Engineering Mathematics Exam Prep: Problems and Solutions

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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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