
Engineering Mathematics Exam Prep
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- Cover
- Half-Title
- Title
- Copyright
- Contents
- Chapter 1: Linear Algebra
- 1.1 Matrices and Their Types
- 1.1.1 Definition of a Matrix
- 1.1.2 Types of Matrices
- 1.2 Algebra of Matrices
- 1.2.1 Negative, Sum, and Differences of Matrices
- 1.2.2 Multiplication of a Matrix by a Scalar
- 1.2.3 Transpose of a Matrix
- 1.2.4 Multiplication of Matrices (Product of Matrices)
- 1.3 Determinant of a Square Matrix
- 1.3.1 Definition of Determinant
- 1.3.2 Properties of a Determinant
- 1.3.3 Minors and Cofactors
- 1.4 Adjoint and Inverse of a Matrix
- 1.4.1 Adjoint of a Matrix
- 1.4.2 Inverse of a Matrix
- 1.5 Various Types of Real Square Matrices
- 1.5.1 Symmetric Matrix
- 1.5.2 Skew-Symmetric Matrix
- 1.5.3 Orthogonal Matrix
- 1.5.4 Idempotent Matrix
- 1.5.5 Involutary Matrix
- 1.5.6 Nilpotent Matrix
- 1.6 Complex Matrices and Their Types
- 1.6.1 Complex Conjugate of a Matrix
- 1.6.2 Transposed Conjugate of a Matrix
- 1.6.3 Unitary Matrix
- 1.6.4 Hermitian Matrix
- 1.6.5 Skew-Hermitian Matrix
- 1.7 Rank of a Matrix
- 1.7.1 Elementary Transformations
- 1.7.2 Equivalent Matrices
- 1.7.3 Rank of a Matrix
- 1.7.4 Determination of the Rank of a Matrix
- 1.8 System of Linear Equations and Their Solutions
- 1.8.1 Introduction
- 1.8.2 Methods for Solving Non-Homogeneous System of Linear Equations
- 1.8.2.1 Cramer's Rule
- 1.8.2.2 Matrix Method
- 1.8.2.3 Rank Method
- 1.8.3 Homogeneous System of Linear Equations
- 1.9 Eigenvalues and Eigenvectors
- 1.9.1 Characteristic Roots (Eigenvalues) of a Matrix
- 1.9.2. Trace of a Matrix
- 1.9.3. Eigenvectors or Characteristic Vectors
- 1.10 Vectors
- 1.10.1 Introduction
- 1.10.2 Linear Dependence and Linear Independence
- 1.10.3 Inner Product and Norm of Vectors
- 1.10.4 Orthogonal and Orthonormal Vectors
- 1.10.5 Basis and Dimension
- Fully Solved MCQs (Level-I)
- Answer Key
- Explanation
- Fully Solved MCQs (Level-II)
- Answer Key
- Explanation
- Previous Years Solved Papers (2000-2018)
- Answer Key
- Answer Key
- Explanation
- Questions for Practice
- Answer Key
- Hints
- Chapter 2: Calculus
- 2.1 Functions and Limits
- 2.1.1 Definition of a Function
- 2.1.2 Some Special Functions
- 2.1.3 Introduction to Limits
- 2.1.4 Definition of Limit
- 2.1.5 Fundamental Theorems on Limits
- 2.1.6 Fundamental Formulas on Limits
- 2.1.7 The Sandwich Theorem
- 2.1.8 Infinite Limits
- 2.1.9 Limits at Infinity
- 2.1.10 Infinite Limits at Infinity
- 2.2 Continuity and Differentiability
- 2.2.1 Continuity
- 2.2.2 Discontinuity
- 2.2.3 Derivative
- 2.2.4 Computation of Derivatives
- 2.3 Indeterminate Forms
- 2.3.1 Introduction
- 2.3.2 The L'Hospital Rule
- 2.4 Mean Value Theorems
- 2.4.1 Rolle's Theorem
- 2.4.2 Lagrange's Mean Value Theorem
- 2.4.3 Cauchy's Mean Value Theorem
- 2.5 Increasing and Decreasing Functions
- 2.6 Maxima and Minima of Functions of a Single Variable
- 2.6.1 First Derivative Test
- 2.6.2 Second Derivative Test
- 2.6.3 Higher Order Derivative Test
- 2.7 Infinite series and Expansion of Functions
- 2.7.1 Infinite Series
- 2.7.2 Test for Convergence of Infinite Series
- 2.7.3 Taylor's Theorem With Lagrange's Form of Remainder
- 2.7.4 The Taylor Series
- 2.7.5 Maclaurin's Series
- 2.8 Indefinite and Definite Integrals
- 2.8.2 Fundamental Formulas of Indefinite Integral
- 2.8.3 Advanced Formulas of Indefinite Integrals
- 2.8.4 Definite Integral
- 2.8.5 Properties of Definite Integral
- 2.8.6 Definite Integral as a Limit of Sum
- 2.8.7 Differentiation Under the Sign of Integration
- 2.9 Improper Integrals, Beta, and Gamma Functions
- 2.9.1 Improper Integral
- 2.9.2 Evaluation of Improper Integrals
- 2.9.3 Beta Function
- 2.9.4 Gamma Function
- 2.10 Functions of Several Variables and Partial Derivatives
- 2.10.1 Functions of Two Variables
- 2.10.2 Limit of Functions of Two Variables
- 2.10.3 Continuity of Functions of Two Variables
- 2.10.4 Partial Derivatives
- 2.10.5 Homogeneous Function
- 2.10.6 Euler's Theorem
- 2.10.7 Total Differential and Total Derivative
- 2.10.8 Jacobian
- 2.11 Maxima and Minima of Functions of two Variables
- 2.11.1 Introduction
- 2.11.2 Working Rule to Find the Maximum and Minimum Values of f(x, y)
- 2.11.3 Lagrange's Method for Undetermined Multipliers
- 2.12 Change of Order of Integration
- 2.13 Double and Triple Integrals
- 2.13.1 Double Integrals
- 2.13.2 Triple Integrals
- 2.14 Arc Length of a Curve
- 2.15 Volumes of Solids of Revolution
- 2.15.1 Working Formulas
- 2.16 Surface Areas of Solids of Revolution
- Fully Solved MCQs (Level-I)
- Answer Key
- Explanation
- Fully Solved MCQs (Level-II)
- Answer Key
- Explanation
- Previous Years Solved Papers (2000-2018)
- Answer Key
- Explanation
- Questions for Practice
- Answer Key
- Explanation
- Chapter 3: Vectors
- 3.1 Basic Concepts
- 3.1.1 Scalars and Vectors
- 3.1.2 Position Vector
- 3.1.3 Equal Vectors
- 3.1.4 Negative of a Vector
- 3.1.5 Unit Vectors
- 3.1.6 Sum and Difference of Two Vectors
- 3.1.7 Triangle Law of Addition
- 3.1.8 Product of a Vector with a Scalar
- 3.1.9 Collinear Vectors
- 3.1.10 Coplanar Vectors
- 3.1.11 Section Formula
- 3.2 Product of Vectors
- 3.2.1 Scalar Product (Dot Product)
- 3.2.2 Vector Product (Cross Product)
- 3.2.3 Scalar Triple Product
- 3.2.4 Vector Triple Product
- 3.3 Vector Differentiation and Integration
- 3.3.1 Derivative of a Vector Function
- 3.3.2 General Rules for Vector Differentiation
- 3.3.3 Velocity and Acceleration
- 3.3.4 Vector Integration
- 3.4 Gradient, Divergence and Curl
- 3.4.1 Del Operator
- 3.4.2 Gradient of a Scalar Point Function
- 3.4.3 Divergence of a Vector Point Function
- 3.4.4 Curl of a Vector Point Function
- 3.4.5 Vector Identities
- 3.4.6 Directional Derivative
- 3.5 Line, Surface, and Volume Integrals
- 3.5.1 Line Integral
- 3.5.2 Surface Integral
- 3.5.3 Volume Integral
- 3.6 Green's, Stokes', and Gauss Divergence Theorem
- 3.6.1 Greens Theorem (in a Plane)
- 3.6.2 Stokes' Theorem
- 3.6.3 Gauss Divergence Theorem
- Fully Solved MCQs
- Answer Key
- Explanation
- Previous Years Solved Papers (2000-2018)
- Answer Key
- Explanation
- Questions for Practice
- Answer Key
- Chapter 4: Ordinary Differential Equations
- 4.1 Basic Concepts
- 4.1.1 Definition of a Differential Equation
- 4.1.2 Classification of Differential Equations
- 4.1.3 Order of a Differential Equation
- 4.1.4 Degree of a Differential Equation
- 4.1.5 Formation of a Differential Equation
- 4.1.6 Solution of a Differential Equation
- 4.2 Linearly Dependent and Linearly Independent Solutions
- 4.2.1 Wronskian
- 4.2.2 Linearly Dependent Solutions
- 4.2.3 Linearly Independent Solutions
- 4.3 Differential Equations of 1st Order and 1st Degree
- 4.3.1 General Form
- 4.3.2 Solution by Separation of Variables
- 4.3.3 Homogeneous Differential Equation
- 4.3.4 Exact Differential Equations
- 4.3.5 Linear Differential Equations
- 4.4 Linear Differential Equations of 2nd Order
- 4.4.1 General Form
- 4.4.2 Complementary Function (C.F)
- 4.4.3 Particular Integral (P.I)
- 4.4.4 Complete (General) Solution
- 4.4.5 Homogeneous Linear Differential Equations of Order Two
- Fully Solved MCQs
- Answer Key
- Explanations
- Fully Solved MCQs
- Answer Key
- Explanations
- Previous Years Questions (2000-18)
- Answer Key
- Explanations
- Questions For Practice
- Answer Key
- Hints
- Chapter 5: Partial Differential Equations
- 5.1 Basic Concepts
- 5.1.1 Introduction
- 5.1.2 Order and Degree
- 5.1.3 Linear and No-Linear Partial Differential Equations
- 5.1.4 Formation of Partial Differential Equations
- 5.2 Classification of 2nd Order Partial Differential Equation
- 5.3 Heat, Wave, and Laplace Equations
- 5.3.1 Solution by Separation of Variables
- 5.3.2 One-Dimensional Heat (Diffusion) Equation and Its Solution
- 5.3.3 One-Dimensional Wave Equation and Its Solution
- 5.3.4 The Laplace Equation and Its Solution
- Fully Solved MCQs
- Answer key
- Explanation
- Fully Solved MCQs
- Answer key
- Explanation
- Previous Years Solved Papers (2000-2018)
- Answer Key
- Explanation
- Questions for Practice
- Answer Key
- Chapter 6: Laplace Transforms
- 6.1 Basics of Laplace Transforms
- 6.1.1 Definition of the Laplace Transform
- 6.1.2 Linear Property of the Laplace Transform
- 6.1.3 Fundamental Formulas of the Laplace Transform
- 6.1.4 First Shifting Theorem
- 6.1.5 Some Advanced Formulas of the Laplace Transform
- 6.1.6 Change of Scale Property
- 6.2 Laplace Transform on Derivatives
- 6.3 Laplace Transform on Integrals
- 6.4 Laplace Transform on Periodic Functions
- 6.5 Evaluation of Integrals Using Laplace Transforms
- 6.6 Initial and Final Value Theorems
- 6.6.1 Initial Value Theorem
- 6.6.2 Final Value Theorem
- 6.7 Fundamentals of Inverse Laplace Transform
- 6.7.1 Definition of Inverse Laplace Transform
- 6.7.2 Useful Formulas on Inverse Laplace Transforms
- 6.8 Important Theorems on Inverse Laplace Transforms
- 6.9 Unit Step Function and Unit Impulse Function
- 6.9.1 Unit Step Function
- 6.9.2 Second Shifting Theorem
- 6.9.3 Unit Impulse Function
- 6.10 Solving Ordinary Differential Equations
- Fully Solved MCQs (Level-I)
- Answer key
- Explanation
- Fully Solved MCQs (Level-II)
- Answers key
- Explanation
- Previous Years Questions (2000-2018)
- Answers key
- Explanations
- Questions for Practice
- Answers key
- Explanation
- Chapter 7: Numerical Analysis
- 7.1 Errors and Approximations
- 7.1.1 Rounding Off
- 7.1.2 Errors and Their Computation
- 7.2 Calculus of Finite Differences
- 7.2.1 Forward Difference Operator
- 7.2.2 Backward Difference Operator
- 7.2.3 Shift Operator
- 7.3 Interpolation
- 7.3.1 Newton's Forward Difference Interpolation Formula
- 7.3.2 Newton's Backward Difference Interpolation Formula
- 7.3.3 Lagrange's Interpolation Formula
- 7.3.4 Error in Interpolation
- 7.4 Numerical Differentiation
- 7.4.1 Differentiation Formula Based on Newton's Forward Difference Formula
- 7.4.2 Differentiation Formula Based on Newton's Backward Difference Formula
- 7.5 Numerical Integration
- 7.5.1 Trapezoidal Rule
- 7.5.2 Simpson's 1/3rd Rule
- 7.5.3 Weddle's Rule
- 7.5.4 Simpson's 3/8th's Rule
- 7.6 System of Linear Algebraic Equations
- 7.6.1 Gauss Elimination Method
- 7.6.2 LU Decomposition Method
- 7.6.3 Gauss-Seidel Iteration Method
- 7.7 Solution of Algebraic and Transcendal Equations
- 7.7.1 Method of Bisection
- 7.7.2 Regula Falsi Method
- 7.7.3 Newton-Raphson Method
- 7.8 Numerical Solution of Ordinary Differential Equations
- 7.8.1 Euler's Method
- 7.8.2 Modified Euler's Method
- 7.8.3 Runge-Kutta Method
- I. Second-Order Runge-Kutta Method
- II. Fourth-Order Runge-Kutta Method
- 7.8.4 Predictor-Corrector Method
- Fully Solved MCQs
- Answer Key
- Explanation
- Previous Years Solved Papers (2000-2018)
- Answer Key
- Explanation
- Questions for Practice
- Answer Key
- Chapter 8: Complex Analysis
- 8.1 Basics of Complex Analysis
- 8.1.1 Complex Number
- 8.1.2 Modulus and Amplitude of a Complex Number
- 8.1.3 Conjugate of a Complex Number
- 8.1.4 Properties of Modulus, Argument, and Conjugate
- 8.1.5 Sum, Difference, and Product of Two Complex Numbers
- 8.1.6 Cube Roots of Unity
- 8.1.7 De Moivre's Theorem
- 8.1.8 Hyperbolic Functions
- 8.1.9 Logarithm of a Complex Number
- 8.2 Calculus of Complex Valued Functions
- 8.2.1 Function of a Complex Variable
- 8.2.2 Limit of a Complex Valued Function
- 8.2.3 Continuity of a Complex Valued Function
- 8.2.4 Derivative of a Complex Valued Function
- 8.2.5 Analytic Function
- 8.2.6 Cauchy Riemann Equations
- 8.2.7 Conjugate Function
- 8.2.8 Harmonic Function
- 8.2.9 Construction of an Analytic Function (by Milne Thomson's method)
- 8.2.10 Construction of Harmonic Conjugate
- 8.3 Complex Integration
- 8.3.1 Curves
- 8.3.2 Complex Line Integral
- 8.3.3 Cauchy-Goursat Theorem
- 8.3.4 Cauchy's Integral Formula
- 8.3.5 Cauchy's Integral Formula on HigherOrder Derivatives
- 8.4 Taylor and Laurent Series
- 8.4.1 The Taylor Series
- 8.4.2 The Laurent Series
- 8.5 Singularities
- 8.5.1 Singular Point
- 8.5.2 Types of Singularities
- 8.5.2.1 Isolated singularity
- 8.5.2.2 Removable singularity
- 8.5.2.3 Essential singularity
- 8.5.3 Zeros and Poles
- 8.6 Residues
- 8.6.1 Residue at a Simple Pole
- 8.6.2 Residue at a Pole of Order "n"
- 8.6.3 Residue at Infinity
- 8.6.4 Cauchy's Residue Theorem
- Fully Solved MCQ's
- Answer Key
- Explanation
- Fully Solved MCQ's (Level-II)
- Answer Key
- Explanation
- Previous Years Solved Papers (2000-2018)
- Answer Key
- Explanation
- Questions for Practice
- Answer Key
- Explanation
- Chapter 9: Probability and Statistics
- 9.1 Basics of Probability
- 9.1.1 Experiment
- 9.1.2 Random Experiment
- 9.1.3 Sample Space (Event Space)
- 9.1.4 Event
- 9.1.5 Equally Likely Events
- 9.1.6 Mutually Exclusive Events
- 9.1.7 Mutually Exhaustive Events
- 9.1.8 Classical Definition Of Probability
- 9.1.9 Independent Events
- 9.2 Conditional Probability and Bayes' Theorem
- 9.2.1 Conditional Probability
- 9.2.2 Theorem on Total Probability
- 9.2.3 Bayes' Theorem
- 9.3 Random Variable and Probability Distribution
- 9.3.1 Random Variable
- 9.3.2 Types of Random Variable
- 9.3.3 Probability Mass Function (P.M.F)
- 9.3.4 Probability Distribution Function
- 9.3.5 Expectation or Mean
- 9.3.6 Variance and Standard Deviation
- 9.4 Special Types of Probability Distributions
- 9.4.1 Binomial Distribution
- 9.4.2 Poisson Distribution
- 9.4.3 Normal Distribution
- 9.4.4 Geometric Distribution
- 9.4.5 Uniform (Rectangular) Distribution
- 9.4.6 Gamma Distribution
- 9.4.7 Exponential Distribution
- 9.5 Introduction of Statistics
- 9.5.1 Statistics
- 9.5.2 Scopes and limitations of Statistics
- 9.5.3 Frequency Distribution
- 9.5.4 Mean (Arithmetic Mean)
- 9.5.5 Median
- 9.5.6 Mode
- 9.5.7 Standard Deviation (S.D)
- 9.5.8 Correlation
- 9.5.9 Regression
- Fully Solved MCQs
- Answers Key
- Explanations
- Fully Solved MCQs
- Answers Key
- Explanations
- Previous Years Solved Papers (2000-2018)
- Answer Key
- Explanation
- Questions For Practice
- Answer Key
- Explanations
- Chapter 10: Fourier Series
- 10.1 Basics of the Fourier Series
- 10.1.1 Definition of the Fourier Series
- 10.1.2 Dirichlet's Condition
- 10.2 Fourier Series of Even and Odd Functions
- 10.2.1 Fourier Series of Even Function
- 10.2.2 Fourier Series for Odd Function
- 10.3 Half Range Fourier Series
- 10.3.1 Half Range Sine Series
- 10.3.2 Fourier Cosine Series
- Fully Solved MCQs
- Answer Key
- Explanation
- Previous Years Solved Papers (2000-2018)
- Answer Key
- Explanation
- Questions for Practice
- Answer Key
- Explanation
- Chapter 11: Graph Theory
- 11.1 Graphs
- 11.1.1 Definition of a Graph
- 11.1.2 Incidence
- 11.1.3 Loops
- 11.1.4 Parallel Edges
- 11.1.5 Degree of a Vertex
- 11.1.6 Directed Graph
- 11.1.7 In Degree and Out Degree
- 11.1.8 Minimum Degree and Maximum Degree
- 11.2 Different Types of Graphs
- 11.2.1 Mixed Graph
- 11.2.2 Multi Graph
- 11.2.3 Simple Graph
- 11.2.4 Trivial Graph
- 11.2.5 Null Graph
- 11.2.6 K-regular Graph
- 11.2.7 Complete Graph
- 11.2.8 Bipartite Graph
- 11.2.9 Complete Bipartite Graph
- 11.3 Walk and Path
- 11.3.1 Walk
- 11.3.2 Path and Circuit
- 11.4 Matrix Representation of Graphs
- 11.4.1 Adjacent Matrix
- 11.4.2 Incidence Matrix
- 11.4.3 Path Matrix
- 11.5 Planar Graphs and Euler's Formula
- 11.5.1 Planar Graph
- 11.5.2 Euler's Formula
- 11.6 Sub Graphs and Isomorphic Graphs
- 11.6.1 Subgraphs
- 11.6.2 Isomorphic Graph
- 11.7 Connectedness
- 11.7.1 Connected Graph
- 11.7.2 Strongly Connected Graph
- 11.7.3 Weakly Connected Graph
- 11.7.4 Component
- 11.7.5 Eulerian Graph
- 11.7.6 Hamiltonian Graph
- 11.8 Vertex and Edge Connectivity
- 11.8.1 Cut Vertex
- 11.8.2 Cut Edge (Bridge)
- 11.8.3 Cut Set
- 11.8.4 Edge Connectivity
- 11.8.5 Vertex Connectivity
- 11.9 Graph Coloring, Matching, and Covering
- 11.9.1 Vertex Coloring
- 11.9.2 Chromatic Number
- 11.9.3 Matching
- 11.9.4 Covering
- 11.10 Tree
- 11.10.1 Definition
- 11.10.2 Spanning Tree
- Construction of Spanning Trees
- (I) BFS (Breath First Search) Algorithm
- (II) DFS (Depth First Search) Algorithm
- 11.10.3 Minimal Spanning Tree
- (I) Prim's Algorithm
- (II) Kruskal's Algorithm
- 11.10.4 Binary Tree
- 11.10.5 Rooted Tree
- 11.10.6 Traversal of a Tree
- Fully Solved MCQs
- Answer Key
- Explanation
- Previous Years Solved Paper (2000-2018)
- Answer Key
- Explanation
- Questions for Practice
- Answer Key
- Hints
- Appendix A: Gate 2019 Solved Papers
- Appendix B: Gate 2020 Solved Papers
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