GATE CS 2011 – 2 Marks Question

Duration: 11 min

This video lesson is available to enrolled students.

Return to /learn/SSC-JE-IMD-SCIENTIFIC-ASSISTANT-COMPUTER-SCIENCE/engineering-mathematics/calculus/integration/asset-gate-cs-2011-2-marks-question 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:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

The video presents a solution to a definite integral problem from GATE CS 2011 involving complex numbers. The integral is $\int_{0}^{\pi/2} rac{\cos x + i \sin x}{\cos x - i \sin x} dx$. The instructor demonstrates two distinct methods to solve it. Method 1 utilizes Euler's formula to convert trigonometric functions into exponential form, simplifying the integrand to $e^{2ix}$. Method 2 involves rationalizing the denominator using the conjugate, applying double-angle identities, and integrating the resulting real and imaginary parts separately. Both methods yield the final answer $i$.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The instructor introduces the problem statement: "Given $i = \sqrt{-1}$, what will be the evaluation of the definite integral $\int_{0}^{\pi/2} rac{\cos x + i \sin x}{\cos x - i \sin x} dx$?" He identifies it as a GATE CS 2011 question worth 2 marks. He begins Method 1 by writing down Euler's formulas on the right side of the board: $e^{i heta} = \cos heta + i \sin heta$ and $e^{-i heta} = \cos heta - i \sin heta$. He then starts substituting these into the integral expression, pointing to the numerator and denominator to show the correspondence with the exponential forms. He writes the integral with the exponential substitution $\int_{0}^{\pi/2} rac{e^{ix}}{e^{-ix}} dx$. The text "Method 1" is visible in the top left corner.

  2. 2:00 – 5:00 02:00-05:00

    The instructor simplifies the integrand using the exponential forms, writing $ rac{e^{ix}}{e^{-ix}} = e^{2ix}$. He sets up the integral $\int_{0}^{\pi/2} e^{2ix} dx$ and finds the antiderivative $\left[ rac{e^{2ix}}{2i} ight]_0^{\pi/2}$. He evaluates the limits, substituting $x = \pi/2$ to get $e^{i\pi}$ and $x=0$ to get $e^0$. He writes $e^{i\pi} = \cos \pi + i \sin \pi = -1$. The calculation proceeds to $ rac{-1 - 1}{2i} = rac{-2}{2i} = rac{-1}{i}$. He rationalizes this to get the final result $i$. He explicitly writes out the steps $ rac{-1}{i} imes rac{i}{i} = rac{-i}{i^2} = rac{-i}{-1} = i$. The board shows the full integration process from start to finish.

  3. 5:00 – 10:00 05:00-10:00

    The instructor switches to Method 2. He writes the original integral and multiplies the numerator and denominator by the conjugate $(\cos x + i \sin x)$. He expands the numerator to $\cos^2 x + i^2 \sin^2 x + 2 \cos x \cdot i \sin x$. He simplifies the denominator to $\cos^2 x - i^2 \sin^2 x = \cos^2 x + \sin^2 x = 1$. He uses the identities $\cos^2 x - \sin^2 x = \cos 2x$ and $2 \sin x \cos x = \sin 2x$ to rewrite the integrand as $\cos 2x + i \sin 2x$. He then integrates this expression: $\int_{0}^{\pi/2} \cos 2x dx + i \int_{0}^{\pi/2} \sin 2x dx$. He evaluates the antiderivatives $\left[ rac{\sin 2x}{2} ight]_0^{\pi/2} - i \left[ rac{\cos 2x}{2} ight]_0^{\pi/2}$. The text "Method 2" is visible in the top left corner.

  4. 10:00 – 10:32 10:00-10:32

    The instructor completes the evaluation for Method 2. He substitutes the limits into the trigonometric expressions: $\sin \pi - \sin 0$ and $\cos \pi - \cos 0$. He calculates the values: $\sin \pi = 0$, $\sin 0 = 0$, $\cos \pi = -1$, $\cos 0 = 1$. The expression simplifies to $0 - i rac{-1 - 1}{2} = -i rac{-2}{2} = i$. This confirms the result obtained in Method 1. The video concludes with a "THANKS FOR WATCHING" screen.

The video provides a comprehensive tutorial on evaluating a complex definite integral using two distinct mathematical approaches. Method 1 leverages the power of Euler's formula to transform trigonometric functions into exponential form, significantly simplifying the integration process. Method 2 relies on algebraic manipulation, specifically rationalizing the denominator and applying double-angle trigonometric identities. Both methods converge on the same result, $i$, reinforcing the consistency of complex number operations. This dual-method approach is valuable for students preparing for competitive exams like GATE, as it demonstrates flexibility in problem-solving strategies and deepens understanding of complex analysis concepts.

Loading lesson…