GATE CS 2014 – Set 1 – 1 Mark Question

Duration: 7 min

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Module outline

About Course: About the Course
General Aptitude & Reasoning: Number System, Percentage, Ratio and Proportion (Ratios), Average, Profit and Loss, Simple Interest and Compound Interest, Time and Work, Speed Time and Distance, Mixture and Alligation, Data Interpretation, Algebra, Probability, Permutation and Combination, Venn Diagram, HCF LCM, Divisibility Rules, Age Problems, Pipe and Cistern, Partnership, Linear Algebra, Sequence and Series (Series), Statistics, Powers and Exponents (Surds and Indices), Arithmetic, Mensuration and Geometry, Calculus, Trigonometry, Vectors, Speed Mathematics and Basic Mathematics, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Analogy, Series (Number and Letter Series) (Numerical Relations and Reasoning), Puzzle Solving (Puzzles), Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Calendar, Seating Arrangements, Coding Decoding, Blood Relations, Directions (Direction Test), Clock, Analytical Reasoning (Counting Figures Reasoning), Ranking, Order and Sequence, Logical Sequence Of Words, Inequality, Data Sufficiency, Cubes & Dices
Computer Networks & Cybersecurity: 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 Security
Programming, Data Structures & Algorithms: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, OOPS with C++, 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, Advanced Algorithms
AI, Machine Learning, Big Data & Cloud: Emerging Tech Trends, Artificial Intelligence, Big Data Systems, Approaches to AI, Search Algorithms, Knowledge Representation, Cloud Computing
Digital Logic & Computer Organization: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Basics of COA, Floating Point Rep, Cache Memory Organization, Input Output Organisation, Pipelining, Instr Formats & Modes, Register Transfer and Microoperations, Control Unit Design, Programming the Basic Computer
TOC & Compiler: 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, 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
Discrete Mathematics: Set Theory, Relations, Functions, Graph Theory, Group Theory, Propositional and Predicate Logic
Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS, Linux OS, Distributed Systems
Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management, Object Oriented Technology, HTML Basics, JavaScript-Basics, Java, XML
Database Management System: 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, Data Warehousing & Mining, Enhanced Data Models, NoSQL
Other Recurring CS Topics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, Multimedia
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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

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.
2:00 – 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.
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.