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VIKRAMA SIMHAPURI UNIVERSITY::NELLORE

w.e.f. 2015-16 (Revised in April, 2016) B.A./B.Sc. FIRST YEAR MATHEMATICS SYLLABUS

SEMESTER –I, PAPER - 1 DIFFERENTIAL EQUATIONS

60 Hrs

UNIT – I (12 Hours), Differential Equations of first order and first degree : Linear Differential Equations; Differential Equations Reducible to Linear Form; Exact Differential Equations; Integrating Factors.

UNIT – II (12 Hours), Orthogonal Trajectories. Cartesian co-ordinates self orthogonal Family of curves. Orthogonal trajectories : polar co-ordinates. Differential Equations of first order but not of the first degree : Equations solvable for p; Equations solvable for y; Equations solvable for x; Equations that do not

contain. x (or y); Equations of the first degree in x and y – Clairaut‟s Equation. UNIT – III (12 Hours), Higher order linear differential equations-I : Solution of homogeneous linear differential equations of order n with constant coefficients; Solution of

the non-homogeneous linear differential equations with constant coefficients by means of polynomial

operators. General Solution of f(D)y=0

General Solution of f(D)y=Q when Q is a function of x.

1

f D is Expressed as partial fractions.

P.I. of f(D)y = Q when Q= axbe

P.I. of f(D)y = Q when Q is b sin ax or b cos ax. UNIT – IV (12 Hours), Higher order linear differential equations-II :

Solution of the non-homogeneous linear differential equations with constant coefficients.

P.I. of f(D)y = Q when Q= bxk

P.I. of f(D)y = Q when Q= e

ax V

P.I. of f(D)y = Q when Q= xV

P.I. of f(D)y = Q when Q= x m

V

UNIT –V (12 Hours), Higher order linear differential equations-III : Method of variation of parameters (without non constant coefficient equations) ; The Cauchy-Euler Equation ; Legender‟s Equations. Prescribed Text Book :

1. A text book of mathematics for BA/BSc Vol 1 by N. Krishna Murthy & others, published by S. Chand

& Company, New Delhi.

Reference Books : 1. Differential Equations and Their Applications by Zafar Ahsan, published by Prentice-Hall of India Learning Pvt. Ltd. New Delhi-Second edition. 2. Ordinary and Partial Differential Equations Raisinghania, published by S. Chand & Company, New Delhi. 3. Differential Equations with applications and programs – S. Balachandra Rao & HR Anuradha-universities press. 4. Telugu Academy Text Book for Differential Equations. 5. I-B.Sc A text Book of a Mathematics Deepthi Publications. Suggested Activities:

Seminar/ Quiz/ Assignments/ Project on Application of Differential Equations in Real life

BLUE PRINT OF QUESTION PAPER

(INSTRUCTIONS TO PAPER SETTER) B.A./B.Sc. MATHEMATICS SEMESTER-I

(DIFFERENTIAL EQUATIONS)

NOTE :- Paper Setter Must select TWO Short Questions and TWO Easy Questions from Each Unit as Follows :-

UNIT TOPICS 5 MARKS

QUESTIONS 10 MARKS

QUESTIONS

UNIT - I

Linear Equations 1 -

Bernoulli‟s Equations - 1

Integrating Factor 1 -

Exact Equations - 1

UNIT - II

Orthogonal Trajectories 1 1

Solvable for x, y, p. 1 1

UNIT - III

General Solution of f(D)y=0 1 -

f(D)y = Q when Q= axbe 1 1

f(D)y = Q when Q is b

sin ax or b cos ax - 1

UNIT - IV

f(D)y = Q when Q= bxk 1 -

f(D)y = Q when Q= e ax

V 1 1

f(D)y = Q when Q= xV - 1

UNIT - V

Variation of Parameters

(without non constant

coefficient equations)

- 1

Cauchy-Euler Equations 2 -

Legender‟s Equations - 1

VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE.

(w.e.f. 2016-17) B.A./B.Sc. FIRST YEAR MATHEMATICS

SEMESTER-I MODEL QUESTION PAPER-1 (DIFFERENTIAL EQUATIONS)

TIME : 3 Hours Max.Marks : 75

PART – A

I. Answer any FIVE Questions : 5 X 5 = 25M

1. Solve 2

2dy xxy edx

.

2. Find Integrating factor of 3 2 2 42 0xy y dx x y x y dy .

3. Find the Orthogonal trajectories of the family of curves

2

3x

2 2 2

3 3 3x y a where

„a‟ is a parameter.

4. Solve 2 42y xP x P .

5. Solve 4 28 16 0D D y .

6. Solve 2 45 6 xD D y e .

7. Solve 2 4 sinD y x x .

8. Solve 2 34 4D D y x .

9. Solve 2 2 1 logx D xD y x .

10. Find the complementary function yc

of 2 2 23 5 sin logx D xD y x x .

PART - B

II. Answer ALL Questions : 5 X 10 = 50M

UNIT - I

11. (a) Solve 2 3 4 1dy

x y xdx

.

(or)

(b) Solve 2 3 3 0x ydx x y dy .

UNIT - II

12. (a) Find the orthogonal Trajectories of the families of Curves 2

1 cos

ar

when “a” is Parameter.

(or)

(b) 2 22 cotP Py x y .

UNIT - III

13. (a) Solve 2

3 1 1xD y e .

(or)

(b) Solve 2 3 2 cos3 .cos 2D D y x x .

UNIT - IV

14. (a) Solve 2

36 13 8 sin 22

d y dy xy e xdxdx

.

(or)

(b) Solve 2 2 21 cosxD y x e x x .

UNIT - V

15. (a) Solve by the method of variation of parameters 2 1 cosD y ecx .

(or)

(b) Solve 2 21 1 1 4cos log 1x D x D y x

.

VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE. w.e.f. 2015-16 (Revised in April, 2016)

B.A./B.Sc. FIRST YEAR MATHEMATICS SYLLABUS SEMESTER – II, PAPER - 2

SOLID GEOMETRY

60 Hrs UNIT – I (12 hrs) : The Plane :

Equation of plane in terms of its intercepts on the axis, Equations of the plane through the given points,

Length of the perpendicular from a given point to a given plane, Distance between parallel planes, System of

Planes.

Planes bisecting the angles between two Planes. Pair of Planes.

UNIT – II (12 hrs) : The Line :

Equation of a line; Angle between a line and a plane; The condition for a line to lie in a plane, Image of a

point in a plane, Image of point in a line coplanar Lines

Shortest distance between two lines; The length and equations of the line of shortest distance between two

straight lines; Length of the perpendicular from a given point to a given line.

UNIT – III (10 hrs) : Sphere :

Definition and equation of the sphere; the sphere through four given points; Plane sections of a sphere;

Intersection of two spheres; Equation of a circle; great circle, small circle; Intersection of a sphere and a line.

UNIT – IV (10 hrs) : Sphere :

Equation of Tangent plane; Angle of intersection of two spheres; Orthogonal spheres; Radical plane;

Coaxial system of spheres; Limiting Points.

UNIT – V (16 hrs) : Cones :

Definitions of a cone; Equation of the cone with a given vertex and guiding curve; Enveloping cone, to

Find Vertex of a cone, Reciprocal Cone, Right circular cone, Equation of the Right Circular cone one with a given

vertex axis and semi vertical angle the cylinder.

Cylinder :

Definition of a cylinder, Equation to the cylinder, Enveloping cylinder, right circular cylinders equation of

the right circular cylinder.

Note : Concentrate on Problematic parts in all above units.

Prescribed Text Book :

1. V. Krishna Murthy & Others “A text book of Mathematics for BA/B.Sc Vol 1, Published by

S. Chand & Company, New Delhi.

Reference Books : 1. Scope as in Analytical Solid Geometry by Shanti Narayan and P.K. Mittal Published

by S. Chand & Company Ltd. Seventeenth Edition.

Sections :- 2.4, 2.5, 2.6, 2.7, 2.8, 3.1 to 3.7, 6.1 to 6.9, 7.1 to 7.4, 7.6 to 7.8.

2. P.K. Jain and Khaleel Ahmed, “A text Book of Analytical Geometry of Three

Dimensions”, Wiley Eastern Ltd., 1999.

3. Co-ordinate Geometry of two and three dimensions by P. Balasubrahmanyam,

K.Y. Subrahmanyam, G.R. Venkataraman published by Tata-MC Gran-Hill Publishers

Company Ltd., New Delhi.

4. Telugu Academy Text Book for Solid Geometry. 5. I-B.Sc A text Book of a Mathematics Deepthi Publications.


(INSTRUCTIONS TO PAPER SETTER) B.A./B.Sc. MATHEMATICS SEMESTER-II

(SOLID GEOMETRY)


UNIT TOPICS 5 MARKS

QUESTIONS 10 MARKS

QUESTIONS

UNIT - I

Planes Introductions 2 (Prb) -

System of Planes & Bisecting

Planes - 1(Prb)

Pair of Planes - 1(Prb)

UNIT - II

Straight Lines First Part 2 (Prb) -

Image & coplaner Lines - 1(Prb)

Shortest Distance - 1(Prb)

UNIT - III

Sphere Introduction 1(Prb) -

Plane Section of a Sphere 1(Prb) 1(Prb)

Great Circle & Small Circle - 1(Prb)

UNIT - IV

Tangent Plane 1(Prb) -

Angle of Intersection of Two

Spheres & Orthogonal Spheres 1(Prb) 1(Prb)

Limiting Points - 1(Prb)

UNIT - V

Cone 1(Prb) 1(Prb)

Cylinder 1(Prb) 1(Prb)

VIKRAMA SIMHAPURI UNIVERSITY :: NELLORE

B.A./B.Sc. FIRST YEAR MATHEMATICS MODEL QUESTION PAPER

SEMESTER-II

(SOLID GEOMETRY)

Time: 3 Hours Max. Marks : 75

Section-A I. Answer any FIVE of the following Questions : (5 X 5= 25 Marks)

1. Find the Equation of the plane through the point (-1,3,2) and perpendicular to the planes

2 2 5x y z and 3 3 2 8x y z .

2. Find the angles between the planes 2 3 5,x y z 3 3 9x y z .

3. Show that the line 1 2 5

1 3 5

x y z

lies in the plane x+2y-z=0.

4. Find the point of intersection with the plane 3 4 5 5x y z and the line 1 3 2

1 3 2

x y z .

5. Find the centre and radius of the sphere 2 2 22 2 2 2 4 2 1 0x y z x y z .

6. Find the equation of the sphere through the circle 2 2 2 9x y z , 2 3 4 5x y z and the

point (1,2,3)

7. Find the equation of the tangent plane to the sphere 2 2 23 3 3 2 3 4 22 0x y z x y z

at the point (1,2,3)

8. Show that the spheres are orthogonal 2 2 2 6 2 8 0;x y z y z

2 2 2 6 8 4 20 0x y z x y z .

9. Find the equation of the cone which passes through the three co-ordinate axis and the lines

1 2 3

x y z

and 2 1 1

x y z .

10. Find the equation of the cylinder whose generators are parallel to 1 2 3

x y z and which

Passes through the curve 2 2 16, 0x y z .

Section - B

II. Answer any ONE Question from each unit. (5 x 10 =50 Marks)

UNIT - I

11. (a) Find the equation of the plane passing through the intersection of the planes

2 3 4,2 5 0x y z x y z and perpendicular to the plane 6 5 3 8 0z x y .

(OR)

(b) Prove that Equation 2 2 22 6 12 18 2 0x y z yz zx xy represents a pair of planes and

find the angle between them.

UNIT - II

12. (a) Find the image of the point (2,-1,3) in the plane 3x-2y+z=9.

(OR)

(b) Find the length and equation to the line of shortest distance between the lines 2 3 1

,3 4 2

x y z

4 5 2

4 5 3

x y z .

UNIT - III

13. (a) Find the equation of the sphere through the circle 2 2 2 2 3 6 0x y z x y ,

2 4 9 0x y z and the centre of the sphere 2 2 2 2 4 6 5 0x y z x y z .

(OR)

(b) Find whether the following circle is a great circle or small circle 2 2 2 4 6 8 4 0,x y z x y z 3x y z .

UNIT – IV

14. (a) Find the equation of the sphere which touches the plane 3x+2y-z+2=0 at (1,-2,1) and

cuts orthogonally the sphere 2 2 2 4 6 4 0x y z x y .

(OR)

(b) Find limiting points of the co axial system of spheres

2 2 2 20 30 40 29x y z x y z 2 3 4 0x y z .

UNIT – V

15. (a) Find the vertex of the cone 2 2 27 2 2 10 10 26 2 2 17 0x y z zx xy x y z .

(OR)

(b) Find the equation to the right circular cylinder whose guiding circle 2 2 2 9,x y z

3x y z .

VIKRAMA SIMHAPURI UNIVERSITY::NELLORE (w.e.f. 2016-17)

B.A./B.Sc. SECOND YEAR MATHEMATICS SYLLABUS

SEMESTER – III, PAPER - 3

ABSTRACT ALGEBRA

60 Hrs

UNIT – 1 : (10 Hrs) GROUPS : -

Binary Operation – Algebraic structure – semi group-monoid – Group definition and elementary properties Finite and Infinite groups – examples – order of a group. Composition tables with examples.

UNIT – 2 : (14 Hrs) SUBGROUPS : - Complex Definition – Multiplication of two complexes Inverse of a complex-Subgroup definition

– examples-criterion for a complex to be a subgroups.

Criterion for the product of two subgroups to be a subgroup-union and Intersection of subgroups.

Co-sets and Lagrange‟s Theorem :-

Cosets Definition – properties of Cosets – Index of a subgroups of a finite groups–Lagrange’s

Theorem Statement and Proof.

UNIT –3 : (12 Hrs) NORMAL SUBGROUPS : -

Definition of normal subgroup – proper and improper normal subgroup–Hamilton group –

criterion for a subgroup to be a normal subgroup – intersection of two normal subgroups – Sub group of

index 2 is a normal sub group – simple group – quotient group – criteria for the existence of a quotient

group.

UNIT – 4 : (10 Hrs) HOMOMORPHISM : -

Definition of homomorphism – Image of homomorphism elementary properties of

homomorphism – Isomorphism – aultomorphism definitions and elementary properties–kernel of a

homomorphism – fundamental theorem on Homomorphism and applications.

UNIT – 5 : (14 Hrs) PERMUTATIONS AND CYCLIC GROUPS : -

Definition of permutation – permutation multiplication – Inverse of a permutation – cyclic permutations – transposition – even and odd permutations.

Cayley's Theorem and Cyclic Groups :-

Definition of cyclic group – elementary properties.


1. A text book of Mathematics for B.A. / B.Sc. by B.V.S.S. SARMA and others, Published by S.Chand &

Company, New Delhi.

Reference Books : 1. Abstract Algebra, by J.B. Fraleigh, Published by Narosa Publishing house.

2. Modern Algebra by M.L. Khanna. 3. Telugu Academy Text Book for Abstract Algebra. 4. I-B.Sc A text Book of a Mathematics Deepthi Publications.

Suggested Activities: Seminar/ Quiz/ Assignments/ Project on Group theory and its applications in Graphics and Medical image Analysis


(INSTRUCTIONS TO PAPER SETTER) B.A./B.Sc. MATHEMATICS SEMESTER-III

(ABSTRACT ALGEBRA)


UNIT TOPICS 5 MARKS

QUESTIONS 10 MARKS

QUESTIONS

UNIT - I

Group Definition and

Elementary Properties 1 (Th) -

Composition Tables 1 (Prb) -

Problems - 2 (Prb)

UNIT - II

Subgroups Lagrange‟s Theorem 1(Th) 2(Th)

Cosets 1(Th) -

UNIT - III Normal Subgroups 2(Th) 2(Th)

UNIT - IV Homomorphism 1(Prb) + 1 (Th) 2(Th)

UNIT - V

Permutations 2 (Prb) 1(Prb)

Cayley's Theorem & Cyclic

Groups - 1 (Th)


(w.e.f. 2016-17) B.A./B.Sc. SECOND YEAR MATHEMATICS

MODEL QUESTION PAPER SEMESTER – III

(ABSTRACT ALGEBRA)


Section-A

I. Answer any FIVE of the following Questions : (5 X 5= 25 Marks)

1. Prove that in a group G Inverse of any Element is unique.

2. 1,2,3,4,5,6G Prepare composition table and prove that G is a finite abelian group of order

6 with respect to 7

X .

3. If H is any subgroups of G then prove that 1H H .

4. Prove that any two left cosets of a subgroups are either disjoint or identical.

5. Prove that intersection of any two normal subgroup is again a normal subgroup.

6. Define the following :

(a) Normal subgroups (b) Simple Groups.

7. Prove that the homomorphic image of a group is a group.

8. If for a group ,G :F G G is given by 2,f x x x G is a homomorphism then prove

that G is abelian.

9. If 1 2 3 1 2 3

,2 3 1 3 1 2

A B

find AB and BA.

10. Find the inverse of the permutation: 1 2 3 4 5 6

3 4 5 6 1 2

II. Answer ALL the following Questions :

11. (a) Define abelian group. Prove that the set of thn roots of unity under multiplication form a finite

abelian group.

(OR)

(b) Show that the set of all positive rational numbers form on abelian group under the composition

„0‟ defined by 2

abaob .

12. (a) Prove that a non-empty finite subset of a group which is closed under multiplication is a

subgroup of G.

(OR)

(b) Prove that the union of two subgroups of a group is a subgroup if f one is contained in the

other.

13. (a) Prove that a subgroup H of a group G is a normal subgroup of G if f each left coset of H in G

is a right coset of H in G.

(OR)

(b) If G is a group and H is a subgroup of index 2 in G then prove that H is a normal subgroup of

G.

14. (a) ,G and 1,G be two groups 1:f G G is an into homomorphism then prove

(i) 1f e e (ii) 11f a f a

Where e , 1e are then identity elements in G and 1G respectively.

(OR)

(b) State and prove fundamental theorem on Homomorphism of Groups.

15. (a) Examine the following permutation are even (or) odd

(i) 1 2 3 4 5 6 7

3 2 4 5 6 7 1f

(ii) 1 2 3 4 5 6 7 8

7 3 1 8 5 6 2 4g

(OR)

(b) Define cyclic group. Prove that every cyclic group is an abelian group.


(w.e.f. 2016-17) B.A./B.Sc. SECOND YEAR MATHEMATICS SYLLABUS

SEMESTER – IV

REAL ANALYSIS

60 Hrs

UNIT – I (12 hrs) : REAL NUMBERS :

The algebraic and order properties of R, Absolute value and Real line, Completeness property of R, Applications of supreme property; intervals. No. Question is to be set from this portion. Real Sequences: Sequences and their limits, Range and Boundedness of Sequences, Limit of a sequence and Convergent sequence, Monotone sequences, Necessary and Sufficient condition for Convergence of Monotone Sequence, Limit and the Bolzano-weierstrass theorem – (Cauchy Sequences – Cauchey‟s general principle of convergence theorem) No. Question is to be set from this portion. Series : Introduction to series, convergence of series of Non-Negative Terms. 1. P-test

2. Cauchey‟s nth

root test or Root Test.

3. D‟-Alemberts‟ Test or Ratio Test.

4. Alternating Series – Leibnitz Test.

Absolute convergence and conditional convergence, semi convergence.

UNIT – II (12 hrs) : CONTINUITY : Limits : Real valued Functions, Boundedness of a function, Limits of functions. Some extensions

of the limit concept, Infinite Limits. Limits at infinity. No. Question is to be set from this portion.

Continuous functions : Continuous functions, Combinations of continuous functions, Continuous

Functions on intervals.

UNIT – III (12 hrs) : DIFFERENTIATION : The derivability of a function, on an interval, at a point, Derivability and continuity of a function,

Graphical meaning of the Derivative, Problems on Differentiation.

UNIT – IV (12 hrs) : MEAN VALUE THEORMS : Mean value Theorems; Roles Theorem, Langrange‟s Theorem, Cauchhy‟s Mean value Theorem

Statement and their Applications.

UNIT – V (12 hrs) : RIEMANN INTEGRATION : Riemann Integral, Riemann integral functions. Necessary and sufficient condition for R–

integrability, Properties of Integrable functions, Continuous Functions R-Integral, Monotonic Function

R-Intigrable constant function R-Intergrable - Fundamental theorem of integral calculus.


1. A Text Book of B.Sc Mathematics by B.V.S.S. Sarma and others, Published by S. Chand & Company

Pvt. Ltd., New Delhi.

Reference Books : 1. Real Analysis by Rabert & Bartely and .D.R. Sherbart, Published by John Wiley. 2. Elements of Real Analysis as per UGC Syllabus by Shanthi Narayan and Dr. M.D. Raisingkania Published by S. Chand & Company Pvt. Ltd., New Delhi. 3. Telugu Academy Text Book for Real Analysis. 4. I-B.Sc A text Book of a Mathematics Deepthi Publications.

Suggested Activities: Seminar/ Quiz/ Assignments/ Project on Real Analysis and its applications


(INSTRUCTIONS TO PAPER SETTER) B.A./B.Sc. MATHEMATICS SEMESTER-IV

(REAL ANALYSIS)


PAPER TOPICS 5 MARKS

QUESTIONS 10 MARKS

QUESTIONS

UNIT - I

Sequence - 1(Th)

Series 2 (Prb) 1(Th)

UNIT - II Continuity 2 (Prb) 1(Prb) + 1 (Th)

UNIT - III Differentiation 2 (Prb) 2 (Prb)

UNIT - IV Mean Value Theorems 1(Prb) + 1 (Th) 1(Prb) + 1 (Th)

UNIT - V Riemann Integration 1(Prb) + 1 (Th) 2(Th)


(w.e.f. 2016-17) B.A./B.Sc. SECOND YEAR MATHEMATICS

SEMESTER – IV MODEL QUESTION PAPER

(REAL ANALYSIS)


Section-A I. Answer any FIVE of the following Questions : (5 X 5= 25 Marks)

1. Test for convergence 1

2 1n

.

2. State cauchy‟s root test and test for convergence

21

1n

n

.

3. Discuss various types of discontinuity.

4. Examine for continuity of a function 1f n x x at x=0.

5. If 1

1

xf x

xe

if 0x and 0f x if x=0 show that f is not derivable at x = 0.

6. Prove that 12 sin , 0f x x xx

and 0 0f is derivable at the origin.

7. State cauchy‟s Mean value theorem.

8. Find „C‟ of the Lagrange‟s mean value theorem for 1 2 3f x x x x on 0,4 .

9. If 2f x x on 0,1 and 1 2 3

0, , , ,14 4 4

P

compute ,L P f and ,U P f .

10. Prove that a constant function is Reiman integrable on ,a b .

II. Answer ALL the following Questions :

11. (a) State and prove the necessary and sufficient condition for convergence of a monotic sequence.

(OR)

(b) State and prove P-test.

12. (a) Discuss the continuity of

1 1

1 1

x xx e e

f x

x xe e

for 0x and 0 0f at x = 0.

(OR)

(b) If f is continuous on ,a b and ,f a f b having opposite sign then prove that there

exit , 0C a b f c .

13. (a) Show that 1

sin , 0, 0f x x x f xx

when x=0 is continuous but not derivable at x=0.

(OR)

(b) Show that

1

1

1

1

xx e

f x

xe

if 0x and 0 0f is continuous at x=0 but not

derivable at x=0.

14. (a) State and prove Rolle‟s theorem.

(OR)

(b) Using Lagrange‟s theorem show that log 11

xx n

x

if log 1f x x .

15. (a) If : ,f a b R is monotonic on ,a b then f is integrable on ,a b .

(OR)

(b) If ,f R a b and m, M are the infimum and supremum of f on ,a b , then

b

m b a f x dx M b a

a

.

vikrama simhapuri university::nellore cbcs...vikrama simhapuri university::nellore w.e.f. 2015-16...

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