GATE CS 2014 – Set 3 Question

Duration: 8 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 calculus lecture by Yash Jain Sir, focusing on a specific definite integral problem from the GATE CS 2014 exam. The core problem is to evaluate the integral $\int_{0}^{2\pi} |x \sin x| dx$ and express the result in the form $k\pi$ to find the constant $k$. The instructor methodically breaks down the absolute value function by analyzing the sign of the sine function over the interval $[0, 2\pi]$. He splits the integral into two distinct regions where the function behaves differently, applies integration by parts to solve the resulting expressions, and carefully substitutes the limits to arrive at the final numerical answer. This problem serves as a practical application of definite integration techniques and absolute value properties.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card displaying "CALCULUS" amidst a background of complex mathematical formulas. The scene transitions to a whiteboard where the problem is written: "If $\int_{0}^{2\pi} |x \sin x| dx = k\pi$, then the value of $k$ is equal to ______ (GATE CS 2014 (Set 3))". The instructor begins by writing the fundamental definition of the absolute value function on the board: $|x| = egin{cases} x & x \ge 0 \ -x & x < 0 \end{cases}$. He draws a coordinate system to visualize the domain, setting the stage for analyzing the sign of the integrand.

  2. 2:00 – 5:00 02:00-05:00

    The instructor analyzes the sign of $\sin x$ within the integration limits $[0, 2\pi]$. He sketches a graph of the sine wave, noting that $\sin x$ is positive in the interval $[0, \pi]$ and negative in $[\pi, 2\pi]$. Since $x$ is non-negative throughout the entire range, the absolute value $|x \sin x|$ simplifies to $x \sin x$ when $\sin x \ge 0$ and $-x \sin x$ when $\sin x < 0$. He writes this piecewise definition on the board and splits the original integral into two parts: $\int_{0}^{\pi} x \sin x dx + \int_{\pi}^{2\pi} -x \sin x dx$.

  3. 5:00 – 7:38 05:00-07:38

    The instructor solves the integrals using the integration by parts formula, writing "LIATE" on the board to justify choosing $u=x$ and $dv=\sin x dx$. He derives the indefinite integral as $\int x \sin x dx = \sin x - x \cos x$. He then evaluates the definite integrals by substituting the limits. For the first part, $[\sin x - x \cos x]_0^\pi$, he substitutes $\sin \pi = 0$ and $\cos \pi = -1$ to calculate the value as $\pi$. For the second part, $[\sin x - x \cos x]_\pi^{2\pi}$, he substitutes $\sin 2\pi = 0, \cos 2\pi = 1$ and $\sin \pi = 0, \cos \pi = -1$ to calculate the value as $-3\pi$. He combines these results into the expression $\pi - (-3\pi)$, simplifying it to $4\pi$. He circles the final answer $4\pi$, indicating that $k=4$.

The lecture provides a comprehensive walkthrough of solving a definite integral with an absolute value function. The key takeaway is the technique of splitting the integration interval based on the sign changes of the function inside the absolute value. By converting the absolute value problem into a sum of standard integrals and applying integration by parts, the instructor demonstrates a clear and effective method for handling such calculus problems commonly found in engineering entrance exams.

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