Government Science College, Hassan (Autonomous) · Page 3 of 43 Government Science College(Autonomous), Hassan-573201 Department of Mathematics Date:06/07/2017 Proceedings of the

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Government of Karnataka Department of Collegiate Education

Government Science College, Hassan (Autonomous)

Proposed syllabus for B.Sc.

Mathematics Under Choice Based

Credit Scheme (CBCS)

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Aims and objectives of the syllabus:

To set up a mathematical laboratory in order to help students in the

exploration of mathematical concepts through activities and

experimentation.

To enable the teacher to demonstrate, explain and reinforce abstract

mathematical ideas by using concrete objects, models charts, graphs pictures,

posters with the help of FOSS tools on a computer.

To develop a spirit of enquiry and mathematical skills among the

students.

To prepare students to face new challenges in mathematics as per

modern requirement.

To make the learning process student – friendly.

To provide greater scope for individual participation in the process of

learning and becoming autonomous learners.

To foster experimental, problem-oriented and discovery learning of

mathematics.

To help the students to build interest and confidence in learning the

subject.

To remove maths phobia through various illustrative examples and

experiments.

Support from The Govt for Students and Teachers in

Understanding and Learning Foss Tools:

As a national level initiative towards learning FOSS tools, IIT Bombay

for MHRD, government of India is giving free training to teachers

interested in learning open source software’s like scilab, maxima, octave,

geogebra and others.

(Website: http://spoken-tutorial.org;)

(email: [email protected]; [email protected])

Page 3 of 43

Government Science College(Autonomous), Hassan-573201

Department of Mathematics Date:06/07/2017

Proceedings of the BOS(UG) in Mathematics.

The meeting of the Board of Studies in UG mathematics for the academic year

2017-18 was held on Thursday, July 6th 2017 at 10.00 am in the department of

mathematics. The following members attended the meeting.

1. Shashikumar H C Chairman

2. Mayigowda University Nominee

3. Sridevi.M.J Member

4. Shankaralingappa B M Member

5. Dr.Jagadeesha R D Member

6. Dr. Vijay S Member

7. Chandran K External Member

8. Shylashree External Member

Agenda and resolution

Discussion on the syllabus of Mathematics papers of BSc course.

The BOS had a discussion on the draft syllabus for three years of BSc

(six semesters) prepared by teachers and approved the same with a practical component (Mathematics Practicals with FOSS tools for programming). The BOS also resolved to change the list of practical experiments each year.

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Structure of B.Sc/BCA Mathematics papers

Sem

ester Title of the paper Teaching

hrs/week

Dura

tion of

Exam

(hrs)

IA

MARK

S

EXAM

MARKS TO

T

AL

MA

R

KS

Sem

ester

Total

BSc

I Paper I Theory 3 hrs 3 hrs 30 70 100

150 Practical 3 hrs 3 hrs 20 30 50

II Paper II Theory 3 hrs 3 hrs 30 70 100


III Paper III Theory 3 hrs 3 hrs 30 70 100


IV Paper IV Theory 3 hrs 3 hrs 30 70 100


IV Open Elective Theory 3 hrs 3 hrs 30 70 100 100

V Paper V

(Compulsory)

Theory 3 hrs 3 hrs 30 70 100 150

Practical 3 hrs 3 hrs 20 30 50

V Paper VI

(Elective-I)

Theory 3 hrs 3 hrs 30 70 100 150


V Paper VII (Elective-II)

Theory 3 hrs 3 hrs 30 70 100 150


V Paper VII

(Elective-III)

Theory 3 hrs 3 hrs 30 70 100 150


VI Paper IX

(Compulsory)

Theory 3 hrs 3 hrs 30 70 100 150


VI Paper X

(Elective-I)

Theory 3 hrs 3 hrs 30 70 100 150


VI Paper XI (Elective-II)

Theory 3 hrs 3 hrs 30 70 100 150


VI Paper XII

(Elective-III)

Theory 3 hrs 3 hrs 30 70 100 150


BCA

I Paper-I Theory 3 Hrs 3 Hrs 30 70 100 100

II Paper-II Theory 3 Hrs 3 Hrs 30 70 100 100

Page 5 of 43

FIRST SEMESTER MATHEMATICS PAPER – I (ALGEBRA AND CALCULUS-I)

(3 lecture hours per week+3 hours of practical/week /25 per batch) (45 HOURS)

THEORY 70 marks

Unit-I:Matrices (15hrs)

Elementary row operations –Echelon form- Rank of a matrix by using row

reduced echelon form–inverse of a non-singular matrix by elementary row operations. System of ‘m’ linear equations in ‘n’ unknowns – matrices associated with linear equations – trivial and non-trivial solutions – criterion for existence of non-trivial solution of system of equations– Criterion for uniqueness of solutions – Problems.

Eigen values and Eigen vectors of a square matrix – Properties – Diagonalization of a real symmetric matrix – Cayley-Hamilton theorem (Statement only) – Applications of Cayley -Hamilton theorem to determine the powers of square matrices and inverses of non-singular matrices.

Unit-II: Differential Calculus (30 hrs) Successive (higher order) differentiation- nth derivatives of the functions: eax ,

(ax + b)n , log(ax + b), sin(ax + b) , cos(ax + b), eax sin(bx+ c), eax cos(bx + c) – Problems, Leibnitz theorem (with proof) – Monotonic functions – Maxima and Minima – Concavity, Convexity and points of inflection.

Polar coordinates –angle between the radius vector and the tangent at a point on a curve – angle of intersection between two curves – Pedal equations in Cartesian, polar and parametric forms – Derivatives of arc length in Cartesian, polar and parametric

form- Coordinates of centre of curvature – radius of curvature – circle of curvature –

evolutes.

Reference Books: 1. Natarajan, Manicavasagam Pillay and Ganapathy – Algebra 2. LipmanBers – Calculus, Volumes 1 and 2 3. N. Piskunov – Differential and Integral Calculus 4. B S Vatssa, Theory of Matrices, New Delhi: New Age International Publishers, 2005.

5. A R Vashista, Matrices, Krishna Prakashana Mandir, 2003. 6. N P Bali, Differential Calculus, India: Laxmi Publications (P) Ltd.., 2010. 7. Shanti Narayan and P K Mittal, Text book of Matrices, 5th edition, New Delhi, S Chand and Co. Pvt. Ltd.,2013. 8. Shanthi Narayan and P K Mittal, Differential Calculus, Reprint. New Delhi: S

Chand and Co. Pvt. Ltd., 2014.

Page 6 of 43

PRACTICALS – I Mathematics practical with Free and open Source Software(FOSS)

tools for computer programs (3 hours/ week/25 per batch)

LIST OF PROBLEMS 1. Introduction to Scilab and commands connected with matrices. 2. Computations with matrices. 3. Row reduced echelon form. 4. Solving system of linear equations. 5. Introduction to Maxima commands for derivatives and nth derivatives.

6. nth derivative without Leibnitz rule. 7. nth derivative with Leibnitz rule. 8. Scilab/Maxima commands for plotting functions. 9. Plotting of standard Cartesian curves using Scilab/Maxima. 10. Plotting of standard Polar curves using Scilab/Maxima. 11. Plotting of standard parametric curves using Scilab/Maxima. Note: The above list may be changed annually with the approval of the BOS (Mathematics).

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Pattern of question paper

Mathematics: Paper-I

(for the I semester)

Duration :3 hours Max marks:70.

Instructions:

1. All the Sections are compulsory. 2. Answer any 13 questions choosing atleast 4 from section-B and atleast 8

from section-C

3. All questions in sections ‘A’ carry 2 marks. 4. All questions in sections ‘B’ and ’C’ carry 4 marks.

Section-A

1. Answer any 9 questions. Each question carries 2 marks.

1.a to d (4 Questions from Unit-I : Matrices )

1.e to l ( 8 Questions from Unit-II : Differential Calculus.)

Section-B

6 questions from Unit-I : Matrices carrying 4 marks each

2.a to f

Section-C

12 questions from Unit-II : Differential Calculus carrying 4 marks each.

3.a to l

Page 8 of 43

SECOND SEMESTER MATHEMATICS PAPER – II (CALCULUS)


THEORY 70 marks

Unit-I :Limits, Continuity and Differentiability (15 hrs)

Limit and Continuity of a function –definitions, properties and problems– Theorems on continuity (only statements)

Differentiability – definitions, properties and problems –Mean value theorems: Rolle’s, Lagrange’s and Cauchy’s mean value theorems with geometrical meaning– Taylor’s and Maclaurin’s series expansions of functions of one variable (Problems only) - indeterminate forms.

Unit-II:Partial Derivatives (15 hrs)

Functions of two or more variables – Limit and Continuity of a function –Partial

derivatives –Homogeneous functions – Euler’s theorem and its extension – Partial derivatives of implicit and composite functions – Chain rule–Jacobians: Definition, Properties, Related problems- Taylor’s theorem for two variables (problems only)

Unit-III:Integral Calculus (15 hrs)

Reduction formulae for 𝑠𝑖𝑛𝑛𝑥 𝑑𝑥, 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥, 𝑡𝑎𝑛𝑛𝑥 𝑑𝑥, 𝑐𝑜𝑡𝑛𝑥 𝑑𝑥, 𝑠𝑒𝑐𝑛𝑥 𝑑𝑥,

𝑐𝑜𝑠𝑒𝑐𝑛𝑥 𝑑𝑥, 𝑠𝑖𝑛𝑚𝑥 𝑐𝑜𝑠𝑛𝑥 𝑑𝑥, 𝑥𝑛𝑠𝑖𝑛𝑥 𝑑𝑥, 𝑥𝑛𝑐𝑜𝑠𝑥 𝑑𝑥, 𝑥𝑛 𝑒𝑎𝑥𝑑𝑥 𝑎𝑛𝑑 𝑥𝑛 𝑙𝑜𝑔𝑥 𝑑𝑥

With definite limits and related problems.

Books for Reference:

1. Serge Lang – First Course in Calculus

2. Lipman Bers – Calculus Volumes 1 and 2

3. Shanthinarayan – Integral Calculus, New Delhi: S. Chand and Co. Pvt. Ltd.

4. N P Bali, Differential Calculus, India: Laxmi Publications (P) Ltd.., 2010.

5. S. Narayanan & T. K. Manicavachogam Pillay, Calculus, S. Viswanathan

Pvt. Ltd., vol. I & II 1996.

6. G B Thomas and R L Finney, Calculus and analytical geometry, Addison Wesley, 1995

Page 9 of 43

PRACTICALS –II Mathematics practical with FOSS tools for computer programs

(3 hours/ week/25 per batch) LIST OF PROBLEMS 1. Obtaining partial derivatives of some standard functions 2. Verification of Euler’s theorem, its extension and Jacobian. 3. Maxima program to illustrate left hand and right hand limits for functions.

4. Maxima program to illustrate continuity of a function. 5. Maxima program to illustrate differentiability of a function. 6. Maxima program to verify Rolle’s Theorem. 7. Maxima program to verify Lagrange’s theorem. 8. Maxima program to verify Cauchy’s mean value theorem. 9. Evaluation of limits by L’Hospital’s rule using Maxima.

10. Maxima commands for reduction formula with or without limits.

Page 10 of 43

Pattern of question paper Mathematics-paper-II (For the II semester)


Instructions:

1. All the Sections are compulsory. 2. Answer any 13 questions choosing at least 4 from sections ‘B’, ’C’ and ‘D’ 3. All questions in sections ‘A’ carry 2 marks. 4. All questions in sections ‘B’, ’C’ and ‘D’carry 4 marks.

Section-A


1.a to d ( 4 Questions from unit-I: Limits, Continuity and Differentiability.)

1.e to h (4 Questions from unit-II: Partial Derivatives.)

1.i to l (4 Questions from unit-III: Integral Calculus.)

Section-B

6 questions from Unit-I: Limits, Continuity and Differentiability carrying 4 marks

each

2.a to f

Section-C

6 questions from Unit-II: Partial Derivatives carrying 4 marks each.

3.a to f

Section-D

6 questions from Unit-III: Integral Calculus carrying 4 marks each.

4.a to f

Page 11 of 43

THIRD SEMESTER MATHEMATICS PAPER – III (Algebra and Differential Equations)


THEORY 70 marks

Unit-I:Group Theory (30 hrs)

Definition and examples of groups – Some general properties of Groups

(statements only), Subgroups – Cyclic groups problems and theorems. Cosets, Index of a group, Langrange’s theorem, consequences. Normal Subgroups, Quotient groups – Homomorphism – Isomorphism -Automorphism – Fundamental theorem of homomorphism. Permutation Groups-Even and odd permutation, Cyclic permutation, Caley’s theorem(with proof).

Unit-II:Differential Equations (15 hrs)

Differential equations of first order and first degree: Exact differential equations

and reducible to exact form, linear differential equations and Bernoulli’s form. Ordinary Linear differential equations with constant coefficients –

complementary function – particular integral –Cauchy – Euler differential equations – Simultaneous differential equations.

Books for References:

1. Daniel A Murray – Introductory Course to Differential equations

2. I. N. Herstien – Topics in Algebra. 3. Vashista, A First Course in Modern Algebra, 11th ed.: Krishna PrakasanMandir, 1980.

4. R Balakrishan and N.Ramabadran, A Textbook of Modern Algebra, 1st ed. New Delhi, India: Vikaspublishing house pvt.Ltd., 1991. 5. M D Raisinghania, Advanced Differential Equations, S Chand and Co. Pvt.Ltd., 2013.

6. S Narayanan and T K Manicavachogam Pillay, Differential Equations.: S V Publishers Private Ltd., 1981. 7. G F Simmons, Differential equation with Applications and historical notes, 2nd ed.: McGraw- Hill Publishing Company, Oct 1991.

Page 12 of 43

PRACTICALS –III Mathematics practical with FOSS tools for computer programs

(3 hours/ week, 25/per batch) LIST OF PROBLEMS:

1. Verifying whether given operator is binary or not. 2. To find identity element of a group. 3. To find inverse element of a group. 4. Finding all possible subgroups of a finite group. 5. Examples to verify Lagrange’s theorem.

6. Illustrating homomorphism and isomorphism of groups. 7. Verification of Normality of a given subgroup. 8. Verifying Cayley’s theorem and isomorphism theorems. 9. Examples for finding left and right coset and finding the index of a group. 10. Solution of Differential equation using Scilab/Maxima and plotting the Solution-I. 11. Solution of Differential equation using Scilab/Maxima and plotting the solution-II. 12. Solution of Differential equation using Scilab/Maxima and plotting the solution-III.

13. Solution of Differential equations using Scilab/Maxima and Plotting the solution-IV.

Note: The above list may be changed annually with the approval of the BOS

(Mathematics).

Page 13 of 43

Pattern of question paper Mathematics-paper-III (For the III semester)


Instructions:

1. All the Sections are compulsory. 2. Answer any 13 questions from sections B and C choosing atleast 8 from

section ‘B’ and atleast 4 from section ’C’ 3. All questions in sections ‘A’ carry 2 marks. 4. All questions in sections ‘B’ and ’C’ carry 4 marks.

Section-A


1.a to h(8 Questions from Unit-I:Group Theory)

1.i to l (4 Questions from Unit-II: Differential Equations.)

Section-B

12 questions from Unit-I:Group Theory carrying 4 marks each

2. a to l

Section-C

6 questions from Unit-II: Differential Equations carrying 4 marks each.

3.a to f

Page 14 of 43

FOURTH SEMESTER MATHEMATICS

PAPER – IV (Differential Equations and Integral Calculus) (3 lecture hours per week+3 hours of practical/week /25 per batch)

(45 HOURS)

THEORY 70 marks

Unit-I:Linear differential equations (15 hrs):

Solution of ordinary second order linear differential equations with variable

coefficients by the following methods: (i) When a part of complementary function is given. (ii) Changing the independent variable. (iii) Changing the dependent variable (Normal form). (iv) By the method of variation of parameters. (v) Exact equations.

Total differential equations – Necessary and sufficient condition for the equation P dx + Q dy + R dz = 0 to be exact (proof only for the necessary part) – Simultaneous equations.

Unit-II:Partial differential equations (15 hrs):

Basic concepts – Formation of a partial differential equations by elimination of

arbitrary constants and functions – Solution of partial differential equations –by Direct integration, Lagrange’s linear equations of the form Pp + Qq = R, Standard types of first order non-linear partial differential equations – Charpit’s method.

Unit-III:Line and Multiple Integrals (15 hrs):

Definition of a line integral and basic properties – Examples on evaluation of line

integrals – Definition of a double integral– Evaluation of double integrals under given limits - Evaluation of double integrals in regions bounded by given curves - Change of

variables from Cartesian to polar form– Surface areas - Triple integrals: definition and evaluation with the given limits only.

Books for References: 1. G. Stephonson – An Introduction to Partial Differential Equations. 2. B. S. Grewal – Higher Engineering Mathematics. 3. E. Kreyszig – Advanced Engineering Mathematics.

4. E. D. Rainville and P E Bedient – A Short Course in Differential Equations. 5. D. A Murray – Introductory Course in Differential Equations. 6. G. P. Simmons – Differential Equations. 7. F. Ayres – Differential Equations (Schaum Series). 8. Martin Brown – Application of Differential Equations. 9. S.C. Malik –Real Analysis. 10. Integral Calculus by N P Bali.

Page 15 of 43

PRACTICALS –IV Mathematics practical with FOSS tools for computer programs

(3 hours/ week ,25/per batch) LIST OF PROBLEMS:

1. Finding complementary function and particular integral of constant coefficient second and higher order ordinary differential equations. 2. Solving second order linear partial differential equations in two variables with constant coefficient. 3. Solutions to the problems on total and simultaneous differential equations.

4. Solutions to the problems on different types of Partial differential equations. 5. Evaluation of the line integral with constant limits. 6. Evaluation of the line integral with variable limits. 7. Evaluation of the double integral with constant limits. 8. Evaluation of the double integral with variable limits. 9. Evaluation of the triple integral with constant limits. 10. Evaluation of the triple integral with variable limits. 11. Scilab/Maxima programs for area and volume. Note: The above list may be changed annually with the approval of the BOS Mathematics).

Page 16 of 43

Pattern of question paper IV Mathematics-paper-IV (For the IV semester)


Instructions:

1. All the Sections are compulsory. 2. Answer any 13 questions choosing at least 4 from sections ‘B’, ’C’ and ‘D’ 3. All questions in sections ‘A’ carry 2 marks. 4. All questions in sections ‘B’, ’C’ and ‘D’carry 4 marks.

Section-A 1. Answer any 9 questions. Each question carries 2 marks.

1.a to d (4 Questions from Unit-I:Linear differential equations)

1.e to h (4 Questions from Unit-II:Partial differential equations)

1.i to l (4 Questions from Unit-III:Line and Multiple Integrals)

Section-B

6 questions from Unit-I:Linear differential equations carrying 4 marks

each

2. a to f

Section-C

6 questions from Unit-II:Partial differential equations carrying 4 marks each.

3. a to f

Section-D

6 questions from Unit-III:Line and Multiple Integrals carrying 4 marks each.

4.a to f

Page 17 of 43

FIFTH SEMESTER MATHEMATICS PAPER – V (Applied Mathematics-I)

Compulsory Paper (3 lecture hours per week+3 hours of practical/week /25 per batch)

(45 HOURS)

THEORY 70 marks

Unit-I:Laplace Transforms (15 hrs):

Definition and basic properties – Laplace transforms of cosat, sinat, tn, coshat

and sinhat –Laplace transform of eat f(t), tn f(t), f(t)/t, related problems – Laplace transform of derivatives and integrals of functions – Laplace transforms of α- functions – Laplace transform of derivatives of functions.

Unit-II:Inverse Laplace Transforms (15 hrs):

Inverse Laplace transforms – problems. Convolution theorem – Simple initial

value problems – Solution of first and second order differential equations with constant coefficients by Laplace transform method.

Unit-III:Fourier series (15 hrs): Introduction – Periodic functions – Fourier series and Euler formulae (statement

only) –Even and odd functions –Fourier series of functions with period 2π and 2L- Half range series-Half range sine and cosine series – Change of interval.


1. S.C Malik –Real Analysis 2. Murray R Speigel – Laplace Transforms 3. S.C.Malik and Savita Arora, Mathematical Analysis, 2nd ed. New Delhi, India: New Age international (P) Ltd., 1992 4. Richard R Goldberg, Methods of Real Analysis, Indian ed. 5. Asha Rani Singhal and M. K Singhal, A first course in Real Analysis 6. E.Kreyszig- Advanced Engineering Mathematics, Wiely India Pvt. Ltd. 7. Raisinghania M. D., Laplace and Fourier Transforms S. Chand publications.

Page 18 of 43

PRACTICALS –V Mathematics practical with FOSS tools for computer programs

(3 hours/ week ,25/per batch)

LIST OF PROBLEMS: 1. To plot periodic functions with period 2π.

2. To plot periodic functions with period 2L. 3. To find full range trigonometric Fourier series of some simple functions with

period 2π and 2L. 4. Plotting of functions in half-range and including their even and odd extensions. 5. To find the half-range sine series of simple functions. 6. To find the half-range cosine series of simple functions. 7. Finding the Laplace transforms of some standard functions. 8. Finding the inverse Laplace transform of simple functions. 9. Implementing Laplace transform method of solving ordinary linear differential equations of first and second order with constant coefficients.

Note: The above list may be changed annually with the approval of the BOS (Mathematics).

Page 19 of 43

Pattern of question paper Mathematics : paper-V

(For the V semester)


Instructions:

1. All the Sections are compulsory.

2. Answer any 13 questions choosing at least 4 from sections ‘B’, ’C’ and ‘D’ 3. All questions in sections ‘A’ carry 2 marks. 4. All questions in sections ‘B’, ’C’ and ‘D’carry 4 marks.

Section-A 1.Answer any 9 questions. Each question carries 2 marks.

1.a to d (4 questions from unit-I:Laplace Transforms.)

1.e to h (4 questions from Inverse unit-II: Inverse Laplace Transforms.)

1.i to l (4 questions from unit-III: Fourier Series.)

Section-B

6 questions from Unit-I: Laplace Transforms carrying 4 marks each

2. a to f

Section-C 6 questions from Unit-II: Inverse Laplace Transforms 4 marks each.

3. a to f

Section-D

6 questions from Unit-III: Fourier Series carrying 4 marks each.

4. a to f

Page 20 of 43

FIFTH SEMESTER MATHEMATICS

PAPER – VI (Algebra and Vector Calculus) Elective paper-I


THEORY 70 marks

Unit-I:Rings, Fields and Integral Domains (30 hrs):

Rings – Examples – Integral Domains – Division rings – Fields – Subrings-

Subfields –Characteristic of a ring – Imbedding of a ring into another ring – The field of quotients – Ideals – Algebra of Ideals - Maximal ideals – Prime ideals – Principal ideal ring – Divisibility in an integral domain – Units and Associates – Prime Elements – Quotient rings – homomorphism of rings – Kernel of a ring homomorphism – Fundamental theorem of homomorphism- Polynomial rings – Divisibility – Irreducible polynomials – Division Algorithm – Greatest Common Divisors – Euclidean Algorithm – Unique factorization theorem – Prime fields – Properties – Eisenstein’s Criterion of

irreducibility.

Unit-II:Vector differential Calculus (15 hrs):

Scalar and vector point functions –Gradient, Divergence, Curl, Laplacian,

Solenoidal and Irrotational vectors– Directional derivatives - Vector identities.

Books for References:

1. I. N. Herstien – Topics in Algebra. 2. G. D. Birkhoff and S Maclane – A brief Survey of Modern Algebra. 3. T. K. Manicavasagam Pillai and K S Narayanan – Modern Algebra Volume 2 4. J B Fraleigh – A first course in Abstract Algebra. 5. Leadership project – Bombay university- Text book of mathematical analysis 6. B.S Grewal – Higher engineering mathematics. 7. Murray R Spiegel – Theory and problems of vector calculus. 8. Shanthinarayan and J N Kapur – A text book of Vector calculus.

Page 21 of 43

PRACTICALS –VI Mathematics practical with FOSS tools for computer programs

(3 hours/ week ,15 ,25 per batch) LIST OF PROBLEMS:

1. Examples on different types of rings. 2. Examples on integral domains and fields.

3. Examples on subrings, ideals and subrings which are not ideals. 4. Homomorphism and isomorphism of rings- illustrative examples. 5. Solving polynomial equations using Scilab/Maxima. 6. Finding G.C.D of polynomials. 7. Finding units and associates. 8. Test for rational roots. 9. To demonstrate the physical interpretation of gradient, divergence and curl. 10. Using cyclic notations to derive different vector identities. 11. Using cyclic notations to derive some more vector identities.

Note: The above list may be changed annually with the approval of the BOS

(Mathematics).

Page 22 of 43

Pattern of question paper Mathematics-paper-VI



Instructions:

1. All the Sections are compulsory.

2. Answer any 13 questions from sections B and C choosing atleast 8 from sections -B and atleast 4 from section-C.

3. All questions in sections ‘A’ carry 2 marks. 4. All questions in sections ‘B’ and ’C’ carry 4 marks.

Section-A


1.a to h (8 Questions from unit-I: Rings and Fields)

1.i to l (4 Questions from unit-II:Vector Differential Calculus.)

Section-B

12 questions from Unit-I: Rings and Fields carrying 4 marks each

2. a to l

Section-C

6 questions from Unit-II: Vector Differential Calculus carrying 4 marks each.

3.a to f

Page 23 of 43

FIFTH SEMESTER MATHEMATICS Elective PAPER-II (Algebra, Number Theory and Calculus-II, Paper-VII)

(3 lecture hours per week) 45 HOURS

UNIT-I: THEORY OF NUMBERS 15 Hrs

Divisibility, Elementary results, Prime numbers, Fundamental theorem of Arithmetic, Linear

Diophantine equations, Linear congruence’s and its properties, Complete residue system and reduced

residue system, Euler’s function, Euler’s theorem, Fermat’s theorem, Wilson’s theorem.

UNIT-II: THEORY OF EQUATIONS 15 Hrs

Relation between roots and coefficients, Symmetric functions, Transformations, Reciprocal equations,

Descartes rule of signs , Multiple roots, Solving cubic equations by Cardon’s method, Solving

biquadratic equations by Descartes method.

UNIT-III : THE RIEMANN INTEGRAL 15 Hrs Upper and lower sums, Criterion for integrability, Integrability of continuous functions and monotone

functions, Fundamental theorem of Calculus, Change of variables, Integration by parts, Theorems of

Integral Calculus. First and Second Mean Value theorems .

Reference Books

1. Serge Lang – First course in Calculus, springer - 1964, 2013

2. LipmanBers – Calculus, Volumes 1 and 2, IBH 1969

3. Frank Ayres – Advanced Calculus, the Mc Graw Hill 6th

edition - 1999

4. Barnard and Child- Higher Algebra, Nadu press - 2000

5. Text book on Gorakhprasad First course in the – Differntial Calculus, Pothishala Ltd 2014

6. J.V. Uspensky- Theory of equations, Mc Graw Hill Book Co. Ltd 1948

7. Natarajan, Manicavachagam pillay and Ganapathy- Algebra, S. Viswanath Pvt. Ltd 11th

edition 2007.

8. PettoFrezzo and Byrkit- Elements in Number theory, orange club 2nd

edition 1994

9. Niven and Zuckerman –An introduction to Theory of Numbers, John Wiley & sons,Inc1991

10. Hua – Introduction to Number Theory- Springer Verlag 1982

Page 24 of 43

PRACTICALS –VII Mathematics practical with FOSS tools for computer programs

(3 hours/ week ,15 ,25 per batch) LIST OF PROBLEMS:

1. To compute the G.C.D of two numbers.

2. To compute the L.C.M. of two numbers.

3. To check if two numbers are relatively prime.

4. To check if a number is prime.

5. Compute the list of all primes up to n.

6. Finding solutions of quadratic polynomials.

7. Visualizing the concept of the Riemann integral using lower and upper

sums.

8. Describe what happens to the values of the lower and upper sum when you increase the number of rectangles.

Page 25 of 43

Pattern of question paper Elective PAPER-II (Algebra and Calculus-II, Paper-VII)



Instructions:

1. All the Sections are compulsory. 2. Answer any 13 questions choosing at least 4 from sections ‘B’, ’C’ and ‘D’

3. All questions in sections ‘A’ carry 2 marks. 4. All questions in sections ‘B’, ’C’ and ‘D’ carry 4 marks.

Section-A 1.Answer 9 five questions. Each question carries 2 marks.

1.a to d (4 questions from UNIT-I: THEORY OF NUMBERS.)

1.e to h (4 questions from UNIT-II: THEORY OF EQUATIONS.)

1.i to l (4 questions from UNIT-III : THE RIEMANN INTEGRAL.)

Section-B

6 questions from UNIT-I: THEORY OF NUMBERS carrying 4 marks each

2. a to f

Section-C 6 questions from UNIT-II: THEORY OF EQUATIONS carrying 4 marks each.

3.a to f

Section-D

6 questions from UNIT-III : THE RIEMANN INTEGRAL carrying 4 marks each.

4.a to f

Page 26 of 43

FIFTH SEMESTER MATHEMATICS Elective PAPER-III (Statistics and Probability, Paper-VIII)

(3 lecture hours per week) 45 Hours

UNIT-I : STATISTICS – I 15 Hrs

Introduction, Classification of data, Graphical representation - Histogram, frequency polygon,

frequency curve, ogives. Measures of central tendency, measures of dispersion, skewness, coefficient

of skewness.

UNIT-II : STATISTICS – II 15 Hrs

Curve fitting(Least squares method) –Fitting of a straight line y=a+bx, fitting of a second degree

parabola y= a +bx +cx2, fitting of a curve of the form y=ab

x. Correlation and Regression- Correlation

and Correlation coefficient r, formula for r, Regression, Regression lines, Regression coefficients.

UNIT-III : PROBABILITY 15 Hrs

Introduction, definition of Exhaustive and mutually exclusive events. Addition law of probability-

Independent events, multiplication law of probability, Bayes theorem, random variables, probability

density function, cumulative distributive function. Binomial distribution, Poisson distribution.

Reference Books

1. Grewal- Higher Engineering Mathematics, Wiley Eastern Ltd., - 1996.

2. Kreyszig –Advanced Engineering Mathematics, Wiley Eastern Ltd., - 2010.

3. Gupta – Statistics, S Chand and Co. Ltd., - 2015.

4. Goon A.M., Gupta M.K. and Dasgupta B (1986); Fundamentals of Statistics, world press

5. Spiegal M.R (1961); Theory and problems of Statistics, Schaum publishing co.

6. Fruend J.E. and Ronald E. Walpole (1987); Mathematical Statistics, PHI..

7. Introduction to probability and Statistics ( Shaum’s outlines)- Seymour Lipschutz and John

Schiller, Total Mc Graw Hill education pvt Ltd.

8. Walpole, Myres and Keying Ye- Probability and statistics for engineers and scientists, pearson

( eighth edition).

Page 27 of 43

PRACTICALS –VIII Mathematics practical with FOSS tools for computer programs

(3 hours/ week ,25 per batch) LIST OF PROBLEMS:

1. Measures of location and dispersion (absolute and relative).

2. Computation of Correlation coefficient for raw and grouped data.

3. Computation of rank correlation coefficient.

4. Computation of regression equations for raw and grouped data.

5. Curve fitting by the method of least squares

(i) y= ax +b

(ii) y= ax2+bx+c

6. Curve fitting by the method of least squares

(i) y=aebx (ii)y=axb.

7. Fitting of binomial and Poisson distribution and tests of goodness of fit.

8.. Exact tests with respect to mean variance and coefficient of correlation.

Page 28 of 43

Pattern of question paper Elective PAPER-III (Statistics and Probability, Paper-VIII)

(For the V semester) Duration :3 hours Max marks:70.

Instructions:

1. All the Sections are compulsory. 2. Answer any 13 questions choosing at least 4 from sections ‘B’, ’C’ and ‘D’ 3. All questions in sections ‘A’ carry 2 marks. 4. All questions in sections ‘B’, ’C’ and ‘D’ carry 4 marks.

Section-A

1.Answer 9 five questions. Each question carries 2 marks.

1.a to d (4 questions from UNIT-I : STATISTICS – I.)

1.e to h (4 questions from UNIT-II : STATISTICS – II.)

1.i to l (4 questions from UNIT-III : PROBABILITY.)

Section-B 6 questions from UNIT-I : STATISTICS – I carrying 4 marks each

2. a to f

Section-C

6 questions from UNIT-II : STATISTICS – II carrying 4 marks each.

3.a to f

Section-D 6 questions from UNIT-III : PROBABILITY carrying 4 marks each.

4.a to f

Page 29 of 43

SIXTH SEMESTER MATHEMATICS PAPER – IX (Calculus and linear Algebra)

Compulsory Paper (3 lecture hours per week+3 hours of practical/week /25 per batch)

(45 HOURS)

THEORY 70 marks

Unit-I:Real Sequences (15 hrs):

Sequence of real numbers – Bounded and unbounded sequences – Infimum and supremum of a sequence – Limit of a sequence – Sum, product and quotient of limits – Standard theorems on limits – Convergent, Divergent and Oscillatory sequences – Standard properties – Monotonic sequences and their properties – Cauchy’s general principle of convergence.

Unit-II:Infinite Series (15hrs):

Infinite series of real numbers – Convergence – Divergence and Oscillation of series – Properties of convergence – Positive term series – Geometric series –

p series- Comparison tests (proof for limit form only) – D’Alembert’s ratio test –Raabe’s test(statement only) – Cauchy’s root test – Alternating series : Leibnitz’s test for Alternating series-Integral test.

Unit-III:Vector Spaces (15 hrs):

Vector Spaces – Introduction – Examples – Vector subspaces – Criterion for a subset to be a subspace – Algebra of Subspaces – Linear Combination – Linear Span – Linear dependence and linear Independence of vectors – Theorems on linear dependence and linear independence – Basis and Dimension of a vector space with theorems – Quotient spaces - Homomorphism of vector spaces–

Isomorphism of vector spaces – Direct Sums


1. S.C Malik –Real Analysis 2. S.C.Malik and Savita Arora, Mathematical Analysis, 2nd ed. New Delhi, India: New Age international (P) Ltd., 1992

3. Richard R Goldberg, Methods of Real Analysis, Indian ed. 4. Asha Rani Singhal and M. K Singhal, A first course in Real Analysis. 5. Stewart – Introduction to Linear Algebra.

6. T. K. Manicavasagam Pillai and K S Narayanan – Modern Algebra Volume 2. 7. S. Kumaresan – Linear Algebra 8. G. D. Birkhoff and S Maclane – A brief Survey of Modern Algebra. 9. Saymour Lipschitz – Theory and Problems of Linear Algebra.

Page 30 of 43

PRACTICALS –VII

Mathematics practical with FOSS tools for computer programs

(3 hours/ week ,25 per batch)

LIST OF PROBLEMS:

1. Illustration of convergent, divergent and oscillatory sequences using Scilab/Maxima.

2. Using Cauchy’s criterion to determine convergence of a sequence (simple examples). 3. Illustration of convergent, divergent and oscillatory series using Scilab/Maxima. 4. Scilab/Maxima programs to find the sum of the series and its radius of convergence. 5. Using Cauchy’s criterion on the sequence of partial sums of the series to determine convergence of series. 6. Vector space, subspace – illustrative examples. 7.Expressing a vector as a linear combination of given set of vectors.

8. Examples on linear dependence and independence of vectors. 9. Basis and Dimension – illustrative examples. 10. Verifying whether a given transformation is linear. 11. Finding matrix of a linear transformation. 12.Problems on rank and nullity.

Note: The above list may be changed annually with the approval of the BOS (Mathematics).

Page 31 of 43


Mathematics-paper-IX

Compulsory Paper (For the VI semester)


Instructions:

1. All the Sections are compulsory. 2. Answer any 13 questions choosing at least 4 from sections ‘B’, ’C’ and ‘D’ 3. All questions in sections ‘A’ carry 2 marks.

4. All questions in sections ‘B’, ’C’ and ‘D’ carry 4 marks.

Section-A 1.Answer any 9 questions. Each question carries 2 marks.

1.a to d (4 questions from unit-I: Real Sequences.)

1.e to h (4 questions from unit-II: Infinite series.)

1.i to l (4 questions from unit-III: Vector calculus.)

Section-B

6 questions from Unit-I:Real Sequences carrying 4 marks each

2. a to f

Section-C

6 questions from Unit-II: Infinite series carrying 4 marks each.

3.a to f

Section-D 6 questions from Unit-III: Vector Spaces carrying 4 marks each.

4.a to f

Page 32 of 43

SIXTH SEMESTER MATHEMATICS PAPER – X (Complex Analysis and Numerical Analysis) Elective paper-I

(3 lecture hours per week+3 hours of practical/ week ,25/per batch) (45 HOURS)

THEORY 70 marks

Complex Analysis

Unit-I:Functions of a Complex Variable (15 hrs):

Equation to a circle and a straight line in complex form- Limit of a function – Continuity and differentiability – Analytic functions – Singular points – Cauchy-Riemann equations in Cartesian and polar forms – Necessary and sufficient condition for function to be analytic – Harmonic functions – Real and Imaginary parts of an analytic function are harmonic – Construction of analytic function) Milne Thomson Method – ii) using the concept of Harmonic function.

Unit-II:Complex Integration (15 hrs):

The complex Line integral – Examples and Properties – Proof of Cauchy’s Integral

theorem using Green’s Theorem – Direct consequences of Cauchy’s theorem – The Cauchy’s integral formula for the function and the derivatives – Applications to the evaluations of simple line integrals – Cauchy’s Inequality – Liouville’s theorem – Fundamental theorem of Algebra.

Unit-III:Numerical Analysis (15 hrs) Numerical solutions of Algebraic and transcendental equations – Bisection method –The method of false position – Newton – Raphson method. Numerical solutions of first order linear differential equations – Euler – Cauchy method – Euler’s modified method – Runge-Kutta method of fourth – Picard’s method.

Finite differences – Forward and backward differences – shift operator –Interpolation –Newton – Gregory forward and backward interpolation formulae – Lagrange’s interpolation formula Numerical integration – General quadrature formula – Trapezoidal Rule – Simpson’s 1/3 rule – Simpson’s 3/8 th rule, Weddle’s rule.

Books for References: 1. L. V. Ahlfors – Complex Analysis 2. Bruce P. Palica – Introduction to the Theory of Function of a Complex Variable. 3. Shanthinarayan – Theory of Functions of a Complex Variable.

4. S. Ponnuswamy – Foundations of Complex Analysis. 5. A R Vashista, Complex Analysis, Krishna PrakashanaMandir, 2012. 6. H. C Saxena – Finite Difference and Numerical Analysis 7. S. S. Shastry- Introductory Methods of Numerical Analysis 8. B. S. Grewal – Numerical Methods for Scientists and Engineers. 9. M K Jain, S R K Iyengar, and R K Jain, Numerical Methods for Scientific and Engineering Computation, 4thed. New Delhi, India: New Age International, 2012.

Page 33 of 43

PRACTICALS –VIII

Mathematics practical with FOSS tools for computer programs

(3 hours/ week,25/batch)

LIST OF PROBLEMS:

1. Some problems on Cauchy-Riemann equations (polar form). 2. Implementation of Milne-Thomson method of constructing analytic functions (simple examples). 3.Illustrating orthogonality of the surfaces obtained from the real and imaginary

parts of an analytic function.

4. Verifying real and imaginary parts of an analytic function being harmonic (in polar coordinates).

5. Illustrating the angle preserving property in a transformation. 6. Illustrating that circles are transformed to circles by a bilinear transformation. 7. Examples connected with Cauchy’s integral theorem. 8. Scilab/Maxima programs on Interpolations with equal intervals. 9.Scilab/Maxima programs on Interpolations with unequal intervals. 10. Solving algebraic equation (Bisection method). 11.Scilab/Maxima programs to evaluate integrals using Simpson’s rule. 12.Scilab/Maxima programs to evaluate integrals using Simpson’s rule.

13. Solving algebraic equation (Regular-Falsi and Newton-Raphson methods). 14.Solving ordinary differential equation by modified Euler’s method. 15.Solving ordinary differential equation by Runge-Kutta method of 4th order. Note: The above list may be changed annually with the approval of the BOS (Mathematics).

Page 34 of 43


Mathematics-paper-X

Elective paper-I (For the VI semester)


Instructions:




1.a to d (4 questions from unit-I:Functions of a Complex Variable.)

1.e to h (4 questions from unit-II: Complex Integration.)

1.i to l (4 questions from unit-III: Numerical Analysis.)

Section-B

6 questions from Unit-I: Functions of a Complex Variable carrying 4 marks each

2. a to f

Section-C

6 questions from Unit-II: Complex Integration carrying 4 marks each.

3.a to f

Section-D 6 questions from Unit-III Numerical Analysis carrying 4 marks each.

4.a to f

Page 35 of 43

SIXTH SEMESTER MATHEMATICS Paper-XI (Applied Mathematics-II)

Elective PAPER-II (3 lecture hours per week)

45 Hours

UNIT - I : SERIES SOLUTION OF DIFFERENTIAL EQUATIONS AND

SPECIAL FUNCTIONS 15 Hrs

Validity of series solution, Series solution when x=0 is an ordinary point, Bessel’s equation,

Recurrence formulae for Jn(x), Expansions for J0 and J1, Value of J1/2, Generating function for

Jn(x).Equations reducible to Bessel’s equation, Orthogonality of Bessel functions, Fourier – Bessel’s

expansion of f(x).

UNIT - II : FOURIER TRANSFORMS 15 Hrs

Introduction, Definition, A Fourier Sine and Cosine Integrals and Complex form of Fourier transforms,

Finite Fourier Sine and Cosine Transforms, Properties of F-Transforms, Convolution Theorem for F-

Transforms, Parseval’s identity for F-Transforms, Relation between Fourier Laplace Transforms by

method of residues, Application of transforms to boundary value problems

UNIT - III : Z-TRANSFORMS 15 Hrs

Introduction, Definition, Some Standard Z-transforms, Linearity property , Damping rule, Some

Standard results, Shifting un to the right and to the left, Multiplication by n, Two Basic theorems, Some

useful Z-transforms and inverse Z-transforms, Convolution theorems, Convergence of Z-transforms,

Two-sided Z-transform, Evaluation of inverse Z-transforms, Application to Difference equations.

Reference Books

1. Brikhoff and Maclane – Brief survey of Modern Algebra, IBH – 2008.

2. C.L. Liu – Elements of Discrete Mathematics, Tata McGraw Hill, 2nd

edition, 2000.

3. Grewal – Higher Engineering Mathematics, Wiley Eastern Ltd., - 2016

4. Kreyszig – Advanced Engineering Mathematics, Wiley Eastern Ltd., 2010.

5. Dass H.K. Advanced Engineering Mathematics, S. Chand and Co. Ltd., - 2008.

6. Jain Iyengar, Advanced Engineering Mathematics.,Narosa – 2002.

7. Zill Cullen, Advanced Engineering Mathematics.,Narosa – 2010.

8. R. K. Jain, S. R. R. Iyengar, Advanced Enginnering Mathematics, Narosa Publishing House -

2014

9. K. S. Chandrashekar, Engineering Mathematics IV, Sudha Publications – 2011.

Page 36 of 43

PRACTICALS – Paper-XI, Elective PAPER-II Mathematics practical with FOSS tools for computer programs


LIST OF PROBLEMS:

1. To find Fourier transforms of standard functions.

2. To find Fourier sine transforms.

3. To find Fourier cosine transforms.

4. To find inverse Fourier transforms.

5. To find inverse Fourier sine transforms.

6. To find inverse Fourier cosine transforms.

7. Solutions to the problems on different types of partial differential equations.

8. Solution of Cauchy problem for first order partial differential equation.

9. Plotting the characteristics for the first order partial differential equation.

10. Plot the integral surfaces of a given first order partial differential equation with initial data.

Note: The above list may be changed annually with the approval of the BOS in Mathematics

Page 37 of 43

Pattern of question paper Paper-XI (Applied Mathematics-II)

Elective PAPER-II

(For the VI semester)


Instructions:

1. All the Sections are compulsory. 2. Answer any 13 questions choosing at least 4 from sections ‘B’, ’C’ and ‘D’

3. All questions in sections ‘A’ carry 2 marks. 4. All questions in sections ‘B’, ’C’ and ‘D’ carry 4 marks.


1.a to d (4 questions from UNIT - I : SERIES SOLUTION OF DIFFERENTIAL

EQUATIONS AND SPECIAL FUNCTIONS.)

1.e to h (4 questions from UNIT - II : FOURIER TRANSFORMS.)

1.i to l (4 questions from UNIT - III : Z-TRANSFORMS)

Section-B

6 questions from UNIT - I : SERIES SOLUTION OF DIFFERENTIAL

EQUATIONS AND SPECIAL FUNCTIONS carrying 4 marks each

2. a to f

Section-C

6 questions from UNIT - II : FOURIER TRANSFORMS carrying 4 marks each.

3.a to f

Section-D 6 questions from UNIT - III : Z-TRANSFORMS carrying 4 marks each.

4.a to f

Page 38 of 43

SIXTH SEMESTER MATHEMATICS Paper-XII (Linear Programming)

Elective PAPER-III (3 lecture hours per week)

45 Hours Unit-I:

Linear programming Problems, Graphical approach for solving some linear

programming problems, Basic feasible solution(B.F.S),degenerate and non-

degenerate BFS, examples of basic solution which are not feasible. upper bound on

the number of B.F.S. upper bound on the absolute value of the basic variables

.convex sets, supporting and separating hyper planes.

Unit-II:

Theory of Simplex method, optimality and unbondedness, the simplex algorithm,

simplex method in tableau format, degeneracy, introduction to artificial variables.

two-phase method, big-M method and their comparison, revised simplex method.

Unit-III:

Duality, formulation of the dual problem, prime-dual relationships, fundamental

theorem of duality .Farka’s theorem. Complementary slackness theorem. economic

interpretation of the dual, Dual simplex method. sensitivity analysis

Reference Books

1.Kanti Swarup, P.K.Gupta and Manmohan,Operation Research,reprint,New Delhi,India:

Sultan Chand & sons,1994.

2.Mokhtar S Bazaraa,John J Jarvis and Hanif D.Sherali,Linear programming and

Networkflows,2nd

Ed.,John Wiley and Sons,India,2004.

3.F.S.Hillier and G.J.Lieberman Introduction to operations Research,8Th

Ed.,Tata

McGrawHill,Singapore,2004.

4.Hamdy A Taha,Operations Research,An Introduction,8 th Ed., Prentice-Hall India,2006.

5.G Hadley,Linear programming,reprint,NewDelhi:Narosa Publishing House,2002.

Suggested web links:

1.http://www.zweigmedia.com/realworld/summary4.html

2.http://people.brunel.ac.uk/~mastjjb/jeb/or/impore.html

3.http://www2.isye.gatech.edu/~jswann/casestudy/assign.html

4.http://mathworld.wolfram.com/gametheory.html.

Page 39 of 43

PRACTICALS – Paper-XII, Elective PAPER-III Mathematics practical with FOSS tools for computer programs


List of Programmes:

1.Introduction to linear programming.

2.Linear inequalities and bounds-1.

3. Linear inequalities and bounds-2.

4. Linear inequalities and bounds-3.

5.Solution of linear system of equations-1.

6. Solution of linear system of equations-2

7. Solution of linear system of equations-3.

8.finding the feasible solution of an LPP.

9.SolvingLPP by graphical method-1

10. Solving LPP by graphical method-2.

11. Solving LPP by graphical method-3.

12.Finding optimal solution of an LPP.

Note: The above list may be changed annually with the approval of the BOS in Mathematics

Page 40 of 43

Pattern of question paper Paper-XII (Linear Programming)

Elective PAPER-III

(For the VI semester) Duration :3 hours Max marks:70.

Instructions:



Section-A


1.a to d (4 questions from UNIT – I)

1.e to h (4 questions from UNIT - II )

1.i to l (4 questions from UNIT - III )

Section-B 6 questions from UNIT - I : carrying 4 marks each

2. a to f

Section-C

6 questions from UNIT - II : carrying 4 marks each.

3.a to f

Section-D 6 questions from UNIT - III : carrying 4 marks each.

4.a to f

Page 41 of 43

Pattern of question paper Elective PAPER-IV (Probability theory and Special Functions)

(For VI semester) Duration :3 hours Max marks:70.

Instructions:

1. Section ‘A’ is compulsory.

2.Answer any 5 full questions from sections ‘B’,’C’ and ‘D’ without omitting any section.

3.All questions in sections ‘B’,’C’ and ‘D’ carry equal marks.

Section-A


5 Questions from Unit-I Probability theory.

5 Questions from Unit-II Sampling Theory

5 Questions from Unit-III Special Functions

Section-B

Two questions from Unit-I Probability theory carrying 12 marks each

2. a), b), and c).

3. a), b), and c).

Section-C

Three questions from Unit-II Sampling Theory carrying 12 marks each.

4. a), b), and c).

5. a), b), and c).

6. a), b), and c).

Section-D

Three questions from Unit-III III Special Functions carrying 12 marks each.

7. a), b), and c).

8. a), b), and c).

9. a), b), and c).

Page 42 of 43

FOURTH SEMESTER MATHEMATICS Open Elective Paper (Discrete Mathematics)

(3 lecture hours per week) (45 HOURS)

THEORY 70 marks

Unit-I:Number system and Set theory (15 hrs):

The number systems: Natural numbers, Integers, Rational and Irrational numbers, Real numbers, Complex numbers, Prime numbers. The concept of Sets: Representation of sets, Empty sets and equal sets, Subsets, Basic set identities, Power set and universal set, set operations (union, intersection, and difference, Compliment of a set). Venn diagram, DeMorgan’s laws.

Unit-II:Relations and Functions (15 hrs):

Ordered pairs and Cartesian product of two sets, Relations-Definition, Examples,

Types, Equivalence relations. Matrix representation of a relation. Functions-Definition, Examples, types of functions (one-one, onto, many-one functions

with examples), Invertible functions, Examples.

Unit-III: Mathematical Logic (15 hrs):

Introduction, Propositions, logical connectives and compound propositions, logical equivalence, converse, inverse, contrapositive, Tautology and contradiction.


1. Higher Engineering Mathematics by B S Grewal.

2. Discrete math for Computer Science Students by Ken Bogart, Scot Drysdale, Cliff

Stein.

3. Discrete Mathematical Structures by Somusundaram.

4. Discrete Mathematics by Sudhir Dawra.

5. Discrete Mathematics by J K Sharma

Page 43 of 43


Discrete Mathematics (Open Elective)

(For IV semester) Duration :3 hours Max marks:70.

Instructions:



Section-A


1.a to d (4 questions from UNIT-I: Number system and Set theory)

1.e to h (4 questions from UNIT-II: Relations and Functions.)

1.i to l (4 questions from UNIT-III : Mathematical Logic.)

Section-B

6 questions from UNIT-I: Number system and Set theory carrying 4 marks each

2. a to f

Section-C

6 questions from UNIT-II: Relations and Functions carrying 4 marks each.

3.a to f

Section-D

6 questions from UNIT-III : Mathematical Logic carrying 4 marks each.

4.a to f

Documents

Government Science College, Hassan (Autonomous) · Page 3 of 43 Government Science College(Autonomous), Hassan-573201 Department of Mathematics Date:06/07/2017 Proceedings of the