Most viewed posts in Study Materials

1

Download NIELIT PDFs

https://github.com/GATEOverflow/GO-PDFs/releases/tag/NIELIT

Scientist  – ‘B’

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.

Exam Dates Question Papers Answer Keys GO/AO Links Exam Links
03 April 2022 2022(CS) : Set – D Answer Key Questions GO  
05 Dec 2021 2021(CS) : Set – A

Answer Key

Changes in Answer Key

Questions GO  
05 Dec 2021 2021(IT) : Set – B

Answer Key

Changes in Answer Key

Questions GO  
22 Nov 2020

2020(CS) : Set – A

Answer Key

Questions GO  
01 & 02 Dec 2018 2018: Set – B Answer Key Questions GO  
 17 Dec2017 2017: Set – A Answer Key

Section – A

 

Section  – B

 
 22 July 2017 2017(CS): Set  – A Answer Key

Section – A

 

Section  – B

 
 22 July 2017 2017(IT): Set – A Answer Key

Section – A

 

Section  – B

 
4 Dec 2016 2016(CS): Set – A Answer Key

Section – A

 

Section  – B

 
4 Dec 2016 2016(IT): Set – A Answer Key

Section – A

 

Section  – B

12-13 Mar 2016 2016: Set – A Answer Key

Section – A

 

Section  – B

Section – C

 

Scientific/Technical Assistant – ‘A’ 

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.

 Exam Dates Question Papers Answer Keys GO/AO Links Exam Links
05 Dec 2021 2021(IT) : Set  – D

Answer Key

Changes in Answer Key

Questions GO  
22 Nov 2020

2020(CS) : Set – A

Answer Key

 

Questions GO  
17 Dec 2017 2017: Set –  A Answer Key

Section – A

 

Section – B

Take exam
15 Oct 2017 2017(CS): Set – A Answer Key

Section – A

Section – B

Section – C

 
15 Oct 2017 2017(IT): Set –  A Answer Key

Section – A

Section – B

Section – C

 
12-13 Mar 2016 2016(1)- Senior TA Answer Key    
12-13 Mar 2016 2016(2) -Sample TA Answer Key    
12-13 Mar 2016 2016(3)- Junior TA Answer Key    

 

Scientist – ‘C’

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.
Exam Dates Question Papers Answer Keys GO/AO Links Exam Links
26 feb 2023 2023 Answer Key Questions GO  
07th Aug 2022 2022 Answer Key Questions AO  
13/02/2022 2022: Set – A Answer Key Questions AO  
10 Feb 2019 2019: Set – B Answer Key

 

Questions AO

 
12-13 Mar 2016 2016: Set – A Answer Key

Section – A

Section – B

Section – C

 

 

Scientist – ‘D’

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.
Exam Dates Question Papers Answer Keys GO/AO Links Exam Links
26 feb 2023 2023 Answer key Questions GO  
13/02/2022 2022: Set – A Answer Key Questions AO  
10 Feb 2019 2019: Set – B Answer Key Questions AO  
12-13 Mar 2016 2016: Set – A Answer Key Questions AO  

 

The official website references: https://nielit.gov.in/recruitments

2

Hi, I’ll talk about free test series available. If you want to join any coaching class/paid test series you can join ACE, Testbook , Made Easy etc.

Okay, then what to do with test series ? Usually test series provide following type of test

  1. Chapter wise
  2. Subjectwise
  3. Mock Test
  4. Aptitude Tests (Previous year papers from all branches)

Let me be honest, I feel that neither Chapter wise or Subject-wise tests are necessary. Only full length mock tests are needed for GATE preparation. Though at the same time, I will still recommend from where to get these tests.

I’ll tell you how you can do this kind of tests for free.

Chapter Tests -> Doing this tests from  any paid test series is not required, free stuff is enough. You can get chapter wise test questions from Reference books. Newgradiance is  enough for chapter tests for many subjects. They are free. Gateoverflow book by Gateoverflow also contains chapter wise test series. You can find Gateoverflow book at Gatecse.

Subject wise tests -> Doing this tests from  any paid test series is not required, free stuff is enough. You can get whole collection of subjectwise tests free of cost from VirtualGate which is sponsored by Techtud . Doing all tests from NewGradiance may also serve purpose of solving Subject-wise Test series for subject it provides.

Full Length Tests -> For this I believe best strategy would be not to see any questions from few papers. Example -> Suppose I’m preparing for GATE 2017 and I want to have few “Real” mocks, in that case what I’ll do, I wont touch GATE 15 & GATE 16 papers while practicing questions. Then when I’m done with preparation, I have  5 “Real” GATE papers for giving Mock exams. I’ve done this , I’ve used 2 papers of GATE 15 as mock exam, for getting online touch we  can use online calculator and pdf reader. Believe me previous year mocks are best test series possible if you have not seen any question from that paper. In case you have solved few questions from them, it is useless (In my case I used to remember all questions.)

Apart from this VirtualGate which is sponsored by Techtud  is also very nice resource for getting 3+ Full lenght tests. They have already released virtual GATE 2015,2016-1,2016-2 and we can expect at least 2 more in 2017.

About Aptitude Tests -> You should try to solve all previous year aptitude tests papers from all branches. It’ll help you getting idea about Aptitude questions in general. You can use http://aptitudeoverflow.in/ for getting difficult questions.

Link to previous year aptitude questions Till 2015 -> https://drive.google.com/file/d/...

For after 2016 aptitude sets check respective IIT/IISc site for GATE and download papers. Remember all papers for any branches in same time slot have same aptitude questions.

3

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. Introduction to Algorithms (SMA 5503), MIT OCW
  2. Algorithms by Shai Simonson

  3. Algorithmic Toolbox, Coursera

  4. Algorithm Specialization (Stanford)

  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. Theory of Computation by Shai Simonson
  1. 1-4, 6,7, 9, 11-13, 15,16, 18, 19 
Computer Organization
  1. High Performance Computing by Prof. Matthew Jacob IISc

  2. Computer Architecture by Prof. Anshul Kumar IIT Delhi

  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. High Performance Computing by Prof. Matthew Jacob IISc
  2. Operating Systems by Mythili Vutukuru IIT Bombay
  1. covered in CO section or GO playlist
Compilers
  1. Compilers by Prof. Alex Aiken Stanford University
  1. week 2-9 (skip Cool Type Checking in week 6)
Linear Algebra
  1. MIT 18.06 Linear Algebra by Prof. Gilbert Strang
  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. Graph Theory by Dr. L. Sunil Chandran IISc
  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. Introduction to Abstract Group Theory by Krishna Hanumanthu, CMI
  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

4

The following notes are from the standard books and cover over 95% of the syllabus. If you go through these notes, then perhaps there is no need to go through any standard books.

https://iitianmanu.in/

When you download these notes and have a look, please leave a comment so that others can also download them. Following are the screenshots from the notes. I spent 4 to 5 months preparing them so that these notes can help others one day.

From CN notes:

Peace, Happiness & Success,
Manu

5

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):  

https://drive.google.com/file/d/1K5DW0rZolcWfJbAMuZvWkvvTuezs4fEP/view?usp=sharing

 

Peace!

 

🌟Notes of all subjects uploaded🌟

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

Update 2(26/04/2021): Computer Organization and Architecture, Compiler Design 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

6

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

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

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.

7

Hi all, below are some free standard video lectures for GATE DA:

Artificial Intelligence: 

  1.  CS60045 Artificial Intelligence (IIT KGP):

    A semester-length UG/PG level course on AI taken by two of the most renowned professors at IIT Kharagpur.    
    Instructors
     
    Prof. Pallab Dasgupta
    Prof. Partha Pratim Chakrabarti
  2. An Introduction to Artificial Intelligence (IIT Delhi):

    A semester-length UG-level course at IIT Delhi (which has also been added as an NPTEL course) taken by one of the best AI researchers in the country.  
    Instructors
      
    Prof. Mausam

 

Machine Learning:

      1. Machine Learning Specialization

 

The most popular course on ML. Everything is explained in simple beginner-friendly language. (Please note that this course can be audited for free in Coursera)                    
Instructors
         
Prof. Andrew NG

 

 

  1. Introduction to Machine Learning(Course sponsored by Aricent), IIT Madras

    Another great course                    
    Instructors
             
    Prof. Balaraman Ravindran

 

 

Statistics and Data Analytics: 

  1.  CS61061 Data Analytics (IIT KGP):

    A semester-length PG-level course on Data Analytics taken by one of the most renowned professors at IIT Kharagpur.    
    Instructors
     
    Prof. Debasis Samanta

 

Calculus and Optimization:

  1. Basic Calculus for Engineers, Scientists and Economists

  2. Foundations of Optimization, IIT Kanpur

                         
    Instructors
             
    Prof. Joydeep Dutta

Programming, Data Structures and Algorithms:

  1. Programming, Data Structures and Algorithms using Python, Chennai Mathematical Institute

                         
    Instructors
             
    Prof. Madhavan Mukund

 

 

I am not adding resources for the subjects already present in GATE CSE as there are already many blogs on gatecse.in and gateoverflow.in. Also, there are some other topics like Data Warehousing but I think some simple Google searches should be enough to learn those topics.
All the very best and happy learning😊

 

 
8
Plzz post half question which you know here and answers which is according to u...post questions which u know just confirmation
9

Decidability Slides, Parsing Notes and Pipelining Slides/Video have been added. 

Adding all the previous Notes which I have made for GATE CSE here:

 

TOC:

Decidability Slides

Last year Discussions: 

Compiler Design:

DBMS

Combinatorics:

Mathematical Logic

Previous discussions: 

Graph Theory

 

Algorithms

Previous discussions: 

Programming:

Previous Discussions:

CO & Architecture

PIPELINING

https://gatecse.in/cache-misses/

Previous discussions: 

Operating Systems

Multi-level Paging

Previous discussions:

10

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

DOWNLOAD

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. 

11

$\textbf{Initial Version of PDF:}$ https://github.com/GATEOverflow/GO-PDFs/releases/tag/initial-draft

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

               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

1.Database System Concepts(6th Edition)

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

2.Fundamentals Of Database Systems

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

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

  Book Title   Author Chapters relevant for GATE Chapters on GO Questions On GO
1.Computer Networking(A Top-Down Approach)

James F. Kurose and Keith W. Ross(6th Edition)

    Questions from Kurose and Ross

2.Computer Networks

Andrew S. Tanenbaum(5th Edition)

    Questions from Tanenbaum

3.Data Communications And Networking

Behrouz A. Forouzan(4th Edition)   Needs to be added from starting Questions from Forouzan
4.Computer Networks: A Systems Approach S. Davie and Larry L. Peterson     Questions from Peterson-Davie

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

           Book Title   Author Chapters relevant for GATE Chapters on GO Questions On GO
1.Digital Design

M.Morris Mano(3rd Edition)

    Questions from Morris Mano
2.Introduction to Logic Design Alan B. Marcovitz(3rd Edition)   Needs to be added from starting Questions from Marcovitz
3.Switching and Finite Automata Theory Zvi Kohavi(3rd Edition)   Needs to be added from starting Questions from Kohavi

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

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO

Computer Organization

V. Carl Hamacher(6th Edition)   Needs to be added from starting Questions from Hamacher

Computer Architecture

A Quantitative Approach

John L. Hennessy, David A. Patterson(5th Edition)   Needs to be added from starting Questions from Patterson

$$\textbf{12. Probability}$$

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO

1.Introductory Probability

Janko Gravner     Questions from Gravner
2.A First Course in Probability Sheldon M. Ross(8th Edition)   Needs to be added from starting Question from Sheldon Ross
3.Advanced Engineering Mathematics Erwin Kreyszig(10th Edition) $24$ Needs to be added Questions from Erwin Kreyszig
4.Advanced Engineering Mathematics H K Dass $11$ Needs to be added Questions from HK Dass
5.Probability Jim Pitman   Needs to be added from starting Questions from Pitman

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

 Book Title Author Chapters relevant for GATE Chapters on GO Questions On GO

1.Linear Algebra and Its Applications

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
12

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

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

$\textbf{Pythagorean Identities:}$

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

$\textbf{Reciprocal Identities:}$

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

$\textbf{Tangent and Cotangent Identities:}$

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

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

  • $\sin2\theta = 2 \sin\theta \cos\theta$
  • $\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. $\displaystyle{}\int \log_{a} x\: dx =  x\log_{a} x\: – \dfrac{x}{\log a} + C$

$\textbf{Trigonometric Functions}$

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

13
I have started reading Kenneth Rosen, I have found only few solved examples and a hell of Exercise questions. I have tried the exercise questions, but able to solve only a few of them. Everyone says Standard books are Gold for GATE, I want to know how to follow them.

How to develop that problem-solving skill?

Any suggestions are welcomed!
14

Hi,

 

This post specifically asks Arjun Suresh for guidance, though anyone is welcome to help.

 

I am a 2014 passout from a private college with 3.5 years of IT experience. In 2018, I quit my job and decided to prepare for GATE 2019 with the aim to complete Masters (and later, a Ph.D.), and to become a professor later. I followed video lectures only (without any reference textbook) which are recommended at best-video-lectures-for-gate-cse ranging from NPTEL, Shai Simonson’s, Stanford’s, etc. During the examination, I messed up very badly by making blunders and ended up losing around 10-15 marks. Now, I don’t know what my future holds, but I am waiting for the results. I realized that going through these video lectures are beneficial only when studied with a standard textbook.

 

I am now thinking of buying some standard textbooks and maybe prepare for next year, with a job, or otherwise. I could only buy one book per subject. I got the link to the What_to_read_series by Bikram Ballav, but it mentions multiple books per subject. I want a list of the most recommended (“the most appropriate”) book which could not only help me with my preparation for next year, but could also be used in the future for reference.

 

Thanks in advance.

16

$$\textbf{Contents}$$

Web Page

Syllabus: 

 Topic Covered in Videos Video link from GO Youtube channel
  GO Videos
17
hello friends. i need kvs pgt cs 2017 question paper can anyone plz provide me..

.thank you
18

Operating system resources: 

1.http://pages.cs.wisc.edu/~remzi/OSTEP/ [best intuitive book on OS]

A companion video series for above book by prof Mythili of IITB: https://www.cse.iitb.ac.in/~mythili/os/

2.https://www.cse.iitb.ac.in/~mythili/teaching/cs347_autumn2016/  [ it contains in-class materials and programming assignments ]

3. CS - 347 IITB

4. https://pdos.csail.mit.edu/6.828/2016/schedule.html

5. https://pdos.csail.mit.edu/6.828/2016/xv6.html

I hope these will be useful for those who are still in UG college and have started to learn OS. For Gate Preparation  the videos will be a nice revision of basic concepts.

19

This blog is for if anyone found any mistake or doubt in any gatebook test series questions that options may be incorrect or some personal doubt ...Then he can post here it in this blog comment  …

 

just write that question link in the comment...

Why this link ------->We can also  just straightly goes to gatebook test series question in link and post it but then other persons will not be notify about your doubts …… and then  your doubt will be there for a long time maybe as ( RECENT  ACTIVITY  ) option changes very fast, so may be no other person or any VETERAN level users didnt get notify about  your doubt and then there is very much probability that your doubt will be removed when you are doing another subject …….. thats why i am creating this blog .
 AND I AM EXCPECTING VERY  ADMINSTRATORS ,VETERANS LEVEL USERS , BOSS LEVEL AND  @EVERYONE ELSE  WHO ENROLLED IN TEST SERIES TO COMMENT HERE AT STARTING FOR NO REASON BUT FOR FUTURE NOTIFICATIONS ……… 
 
We are just going to add links here nothing about solution here solution will be on test series  questions there
20
HI sir

i am preparing for gate 2020 by self study ..sir i want to buy hard copy of GO book .. upto which month it will be available to buy sir?