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Mathematics – Optional (Preliminary Examination)
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1.
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Algebra
: Elements of Set Theory; Algebra of Real and Complex numbers
including Demovire's theorem; Polynomials and Polynomial equations,
relation between Coefficients and Roots, symmetric functions of roots;
Elements of Group Theory; Sub-Group, Cyclic groups, Permutation, Groups
and their elementary properties.
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Rings, Integral Domains and Fields and
their elementary properties.
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2.
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Vector
Spaces and Matrices : Vector Space, Linear Dependence and
Independence
. Sub-spaces. Basis and Dimensions, Finite Dimensional Vector Spaces.
Linear Transformation of a Finite Dimensional Vector Space, Matrix
Representation. Singular and Nonsingular Transformations. Rank and
Nullity.
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Matrices
: Addition, Multiplication,
Determinants of a Matrix, Properties of Determinants of order, Inverse of
a Matrix, Cramer's rule.
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3.
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Geometry
and Vectors : Analytic Geometry of straight lines and conics in
Cartesian and Polar coordinates; Three Dimensional geometry for planes,
straight lines, sphere, cone and cylinder. Addition, Subtraction and
Products of Vectors and Simple applications to Geometry.
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Calculus :
Functions,
Sequences, Series, Limits, Continuity, Derivatives.
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Application
of Derivatives : Rates of
change, Tangents, Normals, Maxima, Minima, Rolle's Theorem, Mean Value
Theorems of Lagrange and Cauchy, Asymptotes, Curvature. Methods of finding
indefinite integrals, Definite Integrals, Fundamental Theorem of integrals
Calculus. Application of definite integrals to area, Length of a plane
curve, Volume and Surfaces of revolution.
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5.
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Ordinary
Differential Equations : Order and Degree of a Differential Equation,
First order differential Equations, Singular solution, Geometrical
interpretation, Second order equations with constant coefficients.
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6.
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Mechanics
: Concepts of particles-Lamina; Rigid Body; Displacements; force;
Mass; weight; Motion; Velocity; Speed; Acceleration; Parallelogram of
forces; Parallelogram of velocity, acceleration; resultant; equilibrium of
coplanar forces; Moments; Couples; Friction; Centre of mass, Gravity; Laws
of motion; Motion of a particle in a straight line; simple Harmonic
Motion; Motion under conservative forces; Motion under gravity;
Projectile; Escape velocity; Motion of artificial satellites.
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7. |
Elements
of Computer Programming : Binary system, Octal and Hexadecimal
systems. Conversion to and from Decimal systems. Codes, Bits, Bytes and
Words. Memory of a computer, Arithmetic and Logical operations on numbers.
Precisions. AND, OR, XOR, NOT and Shit/Rotate operators, Algorithms and
Flow Charts.
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