Filter
1

https://aofa.cs.princeton.edu/40asymptotic/

When studying algorithms, this chapter covers methods for obtaining approximate solutions to problems or approaching exact answers, which allow us to generate succinct and precise approximations of quantities of interest.

hope you find this helpful :)

2

https://aofa.cs.princeton.edu/20recurrence/

chapter focuses on the underlying mathematical aspects of various forms of recurrence relations, which are typically encountered while studying an algorithm by translating a recursive representation of a programme to a recursive representation of a function characterising its attributes.

3
4

### An MS student is required to complete around 12--16 credits of coursework, clear a comprehensive viva-voce at the end of the course work showing competency in basics of CSE (syllabus given below) and in specific subjects the student has taken during his/her MS, and do research…

5

For visualization

https://visualgo.net/en → animation

https://algorithm-visualizer.org/ → animation with code

6

computer science

23/sep/2021  BARC OCES/DGFS 2021   SET 4

mostly question are previous year of gate.  i recall some of them some of them

if packet transfer from node 1 to node b then which of the following will be change  (ttl  checksum fragment offset).

3-4 question from  boolean simplified form

1 question from mux

1 question of 4 D flip flop  which have an nand gate between 2 LSB flip flop and ask after how much time i can regenerate clock.

10-12 question of c programming

mostly theoretical question of each subject,but easy

5-6 question based on direct mapping associative mapping, find tag, no of line  (tag set block)

3-4 basic of DBMS relational algebra to language

1 question from predicate logic   may be    (all student are------ and -----) i cant remember but easy

paper was easy to  moderate not much hard.

7

# Topic-wise video Lectures (with slide format notes) by  J.F. Kurose [Author]

### ttps://gaia.cs.umass.edu/kurose_ross/online_lectures.htm

Solution (book) → https://1lib.in/book/6040932/94b2fa

## Data Communications and Network

Solution (book) → https://1lib.in/book/1201808/d09f03?dsource=recommend

8

Here are the notes I have written during my preparation for GATE CS/IT. Future aspirants may find it helpful. These are elaborated notes, that one may go through to recapitulate stuff fast. I took help from standard books and online courses to put together these notes. As of now, only some chapters of Discrete Mathematics are uploaded. I will upload notes for all the subjects within a month.

https://gatecsebyjs.github.io/

Sample note(on Combinatorics):

Peace!

### 🌟Notes of all subjects uploaded🌟

Update 1(25/04/2021): Discrete Mathematics completely uploaded

Update 3(27/04/2021): Theory of Computation, Databases uploaded

Update 4(28/04/2021): C Programming, Data Structures uploaded

Update 5(01/05/2021): Digital Logic, Algorithms uploaded

Update 6(02/05/2021): Computer Networks uploaded

Update 7(03/05/2021): Operating System Notes, Engg. Maths. materials uploaded

9
Can anyone upload a DRDO previous year papers/ link.

I don’t remember completely but someone had pasted the link on a blog here a year ago.

Thanks in advance. Gateoverflow is one the most wonderful site I had ever used in my entire Journey to AIR 89.

---------------------------------------------------------------------------------------------------------------------------------------------------
10

Books I read :

Subject Name and Author Relevant Chapters
Algorithm and DS
1. Introduction to Algorithms, by CLRS  (3E)
2. Algorithm Design, Jon Kleinberg and Éva Tardos
1. ch 1-4, 6-9,10, 11.1-11.4, 12.1-21.3, 15, 16.1-16.3, 17, 21-25.2
2. ch 1-6
Discrete Mathematics
1. Discrete mathematics and its applications by Kenneth H. Rosen (Indian 7E)
2. Discrete mathematics with applications by Susanna S. Epp (4E)
3. Concrete Mathematics by Donald Knuth, Oren Patashnik, and Ronald Graham (not required for GATE)
1. ch 1,2, 4-8, 11.1-11.3
2. ch 1, 2.1-2.3, 3, 4(optional), 5.1, 5.5-5.7, 6-10, 11-12(optional)
3. ch 1-3, 5, 7-9
Computer Networks
1. Data Communications and Networking by Behrouz A. Forouzan (5E)

1. 1.1-1.3,  2, 3.6, 8-10, 11.1-2, 12, 13.1-13.2, 17.1, 18-19.2, 20-21.2, 23-24.3, 25.1-25.2, 26
Theory of Computation
1. An Introduction to Formal Languages and Automata by Peter Linz (6E)

1. ch 1.2, 1.3, 2-12, Appendix-A
Digital Logic
1. Digital Logic and Computer Design by M. Morris Mano

1. 1.1-1.8, 2.1-2.7, 3-7
Computer Organization
1. Computer Organisation by Carl Hamacher
2. Computer Organization and Design: the Hardware/Software Interface by David A Patterson and John L. Hennessy (5E)
1.  ch 1.6, 2.1-2.5, 2.9, 2.10, 4.1-4.2,4.4-4.6, 5.1,5.2, 5.4-5.8, 5.9.1, 6.1-6.4, 6.7.1, 7, 8.1-8.5, 8.8,
2. 1, 2, 4.1-4.9, 4.14, 5.1-5.10
C
1. The C Programming Language by Brian Kernighan and Dennis Ritchie (2E)

1.  ch 1-8
Operating System
1. Operating Systems by Avi Silberschatz, Greg Gagne, and Peter Baer Galvin (International 9E)

1.  ch 2.1-2.5, 3, 4.1-4.3, 4.6, 5.1-5.3, 6.1-6.10, 7, 8.1-8.6, 91.-9.6, 9.9, 10, 11.1-11.5, 12.1-12.6
Databases
1. Fundamentals of Database Systems by Ramez Elmasri and Shamkant B. Navathe (7E)
1. ch 1.3-1.6, 2.1-2.3, 3, 5-8, 9.1, 14.1-14.5, 14.6-14.7(just overview), 15.1-15.4, 16.1-16.7, 17.1-17.6, 20.1-20.5, 21.1-21.4, 21.7
1. Read every required topic and solved almost all related exercises from these books.
2. Read every gate-related topic from underlined books, but didn’t solve any exercise problem.
3. referred to other mentioned text for few topics and few exercise problems
4. for all the subjects or topics I left above, didn't read any book.

### Video lectures I studied from :

Subject reference links lecture numbers
Algorithm and DS
1.  1-7, 15-19
2.  1-4, 6-8, 11-15     unofficial link
3. can watch the complete course.
4. it’s amazing, can watch it all.
Discrete Mathematics
1. 1-19 (all except last)  unofficial link
Theory of Computation
1. 1-4, 6,7, 9, 11-13, 15,16, 18, 19
Computer Organization
1. I would suggest watch all from 1-28 or GO playlist
2. 7-12, 13-14(optional), 15,16, 17-22(recommended if you have extra time),  23-32, 34-37

(lots of extra things in IITD videos, you can skip as per your interest)

Operating System
1. covered in CO section or GO playlist
Compilers
1. week 2-9 (skip Cool Type Checking in week 6)
Linear Algebra
1. 1-10, 14,16-21
•   for interviews I suggest you watch it all, it just amazing how intuitively he taught everything.
Probability
1. Probabilistic Systems Analysis and Applied Probability  (best lectures acc. to me)
2. Statistics 110: Probability

1.  1-8, 13-15
2.  GO playlist
Graph Theory
1.  1, 2, 7, 9, 13, 15, 17,
• Please note this is a graduate-level course, if you have less time/interest in this topic avoid watching these lectures, better go with a book.
Group Theory
1. 1-9, 16

I also read these notes by Manu Thakur as a revision and just to check in case if I’m missing any topic, they are nicely compiled and only a few topics are not covered by these notes.

Not everything I mentioned is important for GATE, One should be smart enough to filter out what to read/watch from these references, I haven’t added any extra reference here... I have followed every single reference mentioned, I was enjoying learning specially all these maths lectures, I watched lots of out of the syllabus stuff on these topics. I left it for you to filter the necessary topics at your convenience. One person might need a different approach, go according to what is best for you... don’t restrict yourself just to these... explore more and more.

I solved NPTEL assignments as well, You can get those just by a Google search, If you couldn’t find them let me know, I’ll add links to those as well.

and a special Thanks to GO, most of the resources here were recommended by GO, you can find those here: best-books and best-videos

you can find all my GATE related bookmarks here, download this file and open it with any browser, you’ll get some of the additional links I followed plus links to NPTEL courses from where you can download the assignments and official page for many courses I referred, you can check their tests/assignments as well.

Most of the questions in these NPTEL assignments are of 1 mark level (still worth trying if you have time, you’ll find lots of interesting things), I first solved all pyqs (including TIFR problems) once before touching any extra question.

It’s up to you how you wanna use these assignments.

A nice website collecting most nptel courses and some additional useful links

11

Hello Everyone !!!

I am sharing a document titled "SECURITY OF CYBER SYSTEMS".

This document is relevant for all aspirants appearing for DRDO Scientist-B 2020 examination as well as for cyber security enthusiasts.

Initially, I have compiled a list of popular attacks. Based on your feedback I will extend it for rest part of DRDO syllabus.

12

$$\textbf{Scientist – ‘B’}$$

$\textbf{ELIGIBILITY:}$ B.E/ B.Tech/ DOEACC B-level OR AMIE/ GIETE OR MSc OR MCA OR ME/ M.Tech OR M.Phil Electronics, Electronics and Communication, Computer Sciences, Communication, Computer and Networking Security, Computer Application, Software System, Information Technology, Information Technology Management, Informatics, Computer Management, Cyber law, Electronics and Instrumentation.

 $\textbf{Dates}$ $\textbf{Question Paper}$ $\textbf{Answer Key}$ $\textbf{GO/AO Links}$ $\textbf{Exam Links}$ $\text{APR}\;2022$ 2022(CS) Questions GO $\text{Dec}\; 2021$ 2021(CS) Questions GO $\text{Dec}\; 2021$ 2021(IT): Set B Questions GO $22\:\text{Nov}\: 2020$ Set-A Set -B Set-C Set-D Answer Key Questions GO $01\: \&\: 02\; \text{Dec}\: 2018$ 2018: Set – B Answer Key Questions GO $17\: \text{Dec}\: 2017$ 2017: Set – A Answer Key Section – A Section  – B $22\:\text{July}\: 2017$ 2017(CS): Set  – A Answer Key Section – A Section  – B $22\:\text{July}\: 2017$ 2017(IT): Set – A Answer Key Section – A Section  – B $4\:\text{Dec}\: 2016$ 2016(CS): Set – A Answer Key Section – A Section  – B $4\:\text{Dec}\: 2016$ 2016(IT): Set – A Answer Key Section – A Section  – B $12-13\:\text{Mar}\: 2016$ 2016: Set – A Answer Key Section – A Section  – B Section – C

$$\textbf{Scientific/Technical Assistant – ‘A’ }$$

$\textbf{ELIGIBILITY:}$ B.E/ B.Tech/ M.Sc./ MS/ MCA Electronics, Electronics and Communication, Electronics & Telecommunications, Computer Sciences, Computer and Networking Security, Software System, Information Technology, Informatics.

 $\textbf{Dates}$ $\textbf{Question Paper}$ $\textbf{Answer Key}$ $\textbf{GO/AO Links}$ $\textbf{Exam Links}$ $\text{Dec}\; 2021$ 2021(IT): Set D Questions GO $22\:\text{Nov}\: 2020$ Set-A Set-B Set-C Set-D Answer Key Questions GO $17\: \text{Dec}\: 2017$ 2017: Set –  A Answer Key Section – A Section – B Take exam $15\: \text{Oct}\: 2017$ 2017(CS): Set –  A Answer Key Section – A Section – B Section – C $15\: \text{Oct}\: 2017$ 2017(IT): Set –  A Answer Key Section – A Section – B Section – C $12-13\:\text{Mar}\: 2016$ 2016(1)- Senior TA Answer Key $12-13\:\text{Mar}\: 2016$ 2016(2) -Sample TA $12-13\:\text{Mar}\: 2016$ 2016(3)- Junior TA

$$\textbf{Scientist – ‘C’}$$

$$\textbf{ELIGIBILITY}$$

For fulfilling the eligibility criteria, a candidate should possess one of the Essential Educational Qualifications.

• Bachelor degree in Technology or Bachelor degree in Engineering or Associate Member of Institute of Engineers (A&B) (Computer Science or Computer Engineering or Information Technology or Electronics and Communication or Electronics and Telecommunication ) with five years (six years for Associate Member of Institute of Engineers) of relevant work experience in Ministries or Departments or Attached and Subordinate Offices of the Central Government or Statutory Bodies or Autonomous Bodies or Public Sector Undertakings or Private Sectors.
• Master degree in Science (M.Sc.) (Physics or Electronics or Applied Electronics) with six years of relevant work experience in Ministries or Departments or Attached and Subordinate Offices of the Central Government or Statutory Bodies or Autonomous Bodies or Public Sector Undertakings or Private Sectors.
• Department of Electronics and Accreditation of Computer Courses (DOEACC) B-Level or Graduate Institute of Electronics and Telecommunication Engineers (IETE) with six years of relevant work experience in Ministries or Departments or Attached and Subordinate Offices of the Central Government or Statutory Bodies or Autonomous Bodies or Public Sector Undertakings or Private Sectors.
• Master in Computer Application (MCA) with six years of relevant work experience in Ministries or Departments or Attached and Subordinate Offices of the Central Government or Statutory Bodies or Autonomous Bodies or Public Sector Undertakings or Private Sectors.
 $\textbf{Dates}$ $\textbf{Question Paper}$ $\textbf{Answer Key}$ $\textbf{GO/AO Links}$ $\textbf{Exam Links}$ $10\: \text{Feb}\: 2019$ 2019: Set –  B Questions AO(Old) Questions AO $12-13\:\text{Mar}\: 2016$ 2016: Set –  A Section – A Section – B Section – C

$$\textbf{Scientist – ‘D’}$$

$$\textbf{ELIGIBILITY}$$

For fulfilling the eligibility criteria, a candidate should possess one of the Essential Educational Qualifications.

• Bachelor degree in Technology or Bachelor degree in Engineering or Associate Member of Institute of Engineers (A&B) (Computer Science or Computer Engineering or Information Technology or Electronics and Communication or Electronics and Telecommunication ) with eight years (nine years for Associate Member of Institute of Engineers) of relevant work experience in Ministries or Departments or Attached and Subordinate Offices of the Central Government or Statutory Bodies or Autonomous Bodies or Public Sector Undertakings or Private Sectors.
• Master degree in Science (M.Sc.) (Physics or Electronics or Applied Electronics) with nine years of relevant work experience in Ministries or Departments or Attached and Subordinate Offices of the Central Government or Statutory Bodies or Autonomous Bodies or Public Sector Undertakings or Private Sectors;
• Department of Electronics and Accreditation of Computer Courses (DOEACC) B-Level or Graduate Institute of Electronics and Telecommunication Engineers (IETE) with nine years of relevant work experience in Ministries or Departments or Attached and Subordinate Offices of the Central Government or Statutory Bodies or Autonomous Bodies or Public Sector Undertakings or Private Sectors.
• Master in Computer Application (MCA) with nine years of relevant work experience in Ministries or Departments or Attached and Subordinate Offices of the Central Government or Statutory Bodies or Autonomous Bodies or Public Sector Undertakings or Private Sectors.
 $\textbf{Dates}$ $\textbf{Question Paper}$ $\textbf{Answer Key}$ $\textbf{GO/AO Links}$ $\textbf{Exam Links}$ $10\: \text{Feb}\: 2019$ 2019: Set – B Questions AO $12-13\:\text{Mar}\: 2016$ 2016: Set – A Questions AO
13

total questions recalled: 84

Cn:10

1. Router has settings as per the table:

 10.196.1.0/24 r2 192.10.16.0/22 p1 192.255.240.0/20 r1 default r1

A packet directed to addresses

(4 addresses were given and in option there were 4 destinations like: r1,r2,r1,p1)

1. In cidr with 192.10.16/22 how many hosts are possible:

1. Why is CSMA/CD not used in wireless networks:

A)a station can not differntiate between sending and receeiving signals

B)a station can not send and receive at the same time

C)signal received is very weak in power

D)none of the above

4. if in symmetric key encryption 20 parties want to communicate. What is the total number of keys required?

5. which is false:

a)in dvr the count to infinity problem is due to joining of a new node.

b) in link state only neighbouring topology is known

c) in dvr full network topology is known

d) in dvr only packets from neighbours are exchanged

6. in dvr of A,B,C,D,E,F,G nodes in a network. where from

7. if a 3-kHz line has a signal to noise ratio of 20dB. Then the maximum data speed is:

a)around 19kbits

b)3 kbits

c)6 kbits

d)none

8. ipv6 uses 128 bits address. So, if one million addresses are allocated every picosecond. then after how many time the addresses will exhaust:

a)in power of 10^18 years

b)in power of 10^13 years

c)never

d)very soon

9. which of the following digital signatures provide integrity as well as authenticity:

A)some sha1 based technique

B) sha1

C)rsa

D)none

10.

I.Tcp and UDP have same bits for port numbers.

II.UDP retransmits damaged packets but not lost packets.

Correct option:

A)I true and II true

B)I true and II false

C) I false and II true

D) I false and II false

maths:6

1. If laplace transformation of tf(x) is 1/(s-5). Then the laplace transformation of t3f(x) is:

A)2/(s-5)3

B)6/(s-5)6

C)3/(s-5)7

D)6/(s-5)8

(please recheck the coefficients in the options as the powers of (s-5), I can remember but not the constants)

2. if three boys and 4 girls are to be seated in a row, and all the seating arrangements are equally likely, then what is the probability of no boys together.

3. a matrix 3*3 was given, and an identity matrix I, then which of the equation is correct

(options like 4M-7I =0)

4. pnc. all letters of word  BBABCD are allowed maximum as much times as they are in the word. What is the number of all possible words of all possible lengths?

a)72

b)74

c)76

d)78

5. If X is the power-set of A and given X is subset of B. then:

A)2^(|A|)<=|B|

B)2*|A|=B

C)2*|A|>B

D) 2^(|A|)>=|B|

6.  a lattice is a complete lattice. (assertion reasoning type)

7.number of spanning tree in connected graph:, n nodes

A)2*(n*(n-1)/2)

B)2^(n)

C)n*(n-1)/2

D)n

computer graphics:2

1. find the matrix which correspons to scaling a 3-D object enlarging twice

2. vanishing point of a figure consisting of 2 cubes, one is in Ist quadrant rotated some small angle clockwise and other in 2nd quadrant rotated some small same angle anticlockwise.

artificial intelligence:4

1. 10 nodes in an cnn with 3*3 transformation function, then the number of parameters

2. sigmoid function - activation function question

3. for which of the following job, the clustering algo is used:

a)using the historic data, some conclusion drawn

b)e-mails as spam or not

c)types of customers based on their activity pattern

d)none of these

4. for the accurate prediction of digits in converting paper documents to digitl form:

a).....erosion

b).....dilution

c)boundary ....

d)gaussion …......

(….. Are blank spaces, all options had 2 words, I remember only one in each)

c++:6

1. a class

c{}

and two lines:

I. c a1;

II. c a2=a1;

then, how many functions are activated by default. options:

( included different combinations of default constructer, destructer, copy constructer)

2. c++ question one true,

3. c++ question including static variable and printing values

4.

#include<iostream>

void main()

static int a[]={62,61,9,0};

static int b[][]={{1,2,3,4},{5,6,,78,9}};

int *i=a;

printf("%d",i[0][2]);

output:

5. best way to reuse composite modules is:

A)

B)encapsulation inheritance

C)encapsulation composition

D)composition inheritance

6. which of the following is correct;

A) ->* operator in c++ can be overloaded

B) all operator in c++ can be overloaded

C) both a and b

D) none

7. why are abstract class are used over interfaces

A) to set concrete behaviour by methods that are used as it is like default by the derived classes

B) to instantiate a class

C) to have methods that are to be overridden by base classes

D)none of these

java:1

1. output of program simple system.out function related

c:5

1. #include<stdio.h>

int a,b,c;

print(void);

int main()

int a=0,b,c;

a+=print;

prinf("%d",a);

a++;

print();

prinf("%d",a);

a+=print();

prinf("%d",a);

void print(void)

static int a=10;

prinf("%d",a);

return a++;

output:

2. & unary operator is not applied to which of storage classes

3.

#include<stdio.h>

struct str

int a;

char b;

union {int x; char y;}t;

void main()

str s1;

s1.t.x=0;

si.t.y='a';

printf("%d",s1.t.x);

printf("%c",s1.t.y);}

output

4.

#include<stdio.h>

void main()

{int x=5;

char *c =(char *)&x;

printf("%d",c);

5. the condition in (c!='a'&&z<+2=4) can be rewritten as:

a)!(c!='a'||z+2>4)

b)!(c=='a'||z+2<=4)

c)!(c!='a'||z+2>=4)

d)!(c=='a'||z+2>4)

computer architecture:4

1. 4 stage pipeline, with each stage taking 100, 90, 110,70 nanosecond,, with each stage buffer register 5nanosecond.

if each stage is taking one clock cycle, then after 1000 data processing, what is the time elapsed?

2. if in a microprogrammed architecture, 24 bit microinstructions, 13 bit is for micro-operation, 8 multiplexer lines, then what is

x: bits for address in micro-instruction

y: selection bits

z: total number of instructions

3. cao question of 4 instructions asking the time taken by instructions in number of clock cycles:

R2<-M[4998]

r1<-M[5000]+r2;

R3<-r1

M[5002]<-r3.

Internal diagram of cpu was shown.

4. cao question asking address saved in stack when a set of 4 opertions:

r1<-M[1010]

r2<-M[1020]

r1<- r1+r2;

M[2000]<- r1

during writing to memory an interrupt occured, then what address is at stack. if the given program is starting from 1000.

physics:1

1. reflection and colour of light

Os:14

1. file system:256gb, 200 direct blocks, each word address bits given. what is max. file size in this system?

2. a 2-way small cache of 4 blocks is having a refernece string:    8,0,12,6,8,12

how many page faults if lru is used.

3. if 2-level paging system. memory access time 100ns. TLB is used and tlb hit percentage0 80%. Assume tlb access time 0ns. what is effective access time of the system.

4. if in shortest seek time first the head is at 100 number cylinder. and the requests are 80,85, 90, 100, 105, 110, 134, 155. After how many request services 85 will be served.

5. semaphore, S=3; 7 up operations and 10 down. What is the current vlue of semaphore S?

6. If number of disks in raid 1 is _____  as raid 6 if(some data duplication and consistency property given):

A)same

B)2 more than

C)2 less than

D)1 more than

7. Bankers algorithm safe state sequence, if 24 instances of a resource is given.

 allocated Max need p0 4 14 p1 8 10 p2 10 8

A)p0p2p1

B)p1p2p0

C)p2p0p1

D)p1p0p2

8. Deadlock question. Max number of resources needed if 4 processes has need of 4 instances of a resource.

9. in a two process scenario, deadlock can occur due to:

A)directly knowing of each other by a shared variable

B)indirectly dependent on each other

D) os primitives

10. Priority scheduling, with 0 as the lowest priority. The processes are

 Process with burst time Arrival time priority P1-9 0 2 P2-8 2 1 P3-6 6 3 P4-4 8 0

Using preemptive priority scheduling. Then what is the waiting time of p1:

A)10

B)2

C)12

D)14

11. Choose the incorrect one:

A)paging makes memory access slow

B)best fit is not really the best algorithm for continuous algorithms in all cases.

C)paging suffers from internal segmentation

D)Thrashing can be minimized by global page replacement algorithms.

12. If the reference string with all frames initially empty; (given reference string was same as one of gate questions having 9 page faults in fifo)

1. FIFO has 9 page faults with 3 page frames.

1. FIFO is better than optimal here

Choose the correct option:

A)both true

B)only I

C)both false

D)only II

13. In a disk system with 64 sectors per track, 8 plates with both sides used, 500 cylinders. The addressing of a sector is (a,b,c), where a is the track number from 0-63, b is the plate surface number from 0-15 and c is the cylinder number from 0-499. So, the address (20,8,31) corresponds to which sector in decimal addressing?

14. Fork system call

#include<stdio.h>

Void main()

Int I=1;

Switch(i)

case 1:

Printf(“process”);

Fork();

Case 2:

Printf(“process”);

Fork();

Case 3:

Printf(“process”);

How many times will process be printed?

dbms:6

1. 1. left natural outer join of R and S.

R(A,B)={(1,2),(3,1),(2,3),()}

S(C,D)={(2,3),(2,1),(1,2),(1,3)}.

What is the number of rows returned?

2. a database is supporting where,having,average, order by, group by, sum. Which of these will be performed first?

1.  Query output

1. 2 sql queries were given.and asked if both same, different, cant say, both wrong.

(easy queries involving some where condition and join of two tables)

1. If a table has A,B,C,D has dependency as A->B and B->D. Then the table is in :

A)1st normal form

B)2nd normal form

C)3rd normal form

D)4th normal form

1. Given the database table:

 A B C D a1 b1 c1 d1 a2 b1 c2 d2 a1 b2 c3 d3 a2 b3 c4 d3

Then which of the following functional dependency is incorrect:

A)AC->B

B)C-.D

C)B->D

D)C-.>A

web-technology:1

1. which one of the following is true:

a) to prevent sql injection predefined queries are used.....

b) user validation ensures buffer not overflow

c) forgery of digital signatures is called cross-site.....

d)(something related to keys)

algorithms:9

1. if out of n elements, k- elements are inserted to form a max-heap,and out of the remaining elements each is first replacing the root and then the heap again rearranges to be a max-heap. What is the time complexity in big -oh terms?

2. by definition of asymptotic notations if f(x)=n^(1/2) and g(x)=log(n). then which of the following is true:

a)f=big-oh(g(x)) only

b)f=big-omega(g(x)) only

c)f=theta(g(x)) only

d)f=big-oh(g(x)) and f=big-omega(g(x))

3. if f=big-oh of g(x) then:

a) f is increasing for a large x, never faster than g(x)

b) f is increasing for a large x,  faster than g(x)

c)cant say

d) f is increasing for a large x, sometimes faster or slower than g(x)

1. Best, average and worst Time complexity of sorting algorithms. Match

 insertion N*n,n*n,n*n selection Nlogn,nlogn,nlogn merge n,n*n,n*n quick Nlogn, nlogn,n*n
1.  Which of the following options contains the best sorting algorithm for small elements and large number of elements, respectively:

A)insertion-quick

B)selection-quick

1.  Which of the following are greedy algorithms:

I.bellman ford

II.dijkstra’s

III. Floyd warshall

IV.prims

A)I &II

B)II & IV

C)II & III

D)all of the above

data structure:6

1. if 1.2.3.4 are inserted in that order than which of the following pop sequence is wrong:

2. if there are two stacks with X, having 1,2,3 with 1 on top. There is on eother stack Y. If a value is popped from X, it can either be printed or be pushed to Y. But if a value is popped from Y, then it is printed. Then which of the following print sequence is not possible?

3. B+ tree

4. a binary search tree having 3 internal keyed nodes, then how many ways to represent it?

a)6

b)1

c)5

d)4

5. what is the inorder traversal if postorder and preorder is given

6.  an expression x= (a+b)*(c-d). Is given and in the syntax tree, which of the following options do not corresponds to either postorder, preorder or inorder traversal of the tree:

A)(inorder traversal given)

B) (preorder traversal given)

C) (postorder traversal given)

D)(incorrect one)

digital electronics:10

1. if the symbol * means

A        B        output

F        F        T

F        T        T

T        F        F

T        F        F

then AUB can be represented as:

a)A*~B

b)~A*B

c)~A*~B

d)none

2. in the given circuit diagram if s1s2s3=001 and s2s3s4=010 then the output is:(the circuit was a combination of 4 one select line MUX i.e. 2*1 MUX. )

3. which of the folllowing is correct:

a)the range of representation in mantissa and exponent is larger than floating values

b)(gray code and excess-3 BCD something..)

c)0 has two representations in 2s complement

d)none

4. input lines A,B,C,D are applied, what is the output if:

I.

II.

III.

IV.

Are known.

(there were 4 conditions on the behaviour of A,B,C,D were given: like A is always true,if B is true A is also true )

5. what is the minimized circuit if A,B,C,D are input and output f(x) is related as:

1. B and D always have same values

II.A is always 1 if C is 1

III. A is in always 0 if B is 1

Iv. A is always 1 if D is 0.

6. given f(x)=Pie(0,1,2,3,4,5,6,7,8,10,12,13,14,15)

A)A’B’CD+A’BCD’

B)AB’C’D’+A’B’C’D

C) AB’CD+AB’C’D

D)A’B’CD+ABC’D’

7. The circuit for ((~A.~B)+(~A.B))+A  is:

8. which circuit is same as AUB using only NAND gate, if the inputs A and B in OR gate and X and Y in NAND:

A)x=~A & y =~B

B) x=~A & y =B

C) x=A & y =~B

D)none of these

(though the options was in symbol of logic gates but I have given the equivalent expressions)

9. in a serial input parallel output shift register three bit, if the input is 00011011, then the output after 4 clock cycles is:

A)101

B)001

C)111

D)100

1. RAM is an example of:

A)combinational

B)sequential

C)both a and b

D)none

toc:4

1. the transition table is given as

current state        input        next state

A            0        B

A            1        C

B            0        A

B            1        D

C            0        D

C            1        A

D            0        C

D            1        B

If A is the initial as well as final state, then what is the above finite machine doing:

a)accepting languages with odd 0s and even 1s

b)accepting languages with odd 0s and odd 1s

c)accepting languages with even 0s and even 1s

d)accepting languages with even 0s and odd 1s

2. r= 1(1+0)*  , s=11*0  , t=1*0

then which of the following options is correct:

a)r is subset of s, r is subset of t

b)r is subset of s, t is subset of r

c)s is subset of r, s is subset of t

d)t is subset of r, r is subset of s

3. l= (0+1).(0+1).(0+1).(0+1).(0+1)...n times. then what is the number of states in the miniimal dfa accepting it.

a)n-1

b)n

c)n+1

d) cant say

4. a finite automata on E={0,1} was given accepting the multiples of 3. options:

A)all multiples of 3

B)all odd numbers

C)all multiples of 4

D)all even numbers

Compiler desugn:1

1. Given an sdt, what is the value at root, if the expression 22*10+11.

E->T+E    (E.val = T.val*E.val)

E->T  (E.val=T.val)

T->F*T   (T.val=F.val+T.val))

T->F    (T.value=F.value)

F->num    (F.value=num)

14

You can download the ISI/CMI PDF from the below link.

PS: This is the first version of the PDF and though we have spent a lot of time doing tagging, some of the topics might be given wrong unlike GATE PDF which has gone through multiple revisions.

15

## $$\color{Blue}{\textbf{Calculus}}$$

### $$\color{Magenta}{\textbf{1.Trigonometry}}$$

$\textbf{Pythagorean Identities:}$

• $\sin^{2}\theta + \cos^{2}\theta = 1$
• $1+\tan^{2}\theta = \sec^{2}\theta$
• $1+\cot^{2}\theta = \csc^{2}\theta$

$\textbf{Reciprocal Identities:}$

• $\sin\theta = \dfrac{1}{\csc\theta}$
• $\cos\theta = \dfrac{1}{\sec\theta}$
• $\tan\theta = \dfrac{1}{\cot\theta}$
• $\csc\theta = \dfrac{1}{\sin\theta}$
• $\sec\theta = \dfrac{1}{\cos\theta}$
• $\cot\theta = \dfrac{1}{\tan\theta}$

$\textbf{Tangent and Cotangent Identities:}$

• $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
• $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$

$\textbf{Formulas for twice of angle:}$

• $\sin2\theta = 2 \sin\theta \cos\theta$
• $\cos2\theta = 2\cos^{2}\theta\: – 1 = 1 – 2\sin^{2}\theta$
• $\tan2\theta = \dfrac{2\tan\theta} {1− \tan^{2}\theta} = \dfrac{\sin 2\theta} {1− 2\sin^{2}\theta}$

-------------------------------------------------------------------------------------------------------------------

$\color{\Purple}{\textbf{Some important things, we should know}}$

• ${\color{Red} {\cos 0 = 1,\cos \pi = -1,\cos 2\pi = 1,\dots}}$
• ${\color{Blue}{ \text{In general}\: \cos n\pi = (-1)^{n}\: \text{where}\: n=0,1,2,\dots}}$
• ${\color{Magenta} {\sin 0 = 0,\sin \pi = 0,\sin 2\pi = 0,\dots}}$
• ${\color{Green} {\text{In general}\: \sin n\pi = 0\: \text{where}\: n=0,1,2,\dots}}$
• ${\color{Orange} {\sin(\pi-x)=\sin x,\sin (2\pi-x)=-\sin x,\sin(3\pi-x)=\sin x}}$
• ${\color{Teal} {\text{In general}\: \sin (n\pi-x)=(-1)^{n+1}\sin x\: \text{ where} \: n=0,1,2,\dots}}$
•  ${\color{Orchid} {\cos(\pi-x)=-\cos x,\cos (2\pi-x)=\cos x,\cos (3\pi-x)=-\cos x}}$
• ${\color{purple} {\text{ In general}\: \cos (n\pi-x)=(-1)^{n}\cos x \: \text{where}\: n=0,1,2,\dots}}$

__________________________________________________________

$f(x) = \sin x$

$f(x) = \cos x$

Visualization:

### $$\color{Red}{\textbf{2.Integral & Differential Calculus}}$$

$${\color{Teal}{2.1.\textbf{Some Useful Formulae for Differentiation}}}$$

1. $\dfrac{\mathrm{d} }{\mathrm{d} x}(c) = 0;c = \text{Constant}$
2. $\dfrac{\mathrm{d} }{\mathrm{d} x}(x) = 1$
3. $\dfrac{\mathrm{d} }{\mathrm{d} x}(x^{n}) = nx^{n-1}$
4. $\dfrac{\mathrm{d} }{\mathrm{d} x}(e^{x}) = e^{x}$
5. $\dfrac{\mathrm{d} }{\mathrm{d} x}(a^{x}) = a^{x}\log_{e}a$
6. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\log x) = \dfrac{1}{x}$
7. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\log_{a} x) = \dfrac{1}{x}\log_{a}e$
8. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sin x) = \cos x$
9. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cos x) = -\sin x$
10. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\tan x) = \sec^{2} x$
11. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cot x) = -\csc^{2} x$
12. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\csc x) = \csc x \cot x$
13. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sec x) = \sec x\tan x$
14. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sin^{-1} x) = \dfrac{1}{\sqrt{1-x^{2}}}$
15. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cos^{-1} x) = \dfrac{-1}{\sqrt{1-x^{2}}}$
16. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\tan^{-1} x) = \dfrac{1}{1 + x^{2}}$
17. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cot^{-1} x) = \dfrac{-1}{1 + x^{2}}$
18. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sec^{-1} x) = \dfrac{1}{x \sqrt{x^{2}-1 }}$
19. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\csc^{-1} x) = \dfrac{-1}{x \sqrt{x^{2}-1}}$
20. $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sqrt{x}) = \dfrac{1}{ 2 \sqrt{x}}$

$${\color{Purple}{2.2.\textbf{Some Useful Formulae for Integration}}}$$

$\textbf{Algebraic Functions}$

1. $\displaystyle{}\int k\: dx = kx + C$
2. $\displaystyle{}\int x^{n}\: dx = \dfrac{x^{n+1}}{n+1} + C$
3. $\displaystyle{}\int (ax + b)^{n}\: dx = \dfrac{(ax + b)^{n+1}}{a(n+1)} + C\:;(\text{for}\: n \neq -1)$
4. $\displaystyle{}\int \dfrac{1}{x}\: dx = \ln x + C\:;\text{(for positive values of$x$only)}$
5. $\displaystyle{}\int \dfrac{c}{ax + b}\: dx = \dfrac{c}{a}\ln(ax +b) + C$

$\textbf{Exponential Functions}$

1. $\displaystyle{}\int e^{x} dx = e^{x}+ C$
2. $\displaystyle{}\int a^{x} dx = \dfrac{a^{x}}{\ln a}+ C$

$\textbf{Logarithm Functions}$

1. $\displaystyle{}\int \ln x\: dx = x\ln x\: – x+ C$
2. $\displaystyle{}\int \log_{a} x\: dx = x\log_{a} x\: – \dfrac{x}{\log a} + C$

$\textbf{Trigonometric Functions}$

$\displaystyle{}\int \sin x\: dx = -\cos x + C$

16

### $$\color{Blue} {\textbf{Linear Algebra}}$$

$\textbf{1.Properties of determinants:-}$

The determinant is only valid for the square matrix.

1. $\mid A^{T}\mid = \mid A \mid$
2. $\mid AB \mid = \mid A \mid \mid B \mid$
3. $\mid A^{n} \mid = \big(\mid A \mid\big)^{n}$
4. $\mid kA\mid = k^{n} \mid A \mid$, here $A$ is the $n\times n$ matrix.
5. If two rows (or two columns) of a determinant are interchanged, the sign of the value of the determinant changes.
6. If in determinant any row or column is completely zero, the value of the determinant is zero.
7. If two rows (or two columns) of a determinant are identical, the value of the determinant is zero.

$\textbf{2.Matrix Multiplication:-}$

It is valid for both square and non-square matrix.

Let $\mathbf{A_{m\times n}}$ and $\mathbf{B_{n\times p}}$ are two matrices then, the resultant matrix is $\mathbf{(AB)_{m\times p}}$, has

1. Number of elements $=mp$
2. Number of multiplication $= (mp)n = mnp$
3. Number of addition $= mp(n-1)$

_________________________________________________________________

$\color{Red} { \textbf{Key Points:-}}$

1. $(Adj\: A)A = A(Adj\:A) = \mid A \mid I_{n}$
2. $Adj(AB) = (Adj\:B)\cdot (Adj\: A)$
3. $(AB)^{-1} = B^{-1}\cdot A^{-1}$
4. $(AB)^{T} = B^{T}\cdot A^{T}$
5. $(A^{T})^{-1} = (A^{-1})^{T}$
6. $A\cdot A^{-1} = A^{-1} \cdot A = I$
7. $Adj(Adj\:A) = \mid A\mid ^{n-2}\cdot A$
8. $\mid Adj\: A \mid = \mid A \mid ^{n-1}$
9. $\mid Adj(Adj\: A) \mid = \mid A \mid ^{{(n-1)}^{2}}$
10. $Adj(A^{m}) = (Adj\:A)^{m}$
11. $Adj(kA) = k^{n-1}(Adj \:A),k\in \mathbb{R}$

_________________________________________________________________

${\color{Magenta}{\textbf{Some More Points:-}} }$

1. Minimum number of zeros in a diagonal matrix of order $n$ is $n(n-1).$
2. $AB = \text{diag}(a_{1},a_{2},a_{3})\times \text{diag}(b_{1},b_{2},b_{3}) = \text{diag}(a_{1}b_{1},a_{2}b_{2},a_{3}b_{3})$
3. For diagonal and triangular matrix (upper triangular or lower triangular) the determinant is equal to product of leading diagonal elements.
4. The matrix which is both symmetric and skew-symmetric must be a null matrix.
5. All the diagonal elements of the Skew Hermitian matrix are either zero or pure imaginary.
6. All the diagonal elements of the Hermitian matrix are real.
7. The determinant of Idempotent matrix is either $0$ or $1.$
8. Determinant and Trace of the nilpotent matrix is zero.
9. The inverse of the nilpotent matrix does not exist.
10. $\color{green}{\checkmark}$ A square matrix whose all eigenvalues are zero is a nilpotent matrix

$\color{green}{\checkmark}$ In linear algebra, a nilpotent matrix is a square matrix $A$ such that ${\displaystyle A^{k}=0\,}$ for some positive integer ${\displaystyle k}.$ The smallest such ${\displaystyle k}$ is sometimes called the index of ${\displaystyle A}$

Example$:$ the matrix $A = \begin{bmatrix} 0& 0\\1 &0 \end{bmatrix}$ is nilpotent with index $2,$since $A^{2} = 0$

$\color{Blue}{\textbf{Eigenvalues:-}}$ The number is an eigenvalue of $A$ if and only if $A-\lambda I$ is singular: $\text{det}(A-\lambda I) = 0$

This “characteristic equation” $\text{det}(A-\lambda I) = 0$ involves only $\lambda$, not $x.$ When $A$ is $n \times n,$ the equation has degree $n.$ Then $A$ has $n$ eigenvalues and each leads to $x:$ For each $\lambda$ solve $\text{det}(A-\lambda I)x = 0$ or $Ax = \lambda x$ to find an eigenvector $x.$

${\color{Orange}{\textbf{Properties of Eigen Values (or) Characteristics roots (or) Latent roots:-}} }$

$\color{green}\checkmark\:$ Eigenvalue and Eigenvector are only valid for square matrix.

1. The sum of eigen values of a matrix is equal to the trace of the matrix, where the sum of the elements of principal diagonal of a matrix is called the trace of the matrix. $$\sum_{i=1}^{n}(\lambda_{i}) = \lambda_{1} + \lambda_{2} + \lambda_{3} + \dots \lambda_{n} = \text{Trace of the matrix}$$
2. The product of eigen values of a matrix $A$ is equal to the determinant of matrix $A.$ $$\prod_{i=1}^{n} = \lambda_{1} \cdot \lambda_{2} \cdot \lambda_{3} \dots \lambda_{n} = \mid A \mid$$
3. For Hermitian matrix every eigen value is real.
4. The eigenvalues of a skew-Hermitian matrix are all purely imaginary or zero.
5. Every eigenvalue of a Unitary matrix has absolute value. i.e. $\mid \lambda \mid = 1$
6. Any square matrix $A$ and its transpose $A^{T}$ have same eigenvalues.
7. If $\lambda_{1},\lambda_{2},\lambda_{3},\dots ,\lambda_{n}$ are eigenvalues of matrix $A$, then eigenvalues of

$\color{green}\checkmark\: kA$ are $k\lambda_{1},k\lambda_{2},k\lambda_{3},\dots, k\lambda_{n}$

$\color{green}\checkmark\:A^{m}$ are $\lambda_{1}^{m},\lambda_{2}^{m},\lambda_{3}^{m},\dots ,\lambda_{n}^{m}$

$\color{green}\checkmark\:A^{-1}$ are $\frac{1}{\lambda_{1}},\frac{1}{\lambda_{2}},\frac{1}{\lambda_{3}},\dots ,\frac{1}{\lambda_{n}}$

$\color{green}\checkmark\:A+kI$ are $\lambda_{1}+k,\lambda_{2}+k,\lambda_{3}+k,\dots ,\lambda_{n}+k$

1. If $\lambda$ is an eigenvalue of an Orthogonal matrix $A$, then $\dfrac{1}{\lambda}$ is also an eigenvalue of matrix $A(A^{T} = A^{-1})$
2. The eigenvalue of a symmetric matrix are purely real.
3. The eigenvalue of a skew-symmetric matrix is either purely imaginary or zero.
4. Zero is an eigenvalue of a matrix iff matrix is singular.
5. If all the eigenvalues are distinct then the corresponding eigenvectors are independent.
6. The set of eigenvalues is called the spectrum of $A$ and the largest eigenvalue in magnitude is called the spectral radius of $A.$ Where $A$ is the given matrix.

${\color{Orchid}{\textbf{Properties of Eigen Vectors:-}} }$

1. For every eigenvalue there exist at-least one eigenvectors.
2. If $\lambda$ is an eigenvalue of a matrix $A,$ then the corresponding eigenvector $X$ is not unique.i.e., we have infinites number of eigenvectors corresponding to a single eigenvalue.
3. If $\lambda_{1},\lambda_{2},\lambda_{3},\dots ,\lambda_{n}$ be distinct eigen values of a $n\times n$ matrix, then corresponding eigen vectors $=X_{1},X_{2},X_{3},\dots,X_{n}$ form a linearly independent set.
4. If two or more eigenvalues are equal then eigenvectors are linearly dependent.
5. Two eigenvectors $X_{1}$ and $X_{2}$ are called orthogonal vectors if $X_{1}^{T}X_{2}=0.$
6. $\textbf{Normalized eigenvectors:-}$ A normalized eigenvector is an eigen vector of length one. Consider an eigen vector $X = \begin{bmatrix}a \\ b\end{bmatrix}_{2\times 1}$, then length of this eigen vector is  $\left \| X \right \| = \sqrt {a^{2} + b^{2}}.$ Normalized eigenvector is $\hat{X} = \dfrac{X}{\left \| X \right \|}=\dfrac{\text{Eigen vector}}{\text{Length of eigen vector}} =\begin{bmatrix}\dfrac{a}{\sqrt{a^{2} + b^{2}}} \\ \dfrac{b}{\sqrt{a^{2} + b^{2}}} \end{bmatrix}_{2\times 1}$
7. Length of the normalized eigenvector is always unity.

$\color{Orange}\checkmark\:$The eigenvectors corresponding to distinct eigenvalues of a real symmetric matrix are orthogonal.

$\textbf{3.Rank of the Matrix:-}$

The rank of a matrix $A$ is the maximum number of linearly independent rows or columns. A matrix is full rank matrix, if all the rows and columns are linearly independent. Rank of the matrix $A$ is denoted by $\rho{(A)}.$

${\color{Purple}{\textbf{Properties of rank of matrix:-}} }$

1. The rank of the matrix does not change by elementary transformation, we can calculate the rank by elementary transformation by changing the matrix into echelon form. In echelon form, the rank of matrix is number of non-zero row of matrix.
2. The rank of the matrix is zero, only when the matrix is a null matrix.
3. $\rho(A)\leq \text{min(row, column)}$
4. $\rho(AB)\leq \text{min}[\rho(A), \rho(B)]$
5. $\rho(A^{T}A) = \rho(AA^{T}) = \rho(A) = \rho(A^{T})$
6. If $A$ and $B$ are matrices of same order, then $\rho(A+B)\leq \rho(A) + \rho(B)$ and $\rho(A-B)\geq \rho(A)-\rho(B)$
7. If $A^{\theta}$ is the conjugate transpose of $A,$ then $\rho(A^{\theta})= \rho(AA^{\theta}) = \rho(A^{\theta}A) = \rho(A)$
8. The rank of the skew-symmetric matrix cannot be one.
9. If $A$ and $B$ are two $n$-rowed square matrices, then $\rho(AB)\geq \rho(A) + \rho(B) – n$
10. Rank of $A_{n*n}=2$ iff $n-2$  eigenvalues are zero.

$\textbf{4.Solution of Linear Simultaneous Equations:-}$

There are two types of linear simultaneous equations:

1. Linear homogeneous equation$:AX = 0$
2. Linear non-homogeneous equation$:AX = B$

Steps to investigate the consistency of the system of linear equations.

1. First represent the equation in the matrix form as $AX = B$
2. System equation $AX = B$ is checked for consistency as to make Augmented matrix $[A:B].$

$\textbf{Augmented Matrix}\: \mathbf{[A:B]:-}$

1. $\rho(A)\neq \rho([A:B])$ inconsistent $\color{Red} {\textbf{(No solution)}}$
2. $\rho(A) = \rho([A:B])$ consistent $\color{green} {\textbf{(Always have a solution)}}$

$\color{green}\checkmark\:\rho(A) = \rho([A:B]) = \text{Number of unknown variables}\:\: \color{Cyan} {\textbf{(Unique solution)}}$

$\color{green}\checkmark\:\rho(A) = \rho([A:B]) < \text{Number of unknown variables}\:\: \color{Salmon} {\textbf{(Infinite solution)}}$

______________________________________________________________

$\color{green}\checkmark$ Linear homogeneous equation$:AX = 0$ is always consistent.

If $A$ is a square matrix of order $n$ and

$\color{green}\checkmark\:\mid A \mid = 0$, then the rows and columns are $\color{Teal} { \text{linearly dependent}}$ and system has a $\color{Magenta} { \text{non-trivial solution or infinite solution.}}$

$\color{green}\checkmark\:\mid A \mid \neq 0$, then the rows and columns are $\color{purple} {\text{linearly independent}}$ and system has a $\color{green} {\text{trivial solution or unique solution.}}$

______________________________________________________________

$\color{Brown} {\textbf{5.Cayley-Hamilton Theorem:-}}$ According to Cayley-Hamilton Theorem, “Every square matrix satisfies it’s own characteristic equation.”

$\color{green}\checkmark$This theorem is only applicable for square matrix. This theorem is used to find the inverse of the matrix in the form of matrix polynomial.

If $\mathbf{A}$ be $n\times n$ matrix and it’s characteristic equation is, $a_{0}\lambda^{n} + a_{1}\lambda^{n-1} + \dots + a_{n} = 0$, then according to Cayley-Hamilton Theorem, $a_{0}\mathbf{A}^{n} + a_{1}\mathbf{A}^{n-1} + \dots + a_{n}\mathbf{I_{n}} = 0$

$\color{Teal} {\textbf{For finding the roots of}\: A_{3\times 3}:-}$

$\lambda^{3} – (\text{Trace})\lambda^{2} + \text{(Sum Of Principal Cofactor)}\lambda \:– \mid A \mid = 0$

______________________________________________________________

$\color{Red} {\textbf{6.Types of Matrices According to Dimensions(R,C):-}}$

Rows and columns are all together said to be the dimensions of the matrix, according to the dimensions there are two types of matrix, rectangular matrix and square matrix.

$\textbf{1.Rectangular Matrix:-}$ A matrix in which the number of rows is not equal to the number of columns is known as a rectangular matrix $\mathbf{(R\neq C \:\text{(or)}\: m\neq n)}.$

$\color{green}\checkmark\mathbf{A=[a_{ij}]_{m\times n}\:\:; m \neq n}$

$\textbf{2.Square Matrix:-}$ A matrix in which the number of rows are equal to the number of columns is known as a square matrix $\mathbf{(R\neq C)}.$

$\color{green}\checkmark\mathbf{A=[a_{ij}]_{n\times n}\:\:; n \neq n}$

$$\color{Magenta} {\textbf{Types of Square Matrix:-}}$$

$\textbf{1.Diagonal Matrix:-}$ A square matrix in which all the elements except leading diagonal elements are zero is known a a diagonal matrix.

$\textbf{Example:} \: A = \begin{bmatrix} 3 &0 &0 \\ 0 &6 &0 \\ 0 &0 &1 \end{bmatrix}_{3\times 3}$  (or) $A = \text{diag(3,6,1)}$

$\color{green}\checkmark$Minimum number of zeros in a diagonal matrix of order $n$ is $n(n-1).$

$\color{green}\checkmark AB = \text{diag}(a_{1},a_{2},a_{3})\times \text{diag}(b_{1},b_{2},b_{3}) = \text{diag} (a_{1}b_{1},a_{2}b_{2},a_{3}b_{3})$

$\textbf{2.Scalar Matrix:-}$ A diagonal matrix in which all the diagonal elements are equal, is known as a scalar matrix.

$\textbf{Example:} \: A = \begin{bmatrix} 5 &0 &0 \\ 0 &5 &0 \\ 0 &0 &5 \end{bmatrix}_{3\times 3}$  (or) $A = \text{diag(5,5,5)}$

$\textbf{3.Unit Matrix:-}$  A diagonal matrix in which all the diagonal elements are unity is known as unit matrix or identity matrix. The matrix of order $n$ is denoted by $\mathbf{I_{n}}.$

$\textbf{Example:} \: I_{n} = \begin{bmatrix} 1 &0 &0 \\ 0 &1 &0 \\ 0 &0 &1 \end{bmatrix}_{3\times 3}$

$\textbf{4.Upper Triangular Matrix:-}$ A square matrix $A = [a_{ij}]$ is said to be upper triangular matrix if $a_{ij} = 0$ whenever $i>j.$

$\textbf{Example:} \: A = \begin{bmatrix} 5 &2 &3 \\ 0 &1 &6 \\ 0 &0 &8 \end{bmatrix}_{3\times 3}$

$\textbf{5.Lower Triangular Matrix:-}$ A square matrix $A = [a_{ij}]$ is said to be upper triangular matrix if $a_{ij} = 0$ whenever $i<j.$

$\textbf{Example:} \: A = \begin{bmatrix} 5 &0 &0 \\ 4 &9 &0 \\ 3 &2 &1 \end{bmatrix}_{3\times 3}$

$\color{green}\checkmark$For diagonal and triangular matrix (upper triangular or lower triangular) the determinant is equal to product of leading diagonal elements.

$\textbf{6.Symmetric Matrix:-}$ A square matrix is said to be symmetric , if $\mathbf{A^{T} = A},$ where $A^{T}$ or $A’$ is transpose of matrix $A.$ In transpose of matrix th rows and columns are interchanged.

$\textbf{Example:} \: A = \begin{bmatrix} 1 &2 &3 \\ 2 &4 &5 \\ 3 &5 &6 \end{bmatrix}_{3\times 3}\implies A^{T} = \begin{bmatrix} 1 &2 &3 \\ 2 &4 &5 \\ 3 &5 &6 \end{bmatrix}_{3\times 3}$

$\textbf{7.Skew Symmetric Matrix:-}$ A square matrix is said to be symmetric , if $\mathbf{A^{T} = – A},$ where $A^{T}$ or $A’$ is transpose of matrix $A.$ In transpose of matrix th rows and columns are interchanged.

$\textbf{Example:} \: A = \begin{bmatrix} 0 &-2 &-3 \\ 2 &0&-5 \\ 3 &5 &0 \end{bmatrix}_{3\times 3}\implies A^{T} = \begin{bmatrix} 0 &2 &3 \\ -2 &0 &5 \\ -3 &-5 &0 \end{bmatrix}_{3\times 3} = -A$

${\color{Teal}{\textbf{Properties of Symmetric Matrix:-}}}$

1. If $A$ is a square matrix then $A+A^{T},AA^{T},A^{T}A$ are symmetric matrices, while $A – A^{T},A^{T}-A$ are skew symmetric matrix.
2. If $A$ is a symmetric matrix, $k$ any real scalar, $n$ any integer, $B$ square matrix of order that of $A$, then $– A,kA,A^{T},A^{n},A^{-1},B^{T}AB$ are also symmetric matrices. All positive integral power of a symmetric matrix are symmetric.
3. If $A, B$ are two symmetric matrices, then

$\color{green}\checkmark A\pm B, AB + BA$ are also symmetric matrices.

$\color{green}\checkmark AB - BA$ are skew-symmetric matrices.

$\color{green}\checkmark AB$ is a symmetric matrix when $AB=BA$ otherwise $AB$ or $BA$ may not be symmetric.

$\color{green}\checkmark A^{2},A^{3},A^{4},B^{2},B^{3},B^{4},A^{2}\pm B^{2},A^{3}\pm B^{3}$ are also symmetric matrices.

${\color{Orchid}{\textbf{Properties of Skew Symmetric Matrix:-}}}$

$\color{green}\checkmark$If $A$ is a skew symmetric matrix, then

• $A^{2n}$ is a symmetric matrix for $n$ positive integer.
• $A^{2n+1}$ is a skew symmetric matrix for $n$ positive integer.
• $kA$ is also a skew symmetric matrix, where $k$ is a real scalar.
• $B^{T}AB$ is also skew symmetric matrix where $B$ is a square matrix of order that of $A.$

$\color{green}\checkmark$All positive odd integral power of a skew symmetric matrix are skew symmetric matrix and positive even integral powers of a skew symmetric matrix are symmetric matrix.

$\color{green}\checkmark$If $A,B$ are two skew symmetric matrices, then

• $A\pm B,AB-BA$ are skew symmetric matrices.
• $AB+BA$ is symmetric matrix.

$\color{green}\checkmark$If $A$ is a skew symmetric matrix and $C$ is a column matrix then $C^{T}AC$ is a zero matrix.

$\color{green}\checkmark$If $A$ is any square matrix then $A+A^{T}$ is symmetric matrix and $A-A^{T}$ is a skew symmetric matrix.

$\color{green}\checkmark$The matrix which is both symmetric and skew symmetric must be a null matrix.

$\color{green}\checkmark$If $A$ is symmetric and $B$ is skew symmetric, then $\textbf{trace(AB)}=0.$

$\color{green}\checkmark$Any real square matrix $A$ may be expressed as the sum of a symmetric matrix $A_{S}$ and a skew symmetric matrix $A_{AS}.$

$$\color{Cyan}\checkmark A = \dfrac{1}{2}\bigg[A + A^{T}\bigg] + \dfrac{1}{2}\bigg[A – A^{T}\bigg] = A_{S} + A_{AS}$$

$\color{Green}\checkmark$ The determinant of an $n \times n$ Skew-Symmetric matrix is zero if $n$ is odd.

$\textbf{Proof:-}$ $A$ is skew-symmetric means $A^{T}= -A$. Taking determinant both sides $$\det(A^T)=\det(-A)\implies \det A =(-1)^n\det A \implies \det A =-\det A\implies \det A=0$$

$\textbf{8.Singular Matrix (or) Non-Invertible Matrix:-}$ A singular matrix is a square matrix that is not invertible i.e., it does not have an inverse. A matrix is singular (or) degenerate if and only if (or) iff  its determinant is zero.

$$\color{Cyan}\checkmark\mathbf{\mid A \mid _{n\times n} = 0}$$

$\textbf{9.Non – Singular Matrix (or) Invertible Matrix:-}$ A  square matrix is non – singular (or) invertible if its determinant is non – zero.

$$\color{Cyan}\checkmark\mathbf{\mid A \mid _{n\times n} \neq 0}$$

$\color{Cyan}\checkmark$A non - singular matrix has a matrix inverse.

$\textbf{10.Orthogonal Matrix:-}$ A square matrix is said to be orthogonal if $\mathbf{A\cdot A^{T} = I.}$ In other words the transpose of orthogonal matrix is equal to the inverse of the matrix, i.e. $\mathbf{A^{T} = A^{-1}.}$

$\textbf{Example:} \text{If} \: A = \dfrac{1}{3}\begin{bmatrix} 1 &2 &2 \\ 2 &1 &-2 \\ -2 &2 &-1 \end{bmatrix}_{3\times 3} \;\:\: ,\text{then} \:\:\: A^{T} = \dfrac{1}{3}\begin{bmatrix} 1 &2 &-2 \\ 2 &1 &2 \\ 2 &-2 &-1 \end{bmatrix}_{3\times 3}$

and $A\cdot A^{T} =\begin{bmatrix} 1 &0 &0 \\ 0 &1 &0 \\ &0 &1 \end{bmatrix}_{3\times 3}\:\:\:, \:\: A^{-1} = A^{T} = \dfrac{1}{3}\begin{bmatrix} 1 &2 &-2 \\ 2 &1 &2 \\ 2 &-2 &-1 \end{bmatrix}_{3\times 3}$

$\color{Cyan}\checkmark$If matrix $A$ is orthogonal then,

• Its inverse and transpose are also orthogonal.
• Its determinant is unity, i.e. $\mid A \mid = \pm 1.$
• $\mid A \mid \mid A^{T} \mid = 1$

$\textbf{11.Hermitian Matrix:-}$ A square matrix is said to be hermitian if $\mathbf{A = A^{\theta}}.$ Where $A^{\theta}$ is the transpose of conjugate of matrix $A,$ i.e. $A^{\theta} = \big(\overline{A}\big)^{T}$

$\textbf{Example:}\: \text{If} \: A = \begin{bmatrix} 1 &3-2i &2+3i \\ 3+2i &2 &i \\ 2-3i &-i &3 \end{bmatrix}_{3\times 3}$

$\text{then Conjugate of A} = \overline{A} = \begin{bmatrix} 1 &3+2i &2-3i \\ 3-2i &2 &-i \\ 2+3i &i &3 \end{bmatrix}_{3\times 3}$

$\implies A^{\theta} = \big(\overline{A}\big)^{T} = \begin{bmatrix} 1 &3-2i &2+3i \\ 3+2i &2 &i \\ 2-3i &-i &3 \end{bmatrix}_{3\times 3} = A$

$\textbf{12.Skew Hermitian Matrix:-}$ A square matrix $A$ is said to be skew hermitian if $\mathbf{A =\: – A^{\theta}.}$

$\textbf{Example:}\: \text{If} \: A = \begin{bmatrix} i &2 – 3i &4+5i \\ – 2 – 3i &0& 2i \\ – 4 + 5i &2i &-3i \end{bmatrix}_{3\times 3}$

$\text{then Conjugate of A} = \overline{A} = \begin{bmatrix} -i &2 + 3i &4-5i \\ – 2 + 3i &0& -2i \\ – 4 - 5i &-2i &3i \end{bmatrix}_{3\times 3}$

$\implies A^{\theta} = \big(\overline{A}\big)^{T} = \begin{bmatrix} -i &-2 + 3i &-4-5i \\ 2 + 3i &0& -2i \\ 4 - 5i &-2i &3i \end{bmatrix}_{3\times 3} =\: – A$

$\color{Cyan}\checkmark$All the diagonal elements of Skew Hermitian matrix are either zero (or) pure imaginary.

$\color{Cyan}\checkmark$All the diagonal elements of Hermitian matrix are real.

$\color{Cyan}\checkmark$In Hermitian matrix upper and lower diagonal elements should be complex conjugate pair.

$\color{Cyan}\checkmark$In Skew  Hermitian matrix upper and lower diagonal elements should be same but real value

sign are  opposite.

$\textbf{13.Unitary Matrix:-}$ A square matrix is said to be unitary if $\mathbf{A\cdot A^{\theta} = I},$ where $A^{\theta}$ is transpose of conjugate matrix $A,$ i.e. $A^{\theta} = \big(\overline{A}\big)^{T}$

$\textbf{Example:}\: \text{If} \: A = \begin{bmatrix} \dfrac{1+i}{2} &\dfrac{-1+i}{2} \\ \dfrac{1 - i}{2}& \dfrac{-1- i}{2} \end{bmatrix}_{2\times 2}$

$\implies A^{\theta} = \big(\overline{A}\big)^{T} = \begin{bmatrix} \dfrac{1-i}{2} &\dfrac{1+i}{2} \\ \dfrac{-1 - i}{2} & \dfrac{-1+ i}{2} \end{bmatrix}_{2\times 2}$

$\implies A\cdot A^{\theta} = \begin{bmatrix} 1 &0 \\ 0 &1 \end{bmatrix}_{2\times 2}$

$\color{Cyan}\checkmark$If $A$ is unitary matrix then,

• Its inverse and transpose is also unitary.
• Its determinant is also unity, i.e. $\mid A \mid = \pm 1.$
• $\mid A \mid \mid A^{\theta} \mid = 1$

$\textbf{14.Periodic Matrix:-}$ A square matrix is said to be a periodic if $\mathbf{A^{k+1} = A,}$ where $k$ is positive integer. $k$ is also known as period of the matrix.

$\textbf{15.Involutory Matrix:-}$ A matrix is said to be involutory if $\mathbf{A^{2} = I}.$

$\textbf{16.Idempotent Matrix:-}$ An idempotent matrix is a square matrix which, when multiplied by itself i.e. $\mathbf{A^{2}= A}.$

A periodic matrix is said to be idempotent when the positive integer $k$ is unity, i.e. $$A^{k+1} = A\implies A^{1 + 1} = A \implies A^{2} = A$$

$\textbf{17.Nilpotent Matrix:-}$ A square matrix is called a nilpotent matrix if there exists a positive integer $k$ such that $\mathbf{A^{k} = 0}.$

$\color{Cyan}\checkmark$The least positive value of $k$ is called the index of nilpotent matrix $A.$

$\color{Cyan}\checkmark$Determinant of Idempotent matrix is either $0$ (or) $1.$

$\color{Cyan}\checkmark$Determinant and Trace of nilpotent matrix is zero.

$\color{Cyan}\checkmark$Inverse of nilpotent matrix does not exist.

$\textbf{18.Invertible Matrix:-}$ A matrix  $A$ is said to be invertible (or) non-singular (or) non-degenerate if there exists a matrix $B$ such that $\mathbf{AB = BA = I_{n}}.$

$\color{Teal}\checkmark$If matrix $A$ is invertible, then the inverse is unique.

$\color{Teal}\checkmark$ If matrix $A$ is invertible, then $A$ cannot have a row (or) column consisting of only zeros.

$\textbf{19.Rotation Matrix:-}$ A rotation matrix in $n$–  dimensions is a $n\times n$ special orthogonal matrix, that is an orthogonal matrix whose determinant is $1.$ i.e. $R_{T} = R_{-1}, \mid R \mid = 1$

$\textbf{Example:}\:\: A = \begin{bmatrix} \cos\theta &-\sin\theta \\ \sin\theta &\cos\theta \end{bmatrix}_{2\times 2}$

$\textbf{20.Normal Matrix:-}$ A matrix is normal if it commutes with its conjugate transpose.

$\color{Teal}\checkmark$A complex square  matrix $A$ is normal if $\mathbf{\big(\overline{A}\big)^{T}\cdot A = A\cdot \big(\overline{A}\big)^{T} = A^{\theta}\cdot A = A\cdot A^{\theta}}.$

$\color{Teal}\checkmark$A real square matrix $A$ is normal if $\mathbf{A^{T}\cdot A = A\cdot A^{T}},$ since a real matrix satisfies $\mathbf{\overline{A} = A}.$

__________________________________________________________________

$\color{Red}{\text{Question:}}$ Prove that Let the $A_{n\times n}\:,$ then $$\color{Magenta}{\sum_{i=1}^{n}\sum_{j=1}^{n} A_{ij}^2 = \text{Trace of} \:\: \left( A\:A^{T}\right ) }$$

__________________________________________________________________

The given matrix is called square Vandermonde Matrix and it has the form :

$\begin{bmatrix} 1 & \alpha_1 & \alpha_1^2 & .... & \alpha_1^{n-1} \\ 1 & \alpha_2 & \alpha_2^2 & .... & \alpha_2^{n-1} \\ \vdots & \vdots & \vdots &\ddots & \vdots \\ 1 & \alpha_n & \alpha_n^2 & .... & \alpha_n^{n-1} \\ \end{bmatrix}$

and its determinant is given by : $\prod_{1 \leq i < j \leq n} (\alpha_j - \alpha_i)$. Proof is given here using induction.

$\Rightarrow$ For a general $4\times 4$ matrix of the form :

$$\begin{bmatrix} 1 &a &a^2 &a^3 \\ 1 &b &b^2 &b^3 \\ 1 &c &c^2 &c^3 \\ 1 &d &d^2 &d^3 \end{bmatrix}$$

Determinant is given by : $(b-a)*(c-a)*(d-a)*(c-b)*(d-b)*(d-c)$ and similarly, for this type of matrix, we can find determinant of any order.

17

$$\textbf{1.Theory Of Computation}$$

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO 1.Introduction to the Theory Of Computation Michael Sipser(3rd Edition) $0,1,2,3,4,5$ $0,1,2,3,4,5$ Questions from Sipser 2.Introduction To Automata Theory, Languages, and Computation Jeffrey D. Ullman(3rd Edition) $1,2,3,4,5,6,7,8,9$ $1,2,3,4,5,6,7,8,9$ Questions from Ullman 3.An Introduction to Formal Languages and Automata Peter Linz(4th and 5th Edition) $1,2,3,4,5,6,7,8,9,10,11,12$ Edition$4: 1,2,3,4,5,6,7,8,$ Till Exercise $8.1$ Question $8$ (Page No. $212$) Edition$5:1,2,3,4,5,6,7,8,$ Till Exercise $8.1$ Question $7(k)$ (Page No. $212$) Question from Peter Linz Edition 4 Question from Peter Linz Edition 5

$$\textbf{2.Compiler Design}$$

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO Compilers Principles, Techniques, & Tools Jeffrey D. Ullman(2nd ​​​​​ Edition) $1,2,3,4,5,6$ $1,2,3,4,5,6$ Questions from Ullman

$$\textbf{3.Programming}$$

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO The C Programming Language Brian Kernighan and Dennis Ritchie(2nd Edition) $1,2,3,4,5,6$ Needs to be added from starting Questions from Dennis Ritchie

$$\textbf{4.Data Structures}$$

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO Introduction to Algorithms Thomas H. Cormen(3rd Edition) $10,11,12,18,21,22$ Questions from Cormen

$$\textbf{5.Algorithms}$$

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO Introduction to Algorithms Thomas H. Cormen(3rd Edition) $1,2,3,4,6,7,8,15,16,17,23,24,25$ Questions from Cormen

$$\textbf{6.Discrete Mathematics}$$

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO Discrete Mathematics and Its Applications Kenneth H. Rosen(7th Edition) $1,2,6,8,9,10$ $1,2,$ Till  Exercise $2.3$ Question $74$ (Page No. $155$) Questions from Rosen

$$\textbf{7.Databases}$$

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO Henry F. Korth(6th Edition) $1,2,3,4,6,7,8,10,11,12,14,15,16$ Questions from Korth Edition 4 Questions from Korth Edition 6 Shamkant B. Navathe(6th Edition) $1,2,3,4,6,7,15,16,17,18,21,22$ Needs to be added from starting Questions from Navathe 3.Database Management Systems Johannes Gehrke and Raghu Ramakrishnan(3rd Edition) $1,2,3,4,5,8,9,10,15,16,17,19,$ Questions from Raghu Ramakrishnan

$$\textbf{8.Operating System}$$

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO 1.Operating System Concepts Peter Baer Galvin(9th Edition) $1,2,3,4,5,6,7,8,9,10,11,12$ Questions from Galvin 2.Modern Operating Systems Andrew S. Tanenbaum(4th Edition) $1,2,3,4,6$ $1,2,3,4,5,6$ Questions from Tanenbaum

$$\textbf{9.Computer Networks}$$

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$$\textbf{10.Digital Logic}$$

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$$\textbf{11.Computer Organization}$$

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$$\textbf{12.Probability}$$

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$$\textbf{13.Linear Algebra}$$

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO Gilbert Strang(4th Edition) $1,2,3,4,5$ Needs to be added from starting Questions from Gilbert Strang 2.Advanced Engineering Mathematics Erwin Kreyszig(10th Edition) $7,8$ Needs to be added Questions from Erwin Kreyszig 3.Advanced Engineering Mathematics H K Dass $4$ Needs to be added Questions from HK Dass

$$\textbf{14.Calculus}$$

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO Thomas' Calculus George B. Thomas and Maurice D. Weir(11th Edition) Needs to be added from starting Questions from Thomas' Calculus
18

$$\textbf{Contents}$$

Web Page

Syllabus:

 Topic Covered in Videos Video link from GO Youtube channel GO Videos
19

$$\textbf{Contents}$$

Web Page

Syllabus:

 Topic Covered in Videos Video link from GO Youtube channel GO Videos
20

$$\textbf{Contents}$$

Web Page

Syllabus: Connectivity, Matching, Coloring.

$$\small{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline \textbf{Year}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum} \\\hline\textbf{1 Mark Count}&1&1&0&1&0&1&0&0.7&1 \\\hline\textbf{2 Marks Count}&1&1&0&0&0&0&0&0.3&1 \\\hline\textbf{Total Marks}&3&3&0&1&0&1&\bf{0}&\bf{1.3}&\bf{3}\\\hline \end{array}}}$$

 Topic Covered in Videos Video link from GO Youtube channel GO Videos

Topics to be Covered: