Practice Questions

Duration: 9 min

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Module outline

  1. About Course: About the Course
  2. General Aptitude & Reasoning: Number System, Percentage, Ratio and Proportion (Ratios), Average, Profit and Loss, Simple Interest and Compound Interest, Time and Work, Speed Time and Distance, Mixture and Alligation, Data Interpretation, Algebra, Probability, Permutation and Combination, Venn Diagram, HCF LCM, Divisibility Rules, Age Problems, Pipe and Cistern, Partnership, Linear Algebra, Sequence and Series (Series), Statistics, Powers and Exponents (Surds and Indices), Arithmetic, Mensuration and Geometry, Calculus, Trigonometry, Vectors, Speed Mathematics and Basic Mathematics, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Analogy, Series (Number and Letter Series) (Numerical Relations and Reasoning), Puzzle Solving (Puzzles), Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Calendar, Seating Arrangements, Coding Decoding, Blood Relations, Directions (Direction Test), Clock, Analytical Reasoning (Counting Figures Reasoning), Ranking, Order and Sequence, Logical Sequence Of Words, Inequality, Data Sufficiency, Cubes & Dices
  3. Computer Networks & Cybersecurity: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware Basics, Network Security
  4. Programming, Data Structures & Algorithms: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, OOPS with C++, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  5. AI, Machine Learning, Big Data & Cloud: Emerging Tech Trends, Artificial Intelligence, Big Data Systems, Approaches to AI, Search Algorithms, Knowledge Representation, Cloud Computing
  6. Digital Logic & Computer Organization: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Basics of COA, Floating Point Rep, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Register Transfer and Microoperations, Control Unit Design, Programming the Basic Computer
  7. TOC & Compiler: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Complexity Theory, Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment
  8. Discrete Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic
  9. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS, Linux OS, Distributed Systems
  10. Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management, Object Oriented Technology, HTML Basics, JavaScript-Basics, Java, XML
  11. Database Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, Data Warehousing & Mining, Enhanced Data Models, NoSQL
  12. Other Recurring CS Topics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, Multimedia
  13. Live Classes: ISRO Live classes
  14. Full Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video provides a detailed lecture on identifying groups using Cayley tables and modular arithmetic. The instructor systematically evaluates four options at a time to determine if a set with a specific binary operation satisfies the four group axioms: closure, associativity, identity, and inverses. The lesson progresses through examples involving addition and multiplication modulo 4, 5, 7, 8, and 15. Key concepts demonstrated include the additive group of integers modulo n (Z_n) and the multiplicative group of units modulo n (U_n). The instructor uses visual aids, filling out tables and marking correct or incorrect options with checkmarks and crosses to reinforce the criteria for a group.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video begins with the question 'Check out which of the following is a group?' displaying four options involving the set {0,1,2,3}. The instructor first analyzes {0,1,2,3}, +4. He constructs the Cayley table, writing entries like 0+1=1, 1+3=0, and 2+2=0, confirming closure, identity (0), and inverses. He marks this as a group. Next, he examines {0,1,2,3}, *4. He fills the table, showing the first row is all zeros. He points out that 0 has no multiplicative inverse because the equation 0 * x = 1 has no solution in the set, and crosses out this option.

  2. 2:00 – 5:00 02:00-05:00

    The instructor moves to the remaining options from the first set. He crosses out {1,2,3}, +4 and {1,2,3}, *4, noting that {1,2,3}, *4 is not closed because 2*2=4 which is 0 mod 4, and 0 is not in the set. He then introduces a new set of examples modulo 5. He marks {0,1,2,3,4}, +5 as a group. He crosses out {0,1,2,3,4}, *5 because 0 has no inverse. He crosses out {1,2,3,4}, +5 because it lacks the additive identity 0. Finally, he fills the Cayley table for {1,2,3,4}, *5, writing row 2 as 2, 4, 1, 3 and row 3 as 3, 1, 4, 2, confirming it is a group (U(5)).

  3. 5:00 – 9:23 05:00-09:23

    The lecture extends to modulo 7 examples. He marks {0,1,2,3,4,5,6}, +7 as a group and crosses out {0,1,2,3,4,5,6}, *7 due to the presence of 0. He crosses out {1,2,3,4,5,6}, +7 because it lacks the additive identity 0. He marks {1,2,3,4,5,6}, *7 as a group. He then presents {1,3,5,7}, *8. He fills the table completely: row 1 is 1, 3, 5, 7; row 3 is 3, 1, 7, 5; row 5 is 5, 7, 1, 3; row 7 is 7, 5, 3, 1. He confirms this is a group. Finally, he introduces {1,2,4,7,8,11,13,14}, *15, explaining that this set contains numbers less than 15 that are coprime to 15, which forms a group under multiplication modulo 15.

The video effectively teaches the identification of groups by applying the four group axioms to various sets under modular arithmetic. The instructor demonstrates that additive groups Z_n always form a group, while multiplicative groups require the exclusion of zero and non-coprime elements to form U_n. By filling out Cayley tables, the instructor visually verifies closure and the existence of inverses, providing a clear method for students to solve similar problems in abstract algebra.

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