MATHEMATICS (HONS./PG) [ CODE -19]]
A. CLASSICAL ALGEBRA:
1. Integers: Statement of well ordering Principle, first and second
principles of mathematical induction. Proofs of some simple
mathematical results by induction. Divisibility of integers. The division
algorithm. The greatest common divisor of two integers a and b – its
existence and uniqueness. Relatively prime integers. Prime integers.
Euclid’s first theorem; if some prime p divides ab, them p divides a or b.
Euclid’s second theorem: there are infinitely many prime integers.
Unique factorization theorem.
2. Complex numbers: Definition on the basis of ordered paris. Algebra of
complex numbers, Modulus, Amplitudes, Argand Diagram, De-Moivre’s
theorem and its applications, Exponential, Sine, Cosine and Logarithm
of a complex number.
Definition of az ( a ≠ 0), Inverse Circular and Hyperbolic functions.
3. Polynomials with real co-efficeints: Fundamental theorem of classical
algebra (statement only). The nth degree Polynomial equation has
exactly n roots. Nature of roots of an equation (Surd and imaginary)
roots occur in pairs). Statement of Decartes rule of signs and its
applications. Multiple roots. Relation between roots and coefficients.
Symmetric functions of roots. Transformation of equations. Reciprocal
equations. Cardan’s method of solving a cubic equation. Ferrari’s
method of solving a bi-quadratic equation.
4. Inequalities: A. M. ≥ G.M. ≥ H.M. and Cauchy’s inequality – their
simple and direct applications.
B. MODERN ALGEBRA
1. Basic Concepts: Sets, subsets, equality of sets, operations on sets –
Union, Intersection, Complements and Symmetric difference. Properties
including De-Morgan’s laws. Cartesian products, Binary relation from a
set to a set (domain, range, Examples from R x R). Equivalence relation.
Fundamental thereon on Equivalence relation. Partition, Relation of
partial order. Congruence relation modulo n is an equivalence relation..
Congruence classes. Mapping Inijection, Surjection and Bijection.
Inverse and Identity mapping. Composition of mappings and its
associativity.
2. Introduction of Group Theory: Groupoid, Semi-group, Monoid, Group
definition with both sided identity and inverses. Examples of finite and
infinite groups taken from various branches. Additive (multiplicative)
group of integers modulo an integer (resp. a prime). Klein’s 4 group
Integral powers of an element and laws of indices in a group. Order a
group and order of an element of a group. Subgroups. Nec. And Suff.
Condition for a subset of a group to be subgroup. Intersection and Union
of two subgroups. Cosets and Lgrange’s theorem. Cyclic groups –
definition, examples and subgroups of cyclic groups. Generators
Permutations. Cycle. Transposition. Even odd permutations.
Symmetric group. Definition and order of Alternating subgroup.
Normal subgroups of a group -- Definition, examples and
characterizations.
Quotient group of a group by a normal subgroup. Homomorphism and
Isomorphism of groups. Kernel of homomorphism. Fundamental
theorem of homomorphism. An infinite cyclic group is isomorphic to
(z, +) and a finite cyclic group of order n is isomorphic to the group or
residue classes modulo n.
3. Introduction to rigns and fields: Ring-definition and example. Ring of
integers modulo n. Properties directly following from the definition.
Integral domain and Field-Definitions and examples. Sub-ring sub-field
& characteristic of a ring.
C. MATRIX THEORY AND LINEAR ALGEBRA:
1. Matrices of Real and Complex Numbers: Definition, examples, equality,
addition, multiplication of matrices, Transpose of a matrix, Symmetric
and Skew-symmetric matrices.
2. Determinants: Definition of a determinant of a square matrix, Basic
properties, Minors and Cofactors, equations by Cramer’s rule. Problems
of determinants up to order 3.
3. Rank of a Matrix: Adjoint of a square matrix. For a square matrix A of
order n, A. Adj. A – Adj A. A = det A. Singular, non-singular and
invertible matrices. Elementary operations. Rank of matrix and its
determination. Normal forms: Elementary matrices; The normal form
equivalence of matrices. Congruence of Matrices. Diagonalisation of
matrices. Real quadratic from involving three variable. Reduction to
Normal form.
4. Vector/ Linear Space Over a Field: Definition and example of vector
space. Subspace. Union, Intersection and sum of vector spaces. Linear
span. Generators and basis of a vector spaces. Formation of basis from
linearly independent subset. Special emphasis of R.
5. Row-space and column-space of a matrix: Definitions of row-space and
column-space of a matrix. Row rank, column rank and Rank of a matrix.
6. System of Linear Equations: Solution space of a homogeneous system as
a subspace. Condition for the existence of non-trivial solution of a
system of linear homogeneous equations. Necessary and sufficient
conditions for the consistency of a system of non-homogeneous equations.
Solution of system of equations by matrix method.
7. Linear Transformation on Vector Spaces: Definition of linear
transformation. Null space, Range space, Rank and Nullity of linear
transformation. Sylvester’s law of Nullity. (Inverse of linear
transformation relative to ordered bases of finite dimensional vector
spaces.)
8. Inner product space: Definition and examples. Norm. Euclidean Vector
space – Triangle inequality and Cauchy – Schwarz inequality in
Euclidean vector space. Orthogonality of vectors. Orthonormal basis.
Gram-Schmidt process of Orthonormalization.
9. Eigen value and Eigen vector, Characteristics equation of a square
matrix. Caley-Hamilton’s Theorem. Simple properties of Eigen values
and Eigen vectors.
II. REAL ANALYSIS
1. Real-Number: Geometric representation and Cantor, Dedekind Axiom.
Salient properties taken as axioms Bounded set. Least upper bound
axioms. Archimedean property. Decimal representation of real
numbers.
2. Points Sets in R1 and R2: Elementary properties and union of atmost
denumerable sets. Denumberability of rational numbers and
non-denumberability of real numbers and of an interval. Neighbourhood
of a point, interior point, of linear point set, open and closed sets, limit
point of a set in R1 and R2 concepts and simple properties. Union,
intersection and complement of open and closed sets and
Bolzano-Weiestrass theorem in R1. Covering by open intervals of linear
point set, Lindeloff covering theorem and Heine Borel theorem
(statements only) and compact sets in R1.
3. Real-valued functions defined on intervals: Bounded and monotonic
functions. Limits, Algebra of limits. Sandwich rule, condition for the
existence of a finite limit. Important limits like.
sin x, Log(l +x), ex – 1 as x → 0 etc.
n x x
4. Sequence of Points in One Dimension: Bounds, limits, convergence and
divergence. Operation on limits. Sandwich rule. Monotone sequence.
Nested interval therem. Cauchy’s General Principle of convergence.
Cauchy sequence, Limits of important sequence. Definition of e.
Cauchy’s first and second limit theorem. Subsequence.
5. Infinite Series of Constant Terms: Convergene and divergene. Cauchy’s
criterion. Abel-Pringsheim’s Test. Tests (Comparison test, Root Test)
convergence of series of non-negative terms.
Series of arbitrary terms. Absolutely convergent and conditionally
convergent series. Alternative series. Leibnitz test. Root and Ratio
Tests. Non-absolute convergence --- Abel’s and Dirichlet’s tests
(statement and applications)
6. Continuity of a function at a point and on an interval: Continuity of
some standard functions, continuity of composite functions. Piecewise
continuous functions. Uniform continuity. Discontinuities of different
kinds. Properties of continuous functions on a closed interval. Existence
of inverse functions of a strictly monotone function and its continuity.
7. Concept of Differentiability and differential: Chain rule. Sign of
derivative. Successive derivatives. Leibnitz theorem. Theorms on
Derivatives : Darbox theorem, Rolle’s theorem. Mean value theorems of
Lagrange and Cauchy. Taylor’s theorem.
Maclaurin’s series. Expansion of ex, ax, a > 0, log (l+x) (l+x)m, Sinx,
Cosx etc. with their respective ranges of validity.
8. Indeterminate forms: L Hospital’s rule and its consequences.
9. Maxima and Minima: Points of local extremum of a functions in an
interval. Sufficient condition for the existence of a local
miximum/minimum of a function at a point. Applications in Geometrical
and Physical problems.
10. Tangents and Normals: Pedal equation, Peadal of a curve, Rectilinear
Asympotes (Cartesian and parametric form). Curvature- radius and
centre of curvature. Chord of curvature. Curve-Tracing (familiarity
with well-known curves)
11. Indefinite and Suitable Corresponding Definite integrals for the
functions,
sinmx, cosnx, sinnx, sinmx , tannx, secnx
cosnx
cosmx, sinnx etc. Icosx + m sin x
pcos+qsinx
1______ 1________where 1, m, p, q, n are positive integers
(a + cosx)n (n2+ a2)n
12. Area enclosed by a curve, length of a curve.
13. Sequence of functions: Pointwise and uniform convergence. Cauchy’s
criterion of uniform convergence. Limit function:
Boundness, Repeated limits, continuity and differentiability.
14. Series of functions: Pointwise and uniform convergence. Tests of
convergence statements of Abel’s and Dirichlet’s tests and their
applications.
Passage to the limit term-by-term; boundedness, continuity,
integrability and differentiability of a series of functions in case of
uniform convergence.
15. Power Series: Radius of convergence of its existence, Cauchy Hadamard
theorem. Uniform and absolute convergence. Properties of sum
function. Abet’s limit theorems. Uniqueness of P. S. having the same
sum function, Exponential. Logarithm and trigonometric functions
defined by power series and deduction of their salient properties.
16. Riemann integration: Upper sum and lower sum. Upper and lower
integral. Refinement of partitions and associated results. Darboux
theorem. Necessary and sufficient condition of integrability.
Integrability of sum, product, quotient and modulus. Integral on the
limit of a sum. Integrability monotone function, continuous function and
piece wise continuous function. Primitive, properties of definite integral,
Fundamental theorem of integral calculus First and second mean-value
theorem of integral calculus (statements and applications only).
17. Improper Integrants: Tests of convergene : comparison and r-test
(statement only). Absolute and non-absolute convergence-corresponding
test (statement only). Working knowledge of Beta and Gamma functions
and their interrelations.
18. Functions of two variable: Limit, continuity, partial derivatives.
Functions on R2 differentiability, differential. Chain rule. Euler’s
theorem, commutativity of partial derivatives statement of Schwarz and
Young theorems.
III. DIFFERENTIAL EQUATIONS
1. Significance of ordinary differential Equations: Geometrical and
physical consideration. Formation of differential equation by
elimination of arbitrary constants. Meaning of the solution of ordinary
differential equation. Concepts of linear and non-linear differential
equations.
2. Equations of first order and first degree: Statement of existence
theorem. Separable, homogeneous and exact equations, condition of
exactness, integrating factor. Equations reducible to first order linear
equations.
3. First order linear equations: Integrating factor. Equations reducible to
first order linear equations.
4. Equations of first order but not of first Degree: Clairaut’s equation,
singular solution.
5. Applications: Geometric applications, Orthogonal trajectories.
6. Higher order linear equations with constant coefficients:
Complementary function. Particulars integral, Symbolic operator. D.
Method of variation of parameters. Euler Equations – reduction to an
equation of constant coefficients.
IV. ANALYTICAL GEOMETRY OF TWO AND THREE DIMENSIONS
A. TWO DIMENSIONS
1. Transformations of rectangular Axes: Translation, Rotation and their
combinations. Theory of Invariants.
2. General Equations of Second Degree in two variables: Reduction to
canon.
3. Paris of straight lines: Condition that the general equation of second
degree in two variable may represent two straight lines. Point of
intersection of two intersection straight lines. Angle between two lines
given by ax2 + 2hxy +by2 =0 Angle bisectors. Equation of two lines
joining the origin to the points in which a line meets a conic.
4. Circle, parabola, ellipse and phyperbola : Equations of pair of tangents
from an external point, chord of contact, Poles and Polars. Conjugate
point and conjugate line.
5. Polar Equations: Polar equations of straight lines, circles and conic
referred to a focus as pole, Equations of tangent, normal and chord of
contact.
B. THREE DIMENSIONS
1. Rectangular cartesion co-ordinate in space: half and octants concept of a
geometric vector (directed line segment projection of a vector on
co-ordinate axis. Inclination of a projection of a vector on co-ordinate
axis. Inclination of a projection of a vector on co-ordinate axis.
Inclination of a vector with an axix. Co-ordinates of a vector. Direction
cosine of a vector. Distance between two points. Division of a directed
segment in a given ratio.
2. Equation of plane: General form, intercept and Normal forms. The
sides of a plane signed distance of a point from a plane. Equation of a
plane passing through the intersection of two planes. Angle between
intersection planes, Besectors of angels between two intersecting planes.
Parallelism and perpendicularity of two planes.
3. Straight lines in space: Equation (symmetric and parametric form)
Direction ratio and Direction cosines. Canonical equation of the line of
intersection to two intersecting plane. Angle between two lines.
Distance of a point from a line. Condition of coplanarity of two lines.
Equations of skewlines. Shortest distance between two skew lines.
4. Sphere: General equation, circle, sphere-through the inter section of
two-spheres. Radical Plane. Tangent, Normal.
5. General equation of 2nd degree in 3 variable. Reduction to canonical
forms. Classification of quadrics.
V. VECTOR ALGEBRA & ANALYSIS
1. Vector Algebra: Vector (directed line segment) Equality of two free vectors.
Addition. Multiplication by a scalar. Position Vector: Point of division.
Conditions of collinearity of 3 points and co planarity of 4 points.
Rectangular components of a vector in two and three dimensions, product of
two or more vectors: scalar and vector products, Scalar triple products and
vector triple products. Products of four vectors.
Direct applications of vector algebra in (i) Geometrical, trigonometrically
problems, (ii) Work done by a force. Moment of a force about a point,
vectorial equations of straight lines and planes. Volume of trahedron.
Shortest distance between two skew lines.
2. Vector Analysis: Vector differentiation with reference to a sector variable.
Vector functions of one scalar variable. Derivative of a vector. Second
derivative of a vector. Derivatives of sums and products. Velocity and
Acceleration as derivative.
VI. MECHANICS - I
1. Composition and Resolution of coplanar concurrent forces. Resolution of
forces. Moments and Couples.
2. Reduction of a system of coplanar forces. Conditions of equilibrium of
coplanar forces.
3. Fundamental ideas and principles of Dynamics. Laws of motion. Impulse
and impulsive forces. Work, power and energy, principles of conservation of
energy and momentum.
4. Motion in a straight line under variable acceleration. Motion under inverse
square law. Composition of two S. H. M’s of nearly equal frequencies.
Motion of a particle tied to one end of an elastic string. Rectilinear motion
in a resisting medium. Damped forced oscillation. Motion under gravity
where the resistance varies as some integral (nth) power of velocity.
Terminal velocity.
5. Impact of elastic bodies. Newton’s experimental law of elastic impact. Loss
of K. E. in a direct impact.
6. Expressions for velocity and acceleration of a particle moving on a plane in
Cartesian and Polar co-ordinates. Motion of a particle moving in a plane in
Cartesian and Polar co-ordinate.
7. Central forces and central orbits. Characteristics of central orbits.
8. Tangential and Normal accelerations. Circular motions.
9. Motion of a particle in a plane under different laws of resistance. Motion of
a projectile in a resisting medium in which the resistance varies the
velocity.
10. Laws of friction, cone of friction. To find the positions of equilibrium of a
particle lying on a (i) rough plane curve, (ii) rough surface under the action
of any given forces.
11. General formula for the determination of centre of gravity.
VII LINEAR PROGRAMMING PROBLEM (L.P.P.)
1. Definition of L.P.P. Formation of L.P.P. from daily life involving
inequations. Graphical solution of L.P.P.
2. Basic solution and Basic Feasible solution (BFS) with reference to L.P.P.
Matrix formulation of L.P.P. Degenerate and non-degenerate B.F.S.
Hyperplane, convex set, Cone, Extreme points. Convex hull and convex
polyhedron. Supporting and separating hyperplane. Simple results on
convex sets like the collection of all feasible solutions of an L.P.P.
constitutes a convex set.
The extreme points of the convex set of feasible solutions correspond to its
B. F.S. (no proof). The objective function has its optimal value at an
extreme point of the convex polyhedron generated by the act of feasible
solutions (no proof). Fundamental theorem (no proof). Reduction of a F.S.
to a B.F.S.
3. Slack and Surplus variables. Standard form of L.P.P. theory of simplex
method. Feasibility and optimality conditions.
4. The algorithm. Two phase method. Degeneracy in L.P.P. and its
resolution.
5. Duality Theory: The dual of the dual to the Primal.
Relation between the objective values of dual and the primal problems.
Relation between their optimal values. Complementary slackness. Duality
and simplex method and their applications.
6. Transporation and Assignment problems, and that optimal solution.
VIII. MECHANICS - II
1. Laws of friction, cone of friction. To find the positions of equilibrium of a
particle lying on a (a) rough plane curve, (ii) rough surface under the action
of any given forces.
2. General formula for the determination of centre of gravity.
3. Astatic equilibrium, Astatic Centre. Positions of equilibrium of a Particle
lying on a smooth plane curve under action of given forces.
4. Virutal work: Principle of virtural work for a single particle. Deduction of
the conditions of equilibrium of a particle under coplanar forces from the
principle of virtual work. The principle of virtual work for a rigid body.
Forces which do not appear in the equation of virtual work. Forces which
appear in the equation of virtual work. The principle of virtual work for
any system of coplanar force acting on a rigid body.
Converse of principle of virtual work.
5. Forces in 3-dim: Moment of a force about a line. Axis of couple. Resultant
of any number of couples acting on a rigid body. Reduction of a system of
forces acting on a rigid body. Poinsot’s Central axix. Wrench, Pitch,
Intensity and screw. Invariant and equation of the central axis of a given
system of forces.
6. Motions under inverse square law in a plane. Escape velocity. Planetary
motions and Keplar’s Laws. Artificial satellite Motion. Slightly disturbed
orbit. Conservative field of force and principles of conservation of energy,
Motion under rough curve (circle, parabola, ellipse, Cycliod) under gravity
7. RIGID DYNAMICS:
Moments and products of inertia. Theorem of parallel and perpendicular
axes. Principles axes of inertia, momental ellipsoid Equimomental system.
D’Alembert’s principle. Equation of Motion. Principles of moments.
Principle of conservation of linear and angular momentum. Principles of
energy.
Equation of Motion of a rigid body about a fixed axis.
Expression for K.E. and moment of momentum of a rigid body moving about
a fixed axis. Compound pendulum.
Equation of Motion of a rigid body moving in 2-dim. Expression for K.
E.and angular momentum about the origin of a rigid body moving in 2 dim.
Motion of a solid revolution moving on a rough horizontal & inclined plane..
Conditions for pure rolling.
Impulsive action.
Generalised coordinates, momentum
Lagrangian, Cyclic coordinates, Ronthian
IX. A. MATHEMATICAL THEORY OF PROBABILITY
Random experiments. Simple and compound events. Event space. Classical
and frequency definitions of probability and their drawbacks. Axioms of
probability, Statistical regularity. Multiplication rule of probabilities.
Bayes theorem. Independent events. Independent random experiments.
Independent trials. Bernoulli trails and law. Multinominal law. Random
variables, Probability distribution. Distribution function, descrete and
continuous distributions. Bimominal, Poison, Uniform, normal distribution.
Cauchy gamma distributions. Beta distribution of the first and of the
second kind. Poison process. Transformation of random variables.
Two-dim, prob. Distribution. Discrete and continuous distributions in two
dimensions. Uniform distributions, and two-dimensional normal
distribution. Conditional distributions. Transformation of random
variables in two dimensions. Mathematical expectation. Mean, variance,
moment, central moments. Measures of location, dispersion, skewness and
Kurtosts. Median, Mode, quartiles, Moment-generating function
characteristics function. Two dimensional expectation. Covariance.
Co-relation Co-efficient. Joint characteristic function. Multiplication rule
for expectations, conditional expectations, Regression curves, least square
regression lines and parabolas. Chi square and distributions and their
important properties, inequality Convergence in probability. Bermouli’s
limit theorem. Law of large numbers. Poissons approximation to binomial
distribution. Normal approximation to binomial distribution. Concept of
asymptotically normal distributions. Statement of central limit theorem in
the case of equal components and of limit theorem for characteristic
functions and in applications. (Stress should be more on distributive
function theory than on combinational problems. Different combinatorial
problems should be avoided).
B. MATHEMATICAL STATISTICS:
Random samples. Distribution of the sample. Tables and graphical
representations. Grouping of data. Sample characteristic and their
computation. Sampling distribution of a statistic. Estimates of a
population characteristic or parameter. Unbiased consistent estimates.
Sample characteristics as estimates of the corresponding population
characteristics. Sampling distributions of the sample mean and variance.
Exact sampling distributions for the normal populations.
Bivariate samples. Scatter diagram. Sample correlation coefficient. Least
square regression lines and parabolas. Estimation of parameters. Method
of maximum likelihood. Applictions to binomial. Normal populations.
Confidence intervals. Such intervals for the parameters of the normal
populations. Approximate confidence interval for the paratmer of a
binomial population. Statistical hypothesis. Simple and composite
hypothesis. Best critical region of a test. Neyman Pearson theorem and its
applications to normal populations. Likelihood ratio testing and its
applications to normal population.
X.NUMERICAL ANALYSIS
1. Computational Errors: Round-off errors, significant digits, errors in
arithmetical operations, guard figures in calculations.
2. Interpolation: Polynomial Interpolation, remainder, Equally-spaced
interpolating points-difference, difference table, propagation of errors;
Newton’s forward and backward, Stirling and Bessel interpolation formulae,
divided differences, divided difference, formula, confluent divided differences,
inverse interpolation.
3. Numerical Differentiation: Error in numerical differentiation. Newton’s
forward and backward and Lagrange’s numerical differentiation formula.
4. Numerical Integration: Degree of precision, open & closed formulae,
composite rules. Newton-Cotes (closed-type) formula – Trapeezoidal,
Simpson’s one third and Weddle’s rules, error formulae in terms of ordinates
(proofs not necessary).
5. Numerical Solutions of Equations: Initial approximation by methods of
tabulation and graph, methods of bisection, fixed point iteration with
condition of convergence. Newton – Raphson & Regula-falsi methods,
computable estimate of the error in each method.
6. Solution of ODE:
First Order First degree: By Euler, RK4 and Milne’s method.
A. CLASSICAL ALGEBRA:
1. Integers: Statement of well ordering Principle, first and second
principles of mathematical induction. Proofs of some simple
mathematical results by induction. Divisibility of integers. The division
algorithm. The greatest common divisor of two integers a and b – its
existence and uniqueness. Relatively prime integers. Prime integers.
Euclid’s first theorem; if some prime p divides ab, them p divides a or b.
Euclid’s second theorem: there are infinitely many prime integers.
Unique factorization theorem.
2. Complex numbers: Definition on the basis of ordered paris. Algebra of
complex numbers, Modulus, Amplitudes, Argand Diagram, De-Moivre’s
theorem and its applications, Exponential, Sine, Cosine and Logarithm
of a complex number.
Definition of az ( a ≠ 0), Inverse Circular and Hyperbolic functions.
3. Polynomials with real co-efficeints: Fundamental theorem of classical
algebra (statement only). The nth degree Polynomial equation has
exactly n roots. Nature of roots of an equation (Surd and imaginary)
roots occur in pairs). Statement of Decartes rule of signs and its
applications. Multiple roots. Relation between roots and coefficients.
Symmetric functions of roots. Transformation of equations. Reciprocal
equations. Cardan’s method of solving a cubic equation. Ferrari’s
method of solving a bi-quadratic equation.
4. Inequalities: A. M. ≥ G.M. ≥ H.M. and Cauchy’s inequality – their
simple and direct applications.
B. MODERN ALGEBRA
1. Basic Concepts: Sets, subsets, equality of sets, operations on sets –
Union, Intersection, Complements and Symmetric difference. Properties
including De-Morgan’s laws. Cartesian products, Binary relation from a
set to a set (domain, range, Examples from R x R). Equivalence relation.
Fundamental thereon on Equivalence relation. Partition, Relation of
partial order. Congruence relation modulo n is an equivalence relation..
Congruence classes. Mapping Inijection, Surjection and Bijection.
Inverse and Identity mapping. Composition of mappings and its
associativity.
2. Introduction of Group Theory: Groupoid, Semi-group, Monoid, Group
definition with both sided identity and inverses. Examples of finite and
infinite groups taken from various branches. Additive (multiplicative)
group of integers modulo an integer (resp. a prime). Klein’s 4 group
Integral powers of an element and laws of indices in a group. Order a
group and order of an element of a group. Subgroups. Nec. And Suff.
Condition for a subset of a group to be subgroup. Intersection and Union
of two subgroups. Cosets and Lgrange’s theorem. Cyclic groups –
definition, examples and subgroups of cyclic groups. Generators
Permutations. Cycle. Transposition. Even odd permutations.
Symmetric group. Definition and order of Alternating subgroup.
Normal subgroups of a group -- Definition, examples and
characterizations.
Quotient group of a group by a normal subgroup. Homomorphism and
Isomorphism of groups. Kernel of homomorphism. Fundamental
theorem of homomorphism. An infinite cyclic group is isomorphic to
(z, +) and a finite cyclic group of order n is isomorphic to the group or
residue classes modulo n.
3. Introduction to rigns and fields: Ring-definition and example. Ring of
integers modulo n. Properties directly following from the definition.
Integral domain and Field-Definitions and examples. Sub-ring sub-field
& characteristic of a ring.
C. MATRIX THEORY AND LINEAR ALGEBRA:
1. Matrices of Real and Complex Numbers: Definition, examples, equality,
addition, multiplication of matrices, Transpose of a matrix, Symmetric
and Skew-symmetric matrices.
2. Determinants: Definition of a determinant of a square matrix, Basic
properties, Minors and Cofactors, equations by Cramer’s rule. Problems
of determinants up to order 3.
3. Rank of a Matrix: Adjoint of a square matrix. For a square matrix A of
order n, A. Adj. A – Adj A. A = det A. Singular, non-singular and
invertible matrices. Elementary operations. Rank of matrix and its
determination. Normal forms: Elementary matrices; The normal form
equivalence of matrices. Congruence of Matrices. Diagonalisation of
matrices. Real quadratic from involving three variable. Reduction to
Normal form.
4. Vector/ Linear Space Over a Field: Definition and example of vector
space. Subspace. Union, Intersection and sum of vector spaces. Linear
span. Generators and basis of a vector spaces. Formation of basis from
linearly independent subset. Special emphasis of R.
5. Row-space and column-space of a matrix: Definitions of row-space and
column-space of a matrix. Row rank, column rank and Rank of a matrix.
6. System of Linear Equations: Solution space of a homogeneous system as
a subspace. Condition for the existence of non-trivial solution of a
system of linear homogeneous equations. Necessary and sufficient
conditions for the consistency of a system of non-homogeneous equations.
Solution of system of equations by matrix method.
7. Linear Transformation on Vector Spaces: Definition of linear
transformation. Null space, Range space, Rank and Nullity of linear
transformation. Sylvester’s law of Nullity. (Inverse of linear
transformation relative to ordered bases of finite dimensional vector
spaces.)
8. Inner product space: Definition and examples. Norm. Euclidean Vector
space – Triangle inequality and Cauchy – Schwarz inequality in
Euclidean vector space. Orthogonality of vectors. Orthonormal basis.
Gram-Schmidt process of Orthonormalization.
9. Eigen value and Eigen vector, Characteristics equation of a square
matrix. Caley-Hamilton’s Theorem. Simple properties of Eigen values
and Eigen vectors.
II. REAL ANALYSIS
1. Real-Number: Geometric representation and Cantor, Dedekind Axiom.
Salient properties taken as axioms Bounded set. Least upper bound
axioms. Archimedean property. Decimal representation of real
numbers.
2. Points Sets in R1 and R2: Elementary properties and union of atmost
denumerable sets. Denumberability of rational numbers and
non-denumberability of real numbers and of an interval. Neighbourhood
of a point, interior point, of linear point set, open and closed sets, limit
point of a set in R1 and R2 concepts and simple properties. Union,
intersection and complement of open and closed sets and
Bolzano-Weiestrass theorem in R1. Covering by open intervals of linear
point set, Lindeloff covering theorem and Heine Borel theorem
(statements only) and compact sets in R1.
3. Real-valued functions defined on intervals: Bounded and monotonic
functions. Limits, Algebra of limits. Sandwich rule, condition for the
existence of a finite limit. Important limits like.
sin x, Log(l +x), ex – 1 as x → 0 etc.
n x x
4. Sequence of Points in One Dimension: Bounds, limits, convergence and
divergence. Operation on limits. Sandwich rule. Monotone sequence.
Nested interval therem. Cauchy’s General Principle of convergence.
Cauchy sequence, Limits of important sequence. Definition of e.
Cauchy’s first and second limit theorem. Subsequence.
5. Infinite Series of Constant Terms: Convergene and divergene. Cauchy’s
criterion. Abel-Pringsheim’s Test. Tests (Comparison test, Root Test)
convergence of series of non-negative terms.
Series of arbitrary terms. Absolutely convergent and conditionally
convergent series. Alternative series. Leibnitz test. Root and Ratio
Tests. Non-absolute convergence --- Abel’s and Dirichlet’s tests
(statement and applications)
6. Continuity of a function at a point and on an interval: Continuity of
some standard functions, continuity of composite functions. Piecewise
continuous functions. Uniform continuity. Discontinuities of different
kinds. Properties of continuous functions on a closed interval. Existence
of inverse functions of a strictly monotone function and its continuity.
7. Concept of Differentiability and differential: Chain rule. Sign of
derivative. Successive derivatives. Leibnitz theorem. Theorms on
Derivatives : Darbox theorem, Rolle’s theorem. Mean value theorems of
Lagrange and Cauchy. Taylor’s theorem.
Maclaurin’s series. Expansion of ex, ax, a > 0, log (l+x) (l+x)m, Sinx,
Cosx etc. with their respective ranges of validity.
8. Indeterminate forms: L Hospital’s rule and its consequences.
9. Maxima and Minima: Points of local extremum of a functions in an
interval. Sufficient condition for the existence of a local
miximum/minimum of a function at a point. Applications in Geometrical
and Physical problems.
10. Tangents and Normals: Pedal equation, Peadal of a curve, Rectilinear
Asympotes (Cartesian and parametric form). Curvature- radius and
centre of curvature. Chord of curvature. Curve-Tracing (familiarity
with well-known curves)
11. Indefinite and Suitable Corresponding Definite integrals for the
functions,
sinmx, cosnx, sinnx, sinmx , tannx, secnx
cosnx
cosmx, sinnx etc. Icosx + m sin x
pcos+qsinx
1______ 1________where 1, m, p, q, n are positive integers
(a + cosx)n (n2+ a2)n
12. Area enclosed by a curve, length of a curve.
13. Sequence of functions: Pointwise and uniform convergence. Cauchy’s
criterion of uniform convergence. Limit function:
Boundness, Repeated limits, continuity and differentiability.
14. Series of functions: Pointwise and uniform convergence. Tests of
convergence statements of Abel’s and Dirichlet’s tests and their
applications.
Passage to the limit term-by-term; boundedness, continuity,
integrability and differentiability of a series of functions in case of
uniform convergence.
15. Power Series: Radius of convergence of its existence, Cauchy Hadamard
theorem. Uniform and absolute convergence. Properties of sum
function. Abet’s limit theorems. Uniqueness of P. S. having the same
sum function, Exponential. Logarithm and trigonometric functions
defined by power series and deduction of their salient properties.
16. Riemann integration: Upper sum and lower sum. Upper and lower
integral. Refinement of partitions and associated results. Darboux
theorem. Necessary and sufficient condition of integrability.
Integrability of sum, product, quotient and modulus. Integral on the
limit of a sum. Integrability monotone function, continuous function and
piece wise continuous function. Primitive, properties of definite integral,
Fundamental theorem of integral calculus First and second mean-value
theorem of integral calculus (statements and applications only).
17. Improper Integrants: Tests of convergene : comparison and r-test
(statement only). Absolute and non-absolute convergence-corresponding
test (statement only). Working knowledge of Beta and Gamma functions
and their interrelations.
18. Functions of two variable: Limit, continuity, partial derivatives.
Functions on R2 differentiability, differential. Chain rule. Euler’s
theorem, commutativity of partial derivatives statement of Schwarz and
Young theorems.
III. DIFFERENTIAL EQUATIONS
1. Significance of ordinary differential Equations: Geometrical and
physical consideration. Formation of differential equation by
elimination of arbitrary constants. Meaning of the solution of ordinary
differential equation. Concepts of linear and non-linear differential
equations.
2. Equations of first order and first degree: Statement of existence
theorem. Separable, homogeneous and exact equations, condition of
exactness, integrating factor. Equations reducible to first order linear
equations.
3. First order linear equations: Integrating factor. Equations reducible to
first order linear equations.
4. Equations of first order but not of first Degree: Clairaut’s equation,
singular solution.
5. Applications: Geometric applications, Orthogonal trajectories.
6. Higher order linear equations with constant coefficients:
Complementary function. Particulars integral, Symbolic operator. D.
Method of variation of parameters. Euler Equations – reduction to an
equation of constant coefficients.
IV. ANALYTICAL GEOMETRY OF TWO AND THREE DIMENSIONS
A. TWO DIMENSIONS
1. Transformations of rectangular Axes: Translation, Rotation and their
combinations. Theory of Invariants.
2. General Equations of Second Degree in two variables: Reduction to
canon.
3. Paris of straight lines: Condition that the general equation of second
degree in two variable may represent two straight lines. Point of
intersection of two intersection straight lines. Angle between two lines
given by ax2 + 2hxy +by2 =0 Angle bisectors. Equation of two lines
joining the origin to the points in which a line meets a conic.
4. Circle, parabola, ellipse and phyperbola : Equations of pair of tangents
from an external point, chord of contact, Poles and Polars. Conjugate
point and conjugate line.
5. Polar Equations: Polar equations of straight lines, circles and conic
referred to a focus as pole, Equations of tangent, normal and chord of
contact.
B. THREE DIMENSIONS
1. Rectangular cartesion co-ordinate in space: half and octants concept of a
geometric vector (directed line segment projection of a vector on
co-ordinate axis. Inclination of a projection of a vector on co-ordinate
axis. Inclination of a projection of a vector on co-ordinate axis.
Inclination of a vector with an axix. Co-ordinates of a vector. Direction
cosine of a vector. Distance between two points. Division of a directed
segment in a given ratio.
2. Equation of plane: General form, intercept and Normal forms. The
sides of a plane signed distance of a point from a plane. Equation of a
plane passing through the intersection of two planes. Angle between
intersection planes, Besectors of angels between two intersecting planes.
Parallelism and perpendicularity of two planes.
3. Straight lines in space: Equation (symmetric and parametric form)
Direction ratio and Direction cosines. Canonical equation of the line of
intersection to two intersecting plane. Angle between two lines.
Distance of a point from a line. Condition of coplanarity of two lines.
Equations of skewlines. Shortest distance between two skew lines.
4. Sphere: General equation, circle, sphere-through the inter section of
two-spheres. Radical Plane. Tangent, Normal.
5. General equation of 2nd degree in 3 variable. Reduction to canonical
forms. Classification of quadrics.
V. VECTOR ALGEBRA & ANALYSIS
1. Vector Algebra: Vector (directed line segment) Equality of two free vectors.
Addition. Multiplication by a scalar. Position Vector: Point of division.
Conditions of collinearity of 3 points and co planarity of 4 points.
Rectangular components of a vector in two and three dimensions, product of
two or more vectors: scalar and vector products, Scalar triple products and
vector triple products. Products of four vectors.
Direct applications of vector algebra in (i) Geometrical, trigonometrically
problems, (ii) Work done by a force. Moment of a force about a point,
vectorial equations of straight lines and planes. Volume of trahedron.
Shortest distance between two skew lines.
2. Vector Analysis: Vector differentiation with reference to a sector variable.
Vector functions of one scalar variable. Derivative of a vector. Second
derivative of a vector. Derivatives of sums and products. Velocity and
Acceleration as derivative.
VI. MECHANICS - I
1. Composition and Resolution of coplanar concurrent forces. Resolution of
forces. Moments and Couples.
2. Reduction of a system of coplanar forces. Conditions of equilibrium of
coplanar forces.
3. Fundamental ideas and principles of Dynamics. Laws of motion. Impulse
and impulsive forces. Work, power and energy, principles of conservation of
energy and momentum.
4. Motion in a straight line under variable acceleration. Motion under inverse
square law. Composition of two S. H. M’s of nearly equal frequencies.
Motion of a particle tied to one end of an elastic string. Rectilinear motion
in a resisting medium. Damped forced oscillation. Motion under gravity
where the resistance varies as some integral (nth) power of velocity.
Terminal velocity.
5. Impact of elastic bodies. Newton’s experimental law of elastic impact. Loss
of K. E. in a direct impact.
6. Expressions for velocity and acceleration of a particle moving on a plane in
Cartesian and Polar co-ordinates. Motion of a particle moving in a plane in
Cartesian and Polar co-ordinate.
7. Central forces and central orbits. Characteristics of central orbits.
8. Tangential and Normal accelerations. Circular motions.
9. Motion of a particle in a plane under different laws of resistance. Motion of
a projectile in a resisting medium in which the resistance varies the
velocity.
10. Laws of friction, cone of friction. To find the positions of equilibrium of a
particle lying on a (i) rough plane curve, (ii) rough surface under the action
of any given forces.
11. General formula for the determination of centre of gravity.
VII LINEAR PROGRAMMING PROBLEM (L.P.P.)
1. Definition of L.P.P. Formation of L.P.P. from daily life involving
inequations. Graphical solution of L.P.P.
2. Basic solution and Basic Feasible solution (BFS) with reference to L.P.P.
Matrix formulation of L.P.P. Degenerate and non-degenerate B.F.S.
Hyperplane, convex set, Cone, Extreme points. Convex hull and convex
polyhedron. Supporting and separating hyperplane. Simple results on
convex sets like the collection of all feasible solutions of an L.P.P.
constitutes a convex set.
The extreme points of the convex set of feasible solutions correspond to its
B. F.S. (no proof). The objective function has its optimal value at an
extreme point of the convex polyhedron generated by the act of feasible
solutions (no proof). Fundamental theorem (no proof). Reduction of a F.S.
to a B.F.S.
3. Slack and Surplus variables. Standard form of L.P.P. theory of simplex
method. Feasibility and optimality conditions.
4. The algorithm. Two phase method. Degeneracy in L.P.P. and its
resolution.
5. Duality Theory: The dual of the dual to the Primal.
Relation between the objective values of dual and the primal problems.
Relation between their optimal values. Complementary slackness. Duality
and simplex method and their applications.
6. Transporation and Assignment problems, and that optimal solution.
VIII. MECHANICS - II
1. Laws of friction, cone of friction. To find the positions of equilibrium of a
particle lying on a (a) rough plane curve, (ii) rough surface under the action
of any given forces.
2. General formula for the determination of centre of gravity.
3. Astatic equilibrium, Astatic Centre. Positions of equilibrium of a Particle
lying on a smooth plane curve under action of given forces.
4. Virutal work: Principle of virtural work for a single particle. Deduction of
the conditions of equilibrium of a particle under coplanar forces from the
principle of virtual work. The principle of virtual work for a rigid body.
Forces which do not appear in the equation of virtual work. Forces which
appear in the equation of virtual work. The principle of virtual work for
any system of coplanar force acting on a rigid body.
Converse of principle of virtual work.
5. Forces in 3-dim: Moment of a force about a line. Axis of couple. Resultant
of any number of couples acting on a rigid body. Reduction of a system of
forces acting on a rigid body. Poinsot’s Central axix. Wrench, Pitch,
Intensity and screw. Invariant and equation of the central axis of a given
system of forces.
6. Motions under inverse square law in a plane. Escape velocity. Planetary
motions and Keplar’s Laws. Artificial satellite Motion. Slightly disturbed
orbit. Conservative field of force and principles of conservation of energy,
Motion under rough curve (circle, parabola, ellipse, Cycliod) under gravity
7. RIGID DYNAMICS:
Moments and products of inertia. Theorem of parallel and perpendicular
axes. Principles axes of inertia, momental ellipsoid Equimomental system.
D’Alembert’s principle. Equation of Motion. Principles of moments.
Principle of conservation of linear and angular momentum. Principles of
energy.
Equation of Motion of a rigid body about a fixed axis.
Expression for K.E. and moment of momentum of a rigid body moving about
a fixed axis. Compound pendulum.
Equation of Motion of a rigid body moving in 2-dim. Expression for K.
E.and angular momentum about the origin of a rigid body moving in 2 dim.
Motion of a solid revolution moving on a rough horizontal & inclined plane..
Conditions for pure rolling.
Impulsive action.
Generalised coordinates, momentum
Lagrangian, Cyclic coordinates, Ronthian
IX. A. MATHEMATICAL THEORY OF PROBABILITY
Random experiments. Simple and compound events. Event space. Classical
and frequency definitions of probability and their drawbacks. Axioms of
probability, Statistical regularity. Multiplication rule of probabilities.
Bayes theorem. Independent events. Independent random experiments.
Independent trials. Bernoulli trails and law. Multinominal law. Random
variables, Probability distribution. Distribution function, descrete and
continuous distributions. Bimominal, Poison, Uniform, normal distribution.
Cauchy gamma distributions. Beta distribution of the first and of the
second kind. Poison process. Transformation of random variables.
Two-dim, prob. Distribution. Discrete and continuous distributions in two
dimensions. Uniform distributions, and two-dimensional normal
distribution. Conditional distributions. Transformation of random
variables in two dimensions. Mathematical expectation. Mean, variance,
moment, central moments. Measures of location, dispersion, skewness and
Kurtosts. Median, Mode, quartiles, Moment-generating function
characteristics function. Two dimensional expectation. Covariance.
Co-relation Co-efficient. Joint characteristic function. Multiplication rule
for expectations, conditional expectations, Regression curves, least square
regression lines and parabolas. Chi square and distributions and their
important properties, inequality Convergence in probability. Bermouli’s
limit theorem. Law of large numbers. Poissons approximation to binomial
distribution. Normal approximation to binomial distribution. Concept of
asymptotically normal distributions. Statement of central limit theorem in
the case of equal components and of limit theorem for characteristic
functions and in applications. (Stress should be more on distributive
function theory than on combinational problems. Different combinatorial
problems should be avoided).
B. MATHEMATICAL STATISTICS:
Random samples. Distribution of the sample. Tables and graphical
representations. Grouping of data. Sample characteristic and their
computation. Sampling distribution of a statistic. Estimates of a
population characteristic or parameter. Unbiased consistent estimates.
Sample characteristics as estimates of the corresponding population
characteristics. Sampling distributions of the sample mean and variance.
Exact sampling distributions for the normal populations.
Bivariate samples. Scatter diagram. Sample correlation coefficient. Least
square regression lines and parabolas. Estimation of parameters. Method
of maximum likelihood. Applictions to binomial. Normal populations.
Confidence intervals. Such intervals for the parameters of the normal
populations. Approximate confidence interval for the paratmer of a
binomial population. Statistical hypothesis. Simple and composite
hypothesis. Best critical region of a test. Neyman Pearson theorem and its
applications to normal populations. Likelihood ratio testing and its
applications to normal population.
X.NUMERICAL ANALYSIS
1. Computational Errors: Round-off errors, significant digits, errors in
arithmetical operations, guard figures in calculations.
2. Interpolation: Polynomial Interpolation, remainder, Equally-spaced
interpolating points-difference, difference table, propagation of errors;
Newton’s forward and backward, Stirling and Bessel interpolation formulae,
divided differences, divided difference, formula, confluent divided differences,
inverse interpolation.
3. Numerical Differentiation: Error in numerical differentiation. Newton’s
forward and backward and Lagrange’s numerical differentiation formula.
4. Numerical Integration: Degree of precision, open & closed formulae,
composite rules. Newton-Cotes (closed-type) formula – Trapeezoidal,
Simpson’s one third and Weddle’s rules, error formulae in terms of ordinates
(proofs not necessary).
5. Numerical Solutions of Equations: Initial approximation by methods of
tabulation and graph, methods of bisection, fixed point iteration with
condition of convergence. Newton – Raphson & Regula-falsi methods,
computable estimate of the error in each method.
6. Solution of ODE:
First Order First degree: By Euler, RK4 and Milne’s method.
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