Unit – I:
SELF-AWARENESS(WHO) & POSITIVE THINKING(UNICEF)
Unit – II:
EMPATHY
Unit – III:
CRITICAL & CREATIVE THINKING
Unit – IV:
Part of Speech
4.1 Articles
4.2 Noun
4.3 Pronoun
4.4 Verb
4.5 Adverb
4.6 Adjective
4.7 Preposition
Unit – V:
Paragraph and Essay Writing
5.1 Descriptive
5.2 Expository
5.3 Persuasive
5.4 Narrative
Reading Comprehension
Unit I
Reciprocal Equations-Standard form–Increasing or decreasing the roots of a given equation- Removal of terms–Approximate solutions of roots of polynomials by Horner’s method – Related Problems.
Unit II
Summation of Series: Binomial– Exponential –Logarithmic series (Theorems without proof) –Related Problems.
Unit III
Characteristic equation – Eigen values and Eigen VectorsSimilar matrices – Cayley – Hamilton Theorem (Statement only) – Finding powers of square matrix–Inverse of a square matrix up to
order 3– Diagonalization of square matrices –Related Problems
Unit IV
Expansions of sin nθ, cos nθ in powers of sinθ, cosθ – Expansion of tan nθ in terms of tan θ– Expansions of cos^(n) θ, sin^(n)θ, cos^(m) θ sin^(n)θ –Expansions of tan(θ1 +θ2 +,…,+θn ) – Expansions of sinθ, cosθ and tanθ in terms of θ –Related Problems.
Unit V
Hyperbolic functions – Relation between circular and hyperbolic functions–Formulas in hyperbolic functions, Inverse hyperbolic functions– Logarithm of complex quantities, Summation of trigonometric series –Related Problems.
Unit I
Successive Differentiation: Introduction (Review of basic concepts) – The nth derivative – Standard Results – Fractional Expressions – Trigonometrical Transformation – Formation of Equations Involving Derivatives – Leibnitz Formula for n th Derivative of a Product (Without proof)
Unit II
Partial Differentiation: Partial Derivatives – Successive Partial Derivatives – Function of a Function Rule – Total Differential Coefficient – A special case – Implicit Functions
Unit III
Partial Differentiation (Continued): Homogeneous Functions – Partial Derivatives of a Function of Two Variables – Maxima And Minima of Functions of Two Variables – Lagrange’s Method of Undetermined Multipliers.
Unit IV
Envelope: Method of Finding Envelope – Another Definition of Envelope – Envelope of Family of Curves Which are Quadratic in the Parameter
Unit V
Curvature: Definition of a Curvature – Circle, Radius and Centre of Curvature – Evolutes and Involutes – Radius of Curvature in Polar Coordinates, p – r equations; pedal equation of a curve
Unit I
The Solutions of Numerical Algebraic and Transcendental Equations: Introduction – Bisection method – Iteration method – Regula Falsi method – Newton – Raphson method – Horner’s Method
Unit II
Simultaneous Linear Algebraic equations: Introduction – Gauss Elimination method –Computation of the inverse of a matrix using Gauss Elimination method – Method of
Triangularisation – Iterative methods
Unit III
Finite Differences: Backward differences – central difference notations – Properties of the Operator △ – Difference of polynomials – Factorial polynomials – The Operator E –
Relation between E and △ – Relation between D and △ – Relation between the operators – Summation of Series
Unit IV
Central Difference Interpolation Formulae: Gauss forward and backward interpolation formula – Stirling’s formula – Bessel’s formula
Unit V
Interpolation with unequal intervals; Divided differences – properties of divided differences – Newton’s interpolation formula for unequal intervals – Lagrange’s formula for interpolation
Financial Mathematics
Unit I
Time Value of Money: Simple & Compound Interest, Present Value & Future Value, Annuities & Perpetuities
Unit II
Bonds: Net Present Value and Internal Rate of Return, Price and Yield of a Bond, Term Structure, Duration, Immunization
Unit III
Stocks: Common Stock Valuation, Preferred Stock Valuation, Stock Price Validity
Unit IV
Stock Price Models: Geometric Brownian Motion, Binomial Tree
Unit V
Options: Option Basics, Option Pricing Models, Option Trading Strategies
Unit I
Algebra: Binomial theorem, General term, middle term, problems based on these concepts.
Unit II
Sequences and series (Progressions). Fundamental principle of counting. Factorial n.
Unit III
Permutations and combinations, Derivation of formulae and their connections, simple applications, combinations with repetitions, arrangements within groups, formation of groups.
Unit IV
Trigonometry: Introduction to trigonometric ratios, proof of sin(A+B), cos(A+B), tan(A+B) formulae, multiple and sub multiple angles, sin(2A), cos(2A), tan(2A) etc., transformations sum into product and product into sum formulae, inverse trigonometric functions, sine rule and cosine rule
Unit V
Calculus: Limits, standard formulae and problems, differentiation, first principle, uv rule, u/v rule, methods of differentiation, application of derivatives, integration – product rule and substitution method.
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Unit I
RESILIENCE
Unit II
DECISION MAKING
Unit III
PROBLEM SOLVING
Unit IV
Tenses
4.1 Present
4.2 Past
4.3 Future
4.4 Concord
Unit V
English in the Workplace
5.1 E-mail – Invitation, Enquiry, Seeking
Clarification
5.2 Circular
5.3 Memo
5.4 Minutes of the Meeting
Unit I
Polar and pole, conjugate points and conjugate lines-diameters – conjugate diameters of an ellipse.- semi diameters-conjugate diameters of hyperbola
Unit II
Polar coordinates: General polar equation of straight line – Polar equation of a circle given a diameter, Equation of a straight line, circle, conic – Equation of chord, tangent, normal. Equations of the asymptotes of a hyperbola
Unit III
The plane – Transformation to the normal form – Determination of a plane under given conditions – System of Planes – Two sides of a plane – Length of the perpendicular from a point to a plane – Joint equation of two planes – Orthogonal projection on a plane
Unit IV
Representation of line – line and a plane – co-planar lines – constants in the equations of a straight line – the shortest distance Between two skew lines- Length of the perpendicular from a point to a line – intersection of three planes.
Unit V
Equation of a sphere – Definition – the sphere through four given points – Section of a sphere by a plane – equation of a circle – tangent plane – angle of intersection of two spheres- condition for the orthogonality of two spheres – radical plane.
Unit I
Reduction formulae -Types, integration of product of powers of algebraic and trigonometric functions, integration of product of powers of algebraic and logarithmic functions – product of
powers of exponential and trigonometric functions – Bernoulli’s formula
Unit II
Multiple Integrals – definition of double integrals – evaluation of double integrals – double integrals in polar coordinates.
Unit III
Triple integrals –applications of multiple integrals – volumes of solids of revolution – Volume of solids as double intergrals – Volume as a triple integral – areas of curved surfaces–change of
variables – Jacobian – change of variable in the case of two varibles, three variables – transformation from cartesian to polar coordinates and cartesian to spherical polar coordinates
Unit IV
Beta and Gamma functions – infinite integral – Definitions – recurrence formula of Gamma functions – properties of Beta functions- relation between Beta and Gamma functions –
Applications
Unit V
Geometric Applications of Integrations: Areas in polar co-ordinate, Trapezoidal Rule, Simpson’s Rule, Length of a curve – Cartesian co-ordinate – Polar co-ordinate– Area of surface of revolution.
Unit I
Numerical differentiation; Derivatives using Newton’s forward and backward difference formulae – derivatives using Sterling’s formula – derivatives using divided difference formula – Simple Problems
Unit II
Numerical Integration; General quadrature formula – Trapezoidal rule – Simpson’s one third rule – Simpson’s three- eight rule – Weddle’s rule – Simple Problems
Unit III
Difference equation: Definition – order and degree of a difference equation – Linear difference equation – Complementary function and particular integral of f(E) yx = ϕ(x).
Unit IV
Numerical solution of ordinary differential equations(I order only) Taylor’s series method – Picard’s method – Eulers’ method – Simple Problems
Unit V
Numerical solution of ordinary differential equations (I order only) Modified Euler’s method – Runge – kutta method forth order only – Simple Problems
Basic Data Analysis Using Excel
Unit I
Introduction to Excel: Spreadsheet window pane, Title Bar, Menu Bar, Standard toolbar, Formatting toolbar, the ribbon, file tab and backstage view, Formula bar, Workbook window, Status bar, Task pane, workbook and sheets, columns and rows, selecting rows and columns, changing column width and row height, auto fitting rows and columns, hiding/unhiding columns and rows, inserting and deleting rows and columns, cell address of a cell, components of a cell, format value, formula, use of paste and paste special.
Unit II
Creating formula, using formula, formula function, sum, average, if, count, max, min, proper, upper, lower, using Autosum, Advance formulas – concatenate, Vlookup, Hlookup, Match, Countif, Text, Trim Functions.
Unit III
Data Handling Wizards, Sort, Filter, Text to Columns, Remove Duplicates, Consolidate, Data validation.
Unit IV
Creating pivot tables, manipulating a pivot table, using the pivot table toolbar, changing data field, properties, displaying a pivot chart, setting pivot table options, adding subtotals to pivot tables.
Unit V
Creating charts, different types of charts, formatting chart objects, changing chart types, showing and hiding the legend, showing and hiding the data table.
Unit I
Preamble : Motivation – Running LaTeX – Resources – Basic LaTeX – Sample Document and Key Concepts – Type Style – Environments – Lists – Centering – Tables – Verbatim – Vertical
and Horizontal Spacing
Unit II
Typesetting Mathematics – Examples – Equation Environments – Fonts, Hats, and Underlining – Braces -Arrays and Matrices – Customized Commands -Theorem-like Environments – Math Miscellany – Math Styles – Bold Math – Symbols for Number Sets – Binomial Coefficient
Unit III
Further Essential LaTeX: Document Classes and the Overall Structure – Titles for Documents – Sectioning Commands – Miscellaneous Extras – Spacing – Accented Characters – Dashes &
Hyphens – Quotation Marks – Troubleshooting – Pinpointing the Error – Common Errors – Warning Messages
Unit IV
Packages – Inputting Files – Inputting Pictures – Making a Bibliography – Making an Index –Latex through the years
Unit V
Sample Article –Sample Report – Sample presentation – Sample Poster – Internet Resources
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Unit I
ACTIVE LISTENING
Unit II
INTERPERSONAL RELATIONSHIPS
Unit III
COPING WITH STRESS
Unit IV
Grammar
4.1 Phrasal Verbs & Idioms
4.2 Modals and Auxiliaries
4.3 Verb Phrases – Gerund, Participle, Infinitive
Unit V
Composition/ Writing Skills
5.1 Official Correspondence – Leave Letter , Letter of Application, Permission Letter
5.2 Drafting Invitations
5.3 Brochures for Programmes and Events
Unit I
Scalar and Vector point function – level surfaces –Directional Derivative of a scalar point functions – Gradiant of a scalar pointfunction –Summation notation for gradiant – Gradiant of f(r)
Unit II
Divergence and curl of a vector point function– Summation notation for divergence and curl – Laplacian differential operators , other differential operators, divergence and curl of a gradiant and divergence and curl of a curl – Examples
Unit III
Line integrals, independence of path of integration, conservative field and scalar potential, line integral of a conservative vector.
Unit IV
Surface integrals – Volume integrals – Cylindrical and spherical polar coordinates.
Unit V
Integral theorems, Gauss’ divergence Theorem, Integral theorems derived from the divergence theorem, Green’s theorem in plane – Stoke’s Theorem – simple problems
Unit I
Ordinary Differential Equations: Variable Separable – Homogeneous Equation – Non-Homogeneous Equation of First Degreein two Variables – Linear Equation – Bernoulli’s Equation – Exact Differential Equations
Unit II
Equation Of First Order But Not Of Higher Degree: Equation Solvable for dy/dx – Equation Solvable for y – Equation Solvable for x – Clairauts’ Form – Linear Equations with Constant Coefficients – Particular Integrals of Algebraic, Exponential, Trigonometric Functions and Their Products.
Unit III
Simultaneous Linear Differential Equations: Linear Equation of the Second Order – Complete Solution in Terms of a KnownIntegrals – Reduction to Normal Form – Change of the Independent Variable – Method of Variation of Parameters
Unit IV
Partial Differential Equations: Complete Integral – Singular Integral – General Integral – Formation of PDE by Eliminating Arbitrary Constants and Arbitrary Functions – Lagrange’s Linear Equations – Simple Applications
Unit V
Special Methods: Standard Forms – Charpit’s Method –Simple Applications
Unit I
Random variables (discrete and continuous), Distribution function – expected values and Moments – Moment generating function Characteristic function – Uniqueness Theorem (Statement Only) Chebychev’s inequality – Simple problems
Unit II
Concepts of bivariate distributions – Correlation & Regression – Rank Correlation Coefficient – Simple Problems
Unit III
Standard Distributions Normal – Uniform distributions – Sampling Theory – Sampling Distributions – Concept of Standard error – Sampling Distribution based on Normal, t, Chi-Square and F distributions.
Unit IV
Estimation – Introduction – Concept of Unbiasedness – Consistency, Efficiency And Sufficiency – Cramer Rao Inequality – Method of Estimation – Maximum likelihood Function
Unit V
Test of Significance – Tests of Hypothesis – Type I and Type II Errors – Large Sample Test – Exact test based on Normal, t, Chi-Square and F distributions with respect to Population Mean and Variance – Test of Independence of Attributes based on contingency tables – Goodness of fit based on Chi-Square – Simple Problems
Unit I
Introduction to Environmental Studies
Multidisciplinary nature of environmental studies;
Scope and importance; concept of sustainability and sustainable development.
Unit II
Ecosystem
What is an ecosystem? Structure and function of ecosystem; Energy flow in an
ecosystem:
Food chains, food webs and ecological succession, Case studies of the following
ecosystem:
a) Forest ecosystem
b) Grassland ecosystem
c) Desert ecosystem
d) Aquatic ecosystem (ponds, stream, lakes, rivers, ocean, estuaries)
Unit III
Natural Resources : Renewable and Non – renewable Resources
Land resources and land use change: Land degradation, soil erosion and desertification.
Deforestation : Causes and impacts due to mining, dam building on environment,
forests, biodiversity and tribal populations.
Water : Use and over – exploitation of surface and ground water, floods, droughts,
conflicts over water ( international and inter-state).
Energy resources : Renewable and non renewable energy sources, use of alternate
energy sources, growing energy needs, case studies.
Unit IV
Biodiversity and Conservation
Levels of biological diversity: genetics, species and ecosystem diversity,
Biogeographic zones of India: Biodiversity patterns and global biodiversity hot spots
India as a mega- biodiversity nation, Endangered and endemic species of India.
Threats to biodiversity: Habitat loss, poaching of wildlife, man- wildlife conflicts,
biological invasions; Conservations of biodiversity: In-situ and Ex-situ Conservation
of biodiversity.
Ecosystem and biodiversity services: Ecological, economic, social, ethical, aesthetic
and Informational value.
Unit V
Environmental Pollution
Environmental pollution: types, causes, effects and controls: Air, Water, soil and noise
Pollution.
Nuclear hazards and human health risks
Solid waste management: Control measures of urban and industrial waste
Pollution case studies.
Unit VI
Environmental Policies & Practices
Climate change, global warming, ozone layer depletion, acid rain and impacts on
human communities and agriculture
Environment Laws: Environment Protection Act, Air (Prevention & Control of
Pollution) Act; Water (Prevention and Control of Pollution ) Act; Wildlife Protection
Act; Forest Conservation Act. International agreements: Montreal and Kyoto
protocols and Convention on Biological Diversity (CBD).
Nature reserves, tribal populations and rights, and human Wildlife conflicts in Indian
context.
Unit VII
Human Communities and the Environment
Human population growth, impacts on environment, human health and welfare.
Resettlement and rehabilitation of projects affected persons; case studies.
Disaster management: floods, earthquake, cyclone and landslides.
Environmental movements : Chipko, Silent Valley, Bishnois of Rajasthan.
Environmental ethics : Role of Indian and other religions and cultures in
environmental conservation.
Environmental communication and public awareness, case studies(e.g. CNG Vehicles
in Delhi)
Unit VIII
Field Work
Visit to an area to document environmental assets: river / forest/ flora/ fauna etc.
Visit to a local polluted site – Urban / Rural/ Industrial/ Agricultural.
Study of common plants, insects, birds and basic principles of identification.
Study of simple ecosystem- pond, river, Delhi Ridge etc.
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Unit I
GOAL SETTING (UNICEF)
Unit II
INTEGRITY
Unit III
COPING WITH EMOTIONS
Unit IV
Language Competency Sentences
4.1 Simple Sentences
4.2 Compound Sentences
4.3 Complex Sentences
Direct and Indirect Speech
Unit V
Unit I
Linear programming: Formulation – graphical solution. Simplex method. Big-M method
Unit II
Transportation Problem: Mathematical Formulation. Basic Feasible solution. North West Corner rule– Least Cost Method– Vogel’s approximation– Optimal Solution– Unbalanced Transportation Problems– Degeneracy in Transportation problems.
Assignment Problem: Mathematical Formulation. Comparison with Transportation Model. Hungarian Method. Unbalanced Assignment Problems.
Sequencing Problem: n jobs on 2 machines – n jobs on 3 machines – two jobs on m machines – n jobs on m machines.
Unit III
Simulation: Monte Carlo Method – Definition, Types, Advantages and Disadvantages and Limitations, Phases. Generation of Random Numbers – Mid-Square method. Monte Carlo method of Simulation and Applications.
Unit IV
The Acceptance Sampling Problem – Advantages and Disadvantages of Sampling – Types of Sampling Plans – Lot Formation – Random Sampling – Guidelines for using Acceptance Sampling.
Unit V
Acceptance Sampling by attributes: Single Sampling Plan for attributes – Definition of a single sampling plan – The OC Curve – Designing a single sampling plan with a specified OC curve – Rectifying inspection.
Unit I
Sets and Functions: Sets and Elements – Operations on Sets – Functions – Real Valued Functions – Equivalence – Countability – Real Numbers-least Upper Bounds.
Unit II
Sequences of Real Numbers: Definition of a Sequence and Subsequence – Limit of a Sequence – Convergent Sequence – Divergent Sequences – Bounded Sequences – Monotone Sequences.
Unit III
Operations on Convergent Sequences – Operations on Divergent Sequences – Limit Superior and Limit Inferior – Cauchy Sequences.
Unit IV
Series of Real Numbers: Convergence – Divergence – Series With Non-Negative Terms – Alternating Series – Conditional Convergence and Absolute Convergence – Tests for Absolute Convergence
Unit V
Limits and Metric Spaces: Limit of a Function on a Real Line – Metric Spaces – Limits in Metric Spaces – Continuous Functions on Metric Spaces – Function Continuous at a Point on the Real Line – Function Continuous on a Metric Space
Unit I
Laplace Transforms: Definition – Sufficient Condition for the Existence of Laplace Transforms (Without Proof) – Laplace Transform of Periodic Functions – Some General Theorems – Evaluation of Integrals Using Laplace Transform – Problems
Unit II
The Inverse Laplace Transforms: Application of Laplace Transforms to Ordinary Differential Equations with Constant Coefficients and Variable Coefficients – Simultaneous Equations – Equations Involving Integrals – Problems
Unit III
Fourier Series: Fourier Series – Expansion of Periodic Functions of Period 2 – Expansion of Odd and Even Functions – Half Range Fourier Series – Change of Intervals – Problems
Unit IV
Fourier Transforms: Fourier Transform – Infinite Fourier Transform (Complex Form) – Properties of Fourier Transforms.
Unit V
Fourier Transforms (Continued): Fourier Cosine and Sine Transform – Properties – Parseval’s Identity – Convolution Theorem – Problems
Unit I
Introduction to Environmental Studies
Multidisciplinary nature of environmental studies;
Scope and importance; concept of sustainability and sustainable development.
Unit II
Ecosystem
What is an ecosystem? Structure and function of ecosystem; Energy flow in an
ecosystem:
Food chains, food webs and ecological succession, Case studies of the following
ecosystem:
a) Forest ecosystem
b) Grassland ecosystem
c) Desert ecosystem
d) Aquatic ecosystem (ponds, stream, lakes, rivers, ocean, estuaries)
Unit III
Natural Resources : Renewable and Non – renewable Resources
Land resources and land use change: Land degradation, soil erosion and desertification.
Deforestation : Causes and impacts due to mining, dam building on environment,
forests, biodiversity and tribal populations.
Water : Use and over – exploitation of surface and ground water, floods, droughts,
conflicts over water ( international and inter-state).
Energy resources : Renewable and non renewable energy sources, use of alternate
energy sources, growing energy needs, case studies.
Unit IV
Biodiversity and Conservation
Levels of biological diversity: genetics, species and ecosystem diversity,
Biogeographic zones of India: Biodiversity patterns and global biodiversity hot spots
India as a mega- biodiversity nation, Endangered and endemic species of India.
Threats to biodiversity: Habitat loss, poaching of wildlife, man- wildlife conflicts,
biological invasions; Conservations of biodiversity: In-situ and Ex-situ Conservation
of biodiversity.
Ecosystem and biodiversity services: Ecological, economic, social, ethical, aesthetic
and Informational value.
Unit V
Environmental Pollution
Environmental pollution: types, causes, effects and controls: Air, Water, soil and noise
Pollution.
Nuclear hazards and human health risks
Solid waste management: Control measures of urban and industrial waste
Pollution case studies.
Unit VI
Environmental Policies & Practices
Climate change, global warming, ozone layer depletion, acid rain and impacts on
human communities and agriculture
Environment Laws: Environment Protection Act, Air (Prevention & Control of
Pollution) Act; Water (Prevention and Control of Pollution ) Act; Wildlife Protection
Act; Forest Conservation Act. International agreements: Montreal and Kyoto
protocols and Convention on Biological Diversity (CBD).
Nature reserves, tribal populations and rights, and human Wildlife conflicts in Indian
context.
Unit VII
Human Communities and the Environment
Human population growth, impacts on environment, human health and welfare.
Resettlement and rehabilitation of projects affected persons; case studies.
Disaster management: floods, earthquake, cyclone and landslides.
Environmental movements : Chipko, Silent Valley, Bishnois of Rajasthan.
Environmental ethics : Role of Indian and other religions and cultures in
environmental conservation.
Environmental communication and public awareness, case studies(e.g. CNG Vehicles
in Delhi)
Unit VIII
Field Work
Visit to an area to document environmental assets: river / forest/ flora/ fauna etc.
Visit to a local polluted site – Urban / Rural/ Industrial/ Agricultural.
Study of common plants, insects, birds and basic principles of identification.
Study of simple ecosystem- pond, river, Delhi Ridge etc.
Unit I
Value education-its purpose and significance in the present world – Value system – The role
of culture and civilization – Holistic living – balancing the outer and inner – Body, Mind
and Intellectual level – Duties and responsibilities.
Unit II
Salient values for life – Truth, commitment, honesty and integrity, forgiveness and love,
empathy and ability to sacrifice, care, unity, and inclusiveness, Self esteem and self
confidence, punctuality – Time, task and resource management – Problem solving and
decision making skills – Interpersonal and Intra personal relationship – Team work –
Positive and creative thinking.
Unit III
Human Rights – Universal Declaration of Human Rights – Human Rights violations –
National Integration – Peace and non-violence – Dr.A P J Kalam’s ten points for
enlightened citizenship – Social Values and Welfare of the citizen – The role of media in
value building.
Unit IV
Environment and Ecological balance – interdependence of all beings – living and non-living.
The binding of man and nature – Environment conservation and enrichment.
Unit V
Social Evils – Corruption, Cyber crime, Terrorism – Alcoholism, Drug addiction – Dowry –
Domestic violence – untouchability – female infanticide – atrocities against women – How
to tackle them.
Unit I
Introduction to groups- Subgroups- cyclic groups – Lagrange’s Theorem-A counting principle –Examples
Unit II
Normal subgroups and Quotient group- Homomorphism Automorphism -Examples
Unit III
Cayley’s Theorem-Permutation groups – Examples
Unit IV
Definition and examples of ring- Some special classes of rings- homomorphism of rings- Ideals and quotient rings- More ideals and quotient rings.
Unit V
The field of quotients of an integral domain-Euclidean Rings The particular Euclidean Ring – Examples
Unit I
Continuous Functions on Metric Spaces, Open Sets – Closed Sets – Discontinuous Functions on R1 – Connectedness, Completeness and Compactness – More About Open Sets – Connected Sets.
Unit II
Bounded Sets and Totally Bounded Sets, Complete Metric Spaces – Compact Metric Spaces – Continuous Functions on a Compact Metric Space – Continuity of the Inverse Functions – Uniform Continuity
Unit III
Calculus: Sets of Measure Zero – Definition of Riemann Integral – Existence of Riemann Integral – Properties of Riemann Integral
Unit IV
Derivatives: Rolle’s Theorem – The Law of Mean – Fundamental Theorems of Calculus
Unit V
Taylor’s Theorem – Pointwise Convergence of Sequence of Functions – Uniform Convergence of Sequence of Functions.
Unit I
Mathematical Modelling: Simple situations requiring mathematical modelling – The Technique –Classification and some characteristics of mathematical models.
Unit II
Mathematical Modelling through differential equations: Linear Growth and Decay Models. Non-Linear growth and decay models, Compartment models
Unit III
Mathematical Modelling, through system of Ordinary differential equations of first order: Prey-predator models, Competition models, Model with removal and model with immigrations. Epidemics: simple epidemic model, Susceptible-infected- susceptible (SIS) model, SIS model with constant number of carriers. Medicine: Model for Diabetes Mellitus.
Unit IV
Introduction to difference equations.
Unit V
Mathematical Modelling through difference equations: Harrod Model, cob web model application to Actuarial Science
Unit – I
Games And Strategies – Two – Person Zero Sum Game – Some Basic Terms – The Maximin-Minimax Principle – Games Without Saddle Points – Mixed Strategies – Graphical Solution Of 2xn And mx2 Games – Arithmetic Method For nxn Games
Unit – II
Inventory Control – Types of inventories – reasons for carrying inventories – the inventory decisions – objectives of scientific inventory control – costs associated with inventories – factors affecting inventory control – an inventory control problem – the concept of EOQ – Deterministic inventory problem with no shortages – Deterministric inventory problem with shortages – One period problem without setup cost.
Unit – III
Queueing Theory – Queueing system – Elements of queueing system – operating characteristics of queueing system – deterministic queueing system – probability distributions in queueing systems – classification of queueing models – definition of transient and steady states – models M/M/1:/FIFO and M/M/1: N/FIFO
Unit – IV
Network Scheduling( PERT / CPM) – Network – Basic components – logical sequencing – rules of network construction – concurrent activities – critical path analysis – probability
considerations in PERT – distinction between PERT and CPM
Unit – V
Information Theory: Communication Process – A Measure of Information – Measures of Other Information Quantities – Channel Capacity, Efficiency and Redundancy – Encoding – Shannon-Fano Encoding Procedure – Necessary and Sufficient Condition for Noiseless Encoding.
Unit – I
Propositional Logic: Definition, Statements & Notation, Truth Values, Connectives, Statement Formulas & Truth Tables, Well formed Formulas, Tautologies, Equivalence of Formulas, Duality Law, Tautological Implications, Examples
Unit – II
Predicate Logic: Definition of Predicates; Statement functions, Variables, Quantifiers, Predicate Formulas, Free & Bound Variables; The Universe of Discourse, Examples, Valid Formulas &
Equivalences, Examples.
Unit – III
Lattices & Boolean Algebra: Properties of lattices – Lattice as Algebraic System-Sub lattices- lattice Homomorphism- Special Lattices – Boolean Algebra- sub algebra- Boolean Expression and
Boolean functions- expression of a Boolean function in canonical form logic Gates- Karnaugh Map Method
Unit – IV
Basics of Counting: The Pigeonhole Principle, Permutations and Combinations, Binomial Coefficients, Generalized Permutations and Combinations, Generating Permutations and
Combinations, Inclusion-Exclusion Principle.
Unit – V
Formal Language: Introduction- Phrase –Structure Grammar- Types – BNF- Finite state Machine – Input output strings Finite state Automata
Unit I
Vector spaces – Subspaces – Linear Combinations and Linear span – System of linear equations – Elementary Matrices
Unit II
Linear Dependence and Linear independence – Bases – Dimensions – Homogenous Equations – Non-homogenous equations Row reduced – Echelon form.
Unit III
Linear transforms, null spaces and ranges – Matrix representation of a linear transformation – Invertibility and isomorphisms – Dual spaces.
Unit IV
Eigen values, Eigen vectors, Diagonalizability – Invariant subspaces – Cayley – Hamilton theorem.
Unit V
Inner Products Space: Inner Products and norms Gram-Schmidt Orthogonalization Process – Orthogonal complements
Unit I
Analytic functions: Functions of a Complex variable –Limits –Theorem on limits –Continuity – Derivatives – Differentiation formulas – Cauchy Riemann equation – Sufficient conditions for
differentiability – Polar coordinates– Analytic functions– ExamplesHarmonic functions.
Unit II
Mappings – Mapping with exponentail function – Linear transformation – The transformation w= 1 – Mappings by 1 – Linear fractional transformations (bilinear) – An Implicit form.
Unit III
Complex Integration: Contour integrals–Examples – Simply and Multiply connected domains– Cauchy integral formula –Derivatives of Analytic Functions– Liouville’stheorem and Fundamental theorem of Algebra– Maximum modulus principle.
Unit IV
Sequences and Series: Convergence of sequences – Convergence of series– Taylor series –Examples- Laurent series– Examples- Absolute and uniform convergence of power Series
Unit V
Residues and Poles:– Residues – Cauchy Residue theorem – The three types of isolated singular points – Residues at poles- Examples – Zeros of analytical functions – Zeros and poles – Evaluation of Improper Integrals.
Unit I
Force: Newton’s laws of motion – Resultant of two forces ona particle – Equilibrium of a Particle: Equilibrium of a particle – Limiting equilibrium of a particle on an inclined plane.
Unit II
Forces on a Rigid Body: Moment of a Force – General motion of a body – Equivalent systems of forces- ParallelForces – Couples. A Specific reduction of Forces: Reduction of coplanar forces into a force and couple – Problems involving frictional forces.
Unit III
Work, Energy and Power: Work – Conservative field of force – Power. Rectilinear Motion under Varying Force: Simple Harmonic Motion – along a horizontal line – along a vertical line.
Unit IV
Projectiles: Forces on a projectile – Projectile projected on an inclined plane.
Unit V
Central Orbits: General orbits – Central orbit – Conic as a centered orbit
Unit – I
Fuzzy Set Theory: Fuzzy Sets – Definition – Types of Fuzzy Sets–Characteristics of Fuzzy Sets.
Unit – II
Other Important Operations – General Properties – Fuzzy vs Crisp – Operations on Fuzzy Sets – Some Important Theorems.
Unit – III
Extension Principle for Fuzzy Sets – Fuzzy Complements
Unit – IV
Fuzzy Relations and Fuzzy Graphs – Introduction – Projections and Cylindrical Fuzzy – Relations – Composition – Properties of Min-Max Compositions.
Unit – V
Decision Making in Fuzzy Environment – Introduction – Individual Decision Making – Multi Person Decision Making
Unit – I
Computer systems – Python Programming Language Computational Thinking – Python Data Types: Expressions, Operator, Variables, and Assignments – Strings – Lists – Objects &
Classes – Python standard library.
Unit – II
Imperative programming: Python modules – Built-in-function: print() function –eval() function – user-defined function & assignments – parameter passing.
Unit – III
Text Data, Files & Exceptions: Strings, revisited – formatted output – files – errors & Exceptions – Execution control Structures: decision control & the IF statement. For LOOP & Iteration Patterns – two-dimensional list- while loop – more loop patterns – additional iteration control statements – Container and Randomness: Dictionaries – other built-in container types – character encodings & strings – module random. Namespaces – encapsulation in functions – global vs. local namespaces exceptional flow control – modules as namespaces.
Unit – IV
NumPy Basics: Array and Vectorized Computation – A Multidimensional Array Object – Data Processing using Arrays, File Input and Output with Arrays – Linear Algebra – Random Number
Generation.
Unit – V
Pandas – Data Structure – Essential Functionality – Handling Missing Data – Hierarchical Indexing – Data loading, Storage and File formats – Data wragling- Plotting and Visualization
Unit – I
Simplifications – Averages – concepts – problems
Unit – II
Simple Interest – Compound interest – concepts – problems
Unit – III
Time and work -short cuts – concepts – problems
Unit – IV
Profit and Loss – short cuts – concepts – problems
Unit – IV
Problems on numbers – short cuts – concepts – problems