What topics are covered on Page 5 of GATE 2016 CS/IT Set 1?

Page 5 currently contains Questions 53 to 61. Engineering Mathematics topics include functions, combinatorics, eigenvalues, probability density functions and polynomial factors. Operating Systems topics include page replacement, C-LOOK disk scheduling, page-table size and CPU scheduling.

What is the coefficient of x^12 in the GATE 2016 combinatorics question?

The coefficient of x^12 in (x^3 + x^4 + x^5 + x^6 + ...)³ is 10.

What is the determinant of the 3 by 3 real matrix in GATE 2016 Question 55?

Because a real matrix with eigenvalue 2+i must also have eigenvalue 2-i, the three eigenvalues are 2+i, 2-i and 3. Their product gives a determinant of 15.

What is the value of a for the probability density 1/x² over [a,1]?

Normalizing the probability density over the interval gives a = 0.5.

What disk scheduling algorithm is used in GATE 2016 Question 59?

Question 59 uses the C-LOOK disk scheduling algorithm, with the disk head initially moving toward higher-numbered cylinders.

Which scheduling algorithm minimizes average waiting time when all CPU-bound processes arrive together?

When all processes arrive at the same time and their CPU burst lengths are known, shortest remaining time first reduces to shortest-job-first behaviour and minimizes average waiting time.

CS and IT Gate 2016 Set-1 Questions with Answer

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Ques 53 Gate 2016 Set-1

A function f : N⁺ → N⁺, defined on the set of positive integers N⁺, satisfies the following

properties:
f(n) = f(n/2) if nis even
f(n) = f(n+5) if nis odd

Let R = {i|∃ j : f(j) = i} be the set of distinct values that f takes. The maximum possible size of R is .

(a) is the correct answer.

Ques 54 Gate 2016 Set-1

The coefficient of x¹² in (x³ + x⁴ + x⁵ + x⁶ + ...)³ is_________

(a) is the correct answer.

Ques 55 Gate 2016 Set-1

Two eigenvalues of a 3 x 3 real matrix P are (2 + √ -1) and 3. The determinant of P is _____

(a) is the correct answer.

Ques 56 Gate 2016 Set-1

A probability density function on the interval [a, 1] is given by 1 / x² and outside this interval the value of the function is zero. The value of a is :

(a) is the correct answer.

Ques 57 Gate 2016 Set-1

If f(x) = 2x⁷ + 3x - 5 Which of the following is a factor of f(x)?

(x³ + 8)

(x - 1)

(2x - 5)

(x + 1)

(Q) is the correct answer.

Given:
f(x) = 2x⁷ + 3x - 5
To find:
Which of the given options is a factor of f(x).
Solution:
According to the factor theorem, (x - a) is a factor of f(x) if f(a) = 0.
Let me quickly test the simplest linear factors from the options.
Let's check option (B) first, which gives x = 1.
Now substituting x = 1 into f(x):
f(1) = 2(1)⁷ + 3(1) - 5
So we get:
f(1) = 2 + 3 - 5 = 0
Since f(1) = 0, (x - 1) is definitely a factor of f(x).
No need to check the rest, option B works directly.
Answer: (B) (x - 1)

Ques 58 Gate 2016 Set-1

Consider a computer system with ten physical page frames. The system is provided with an access sequence a₁, a₂, ..., a₂₀, a₁, a₂, ..., a₂₀), where each ai 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 __________

(a) is the correct answer.

Ques 59 Gate 2016 Set-1

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______

(a) is the correct answer.

Ques 60 Gate 2016 Set-1

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.

(a) is the correct answer.

Ques 61 Gate 2016 Set-1

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?

Shortest remaining time first

Round-robin with time quantum less than the shortest CPU burst

Uniform random

Highest priority first with priority proportional to CPU burst length

(S) is the correct answer.

Given:
An arbitrary set of CPU-bound processes with unequal burst lengths.
All processes arrive at the same time (arrival time t = 0).
To find:
The scheduling algorithm that minimizes the average waiting time in the ready queue.
Solution:
Okay, let's think about which algorithm gives the minimum average waiting time.
I know that Shortest Job First (SJF) is provably optimal for giving the minimum average waiting time among all scheduling algorithms.
Here, it says all processes are submitted at the same time.
When all processes arrive at time t = 0, no new process arrives later with a shorter burst to cause preemption.
So, Shortest Remaining Time First (SRTF), which is the preemptive version of SJF, behaves exactly like non-preemptive SJF.
Both will run the process with the smallest CPU burst first, then the next smallest, and so on.
Let's check the other options to be completely sure.
Option B is Round-robin, which shares the CPU and increases the average waiting time compared to SJF.
Option C is uniform random scheduling, which is clearly not optimal.
Option D assigns higher priority to processes with longer bursts, which turns it into Longest Job First (LJF), giving the worst average waiting time.
Hence, Shortest Remaining Time First gives the minimum average waiting time.
Answer: (A) Shortest remaining time first

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