UNIT I
Basic Mathematics:
Binomial, Exponential, Logrithmic series, summation
of infinite series and approximation promblems. L Hospitals'
rule, point wise convergence of sequence of functions,
uniform convergence of sequences of funcitons, Consequences
of Uniform convergence, Taylor's series.
Theory of numbers :
Prime and Composite numbers - Decomposition of composite
number, Divisor of N, Euler function (N), Highest power
of prime p contained in N. Divisibility of the product
of r consecutive integers by r! Format's & Wilson's
Theorems.
Vector Spaces & Inner product spaces :
Definitions and euqation of Vector space, subspace,
liner Independence - bases - Dimension, Dual spaces,
Inner products Spaces Orthogonality - Orthogonal complement.
UNIT
II
ANALYTICAL GEOMETRY:
Pairs of Straight lines - Angle between them - related
problems - conditions for second degree equation to
represent pair of straight line or Circle - System of
Circles - Orthogonal and Coaxial system - Radical axis
and radical centre - Limiting point - conics - parabola,
ellipse and hyperbola - polar equations to straight
line, circle and conic.
Dimensions :
Equation of a sphere with given centre and radius -
General form of the euqation of a sphere - Diameter
from - Circular section, tangent plane to a sphere -
Radical plane - Coaxial system of spheres - Orthogonality
- Equation of a Cone with its Vertex at the origin -
Equation of a quadratic cone with given vertix and given
guiding curve - necessary and sufficient condition for
a general second degree equation to represent a cone,
right circular cone - euqation of enveloping cone -
general equation of a cylider - right circular cylinder.
UNIT
III
CALCULUS:
Differential:-
Higher order derivatives Leibnitz's theorem - simple
problems using the above theorem. Maxima and Minima
- conditions for external value - Standard function
only - curvature - radius of curvature (Cartesian Co-ordinates
only)
Partial Differentiation:
Total differentiation Coefficieng, Valvue of dy/dx and
d2y/dx2 in case of implicit functions in x and y in
terms of partial derivatives, Total differential, Jacobians.
Integral:
Methods of integration, Integration of rational and
irrational algebraic functions, Bernaulli's formula
for Integration by parts, reduction formulae - properties
of difinite Integrals.
Evaluation of double and triple integrals, change of
order of integration, Double Integrals in polar Co-ordinates,
application of double & triple Integrals to area,
volume. Evaluation of Define integrals using Beta and
gamma functions.
UNIT
IV
STATICS:
Gradient, Divergence, Curl, solenoidal & irrotational
vectors, Directional derivative, Unit vector normal
to a surface, tangent and normal planes to a surface
2, expansion formulea, Ordinary integrals of Vectors,
line integrals, surface intergrals and volume Integrals.
Gass stock, Green's theorems. Parallelogram and Triangle
laws of force, Lamis theorem, parallel forces, moments,
couples, three forces acting on a rigid body, conditons
for equiliburium of Co-planar forces.
Forces in 3 dimensions, Invariance of F2, Friction,
Centre of Gravity, method of symmetry for uniformbodies
like thin rod, thin parallelogram, Circular ring &
lamina triangular lamina, trapezium lamina.
UNIT
V
REAL ANALYSIS:
Set and functions, sequences of real numbers - Definition,
Limit, Convergent and divergent sequences, bounded sequences,
monotonice sequence, series of real numbers, limit superior,
Limit inferior, Cauchy, Sequence, convergent & divergent
sequence, series with non-negative terms, alternating
series.
Series of real numbers:
rearrangement of series, Tests of absolute Covergence.
Limits & matric spaces:
Limit of a function on the real line, matric spaces,
limits in matric spaces. Continuous functions on matric
spaces, functions continuous at a point on the real
line, reformulation, function continious on a matric
space, open sets, closed sets, Discontinuous functions
on 'R' Connectedness, Complexness and Compactness.
UNIT
VI
OPERATIONS RESEARCH AND LINER PROGRAMMING:
Origin and development of O.R. - Nature and characteristics
of O.R. Models in O.R. General solutions, methods for
O.R. models - uses and limitations of O.R.
Linear Programming:
Formulation of problems, Graphical solution - standard
form. Definition of basic solution. degenerate Simplex
method, Definition of artifical variable.
Tranportation problem:
Definition solutions to transport problem - intial feasible
solution - optimality test - Degenerary - Travelling
sales man problem
Sequencing:
Processing n jobs through m machines.
UNIT
VII
ALGEBRA:
Set theory - Relations - types of relations - Venn diagram
- Groups - Sub group - order of an element - cyclic
groups - normal groups-quotient groups - order of a
Group Lagrange's theorem - homomorphism, automorphims,
Cayley's theorm of permutation groups.
Rings:
Definition, examples - special classes of rings - Homomorphism,
ideals and quotient rings - field of quotients of an
integral domain - Euclidean rings.
Matrices:
Types of matrices - operation on matrices, singular
and non singular matrices - Rank of a matrix and consistence
of equation, eigen values & eigen vectors. Cayley
- Hamilton theorem. Similar matrices, Diagonalisation
of a matrix.
UNIT
VIII
DIFFERENTIAL GEOMETRY:
Curvature, Radius and centre of curvature in Cartesian
Co-ordinates, Evalute - curvature in Polar Co-ordinates,
p-r equations, Angle between radius vector and tangent,
Angle of intersection of two curves. Pedal equation
of a curve, Envelopes, Asymptotes.
Polar Co-ordinates :
Equations of straight line, Circle in polars - equations
of tangent, normal & polar Equations of Conics in
polars - equations of tangent, normal, polar & asymptotes.
UNIT
IX
DIRRERENTIAL EQUATIONS:
Ordinary differential equations - first order but not
of first degree. Total differntial equation Pdx + Qdy
+ Rdz = 0, second order differential equations with
constant Co-efficients. P.I. for the polynomials and
eaxv, where V is Xn, Cos mx, Sin mx, n and m are constants.
Differential equations of second order with variable
Co-efficients. Partial differential equations - formation
of partial differential equations by elimination - Laplace
transforms - Inverse laplace transform.
UNIT
X
Dynamics:
Virtual displacement, Principle of Virtual work.
Kinematic:
Velocity, Acceleration, components of velocity and acceleration
work power, energy, Ractilinear motion - motion with
constant acceleration - motion under gravity - motion
along an inclined plane, motion under gravity in a resisting
medium.
Implusive forces and Impact:
Implusive forces and Impact, Principles of Conservation
of linear momentum, Collison of two smooth spheres -
Direct impact of sphere on a fixed plane.
Projectiles: Two dimensions motion of a particle - projectile,
range on a horizontal plane - range on an inclined plane.
Circular motion of a particle:
Motion of a particle constrained to move along a smooth
verticle circle under gravity - circular pendulum -
simpel pendulum.
Moments of Inertia:
Momentsof Inertia of simple bodies of paralle and perpendicular
axed theorem. Motion of a rigid body about a fixed axis.
UNIT
XI
STATISTICS:
Frequency distributions - Graphs of frequency distribution,
measures of central tendency, measures of dispresion,
normal probability curve, skewness, kur tosis, Probability
- Addition and multiplicaiton theorem. Baye's theorem.
Probability Distributions :
Binomial, Poisson, Normal Bivariate data, Curve fitting
- Method of least squares. Correlation and regression
Coefficient - Regression lines - rank Correlation. Test
of hypothesis - uses of X2 - F tests - Tests involving
means - Variances and proportions test of fit, test
of independence in contingency table.
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