Page 1 of 32 DEPARTMENT OF MATHEMATICS Proposed Syllabus for B.Sc. Mathematics Paper for 6 Semesters under Choice Based Credit Scheme (CBCS) Aims and objectives of introducing new syllabus • To set up a mathematical laboratory in every college 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 in 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 student 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]
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Page 1 of 32
DEPARTMENT OF MATHEMATICS
Proposed Syllabus for B.Sc. Mathematics Paper for 6 Semesters under Choice Based Credit Scheme (CBCS)
Aims and objectives of introducing new syllabus
• To set up a mathematical laboratory in every college 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 in 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 student 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.
Total: 42 Hrs Mathematics practical with Free and open Source Software (FOSS) tools for
computer programs (3 hours/ week per batch of not more than 15 students)
LIST OF PROBLEMS
1. Introduction to Scilab and commands connected with matrices. 2. Computations with matrices. 3. Row reduced echelon form and normal form. 4. Establishing consistency or otherwise and solving system of linear equations. 5. Scilab/Maxima programs to illustrate left hand and right hand limits for discontinuous
functions. 6. Scilab/Maxima programs to illustrate continuity of a function. 7. Scilab/Maxima programs to illustrate differentiability of a function. 8. Scilab/Maxima programs to solve Exact Differential Equations. 9. Scilab/Maxima programs to solve Linear Differential Equations. 10. Scilab/Maxima programs to solve Bernoulli's Differential Equations. 11. Scilab/Maxima programs to solve first order but not of first degree differential
equations -solvable for p only. Text Books:
1. Shanti Narayan and P K Mittal , Text book of Matrices, 5th edition, New Delhi, S Chand and Co. Pvt. lt
2. . Shanthi Narayan and P K Mittal, Differential Calculus, Reprint. New Delhi:SChand and Co. Pvt. Ltd.,
3. www.scilab.org 4. wxmaxima.sourceforge.net 5. www.geogebra.org Reference Books: 1.F.Ayres: Matrices 2.Shanti Narayan: Matrices 3.Shanti Narayan. : Differential Calculus (S.Chand & Co). 4. Murray.R. Spiegel: Advanced Calculus (Schaum Publicity Co). 5. Lipman Bers : Calculus Vol-I and II (IBM). 6. M D Raisinghania: Advanced Differential Equations, S Chand and Co. Pvt. Ltd., 2013. 7. F Ayres: Schaum's outline of theory and problems of Differential Equations, 1st ed. USA: McGraw-Hill, 2010. 8. S Narayanan and T K Manicavachogam Pillay:Differential Equations.: S V Publishers Private Ltd., 1981. 9. G F Simmons: Differential equation with Applications and historical notes, 2nd ed.: McGraw- Hill Publishing Company, Oct 1991. 10.Murry R : An introduction to Differential Equations. 11.F.Chorlton : Ordinary Differential Equations
Total: 42 Hrs Mathematics practical with Free and open Source Software (FOSS) tools for
computer programs (3 hours/ week per batch of not more than 15 students)
LIST OF PROBLEMS
1. To find identity element of a group and inverse element of a group. 2. Finding all possible subgroups of a finite group. 3. Introduction to Maxima and commands for derivatives and nth derivatives and nth derivative with Leibnitz's rule. 4. Finding first and second order partial derivatives for given functions. 5. Verification of Euler's Theorem for given homogeneous function. 6. Verification of Extended Euler's Theorem for for given homogeneous function. 7. Finding Jacobian constant for given two functions. 8. Scilab/Maxima programs to solve Differential Equations with Constant Coefficients by Rule-I to Rule-V. 9. Scilab/Maxima programs to solve Cauchy's and Legendre's Differential Equations.. 10. Scilab/Maxima programs to solve Differential Equations and find particular solutions. Text Books: 1.Shanti Narayan. : Differential Calculus (S.Chand & Co). 2. Murray.R. Spiegel: Advanced Calculus (Schaum Publicity Co). 3. Lipman Bers : Calculus Vol-I and II (IBM). 4. Rudraiah et al : College Mathematics Vol-I (Sapna, Bangalore). 5. Text Book of B.Sc Mathematics- G.K.Ranganath Reference Books: 1. I. N. Herstien – Topics in Algebra. 2. Joseph Gallian – Contemporary Abstract Algebra, Narosa Publishing House, New Delhi, Fourth Edition. 3. G. D. Birkhoff and S Maclane – A brief Survey of Modern Algebra. 4. J B Fraleigh – A first course in Abstract Algebra. 5. Michael Artin – Algebra, 2nd ed. New Delhi, India: PHI Learning Pvt. Ltd., 2011. 6. Vashista- A First Course in Modern Algebra, 11th ed.: Krishna Prakasan Mandir, 1980. 7.Shanthi Narayan and P K Mittal- Integral Calculus, Reprint. New Delhi: S. Chand and Co. Pvt. Ltd., 8. M D Raisinghania: Advanced Differential Equations, S Chand and Co. Pvt. Ltd., 2013. 9. F Ayres: Schaum's outline of theory and problems of Differential Equations, 1st ed. USA: McGraw-Hill, 2010. 10. S Narayanan and T K Manicavachogam Pillay:Differential Equations.: S V Publishers Private Ltd., 1981. 11. G F Simmons: Differential equation with Applications and historical notes, 2nd ed.: McGraw- Hill Publishing Company, Oct 1991.
Page 7 of 32
III Semester Paper III - BSM 3.1T (ALGEBRA-III & CALCULUS-III )
4 Lecture Hours/ Week + 3 Hrs Practical’s/Week
Total: 56Hrs ALGEBRA-III
Unit 1 : Order of an element of a group – properties related to order of an element-
subgroup generated by an element of a group - coset decomposition of a group - Cyclic
groups - properties- modulo relation- index of a group –Lagrange’s theorem-
consequences (14 Hours)
CALUCULUS - III
Unit 2 : Differential Calculus: Polar Co-ordinates - Angle between the Radius Vector and
Tangent - Angle of Intersection of Curves (Polar Forms) - Pedal Equations - Derivative of
an Arc in Cartesian, Parametric and Polar Forms - Curvature of a plane curve – formula in
Cartesian, parametric, polar and pedal forms – centre of curvature - Evolutes.(14Hours)
Unit 3 : Differential Calculus Of Scalar And Vector Fields
Scalar field – gradient of a scalar field - geometrical meaning – directional derivative –
Maximal directional derivative – Angle between two surfaces - vector field – divergence
and curl of a vector field – solenoidal and irrotational fields – scalar and vector potentials –
Laplacian of a scalar field – vector identities - Standard properties - Harmonic functions -
Problems. (14 Hours)
Unit 4 : Integral Calculus :
General rules for tracing of curves - computation of length of arc, plane area , surface area
and volume of solids of revolutions for standard curves in Cartesian and Polar forms
(14 Hours)
Page 8 of 32
PRACTICALS – III BSM 3.1P (ALGEBRI-III & CALCULUS-III )
Total: 42 Hrs Mathematics practical with Free and open Source Software (FOSS) tools for
computer programs (3 hours/ week per batch of not more than 15 students)
LIST OF PROBLEMS
1. Finding all possible Subgroups of the given finite group.
2. Examples to verify Lagrange’s theorem.
3. Plotting of standard Cartesian curves using Scilab/Maxima.
4. Plotting of standard Polar curves using Scilab/Maxima.
5. Plotting of standard parametric curves using Scilab/Maxima
6.To demonstrate the physical interpretation of gradient, divergence and curl.
7. Program to find gradient, divergence, curl and Laplacian in cylindrical coordinates and
spherical co-ordinates.
8. Using cyclic notations to derive different vector identities.
9. Scilab/Maxima programs for Surface area.
10. Scilab/Maxima programs for volume.
Text Boos:
1. Shanti Narayan: Differential Calculus ( S Chand & Co.)
2. Murray R Spiegel: Advanced Calculus (Schaum’s Series)
3. B.Sc Mathematcs - G.K.Ranganath
4. Shanthi Narayan and P K Mittal- Integral Calculus, Reprint. New Delhi: S. Chand and Co.
Pvt. Ltd.,
Reference Books:
1. I. N. Herstien – Topics in Algebra.
2. Joseph Gallian – Contemporary Abstract Algebra, Narosa Publishing House, New Delhi,
Fourth Edition.
3. G. D. Birkhoff and S Maclane – A brief Survey of Modern Algebra.
4. J B Fraleigh – A first course in Abstract Algebra.
5. Michael Artin – Algebra, 2nd ed. New Delhi, India: PHI Learning Pvt. Ltd., 2011.
6. Vashista- A First Course in Modern Algebra, 11th ed.: Krishna Prakasan Mandir, 1980.
7. J Edwards: An elementary treatise on the differential calculus: with applications and
numerous example, Reprint. Charleston, USA:BiblioBazaar, 2010.
8. N P Bali: Differential Calculus, India: Laxmi Publications (P) Ltd.., 2010.
9. S Narayanan & T. K. Manicavachogam Pillay: Calculus.: S. Viswanathan Pvt. Ltd., vol. I &
II1996.
10. Frank Ayres and Elliott Mendelson: Schaum's Outline series of Calculus, 5th ed. USA:
Mc. Graw Hill.,
Page 9 of 32
IV Semester Paper IV - BSM 4.1T (ALGEBRA-IV , CALCULUS-IV & ANALYSIS - I)
Unit 1 : Normal Subgroups - definition and examples - Standard theorems on normal subgroups - Quotient groups - Homomorphism, isomorphism and fundamental theorem of homomorphism. (14 Hours) CALCULUS-IV
Unit 2 : Differential Calculus:
Mean Value Theorems: Rolle’s theorem, Lagrange’s mean value theorem, Cauchy’s mean
value theorem - Taylor’s theorem with Lagrange’s form of the remainder - Taylor’s and
Maclaurin’s series - L’Hospital’s rule and problems.(14 Hours)
ANALYSIS – I
Unit 3 : Sequence of Real Numbers: Definition of a sequence - limit of a sequence -
Algebra of limit of a sequence – Convergent, Divergent and oscillatory sequences -
Problems there on. Bounded sequence - Every convergent Sequence is bounded – converse
is not true – monotonic sequences and their properties - Cauchy’s sequence. (14Hours)
Unit 4 : Infinite Series: Definition of convergent, divergent and oscillatory series –
Standard properties and results - Tests for convergence of series – comparison tests –P-
series - D’Alemberts Ratio test – Raabe’s test – Cauchy’s root test - Absolute and
conditional Convergence - Leibnitz’s test for alternating series. (14Hours)
9. K. Sankar Rao- Introduction to partial differential equations
10. M. D. Raisinghania- Ordinary and partial differential equations
Page 23 of 32
DAVANAGERE UNIVERSITY B.Sc First SEMESTER (Paper - I) MATHAMATICS QUESTION PAPER PATTERN.
Time: 3Hrs Max Marks: 80
Note: All parts are compulsory
PART-A
A. Answer any TEN of the following. 10 x 2 = 20
1) Unit 1 2) Unit 1 3) Unit 1 4) Unit 1 5) Unit 1 6) Unit 1 7) Unit 4 8) Unit 4 9) Unit 4 10) Unit 4 11) Unit 3 12) Unit 3
PART-B
B. Answer any SIX of the following. 6 x 5 =30 13) Unit 1 14) Unit 1
15) Unit 2 16) Unit 2 17) Unit 4 18) Unit 4 19) Unit 3 20) Unit 3
PART - C C. Answer any THREE of the following. 3 x 10 = 30 21) a) Unit 1 b) Unit 3 22) a) Unit 2 b) Unit 3 23) a) Unit 4 b) Unit 3 24) a) Unit 1 b) Unit 4
25) a) Unit 2
b) Unit 1
Page 24 of 32
DAVANAGERE UNIVERSITY B.Sc Second SEMESTER (Paper - II) MATHAMATICS QUESTION PAPER
PATTERN. Time: 3Hrs Max Marks: 80
Note: All parts are compulsory
PART-A
A. Answer any TEN of the following. 10 x 2 = 20 1) Unit 2 2) Unit 2 3) Unit 1 4) Unit 1 5) Unit 1 6) Unit 1 7) Unit 3 8) Unit 3 9) Unit 3 10) Unit 3 11) Unit 4 12) Unit 4
PART-B
B. Answer any SIX of the following. 6 x 5 =30 13) Unit 1 14) Unit 1
15) Unit 2 16) Unit 2 17) Unit 3 18) Unit 3 19) Unit 4 20) Unit 4
PART - C C. Answer any THREE of the following. 3 x 10= 30 21) a) Unit 2 b) Unit 4 22) a) Unit 1 b) Unit 4 23) a) Unit 3 b) Unit 4 24) a) Unit 2
b) Unit 3
25) a) Unit 2
b) Unit 1
Page 25 of 32
DAVANAGERE UNIVERSITY
B.Sc Third SEMESTER (Paper - III) MATHAMATICS QUESTION PAPER PATTERN. Time: 3Hrs Max Marks: 80
Note: All parts are compulsory
PART-A
A. Answer any TEN of the following. 10 x 2 = 20 1) Unit 1 2) Unit 1 3) Unit 2 4) Unit 2 5) Unit 2 6) Unit 2 7) Unit 3 8) Unit 3 9) Unit 3 10) Unit 3 11) Unit 4 12) Unit 4
PART-B
B. Answer any SIX of the following. 6 x 5 =30 13) Unit 1 14) Unit 1
15) Unit 2 16) Unit 2 17) Unit 3 18) Unit 3 19) Unit 4 20) Unit 4
PART - C C. Answer any THREE of the following. 3 x 10 = 30 21) a) Unit 1 b) Unit 4 22) a) Unit 2 b) Unit 4 23) a) Unit 3 b) Unit 4 24) a) Unit 1 b) Unit 3 25) a) Unit 2 b) Unit 1
Page 26 of 32
DAVANAGERE UNIVERSITY B.Sc Fourth SEMESTER (Paper - IV) MATHAMATICS QUESTION PAPER
PATTERN. Time: 3Hrs Max Marks: 80
Note: All parts are compulsory
PART-A
A. Answer any TEN of the following. 10 x 2 = 20 1) Unit 1 2) Unit 1 3) Unit 2 4) Unit 2 5) Unit 3 6) Unit 3 7) Unit 3 8) Unit 3 9) Unit 4 10) Unit 4 11) Unit 4 12) Unit 4
PART-B
B. Answer any SIX of the following. 6 x 5 =30 13) Unit 1 14) Unit 1
15) Unit 2 16) Unit 2 17) Unit 3 18) Unit 3 19) Unit 4 20) Unit 4
PART - C C. Answer any THREE of the following. 3 x 10 = 30 21) a) Unit 1 b) Unit 2 22) a) Unit 3 b) Unit 2 23) a) Unit 2 b) Unit 4 24) a) Unit 3 b) Unit 1 25) a) Unit 1 b) Unit 4
Page 27 of 32
DAVANGERE UNIVERSITY
B.Sc FIFTH SEMESTER (Paper - V) MATHEMATICS QUESTION PAPER PATTERN.
(COMPULSORY)
Time: 3Hrs Note: All parts are compulsory Max Marks: 80
PART-A
A. Answer any TEN of the following. 10 x 2 = 20
1) Unit 1
2) Unit 1
3) Unit 2
4) Unit 2
5) Unit 2
6) Unit 2
7) Unit 3
8) Unit 3
9) Unit 3
10) Unit 3
11) Unit 4
12) Unit 4
PART-B
B. Answer any SIX of the following. 6 x 5 =30
13) Unit 1
14) Unit 1
15) Unit 2
16) Unit 2
17) Unit 3
18) Unit 3
19) Unit 4
20) Unit 4
PART - C
C. Answer any THREE of the following. 3 x 10 = 30
21) a) Unit 1
b) Unit 4
22) a) Unit 2
b) Unit 4
23) a) Unit 3
b) Unit 4
24) a) Unit 1
b) Unit 3
25) a) Unit 2
b) Unit 1
******
Page 28 of 32
DAVANGERE UNIVERSITY B.Sc FIFTH SEMESTER (Paper - VI)(ELECTIVE) MATHEMATICS QUESTION PAPER
PATTERN. Time: 3Hrs Note: All parts are compulsory Max Marks: 80
PART-A
A. Answer any TEN of the following. 10 x 2 = 20 1) Unit 1 2) Unit 1 3) Unit 2 4) Unit 2 5) Unit 2 6) Unit 2 7) Unit 3 8) Unit 3 9) Unit 3 10) Unit 3 11) Unit 4 12) Unit 4
PART-B B. Answer any SIX of the following. 6 x 5 =30 13) Unit 1 14) Unit 1
15) Unit 2 16) Unit 2 17) Unit 3 18) Unit 3 19) Unit 4 20) Unit 4
PART - C C. Answer any THREE of the following. 3 x 10 = 30 21) a) Unit 1 b) Unit 4 22) a) Unit 2 b) Unit 4 23) a) Unit 3 b) Unit 4 24) a) Unit 3 b) Unit 1 25) a) Unit 2 b)Unit 1
********
Page 29 of 32
DAVANGERE UNIVERSITY B.Sc FIFTH SEMESTER (Paper - VII)(ELECTIVE) MATHEMATICS QUESTION PAPER
PATTERN. Time: 3Hrs Note: All parts are compulsory Max Marks: 80
PART-A
A. Answer any TEN of the following. 10 x 2 = 20 1) Unit 1 2) Unit 1 3) Unit 2 4) Unit 2 5) Unit 2 6) Unit 2 7) Unit 3 8) Unit 3 9) Unit 3 10) Unit 3 11) Unit 4 12) Unit 4
PART-B B. Answer any SIX of the following. 6 x 5 =30 13) Unit 1 14) Unit 1
15) Unit 2 16) Unit 2 17) Unit 3 18) Unit 3 19) Unit 4 20) Unit 4
PART - C C. Answer any THREE of the following. 3 x 10 = 30 21) a) Unit 1 b) Unit 4 22) a) Unit 2 b) Unit 4 23) a) Unit 3 b) Unit 4 24) a) Unit 3 b) Unit 1 25) a) Unit 2 b)Unit 1
******
Page 30 of 32
DAVANGERE UNIVERSITY
B.Sc SIXTH SEMESTER (Paper-VIII) MATHEMATICS QUESTION PAPER PATTERN.
(COMPULSORY)
Time: 3Hrs Note: All parts are compulsory Max Marks : 80
PART-A
A. Answer any TEN of the following. 10 x 2 = 20
1) Unit 1
2) Unit 1
3) Unit 2
4) Unit 2
5) Unit 2
6) Unit 2
7) Unit 4
8) Unit 4
9) Unit 4
10) Unit 4
11) Unit 3
12) Unit 3
PART-B
B. Answer any SIX of the following. 6 x 5 =30
13) Unit 1
14) Unit 1
15) Unit 2
16) Unit 2
17) Unit 3
18) Unit 3
19) Unit 4
20) Unit 4
PART - C
C. Answer any THREE of the following. 3 x 10 = 30
21) a) Unit 1
b) Unit 3
22) a) Unit 2
b) Unit 3
23) a) Unit 3
b) Unit 4
24) a) Unit 1
b) Unit 2
25) a) Unit 4
b) Unit 1
Page 31 of 32
DAVANGERE UNIVERSITY
B.Sc SIXTH SEMESTER (Paper-IX)(ELECTIVE) MATHEMATICS QUESTION PAPER
PATTERN.
Time: 3Hrs Note: All parts are compulsory Max Marks: 80
PART-A
A. Answer any TEN of the following. 10x 2 = 20
1) Unit 1
2) Unit 1
3) Unit 2
4) Unit 2
5) Unit 2
6) Unit 2
7) Unit 3
8) Unit 3
9) Unit 4
10) Unit 4
11) Unit 4
12) Unit 4
PART-B
B. Answer any SIX of the following. 6 x 5 =30
13) Unit 1
14) Unit 1
15) Unit 2
16) Unit 2
17) Unit 3
18) Unit 3
19) Unit 4
20) Unit 4
PART - C
C. Answer any THREE of the following. 3 x 10 = 30
21) a) Unit 1
b) Unit 3
22) a) Unit 2
b) Unit 3
23) a) Unit 4
b) Unit 3
24) a) Unit 1
b) Unit 2
25) a) Unit 4
b) Unit 1
********
Page 32 of 32
DAVANGERE UNIVERSITY
B.Sc SIXTH SEMESTER (Paper-X)(ELECTIVE) MATHEMATICS QUESTION PAPER
PATTERN.
Time: 3Hrs Note: All parts are compulsory Max Marks: 80