Gate CS 2014 - Set 1 - 2 Marks Question

Duration: 6 min

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Module outline

  1. About Course: About the Course
  2. Paper-2 | 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
  3. Paper-2 | Databases / DBMS & SQL: 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
  4. Paper-2 | Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos
  5. Paper-2 | Computer Networks & Security: 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 Access Technologies, Network Security
  6. Paper-2 | Digital Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Floating Point Rep
  7. Paper-2 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management
  8. 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
  9. Paper-2 | Computer Organization & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Control Unit Design, Programming the Basic Computer
  10. 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
  11. Paper-1 | General Awareness: Polity, Physics, Biology, Chemistry, Static General Knowledge, GK- Current Affairs, History, Geography, Indian Economy, Environment
  12. Paper-1 | Logical Reasoning: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Data Interpretation, Blood Relations, Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Selection Decision Table (Decision Making), Clock, Calendar, Cubes & Dices, Directions (Direction Test), Analogy, Seating Arrangements, Analytical Reasoning (Counting Figures Reasoning)
  13. Paper-1 | Quantitative Aptitude: Ratio and Proportion (Ratios), Mixture and Alligation, Permutation and Combination, Number System, HCF LCM, Speed Time and Distance, Average, Age Problems, Percentage, Profit and Loss, Time and Work, Pipe and Cistern, Simple Interest and Compound Interest, Powers and Exponents (Surds and Indices), Mensuration and Geometry, Probability, Statistics, Arithmetic, Algebra
  14. Paper-1 | English: Vocabulary, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Sentence Construction, Cloze Test (Cloze Passage)
  15. Coal India Previous Year Paper:
  16. Engineering Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic, Linear Algebra, Calculus
  17. Coal India Mock Test:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video features a lecture by Yash Jain Sir from Knowledge Gate Educator, focusing on a calculus problem from the GATE CS 2014 exam. The problem asks to identify a true statement about a continuous function f(x) defined on the interval [0, 2], given specific values: f(0) = -1, f(2) = -1, and f(1) = 1. The instructor systematically analyzes the four multiple-choice options, ultimately proving that Option A is correct using the Intermediate Value Theorem. He constructs an auxiliary function to demonstrate the existence of a specific point y within the interval (0, 1) that satisfies the condition f(y) = f(y+1).

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video begins with the instructor presenting the problem statement clearly on the screen. The text reads: 'Q. A function is continuous in the interval [0, 2]. It is known that f(0) = f(2) = -1 and f(1) = 1. Which one of the following statements must be true. (GATE CS 2014) (SET-1) (2 MARKS)'. He points to the options A, B, C, and D. He writes 'f(x) [0, 2]' on the whiteboard to establish the domain of the function. He emphasizes the continuity condition and the specific function values provided in the question.

  2. 2:00 – 5:00 02:00-05:00

    The instructor begins analyzing the options, specifically targeting Option A: 'There exist a y in the interval (0, 1), such that f(y) = f(y+1)'. He decides to test this statement by defining a new function, which he labels g(y). He writes the equation 'g(y) = f(y) - f(y+1)' on the board. He explains that since f(x) is continuous, the function g(y) must also be continuous. He writes 'f(x) -> con' and 'g(x) -> con' to denote continuity. He notes that the domain for y is (0, 1), which implies y+1 is in (1, 2), keeping the function within the original domain [0, 2].

  3. 5:00 – 6:26 05:00-06:26

    The instructor proceeds to evaluate the function g(y) at the boundaries of the interval (0, 1). He calculates g(0) by substituting y=0 into the equation: 'g(0) = f(0) - f(1) = -1 - 1 = -2'. Next, he calculates g(1): 'g(1) = f(1) - f(2) = 1 - (-1) = +2'. He draws a coordinate system with the y-axis representing the function value and the horizontal axis representing y. He marks -2 at y=0 and +2 at y=1. He draws a curve connecting these points, showing it must cross the x-axis (where g(y)=0) somewhere between 0 and 1. He concludes that since g(0) is negative and g(1) is positive, by the Intermediate Value Theorem, there exists a y in (0, 1) such that g(y) = 0. This implies f(y) - f(y+1) = 0, or f(y) = f(y+1), confirming Option A is the correct answer.

The lecture effectively demonstrates a problem-solving strategy for calculus questions involving continuous functions. By defining an auxiliary function g(y) = f(y) - f(y+1), the instructor transforms the problem of finding a point where two function values are equal into a root-finding problem. The calculation of g(0) = -2 and g(1) = +2 provides the necessary sign change to apply the Intermediate Value Theorem. This guarantees the existence of a solution y in the interval (0, 1), validating the statement in Option A. The visual aids, including the board writing and graph sketching, reinforce the logical steps taken to reach the conclusion.

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