {0, 1}, the null/empty string λ and the sets of strings X0, X1 and X2 generated by the corresponding non-terminals of a regular grammar. X0, X1 and X2 are related as follows:

```X0 = 1 X1
X1 = 0 X1 + 1 X2
X2 = 0 X1 + {λ}```

Which one of the following choices precisely represents the strings in X0?

 A 10 (0* + (10)*)1 B 10 (0* + (10)*)*1 C 1(0* + 10)*1 D 10 (0 + 10)*1 + 110 (0 + 10)*1

Explanation:

```The smallest possible string by given grammar is "11".
X0 = 1X1
= 11X2  [Replacing X1 with 1X2]
= 11    [Replacing X2 with λ]

The string "11" is only possible with option (C).```

 Question 46

Given below are some algorithms, and some algorithm design paradigms.

```List-I
A. Dijkstra’s Shortest Path
B. Floyd-Warshall algorithm to compute all pairs shortest path
C. Binary search on a sorted array
D. Backtracking search on a graph

List-II
1. Divide and Conquer
2. Dynamic Programming
3. Greedy design
4. Depth-first search

Match the above algorithms on the left to the corresponding design paradigm they follow Codes:

```    A B C D
(a) 1 3 1 5
(b) 3 3 1 5
(c) 3 2 1 4
(d) 3 2 1 5```
 A a B b C c D d

Explanation:

```Dijkstra’s Shortest Path is a Greedy Algorithm.

Floyd-Warshall algorithm is Dynamic Programming.

Binary search is a Divide and Conquer.

Backtracking is Depth-first search```

 Question 47

The number of min-terms after minimizing the following Boolean expression is _________.

`      [D′ + AB′ + A′C + AC′D + A′C′D]′`
 A 1 B 2 C 3 D 4

Explanation:

```Given Boolean expression is:
[D′ + AB′ + A′C + AC′D + A′C′D]′

Step 1 : [D′ + AB′ + A′C + C′D ( A + A')]′
( taking C'D as common )

Step 2 : [D′ + AB′ + A′C + C′D]′
( as, A + A' = 1 )

: [D' + DC' + AB' + A'C]' (Rearrange)

Step 3 : [D' + C' + AB' + A'C]'
( Rule of Duality, A + A'B = A + B )

: [D' + C' + CA' + AB']' (Rearrange)

Step 4 : [D' + C' + A' + AB']'
(Rule of Duality)

: [D' + C' + A' + AB']' (Rearrange)

Step 5 : [D' + C' + A' + B']'
(Rule of Duality)

:[( D' + C' )'.( A' + B')']
(Demorgan's law, (A + B)'=(A'. B'))

:[(D''.C'').( A''.B'')] (Demorgan's law)

:[(D.C).(A.B)] (Idempotent law, A'' = A)

: ABCD

Hence only 1 minterm after minimization.```

 Question 48

Consider the C program below.

 `#include ` `int` `*A, stkTop;` `int` `stkFunc (``int` `opcode, ``int` `val)` `{` `    ``static` `int` `size=0, stkTop=0;` `    ``switch` `(opcode)` `    ``{` `    ``case` `-1:` `        ``size = val;` `        ``break``;` `    ``case` `0:` `        ``if` `(stkTop < size ) A[stkTop++]=val;` `        ``break``;` `    ``default``:` `        ``if` `(stkTop) ``return` `A[--stkTop];` `    ``}` `    ``return` `-1;` `}` `int` `main()` `{` `    ``int` `B[20];` `    ``A=B;` `    ``stkTop = -1;` `    ``stkFunc (-1, 10);` `    ``stkFunc (0, 5);` `    ``stkFunc (0, 10);` `    ``printf` `(``"%d\n"``, stkFunc(1, 0)+ stkFunc(1, 0));` `}`

The value printed by the above program is ___________

 A 9 B 10 C 15 D 17

Explanation:

The code in main, basically initializes a stack of size 10, then pushes 5, then pushes 10. Finally the printf statement prints sum of two pop operations which is 10 + 5 = 15.

```    stkFunc (-1, 10);   // Initialize size as 10
stkFunc (0, 5);    // push 5
stkFunc (0, 10);   // push 10

// print sum of two pop
printf ("%d\n", stkFunc(1, 0) + stkFunc(1, 0));```

 Question 49

The secant method is used to find the root of an equation f(x) = 0. It is started from two distinct estimates xa and xb for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if f(xb) is very small and then xb is the solution. The procedure is given below. Observe that there is an expression which is missing and is marked by? Which is the suitable expression that is to be put in place of? So that it follows all steps of the secant method? Secant

```Initialize: xa, xb, ε, N     // ε = convergence indicator
fb = f(xb) i = 0
while (i < N and |fb| > ε) do
i = i + 1                 // update counter
xt = ?                    // missing expression for
// intermediate value
xa = xb                   // reset xa
xb = xt                   // reset xb
fb = f(xb)                // function value at new xb
end while
if |fb| > ε
then // loop is terminated with i = N
write “Non-convergence”
else
write “return xb”
end if```
 A xb – (fb– f(xa)) fb/ (xb – xa) B xa– (fa– f(xa)) fa/ (xb – xa) C xb – (fb – xa) fb/ (xb – fb(xa) D xa – (xb – xa) fa/ (fb – f(xa))

 Question 50
The number of onto functions (surjective functions) from set X = {1, 2, 3, 4} to set Y = {a, b, c} is ________________
 A 36 B 64 C 81 D 72

Explanation:

A function f from X to Y is called onto if for all ‘y’ in Y there is an ‘x’ in X such that f(x) = y.          In onto functions, all elements in Y are used. Every Surjective or Onto function sends two elements of {1, 2, 3, 4} to the same element of {a, b, c}. There are 4C2 = 6 such pairs of elements. The pairs are {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}. For a given pair {i, j} ⊂ {1, 2, 3, 4}, there are 3! sujective functions such that f(i) = f(j). Hence there are total 6 * 6 = 36 surjective functions.

 Question 51

Consider the following routing table at an IP router

For each IP address in Group-I identify the correct choice of the next hop from Group-II using the entries from the routing table above.

```   List-I               List-II
A. 128.96.171.92     1. Interface 0
B. 128.96.167.151    2. Interface 1
C. 128.96.163.121    3. R2
D. 128.96.165.121    4. R3
5. R4
Codes:
A B C D
(a)  1 3 5 4
(b)  1 4 2 5
(c)  2 3 4 5
(d)  2 3 5 4```
 A a B b C c D d

Explanation:

The next hop is decide according to the longest prefix matching.  Refer following link for details. https://www.youtube.com/watch?v=PSC5omE3pX8&list=PLkHsKoi6eZnzJl1qTzmvBwTxrSJW4D2Jj&index=32.

 Question 52

Consider a processor with byte-addressable memory. Assume that all registers, including Program Counter (PC) and Program Status Word (PSW), are of size 2 bytes. A stack in the main memory is implemented from memory location (0100)16 and it grows upward. The stack pointer (SP) points to the top element of the stack. The current value of SP is (016E)16. The CALL instruction is of two words, the first word is the op-code and the second word is the starting address of the subroutine (one word = 2 bytes). The CALL instruction is implemented as follows:

```   • Store the current value of PC in the stack.
• Store the value of PSW register in the stack.

The content of PC just before the fetch of a CALL instruction is (5FA0)16. After execution of the CALL instruction, the value of the stack pointer is

 A (016A)16 B (016C)16 C (0170)16 D (0172)16

Explanation:

```The current value of SP is (016E)16

The value of SP after following operations is asked
in question

• Store the current value of PC in the stack.
This operation increments SP by 2 bytes as size
of PC is given 2 bytes in question.
So becomes (016E)16 + 2  = (0170)16

• Store the value of PSW register in the stack.
This operation also increments SP by 2 bytes as size
of PSW is also given 2 bytes.
So becomes (0170)16 + 2  = (0172)16

The Load operation doesn't change SP.

So new value of SP is  (016E)16```

 Question 53
Which one of the following assertions concerning code inspection and code walkthrough is True?
 A Code inspection is carried out once the code has been unit tested B Code inspection and code walkthrough are synonyms C Adherence to coding standards is checked during code inspection D Code walkthrough is usually carried out by an independent test team

Explanation:

A is False: It is not necessary that code inspection is carried out once the code has been unit tested. B and D are false: Code walkthrough or walk-through is a form of software peer review in which a designer or programmer leads members of the development team and other interested parties through a software product, and the participants ask questions and make comments about possible errors, violation of development standards, and other problems” Inspection in software engineering, refers to peer review of any work product by trained individuals who look for defects using a well defined process.

 Question 54

Consider the sequence of machine instructions given below:

```  MUL R5, R0, R1
DIV R6, R2, R3
SUB R8, R7, R4```

In the above sequence, R0 to R8 are general purpose registers. In the instructions shown, the first register stores the result of the operation performed on the second and the third registers. This sequence of instructions is to be executed in a pipelined instruction processor with the following 4 stages: (1) Instruction Fetch and Decode (IF), (2) Operand Fetch (OF), (3) Perform Operation (PO) and (4) Write back the Result (WB). The IF, OF and WB stages take 1 clock cycle each for any instruction. The PO stage takes 1 clock cycle for ADD or SUB instruction, 3 clock cycles for MUL instruction and 5 clock cycles for DIV instruction. The pipelined processor uses operand forwarding from the PO stage to the OF stage. The number of clock cycles taken for the execution of the above sequence of instructions is ___________

 A 11 B 12 C 13 D 14

Explanation:

```  1   2   3   4   5   6   7   8   9   10   11   12   13
IF  OF  PO  PO  PO  WB
IF  OF          PO  PO  PO  PO  PO   WB
IF          OF                   PO   WB
IF          OF                    PO   WB```

 Question 55

Suppose you are provided with the following function declaration in the C programming language.

`   int partition (int a[], int n);`

The function treats the first element of a[] as a pivot, and rearranges the array so that all elements less than or equal to the pivot is in the left part of the array, and all elements greater than the pivot is in the right part. In addition, it moves the pivot so that the pivot is the last element of the left part. The return value is the number of elements in the left part. The following partially given function in the C programming language is used to find the kth smallest element in an array a[ ] of size n using the partition function. We assume k ≤ n

 `int` `kth_smallest (``int` `a[], ``int` `n, ``int` `k)` `{` `   ``int` `left_end = partition (a, n);` `   ``if` `(left_end+1==k)` `   ``{` `       ``return` `a [left_end];` `   ``}` `   ``if` `(left_end+1 > k)` `   ``{` `      ``return` `kth_smallest (____________________);` `   ``}` `   ``else` `   ``{` `      ``return` `kth_smallest (____________________);` `    ``}` `}`

The missing argument lists are respectively

 A (a, left_end, k) and (a+left_end+1, n–left_end–1, k–left_end–1) B (a, left_end, k) and (a, n–left_end–1, k–left_end–1) C (a, left_end+1, N–left_end–1, K–left_end–1) and(a, left_end, k) D (a, n–left_end–1, k–left_end–1) and (a, left_end, k)

 Question 56

Consider a simple checkpointing protocol and the following set of operations in the log.

```(start, T4); (write, T4, y, 2, 3); (start, T1); (commit, T4); (write, T1, z, 5, 7);
(checkpoint);
(start, T2); (write, T2, x, 1, 9); (commit, T2); (start, T3); (write, T3, z, 7, 2);```

If a crash happens now and the system tries to recover using both undo and redo operations, what are the contents of the undo list and the redo list

 A Undo: T3, T1; Redo: T2 B Undo: T3, T1; Redo: T2, T4 C Undo: none; Redo: T2, T4, T3; T1 D Undo: T3, T1, T4; Redo: T2

Explanation:

Since T1 and T3 are not committed yet, they must be undone. The transaction T2 must be redone because it is after the latest checkpoint.

 Question 57
A Computer system implements 8 kilobyte pages and a 32-bit physical address space. Each page table entry contains a valid bit, a dirty bit three permission bits, and the translation. If the maximum size of the page table of a process is 24 megabytes, the length of the virtual address supported by the system is _______________ bits
 A 36 B 32 C 28 D 40

Explanation:

```Max size of virtual address can be calculated by
calculating maximum number of page table entries.

Maximum Number of page table entries can be calculated
using given maximum page table size and size of a page
table entry.

Given maximum page table size = 24 MB

Let us calculate size of a page table entry.

A page table entry has following number of bits.
1 (valid bit) +
1 (dirty bit) +
3 (permission bits) +
x bits to store physical address space of a page.

Value of x = (Total bits in physical address) -
(Total bits for addressing within a page)
Since size of a page is 8 kilobytes, total bits needed within
a page is 13.
So value of x = 32 - 13 = 19

Putting value of x, we get size of a page table entry =
1 + 1 + 3  + 19 = 24bits.

Number of page table entries
= (Page Table Size) / (An entry size)
=  (24 megabytes / 24 bits)
=  223

= (Number of page table entries) * (Page Size)
=  223 * 8 kilobits
= 236
Therefore, length of virtual address space = 36```

 Question 58
A half adder is implemented with XOR and AND gates. A full adder is implemented with two half adders and one OR gate. The propagation delay of an XOR gate is twice that of an AND/OR gate. The propagation delay of an AND/OR gate is 1.2 microseconds. A 4-bit ripple-carry binary adder is implemented by using full adders. The total propagation time of this 4-bit binary adder in microseconds is

Explanation:

```
Let us first calculate propagation delay of a single

Propagation Delay by n bit full adder is (2n + 2)
gate delays.
[See this for formula].

Here n = 1, so total delay of a 1 bit full adder
is (2 + 2)*1.2 = 4.8 ms

Delay of 4 full adders is = 4 * 4.8 = 19.2 ms```

 Question 59
Consider a typical disk that rotates at 15000 rotations per minute (RPM) and has a transfer rate of 50 × 106 bytes/sec. If the average seek time of the disk is twice the average rotational delay and the controller’s transfer time is 10 times the disk transfer time, the average time (in milliseconds) to read or write a 512 byte sector of the disk is _____________

Explanation:

```Disk latency = Seek Time + Rotation Time + Transfer Time + Controller Overhead
Seek Time? Depends no. tracks the arm moves and seek speed of disk
Rotation Time? depends on rotational speed and how far the sector is from the head
Transfer Time? depends on data rate (bandwidth) of disk (bit density) and the size of request

Disk latency = Seek Time + Rotation Time +

Average Rotational Time = (0.5)/(15000 / 60) = 2 miliseconds
[On average half rotation is made]

It is given that the average seek time is twice the average rotational delay
So Avg. Seek Time =  2 * 2 = 4 miliseconds.

Transfer Time = 512 / (50 × 106 bytes/sec)
= 10.24 microseconds

Given that controller time is 10 times the average transfer time
Controller Overhead = 10 * 10.24 microseconds
= 0.1 miliseconds

Disk latency = Seek Time + Rotation Time +
= 4 + 2 + 10.24 * 10-3 + 0.1 miliseconds
= 6.1 miliseconds
```

 Question 60
In a connected graph, a bridge is an edge whose removal disconnects a graph. Which one of the following statements is True?
 A A tree has no bridge B A bridge cannot be part of a simple cycle C Every edge of a clique with size ≥ 3 is a bridge (A clique is any complete subgraph of a graph) D A graph with bridges cannot have a cycle

Explanation:

A bridge in a graph cannot be a part of cycle as removing it will not create a disconnected graph if there is a cycle.

 Question 61

Which of the following languages is/are regular?

```L1: {wxwR ⎪ w, x ∈ {a, b}* and ⎪w⎪, ⎪x⎪ >0} wR is the reverse of string w
L2: {anbm ⎪m ≠ n and m, n≥0
L3: {apbqcr ⎪ p, q, r ≥ 0}```
 A L1 and L3 only B L2 only C L2 and L3 only D L3 only

Explanation:

```L3 is simple to guess, it is regular.
Below is DFA for L1.

L1 is interesting. The important thing to note
about L1 is length of x is greater than 0, i.e.,
|x| > 0.
So any string than starts and ends with same character
is acceptable by language and remaining string becomes w.
Below is DFA for L1.
```

 Question 62
Host A sends a UDP datagram containing 8880 bytes of user data to host B over an Ethernet LAN. Ethernet frames may carry data up to 1500 bytes (i.e. MTU = 1500 bytes). Size of UDP header is 8 bytes and size of IP header is 20 bytes. There is no option field in IP header. How may total number of IP fragments will be transmitted and what will be the contents of offset field in the last fragment?
 A 6 and 925 B 6 and 7400 C 7 and 1110 D 7 and 8880

Explanation:

```UDP data = 8880 bytes

Total Size excluding IP Header = 8888 bytes.

Number of fragments  = ⌈ 8888 / 1480 ⌉
= 7
Refer the Kurose book slides on IP (Offset is always scaled by 8)

Offset of last segment = (1480 * 6) / 8 = 1110```

 Question 63
The number of states in the minimal deterministic finite automaton corresponding to the regular expression (0 + 1)*(10) is ____________
 A 2 B 3 C 4 D 5

Explanation:

Below is NFA for the given regular expression (0 + 1)*(10)
Below is DFA for the same.

 Question 64
Let X and Y denote the sets containing 2 and 20 distinct objects respectively and F denote the set of all possible functions defined from X and Y. Let f be randomly chosen from F. The probability of f being one-to-one is _________
 A 0.95 B 0.8 C 0.75 D 0.7

Explanation:

X has 2 elements Y has 20 elements Number of functions from X to Y is 20*20 [Every element can take any of the 20 values] Number of one to one functions from X to Y is 20*19 [Every element takes a different value] So probability of a function being one to one = (20*19) / (20*20) = 380 / 400 = 0.95

 Question 65
Which one of the following well formed formulae is a tautology?
 A A B B C C D D