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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.
BLUE PRINT OF QUESTION PAPER
(INSTRUCTIONS TO PAPER SETTER) B.A./B.Sc. MATHEMATICS SEMESTER-II
(SOLID GEOMETRY)
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
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.
Prescribed Text Book :
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
BLUE PRINT OF QUESTION PAPER
(INSTRUCTIONS TO PAPER SETTER) B.A./B.Sc. MATHEMATICS SEMESTER-III
(ABSTRACT ALGEBRA)
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
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)
VIKRAMA SIMHAPURI UNIVERSITY::NELLORE
(w.e.f. 2016-17) B.A./B.Sc. SECOND YEAR MATHEMATICS
MODEL QUESTION PAPER SEMESTER – III
(ABSTRACT ALGEBRA)
Time: 3 Hours Max. Marks : 75
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.
VIKRAMA SIMHAPURI UNIVERSITY::NELLORE
(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.
Prescribed Text Book :
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
BLUE PRINT OF QUESTION PAPER
(INSTRUCTIONS TO PAPER SETTER) B.A./B.Sc. MATHEMATICS SEMESTER-IV
(REAL ANALYSIS)
NOTE :- Paper Setter Must select TWO Short Questions and TWO Easy Questions from Each Unit as Follows :-
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)
VIKRAMA SIMHAPURI UNIVERSITY::NELLORE
(w.e.f. 2016-17) B.A./B.Sc. SECOND YEAR MATHEMATICS
SEMESTER – IV MODEL QUESTION PAPER
(REAL ANALYSIS)
Time: 3 Hours Max. Marks : 75
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
.
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