Practice Questions

Duration: 5 min

This video lesson is available to enrolled students.

Return to /learn/UP-LT-GRADE-ASSISTANT-TEACHER-2025/digital-logic-circuits-discrete-math-structures/group-theory/classification-of-finite-and-infinite-groups/asset-practice-questions-2328 after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. About Course: About the Course
  2. CS | 01 | Digital Logic and Circuits and Discrete Mathematical Structures: Digital Systems & Boolean Basics, Number System, Number Representation, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Permutation and Combination / क्रमचय और संचय
  3. CS | 02 | Computer Organization and Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Multiprocessors
  4. CS | 03 | Data Structures and Algorithm: 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, Complexity Theory, Advanced Algorithms
  5. CS | 04 | Problem Solving through C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  6. CS | 05 | Object Oriented Techniques: Object Oriented Technology, Java
  7. CS | 06 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Windows OS, Unix OS
  8. CS | 07 | Database Management Systems: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, Relational Algebra, SQL, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, Database Security & Authorization, Query Processing & Optimization, File Organization & Indexing
  9. CS | 08 | Computer Networks: 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
  10. CS | 09 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  11. CS | 10 | Internet Technology, Web Design and Web Technology: Internet Fundamentals, Web Design & Development, Protocols & Services, Web Publishing & Hosting, Server Side & Security, Emerging Technologies, JavaScript-Basics, HTML Basics
  12. CS | 11 | System Analysis And Design: Fundamentals of Systems and Information Systems, SDLC and Development Approaches, Software Documentation and Software Requirements Specification (SRS), SDD and Quality Assurance, Security, Risk Management & Maintenance
  13. CS | 12 | Information Security and Cyber Laws: Cyber Threats and Malware Attacks, Role of Internet and Web Services, Security in Mobile and Wireless Computing, Electronic Data Interchange and E-Commerce, Physical Security and Cryptography Basics, Intrusion Detection Systems (IDS), Cyber Safety and Security
  14. CS | 13 | Computer Graphics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing
  15. GS Paper | 01 | Indian and World Geography / भारत एवं विश्व का भूगोल: Geography / भूगोल
  16. GS Paper | 02 | History / इतिहास: History / इतिहास
  17. GS Paper | 03 | Indian Economy & Social Development / भारतीय अर्थव्यवस्था एवं सामाजिक विकास: Indian Economy / भारतीय अर्थव्यवस्था
  18. GS Paper | 04 | General Science / सामान्य विज्ञान: Biology / जीव विज्ञान, Chemistry / रसायन विज्ञान, Scientific Achievements and Research / वैज्ञानिक उपलब्धियाँ और अनुसंधान
  19. GS Paper | 05 | Elementary Maths / प्रारंभिक गणित: Number System / संख्या पद्धति, Divisibility Rules / विभाज्यता के नियम, HCF and LCM / महत्तम समापवर्तक और लघुत्तम समापवर्त्य, Percentage / प्रतिशत, Profit, Loss and Discount / लाभ, हानि और बट्टा, Simple and Compound Interest / साधारण और चक्रवृद्धि ब्याज, Ratio and Proportion / अनुपात और समानुपात, Average / औसत, Time and Work / समय और कार्य, Time, Speed and Distance / समय, चाल और दूरी, Algebra / बीजगणित, Solid Mensuration / घन क्षेत्रमिति, Data Interpretation / आँकड़ा निर्वचन, Arithmetic / अंकगणित, Statistics / सांख्यिकी, Probability / प्रायिकता, Sequences, Series and Combinatorics / अनुक्रम, श्रेणी और संयोजन, Mixture and Alligation / मिश्रण और एलिगेशन, Calculus / कलन, Trigonometry / त्रिकोणमिति, Vectors / सदिश, Number Systems and Number Theory / संख्या पद्धति और संख्या सिद्धांत, Euclidean Geometry and Mensuration / यूक्लिडीय ज्यामिति और क्षेत्रमिति, Statistics and Probability / सांख्यिकी और प्रायिकता, Coordinate Geometry, Vectors and 3D Geometry / निर्देशांक ज्यामिति, सदिश और त्रिविमीय ज्यामिति, Polynomials, Equations and Complex Numbers / बहुपद, समीकरण और सम्मिश्र संख्याएँ, Plane Mensuration / समतल क्षेत्रमिति, Simplification, Indices and Surds / सरलीकरण, घातांक और करणी, Sets, Relations, Functions and Discrete Mathematics / समुच्चय, संबंध, फलन और विविक्त गणित
  20. GS Paper | 06 | Indian Polity and Governance / भारतीय राजव्यवस्था एवं शासन: Polity / राजव्यवस्था
  21. GS Paper | 07 | Current Events / समसामयिक घटनाएँ: Current Affairs / समसामयिकी
  22. UPLT Live Classes Notes: UPLT Live Classes
  23. UPLT Full Mock Tests Tier 1:
  24. UPLT Full Mock Tests Tier 2:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video provides a focused lesson on Group Theory, specifically targeting the identification of groups and the calculation of inverses within modular arithmetic systems. The instructor begins by evaluating four distinct sets and operations to determine which satisfy the axioms of a group. He then proceeds to solve three multiple-choice problems that require applying the definition of an inverse element (a * a^-1 = e) in different modular groups. The lesson emphasizes checking closure, identity, and inverse properties through concrete examples involving addition and multiplication modulo p.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor begins by presenting four mathematical structures labeled 15 through 18, each defining a set and an operation to test for group properties. Structure 15 is {1, ..., p-1} under multiplication. Structure 16 is {0, ..., p-1} under multiplication. Structure 17 is {1, ..., p-1} under addition. Structure 18 is {0, ..., p-1} under addition. The instructor analyzes each to determine if they form a group. He marks 15 as correct because the set of non-zero residues modulo a prime p forms a group under multiplication. He marks 16 as incorrect because the element 0 does not have a multiplicative inverse. He marks 17 as incorrect because the set lacks the identity element 0 for addition. Finally, he marks 18 as correct, as the set of all residues modulo p forms a group under addition.

  2. 2:00 – 5:00 02:00-05:00

    The lecture transitions to solving specific multiple-choice questions involving inverses. The first question asks to identify the false statement for the group {0, 1, 2, 3, 4, 5} under addition modulo 6. The options involve finding inverses. The instructor explains that for an element 'a', its inverse 'a^-1' must satisfy a + a^-1 = 0 (mod 6). He verifies that 1^-1 = 5 (since 1+5=6=0), 2^-1 = 4 (since 2+4=6=0), and 0^-1 = 0. He identifies option (c) 3^-1 = 6 as false because 6 is not in the set and 3+3=0, so the inverse of 3 is 3. The next question involves the group {1, 2, 3, 4, 5, 6} under multiplication modulo 7. He verifies that 1^-1=1, 2^-1=4 (2*4=8=1), 3^-1=5 (3*5=15=1), and 6^-1=6 (6*6=36=1). All options appear true, suggesting a potential trick or error in the question, but he moves on. The final question is about the group {1, 2, 4, 7, 8, 11, 13, 14} under multiplication modulo 15. He checks the inverses: 2^-1=8 (2*8=16=1), 4^-1=4 (4*4=16=1), 7^-1=13 (7*13=91=1). He calculates 11*14=154, which is 4 mod 15, not 1. Thus, he marks option (d) 11^-1 = 14 as the false statement. A Cayley table is displayed for the first problem to visualize the operation.

The video effectively bridges theoretical definitions with practical problem-solving. It starts by reinforcing the conditions for a set to be a group, highlighting common pitfalls like missing identity elements or zero divisors. The progression to specific MCQs demonstrates how to apply these concepts to find inverses. The instructor uses visual aids like checkmarks and crosses to indicate correctness and performs arithmetic calculations on screen to verify modular results, providing a clear, step-by-step method for students to follow when analyzing group properties.

Loading lesson…