Practice Questions

Duration: 9 min

This video lesson is available to enrolled students.

Return to /learn/SSC-JE-IMD-SCIENTIFIC-ASSISTANT-COMPUTER-SCIENCE/engineering-mathematics/group-theory/classification-of-finite-and-infinite-groups/asset-practice-questions-2335 after enrolling

Inside: a video lesson and guided study material.

Module outline

  1. General Intelligence & Reasoning: Analogy, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Logical Sequence Of Words, Syllogisms, Venn Diagram, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Data Sufficiency, Puzzle Solving (Puzzles), Data Arrangement, Cubes & Dices, Calendar, Clock, Arithmetic, Number System, HCF LCM, Verbal Analogy (Word Based Analogy), Visual Recognition and Figure Completion, Paper Folding and Cutting, Transformation of Shapes, Clocks and Calendars, Direction Sense, Cubes and Dice, Patterns in Two-Dimensional Figures, Seating Arrangements, Analogy and Classification, Logical Deduction and Induction, Alphabet and Word Tests, Mathematical Operations, Series Completion, Simplification and Approximation, Coding and Decoding, Numerical Reasoning, Assembling and Grouping of Shapes, Three-Dimensional Patterns and Visualization, Vocabulary, Number System, Averages, Mixtures and Alligation, Simple and Compound Interest, Percentages, Algebra, Speed, Time and Distance, Seating and Arrangement Puzzles, Traverse Surveying, Order, Ranking and Sequence, Constraint Puzzles, Syllogisms and Venn Diagrams, Reading Comprehension, Critical Reasoning, Permutations and Combinations, Narrative Sequencing, Project Monitoring, Resources and Safety, Profit, Loss and Discount, Ratio and Proportion, Time and Work, Mensuration, Data Interpretation, Powers, Surds and Logarithms, Sequences and Series, Articles and Determiners, Sentence Completion, Data Sufficiency, Blood Relations, Simple Interest and Compound Interest, Mixture and Alligation, Sequence and Series (Series), Speed Time and Distance, Algebra, Profit and Loss, Powers and Exponents (Surds and Indices), Permutation and Combination
  2. General Awareness: Physics, Chemistry, Biology, History, Geography, Polity, Indian Economy, Environment, Static General Knowledge, GK- Current Affairs, Artificial Intelligence, MS Office Applications, Terms & Abbreviations, Keyboard Shortcuts, Evolution of Indian Banking (Banking & Financial Awareness), Emerging Tech Trends
  3. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus, Probability, Statistics, Signal Space and Optimum Detection, Probability, Statistics
  4. Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep, Number Systems and Codes, Registers and Counters, Boolean Algebra and Logic Gates, Microprocessors and Microcontrollers, Logic Families and CMOS Logic, Microprocessors, Logic Gates and Boolean Algebra, Minimization of Boolean Functions, Timing, Hazards and Testing, Logic Families, Combinational Logic Design, Arithmetic Circuits, Waveform Generators and Timer Circuits, Semiconductor Memories and Programmable Logic Devices, Combinational Logic Circuits, Latches and Flip-Flops, Finite State Machines, Sequential Circuits and State Machines, Minimization of Boolean Functions, Arithmetic Circuits, Latches and Flip-Flops, Number Systems and Codes, Registers and Counters
  5. Computer Organisation & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer, Register Transfer and Microoperations, Semiconductor Memories and Programmable Logic, Processor Organization and Instructions, Memory Hierarchy and Input/Output, Registers and Counters
  6. Programming & Data Structures: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, HTML Basics, OOPS with C++, Java, Object Oriented Technology
  7. Databases: 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, Database Security & Authorization
  8. Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  9. 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
  10. 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
  11. Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Security, Windows OS, Distributed Systems, Unix OS, Linux OS
  12. Computer Networks: Introduction to CN, 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, Data Communication, Website, Browser & Services, Web Design & Development, Network Security, Cyber Threats and Malware Attacks, Cyber Safety and Security
  13. Full Length Mock Tests:
  14. Sectional Tests:
  15. 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.

Loading lesson…