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Mathematics

Part A

Algebra: 
Algebra of sets, relations and functions, inverse of a function, composite function, equivalence relation.
Numbers: Integers, rational numbers, real numbers (statement of properties), complex numbers, algebra of complex numbers. 
Groups, sub-groups, normal sub-groups, cyclic and permutation groups, Lagrange's theorem, isomorphism. De-Moivre's theorem for rational index and its simple applications.

Theory of Equations: 
Polynomial equations, transformation of equations, relations between roots and coefficients of a polynomial equation. symmetric functions of roots of cubic and biquadratic equations, location of roots and Newton's method for finding roots.

Matrices: 
Algebra of matrices, determinants-simple properties of determinants, products of determinants andjoint of a matrix, inversion of matrices, rank of matrix, application of matrices to the solution of linear equations (in three unknowns).

Inequalities: 
Arithmetic and geometric means. Cauchy Schewarz inequality (only for finite sums).
Analytical Geometry of two dimensions: Straight lines, pair of straight lines, circles, systems of circles, Ellipse, parabola, hyperbola (referred to principal axis). Reduction of a second degree equation to standard form. Tangents and normals.

Analytical Geometry of three dimensions: 
Planes, straight lines and spheres (Cartesian Co-ordinates only).

Calculus and Differential Equations

Differential Calculus: 
Concept of limit, Continuity and differentiability of a function of one real variable, derivative of standard functions, successive differentiation. Roll's theorem. Mean Value theorem. Muclauri and Taylor series (proof not needed) and their applications; Binomial expansion for rational index, expansion of exponential, logarithmic, trigonometrical and hyperbolic functions. Indeterminate forms. Maxima and Minima of a function of a single variable, geometrical applications such as tangent, normal, sub-tangent, sub-normal, asymptomatic curvature (Cartesian coordinates only). Envelops, partial differentiation. Euler's theorem for homogeneous functions.

Integral Calculus: 
Standard methods of integration. Riemann definition of definite integral of continuous functions. Fundamental theorem of Integral calculus. Rectification, quadrature, volumes and surface area of solids of revolution. Simposon's rule for numerical integral.

Convergence of sequence and series, test of convergence of series with positive terms. Ratio, Root and Gauss tests

Alternating Series
Differential Equations: Solution of standard first order differential equations: Solution of second and higher order linear differential equations with constant coefficients. Simple applications of problems on growth and decay, simple harmonic motion. Simple pendulum and the like.

Part B

Mechanics

Static's:
Representation of a force, parallelogram of forces; composition and resolution of forces and conditions of equilibrium of coplanar and concurrent forces. Triangle of forces. Like and unlike parallel forces Moments: Couples. General conditions for equilibrium of coplanar forces. Centre of gravity of simple bodies. Friction-static and limiting friction, angle of friction, equilibrium of a particle on a rough inclined plane, simple problems, simple machines (lever, system of Pulleys, gear). Virtual work (two dimensions).

Dynamics:
Kinematics-displacement, speed, velocity and acceleration of a particle, relative velocity. Motion in a straight line under constant acceleration. Newton's laws of motion. Central orbits. Simple harmonic motion, motion under gravity (in vacuum). Impulse, work and energy. Conservation of energy and linear momentum. Uniform circular motion.

Astronomy

Spherical Trigonometry: 
Sine and cosine formulae, properties of right-angled spherical triangles.

Spherical Astronomy: 
Celestial sphere, Coordinate systems and their conversion, Diurnal motion. Sidereal and solar times mean solar time, local and standard times, equation of time. Rising and setting of the sun and stars, dip of the horizon. Astronomical refraction. Twilight. Parallax, aberration, procession and nutation. Kepler's laws, Planetary or its and stationary points. Apparent motion of the moon, phases of the moon. Astronomical Instruments-Sextant transmit instrument.

Statistics

Probability: 
Classical and statistical definition of probability, calculation of probability of combinational methods, addition and multiplication theorems, conditional probability. Random variables (discrete and continuous), density function: Mathematical expectation.

Standard distribution:
Binomial-definition, mean and variance, skewness, limiting form, simple applications; Poisson definition, mean and variance, additive property, fitting of Poisson-distribution to given data; Normal-simple properties and simple applications, fitting a normal distribution to given data.

Bivariate distribution: 
Correlation, linear regression involving two variables, fitting of straight line, parabolic and exponential curves, properties of correlation coefficient.

Simple sampling distributions and simple tests of hypothesis: Random sample, Statistics, Sampling distribution and standard error. Simple application of the normal, t, chi2 and F-distribution for test of significance.

Note:
Candidates will be required to answer compulsorily from Part A of the syllabus one question on each of the three topics, viz. (1) Algebra, (2) Analytical Geometry of two and three dimensions, and (3) Calculus and differential equations. From Part B of the syllabus it will be compulsory to answer at least one question on any one of the three topics, viz., (1) Mechanics, (2) Astronomy, and (3) Statistics.

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