GATE CS 2014 – Set 1 – 1 Mark Question

Duration: 7 min

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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.

The video is an educational lecture by Yash Jain Sir from Knowledge Gate Educator, focusing on a calculus problem from the GATE CS 2014 exam. The problem defines a function f(theta) as a 3x3 determinant involving trigonometric terms. The instructor analyzes the properties of this function over the interval (pi/6, pi/3) to determine the truth of two statements regarding its derivative f'(theta). The core of the solution relies on evaluating the function at the endpoints of the interval and applying Rolle's Theorem, followed by determining if the function is constant or not to address the second statement.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a 'CALCULUS' title card before displaying a mathematical problem on a digital whiteboard. The problem defines a function f(theta) as a determinant with sin theta, cos theta, and tan theta in the first row, and fixed values for pi/6 and pi/3 in the subsequent rows. The variable theta is restricted to the interval (pi/6, pi/3). The instructor, Yash Jain Sir, asks the viewer to identify true statements regarding the derivative f'(theta). Statement (I) claims there exists a theta where f'(theta) = 0, and Statement (II) claims there exists a theta where f'(theta) != 0. He notes this is a 1-mark question from GATE CS 2014. He writes the interval endpoints as a and b and explicitly writes 'Rolle's theorem' on the board, signaling the method to be used.

  2. 2:00 – 5:00 02:00-05:00

    The instructor begins solving by testing the boundary conditions required for Rolle's Theorem. He substitutes theta = pi/6 into the function. He points out that the first row becomes sin(pi/6), cos(pi/6), tan(pi/6), which is exactly the same as the second row. He explains the determinant property that if two rows are identical, the value is zero, so f(pi/6) = 0. He repeats this logic for theta = pi/3, where the first row matches the third row, resulting in f(pi/3) = 0. He writes f(a) = f(b) = 0 on the board. To visualize this, he draws a graph showing a curve starting and ending at the same height, illustrating the geometric interpretation of Rolle's Theorem. He also draws a horizontal line labeled 'constant' and writes f'(theta) = 0, considering the possibility that the function might be constant throughout the interval.

  3. 5:00 – 7:06 05:00-07:06

    The instructor finalizes the solution by addressing both statements. He confirms that since f(a) = f(b) = 0 and the function is continuous and differentiable, Rolle's Theorem guarantees at least one point c where f'(c) = 0, validating Statement (I). He then refutes the idea that the function is constant. He explains that expanding the determinant yields a non-zero expression involving sin theta, cos theta, and tan theta, meaning the function varies. Because the function is not constant, its derivative cannot be zero everywhere, which implies there must be points where f'(theta) != 0, validating Statement (II). He circles the option 'Both I and II' and marks it as the correct answer. The video concludes with a black screen displaying 'THANKS FOR WATCHING' in white text.

The lecture demonstrates a classic application of Rolle's Theorem in the context of determinants. By evaluating the function at the boundaries, the instructor establishes the necessary condition f(a) = f(b) to prove the existence of a critical point. The second part of the problem requires recognizing that satisfying the boundary condition does not imply the function is constant, thereby ensuring the derivative is non-zero at other points. This dual analysis confirms both statements are true.

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