## RevisedFYBSc FYBAsyllabus2_1 Syllabus Mumbai University by munotes

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1UNIVERSITY OFMUMBAI

SYLLABUS fortheF.Y.B.A/B.Sc.

Programme: B.A./B.Sc.

Subject: Mathematics

Choice BasedCredit System (CBCS)

witheffectfromtheacademic year2018-19

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2F.Y.B.Sc.(CBCS) Semester I

CALCULUS I

COURSE CODE UNIT TOPICS CREDITS L/W

USMT101 UNIT I Real Number System

2 3 UNIT II Sequences

UNIT III Limits and Continuity

ALGEBRA I

USMT102 UNIT I Integers and Divisibility

2 3 UNIT II Functions and EquivalenceRelation

UNIT III Polynomials

PRACTICALS

USMTP01 UNIT I Practicalsbasedon

USMT101,USMT1022 2

F.Y.B.A.(CBCS) SemesterI

UAMT101 UNIT I Real Number System

3 3UNIT II Sequences

UNIT III Limits and Continuity

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3F.Y.B.Sc.(CBCS) Semester II

CALCULUS II

COURSE CODE UNIT TOPICS CREDITS L/W

USMT201 UNIT I Infinite Series

2 3 UNIT II Continuous functions andDifferentiation

UNIT III Applications ofDifferentiability

ALGEBRA II

USMT202 UNIT I System of Linear Equationsand Matrices

2 3UNIT II Vector Spaces

UNIT III Basis & LinearTransformation

PRACTICALS

USMTP02 UNIT I Practicalsbasedon

USMT201,USMT2022 2

F.Y.B.A.(CBCS) SemesterII

UAMT201 UNIT I Infinite Series

3 3UNIT II Continuous functions andDifferentiation

UNITIIIApplications of

Differentiation

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4TEACHING PATTERN FOR SEMESTER I

1. Threelecturesperweekpercourse..Eachlectureisof48minutesduration.

2. OnePractical(2L)perweekperbatchforcoursesUSMT101, USMT 102

combined (thebatchestobeformedasprescribedby theUniversity).Each

practicalsessionisof48minutesduration.

3.OneTutorial perweekperbatchforthecourse UAMT101 (thebatchestobe

formedasprescribedbythe University). Each tutorial session is of48 minutes

duration.

TEACHING PATTERN FOR SEMESTER II

1. Threelecturesperweekpercourse.Eachlectureisof48minutesduration.

2. OnePractical(2L)perweekperbatchforcoursesUSMT201, USMT202

combined (thebatchestobeformedasprescribedby theUniversity).Each

practicalsessionisof48minutesduration.

3.OneTutorial perweekperbatchfor thecourse UAMT201 (thebatchestobe

formedasprescribedbythe University). Eachtutorial session is of48

minutesduration.

SYLLABUS FOR SEMESTER I

Note: All topics have to be covered with proof in details (unless mentioned otherwise) and

with examples.

USMT101/UAMT101 CALCULUS I

Unit 1 : Real Number System (15 Lectures)

1.Real number system ℝand order properties of ℝ,absolute value||and its

properties.

2.AM-GM inequality, Cauchy -Schwarz inequality, Intervals and neighbourhoods,

Hausdorff property.

3.Bounded sets, statements of I.u.b. axiom and its consequences, Supremum and

infimum, Maximum and minimum, Archimedean property and its applications,

density of rationals.

Unit II: Sequences (15 Lectures)

1.Definition of a sequence and examples, Convergence of sequences, every convergent

sequences is bounded. Limit of a convergent sequence and uniqueness of limit,

Divergent sequences.

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52.Convergence of standard sequences like ∀>0,()∀0<<

1,∀>0,&,

3.Algebra of convergent sequences, sandwich theorem, monotone sequences, monotone

convergence theorem and consequences as convergence of 1+.

4.Definition of subsequence, subsequence of a convergent sequence is convergent and

converges to the same limit, definition of a Cauchy sequences, every convergent

sequences s a Cauchy sequence and converse.

Unit III: Limits and Continuity (15 Lectures)

Brief review: Domain and range of a function, injective function, surjectiove function,

bijective function, composite of two functions, (when defined) Inverse of a bijective function.

1.Graphs of same standard functions such as

2.||,,log ++,,(≥3),sin,cos,tan,sin,sin

over suitable intervals of ℝ.

3.Definition of Limit lim→()of a function (),evaluation of limit of simple

functions using the −definition, uniqueness of limit if it exists, algebra of limits,

limits of composite function, sandwich theorem, left -hand-limitlim→(),right-

hand-limitlim→(),non-existence of limits, lim→(),lim→()and

lim→()=±∞.

4.Continuous functions: Continui ty of a real valued function on a set in terms of limits,

examples, Continuity of a real valued function at end points of domain, Sequential

continuity, Algebra of continuous functions, discontinuous functions, examples of

removable and essential discontin uity.

Reference Book s:

1.R.R. Goldberg, Methods of Real Analysis, Oxford and IBH, 1964.

2.K.G. Binmore, Mathematical Analysis, Cambridge University Press, 1982.

3.R. G. Bartle -D. R. Sherbert, Introduction to Real Analysis, John Wiley & Sons, 1994.

Additional Reference Books

1.T.M. Apostol, Calculus Volume I, Wiley & Sons (Asia) Pte, Ltd.

2.Richard Courant -Fritz John, A Introduction to Calculus and Analysis, Volume I,

Springer.

3.Ajit kumar -S. Kumaresan, A Basic Course in Real Analysis, CRC Press, 2014.

4.James Stewa rt, Calculus, Third Edition, Brooks/ cole Publishing Company, 1994.

5.Ghorpade, Sudhir R. -Limaye, Balmohan V., A Course and Real Analysis, Springer

International Ltd.2000.

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6USMT102 ALGEBRA I

Prerequisites:

Set Theory : Set, subset, union and intersection of two sets, empty set, universal set,

complement of a set, De Morgan’s laws, Cartesian product of two sets, Relations,

Permutations and Combinations .

Complex numbers: Addition and multiplication of complex numbers, modulus, amplitude and

conjugate of a complex number.

Unit I : Integers & Divisibility (15 Lectures)

1.Statements of well -ordering property of non -negative integers, Principle of

finite induction (first and second) as a consequence of well -ordering property,

Binomial theorem for non -negative exponents, Pascal Triangle.

2.Divisibility in integers, division a lgorithm, greatest common divisor (g.c.d)

and least common multiple (L.c.m) of two integers, basic properties of g.c.d

such as existence and uniqueness of g.c.d of integers &and that the g.c.d.

can be expressed as +for some ,∈ℤ,Euclidean algorithm,

Primes, Euclid’s lemma, Fundamental Theorem of arithmetic, The set of

primes is infinite.

3.Congruences, definition and elementary properties, Eulers function,

statements of Eulers theorem, Fermats theorem and Wilson theorem,

Applications.

Unit II:Functions and Equivalence relations (15 Lectures)

1.Definition of function, domain, co -domain and range of a function, composite

functions, examples, Direct image ()and inverse image ()for a

function ,injective, surjective, bijec tive functions, Composite of injective,

surjective, bijective functions when defined, invertible functions, bijective

functions are invertible and conversely examples of functions including

constant, identity, projection, inclusion, Binary operation as a f unction,

properties, examples.

2.Equivalence relation, Equivalence classes, properties such as two equivalences

classes are either identical or disjoint, Definition of partition, every partition

gives an equivalence relation and vice versa.

.

3.Congruence is an equivalence relation on ℤ,Residue classes and partition of ℤ,

Addition modulo ,Multiplication modulo ,examples.

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7Unit III: Polynomials (15 Lectures)

1.Definition of a polynomial, polynomials over the field where=ℚ,ℝor

ℂ,Algebra of polynomials, degree of polynomial, basic properties.

2.Division algorithm in [](without proof), and g.c.d of two polynomials and

its basic properties (without proof), Euclidean algorithm (without proof),

applications, Roots of a polynomial, relation between roots and coefficients,

multiplicity of a root, Remainder theorem, Factor th eorem.

3.A polynomial of degree over has at roots, Complex roots of a polynomial in

ℝ[]occur in conjugate pairs, Statement of Fundamental Theorem of

Algebra, A polynomial of degree in ℂ[]has exactly complex roots counted

with multiplicity, A no n constant polynomial in ℝ[]can be expressed as a

product of linear and quadratic factors in ℝ[],necessary condition for a

rational number to be a root a polynomial with integer coefficients, simple

consequences such as is a irrational number w hereis a prime number,

roots of unity, sum of all the roots of unity.

Reference Books:

1.David M. Burton, Elementary Number Theory, Seventh Edition, McGraw

Hill Education (India) Private Ltd.

2.Norman L.

PRACTICALS FOR F.Y.B.Sc

USMTP01 –Practicals

A. Practicals for US MT101:

1.Application based examples of Archimedean property, intervals,

nieghbourhood. Consequences of l.u.b axiom, infimum and supremum of sets.

2.Calculating limts of sequences, Cauchy sequences, monotone sequences.

3.Limits of functio n and Sandwich theorem, continuous and discontinuous

functions.

4.Miscellaneous Theoretical Questions based on full paper.

B.Practicals for US MT102:

1.Mathematical induction Division Algorithm and Euclidean algorithm in

,primes

and theFundamental theorem of Arithmetic. Convergence and Eulers -

function, Fermat’s little theorem, Euler’s theorem and Wilson’s theorem,

2.Functions ( direct image and inverse image), Injective, surjective, bijective

functions, finding inverses of bijective func tions. Equivalence relation.

3.Factor Theorem, relation between roots and coefficients of polynomials,

factorization and reciprocal polynomials.

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84.Miscellaneous Theoretical Questions based on full paper.

TUTORIALS FOR F.Y.B.A

A. Tutorials for UAMT101:

1.Application based examples of Archimedean property, intervals,

neighbourhood.

2.Consequence of l.u.b axion, infimum and supremum of sets.

3.Calculating limits of sequences

4.Cauchy sequences, monotone sequences

5.Limit of a function and Sandwich theorem.

6.Continuous and discontinuous func tion.

7.Miscellaneous Theoretical Questions based on full paper.

Semester II

USMT 201 CALCULUS II

Unit-I : Series (15 Lectures)

1.Series∑pf real numbers, simple examples of series, Sequence of partial sums

of a series, convergent series, divergent series. Necessary condition : ∑

converges ⇒→0, but converse is not true, algebra of convergent series,

2.Cauchy criterion, diverge nce of harmonic series, convergence of ∑(P>1),

comparison test, limit comparison test, alternating series, Leibnitz’s theorem

(alternating series test) and convergence of ∑(), absolute convergence,

conditional convergence, absolute c onvergence implies convergence but not

conversely, Ratio test (without proof), root test(without proof) and examples.

Unit –II : Limits and Continuity of functions( 15 lectures)

1.Definition of Limit lim→()of a function (), evaluation of limit of simple

functions using the ∈−definition, uniqueness of limit if it exists, algebra of limits,

limit of composite function, sandwich theorem, left -hand-limitlim→(), right

hand limit lim→()non existence of limits , lim→(),lim→(), and

lim→()=∞.

2.Continuous functions: Continuity of real valued function on a set in terms of limits,

examples, continuity of a real valued function at end points of domain, Sequential

continuity, Algebra of continuous functions, Discontinuous functions, examples of

removable and essential discontinuity. Intermediate value theorem and it’s

applications, Bolzano -Weierstrass theorem (statement only): A continuous function

on a closed and bounded interval is bounded and attains its bounds.

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93.Differentiation of real valued function of one variable: Definition of differentiation at

a point of an open interval, examples of differentiable and non differentiable

functions, differentiable functions are continuous but not conversely, chain rule ,

Higher order derivatives, Leibnitz rule, Derivative of inverse functions, Implicit

differentiation (only examples)

Unit –III Applications of differentiation ( 15 lectures)

1.Definition of local maximum and local minimum, necessary condition, stationary points,

second derivative test, examples, Graphing of functions using first and second

derivatives, concave , convex , concave functions, points of inflection.

2.Rolle’s theorem, Lagrange’s and Cauchy’s mean value theorems, applications and

examples, Monotone increasing and decreasing function, examples,

3.L-Hospital rule without proof, examples of intermediate forms, Taylor’s theorem with

Lagrange’s form of remainder with proof. Taylor’s polynomial and applications.

Reference Books:

1.R.R.Goldberg, Methods of Real Analysis, Oxford and IBH, 1964.

2.James Stewart, Calculus, Third Edition, Brooks/ Cole Publishing company, 1994.

3.T.M.Apostol, Calculus, Vol I, Wiley And Sons (Asia) Pte. Ltd.

Additional Reference:

1.Richard Courant -Fritz John, A Introduction to Calculus and A nalysis, Volume -I,

Springer.

2.Ajit Kumar -S.Kumaresan, A Basic course in Real Analysis, CRC Press, 2014.

3.Ghorpade, Sudhir R, -Limaye, Balmohan V, A course in Calculus and Real Analysis,

Springer International Ltd, 2000.

4.K.G. Binmore, Mathematical Analysis, Cambridge University Press, 1982.

5.G.B.Thomas, Calculus, 12 th Edition 2009

USMT202 ALGEBRA I

Unit I System of Equations and Matrices (15 Lectures )

1.Parametric Equation of Lines and Planes , System of homogeneous and non

homogeneous linear Equations, The solution of m homogeneous linear equations in n

unknowns by elimination and their geometrical interpretation for (,)=

(1,2),(1,3),(2,2),(2,2),(3,3); Definition of n -tuple of real numbers, sum of n -tuples

and scalar multiple of n -tuple.

Deduce that the system of m homogeneous linear e quations has a non trivial solution

if m < n.

2.Matrices with real entries; addition, sca lar multiplication of matrices and

multiplication of matrices, transpose of a matrix, types of matrices: zero matrix,

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10identity matrix, scalar matrix,diagonal matrix, upper and lower triangular matrices,

symmetric matrix, skew symmetric matrix, invertible matrix; Identities such as

()=,()=

3.System of linear equations in matrix form , Elemetary row operations , row echelon

matrix, Gausian elimination method.

Unit II Vector Spaces ( 15 Lectures)

1.Definition of real vector space , Examplees such as ,[],(), space of

realvalued functions on a non empty set.

2.Subspace: definition, examples: lines , planes passing through origin as subspaces of

respectively; upper triangular matrices, diagonal matrices, symmetric matrices, skew

–symmetric matrix as subspaces of ()(=2,3);()=++

+⋯+,∈,∀1≤≤as subpac e of[], the space of all

solutions of the system of m homogeneous linear equations in n unknowns as a

subspace of .

3.Properties of a subspace such as necessary and sufficient conditions for a non empty

subst to be a subspace of a vector space, arbitrary intersection of subspaces of a

vectorspace is a subspace, union of two subspaces is a subspace if and only if one is

the subset of other.

4.Finite linear combination of vectors in a vector space; linear span L(S) of a non empty

subet S of a vector space, S is generating set for L(S), L(S) is a vector subspace of V;

Linearly independent/ Linearly Dependent subsets of a vector space, a subset

{,,…} is linearly dependent if and only ∃∈{1,2,…}such that is a linear

combination of other vectors ’s .

Unit-IIIBasis of a Vector Space and Linear Transformation (15 Lectures)

1.Basis of a vector space, dimension of a vector space, m aximal linearly independent

subset of a vector space is a basis of a vector space, any two basis of a vector space

have same number of elements, any set of n linearly independent vectors in an n -

dimensional vector space is a basis, any collection of n+1 ve ctors in an n -dimensional

vector space is linearly dependent.

2.Extending any basis of a subspace W of a vector space V to a basis of the vector space

V.

If,are two subspaces of a vector space V then +is a subspace,

dim(+)=dim()+dim()−dim(∩).

3.Linear Transformations; Kernel, Image of a Linear Transformation T ,Rank T, Nullity

T, properties such as: kernel T is a subspace of domain space of T and Img T is a

subspace of codomain subspace of T. If ={,,…}is a basis of V and

={,,…}any vectors in W then there exists a unique linear transformation

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11:→such that =∀1≤≤, Rank nullity theorem (statement only)

and examples.

Reference Books:

1.Serge Lang, Introduction to Linear Algebra, Second ed ition Springer.

2.S. Kumaresan , Linear Algebra , Prentice Hall of India Pvt limited .

3.K.Hoffmann and R. Kunze Linear Algebra, Tata MacGraw Hill, New Delhi, 1971

4.Gilbert Strang , Linear Algebra and it’s Applications, International Student Edition.

5.L. Smith , Linear Algebra, Springer Verlang

6.A. Ramchandran Rao, P. Bhimashankaran; Linear Algebra Tata Mac Graw Hill.

PRACTICALS FOR F.Y.B.Sc

USMTP02 -Practicals

A.Practicals for UAMT201:

1.Calculating limit of series, Convergence tests.

2.Properties of continuous and differentiable functions. Higher order

derivatives, Leibnitz theorem. Mean value theorems and its applications.

3.Extremevalues, increasing anddecreasing functions. Applications ofTaylor’stheoremand

Taylor ’spolynomials.

4.Miscellaneous Theoretical Questions based on full paper

B.Practicals for UAMT202:

1.Solvinghomogeneous systemofmequations inn unknownsbyelimination

for (m,n)=(1,2),(1,3),(2,2),2,3),(3,3),row echelonform.Solvingsystem

Ax=bbyGausselimination, Solutions ofsystemoflinear Equations.

2.Examples of Vectorspaces , Subspaces

3.Linearspanofannon-emptysubsetofavectorspace, Basis and Dimension of Vector Space

4.Examples of Linear Transformation, Computing Kernel, Image of a linear

map , Verifying Rank Nullity Theorem

5.Miscellaneous Theoretical Questions based on full paper

TUTORIALS FOR F.Y.B.A

A.Tutorials for UAMT201:

1.Calculating limit of series, Convergence tests.

2.Properties of continuous functions.

3.Differentiability, Higher order derivatives, Leibnitz theorem.

4.Mean value theorems and its applications.

5.Extreme values, increasing anddecreasing functions.

6.Applications ofTaylor ’stheoremandTaylor'spolynomials.

7.Miscellaneous Theoretical Questions b ased on full paper

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12B.Tutorials for UAMT202:

1.Solvinghomogeneous systemofmequations innunknowns byelimination for

(m,n)=(1,2),(1,3),(2,2),2,3),(3,3), rowechelonform.

2.SolvingsystemAx=bbyGausselimination, Solutions ofsystemof

linear Equations.

3.Examples of Vector spaces

4.Examples of Subspaces, Linear Span, Linear dependence/ independence of sets.

5.Basis and dimension of a vector space

6.Linear Transformations, Rank Nullity Theorem

7.Miscellaneous Th eoretical Questions based on full paper

SchemeofExamination

I.Semester End Theory Examination s:

Therewillbe a Semester -end external The ory examination of 100 m arks for each of the

coursesUSMT101/UAMT101,USMT102/UAMT102,of Semester Iand

USMT201/UAMT201, semester II to be conducted by the Universi ty.

Duration: The examinations shall be of 3 Hours duration.

Theory QuestionPaperPattern:

a)There shall beFIVEquestions. The first question Q1 shall be of objective type

for 20mark s based on the entire syllabus. The next three questions Q2, Q3,Q4

shallbe of 20marks, each based on the unitsI, II,IIIrespectively. The fifth question

Q5 shallbe of 20 m arks based on the entire syllabus.

b)Allthe questions shall be compuls ory. The questionsQ2, Q3, Q4, Q5 shall have

internalchoices within the questions including the choices, the m arks for each

question shall be 30-32.

c)The questions Q2, Q3, Q4, Q5 m aybe subdivided in to sub -questions as a,b,c,d

& e,etc and the all ocation of m arks depends on the weightage of the topic.(a)

ThequestionQ1m aybe subdivided into10 sub questions of 2m arkseach.

II.Semester End Examinations Practical s:

Atthe end of the Semesters I&II Practical examinations of two hoursduration

and100 m arks shallbeconducted f or the courses USMTP01, USMTP02.

In semester I,the Practical examinations f or USMT101 and USMT102 are held

togetherby the college.

In Semester II, the Practical examinations f or USMT201 and USMT202

are held together by the college.

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13Paperpattern:

There will be Two parts Part A , Part B

USMTP01 -Max Marks 80. Duration-2hours

Part A: Questions from USMT101, Part B : Questions from USMT102

USMTP02 -Max Marks 80. Duration-2 hours

Part A: Questions from USMT201, Part B : Questions from USMT2 02

Eachpart shall have two Sections

Section I

Objective in nature -Attempt any eight out of 12 multiple choice questions.

(8x3= 24)

Section II

Problems -Attempt any two out of Three

(8 x 2 =16)

Marks for Journals and Viva:

For each course USMT101, USMT102 and USMT201 ,USMT202

1. Journals: 5 marks.

2. Viva:5marks.

Each Practical of every course of Semester I& IIshall contain10 (ten ) problemsoutof

which minimum 05 (five) have to be written in the journal. A student must have a certified

journal before appearing for the practical examination.