Practice Questions

Duration: 9 min

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Module outline

  1. About Course: About the Course
  2. Paper 1 | General Studies: State Wise GK
  3. Paper 1 | Information and Communication Technology: Role of Internet and Web Services, Security in Mobile and Wireless Computing
  4. Paper 1 | Current Affairs: GK- Current Affairs
  5. Paper 2 | Discrete Mathematical Structure: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Probability
  6. Paper 2 | Digital Computer Organization And Architecture: Number System, Number Representation, Sequential Circuits, Combinational Circuit, 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. Paper 2 | C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  8. Paper 2 | Programming Languages & Object-Oriented Programming Concepts: Java, OOPS with C++, Java Fundamentals, Array, String, IO & References
  9. Paper 2 | Python: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis
  10. Paper 2 | Data Structure And Algorithms: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Minimum Spanning Trees, Advanced Algorithms
  11. Paper 2 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Deadlock, Threads & Process Creation, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS
  12. Paper 2 | Theory Of Computation: 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
  13. Paper 2 | Compiler Design: 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
  14. Paper 2 | Database Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, Relational Algebra, SQL, Transaction Management, Concurrency Control, Database Recovery, Database Security & Authorization, Enhanced Data Models
  15. Paper 2 | Computer Network And Information Security: Introduction to CN, Data Communication, Application Layer, Hardware basics, Cyber Threats and Malware Attacks, Network Security, 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
  16. Paper 2 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  17. Paper 2 | Current Trends & Technologies: Emerging Tech Trends, Approaches to AI, Search Algorithms, Game Playing, Knowledge Representation, Planning, Multi Agent Systems, Fuzzy Sets, Natural Language Processing, Artificial Neural Networks, Genetic Algorithms, Artificial Intelligence, Data Warehousing & Mining, Electronic Data Interchange and E-Commerce, Terms & Abbreviations
  18. Paper 2 | Full Length Mock Tests:
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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