What topics are covered on Page 5 of GATE 2016 CS/IT Set 2?

Page 5 currently contains Questions 53 to 58. Questions 53 and 54 cover modular arithmetic and linear algebra. Questions 55 to 58 cover Operating Systems topics including counting semaphores, preemptive shortest-remaining-time-first scheduling, two-process synchronization and page replacement.

What is 13^99 modulo 17 in GATE 2016 Set 2?

Using modular arithmetic, 13^99 modulo 17 evaluates to 4.

Which statement about a system of m linear equations in n variables is always correct?

The statement that when m equals n there exists a system having a solution is correct. Having fewer equations than variables does not guarantee consistency, and having more equations than variables does not guarantee inconsistency.

What does the turn-variable synchronization solution violate?

The strict alternation solution preserves mutual exclusion but can violate the progress requirement because one process may be forced to wait even when the other process is not interested in entering its critical section.

Which page replacement algorithm can exhibit Belady anomaly?

FIFO page replacement can exhibit Belady anomaly, where increasing the number of available page frames can unexpectedly increase the number of page faults.

What Operating Systems topics are present on Page 5?

The Operating Systems questions cover semaphore blocking, preemptive SRTF scheduling, critical-section synchronization and FIFO page replacement.

CS and IT Gate 2016 Set-2 Questions with Answer

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Ques 53 Gate 2016 Set-2

The value of the expression 13⁹⁹(mod 17), in the range 0 to 16, is

(Number System) is the correct answer.

PROCESSED

Ques 54 Gate 2016 Set-2

Consider the systems, each consisting of m linear equations in n variables.

I. If m < n, then all such systems have a solution
II. If m > n, then none of these systems has a solution
III. If m = n, then there exists a system which has a solution

Which one of the following is CORRECT?

I, II and III are true

Only II and III are true

Only III is true

None of them is true

(L) is the correct answer.

Given:
A system of m linear equations in n variables.
Statement I: If m < n, then all such systems have a solution.
Statement II: If m > n, then none of these systems has a solution.
Statement III: If m = n, then there exists a system which has a solution.
To find:
The correct statement(s) among I, II, and III.
Solution:
Let me analyze each statement one by one.
First, checking statement I: "If m < n, then all such systems have a solution."
Let me take a counterexample with m = 2 equations and n = 3 variables:
x + y + z = 1
x + y + z = 2
Here, the two parallel planes never intersect, so there is no solution.
Since an inconsistent underdetermined system can exist, not all such systems have a solution.
So statement I is FALSE.
Now, checking statement II: "If m > n, then none of these systems has a solution."
Let me test an overdetermined system where m = 3 and n = 2:
x + y = 2
2x + 2y = 4
3x + 3y = 6
Clearly, this has infinitely many solutions. Even a system like x = 1, y = 1, x + y = 2 has a unique solution.
So overdetermined systems can have solutions.
Thus, statement II is FALSE.
Now, looking at statement III: "If m = n, then there exists a system which has a solution."
This statement just says "there exists", so I only need to show one example.
Let m = n and choose the identity matrix for coefficients:
I_n X = B
Since det(I_n) = 1 ≠ 0, this system always has a unique solution.
So statement III is definitely TRUE.
Therefore, only statement III is correct.
Answer:
(C) Only III is true

Ques 55 Gate 2016 Set-2

Consider a non-negative counting semaphore S. The operation P(S) decrements S, and V(S) increments S. During an execution, 20 P(S) operations and 12 V(S) operations are issued in some order. The largest initial value of S for which at least one P(S) operation will remain blocked is ________.

(a) is the correct answer.

Ques 56 Gate 2016 Set-2

Consider the following processes, with the arrival time and the length of the CPU burst given in milliseconds. The scheduling algorithm used is preemptive shortest remaining-time first.

Process	Arrival Time	Burst Time
P₁	0	10
P₂	3	6
P₃	7	1
P₃	8	3

The average turn around time of these processes is ___________ milliseconds.

(a) is the correct answer.

Ques 57 Gate 2016 Set-2

Consider the following two-process synchronization solution.

Process 0
---------
Entry: loop while (turn == 1);
(critical section)
Exit: turn = 1;

Process 1
----------
Entry: loop while (turn == 0);
(critical section)
Exit: turn = 0;

The shared variable turn is initialized to zero. Which one of the following is TRUE?

This is a correct two-process synchronization solution.

This solution violates mutual exclusion requirement.

This solution violates progress requirement.

This solution violates bounded wait requirement.

(process synchronization) is the correct answer.

PROCESSED

Ques 58 Gate 2016 Set-2

In which one of the following page replacement algorithms it is possible for the page fault rate to increase even when the number of allocated frames increases?

LRU (Least Recently Used)

OPT (Optimal Page Replacement)

MRU (Most Recently Used)

FIFO (First In First Out)

(P) is the correct answer.

Given:
Page replacement algorithms: LRU, OPT, MRU, and FIFO.
Condition: Page fault rate increases even when the number of allocated frames increases.
To find:
The page replacement algorithm that shows this behavior.
Solution:
Okay, I know that this phenomenon is called Belady's Anomaly.
Let me recall which algorithms suffer from Belady's Anomaly.
Stack-based algorithms never exhibit Belady's anomaly because the set of pages in memory for n frames is always a subset of the pages in memory for n + 1 frames.
Both LRU and Optimal (OPT) are stack algorithms, so they never suffer from this issue.
Now looking at FIFO, the First-In-First-Out algorithm does not satisfy the inclusion property.
Because of this, for certain reference strings, increasing the number of page frames actually leads to more page faults in FIFO.
So FIFO is definitely the algorithm that shows this behavior.
Answer: (D) FIFO (First In First Out)

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