MATHEMATICS PAPER-I

SECTION –A

Linear Algebra

• Matrices, row and column reductions, echelon forms. Eigenvalues, eigenvectors and characteristic equation of a matrix.
• Cayley-Hamilton theorem and its applications, rank of a matrix.
• Applications of matrices to solve a system of linear homogeneous /non-homogeneous equations.
• Vector space, linear dependence and independence, Subspaces, Bases, dimensions. Finite dimensional vector spaces.
• Linear transformations, the algebra of linear transformations, isomorphism, representation of transformations by Matrices, linear functionals.
• The double dual and the transpose of a linear transformation.
• Inner product spaces. Cauchy-Schwarz inequality. Orthogonal vectors. Orthogonal complements.
• Orthonormal sets and orthonormal bases. Bessel’s inequality for finite dimensional spaces. Gram-Schmidt orthogonalization process.
• Linear functionals and adjoints.

Calculus

• Real numbers, limits, continuity, differentiability, mean-value theorems. Taylor’s theorem with remainders.
• Indeterminate forms, maxima and minima, asymptotes.
• Curvature, Concavity, Convexity, Points of inflexion and tracing of curves.
• Functions of two variables: continuity, differentiability, partial derivatives, Euler’s theorem for homogeneous functions, Jacobian, maxima and minima.
• Lagrange’s method of multipliers.
• Riemann’s definition of definite integrals. Indefinite integrals, infinite and improper integrals, beta and gamma functions.
• Double and triple integrals. Areas, surface and volumes.

Analytic Geometry

• Cartesian and polar coordinates in two and three dimensions, second degree equations in two and three dimensions, reduction to canonical forms.
• Straight lines, shortest distance between two skew lines.
• Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

SECTION -B

Ordinary Differential Equations

• Formulation of differential equations, order and degree, equations of first order and first degree, integrating factor.
• Equations of first order but not of first degree, Clairaut’s equation, singular solution.
• Higher order linear equations with constant coefficients, complementary function and particular integral, general solution, Euler-Cauchy equation.
• Second order linear equations with variable coefficients, determination of complete solution when one solution is known, method of variation of parameters.
• Solution by Power series method and its basis, solution of Bessel and Legendre’s equations, properties of Bessel and Legendre functions.

Vector Analysis

• Scalar and vector fields, triple products, differentiation of vector function of a scalar variable.
• Gradient, divergence and curl in Cartesian, cylindrical and spherical coordinates and their physical interpretations.
• Higher order derivatives, vector identities and vector equations.
• Applications to Geometry: curves in space, curvature and torsion. Serret-Frenet’s formulae.
• Gauss’ and Stokes’ theorems, Green’s identities.

Statics

• Analytical conditions of equilibrium of coplanar forces, virtual work.
• Forces in three dimensions, Poinsot’s central axis, Wrenches, Null lines and planes, Stable and unstable equilibrium.

Dynamics

• Simple harmonic motion, motion on rough curve, tangential & normal accelerations, motion in a resisting medium, motion when the mass varies.
• Velocity along radial and transverse directions, central orbits.
• Kepler’s laws of motion, motion of a particle in three dimensions, acceleration in terms of Polar and Cartesian co-ordinate systems.

MATHEMATICS PAPER-II

Note: Use of Scientific non-programmable calculators will be allowed in this paper for numerical analysis part.

SECTION –A

Abstract Algebra

• Mappings, elementary properties of integers.
• Definition of a Group and Subgroup their examples and properties. Normal subgroups, Quotient Groups.
• Homomorphism, Group-automorphisms, Cayley’s theorem, permutation Groups.

Real Analysis

• The Riemann integral: Definition and existence of integral, refinement of partitions, Darboux’s theorem, condition of integrability.
• Integrability of the sum and difference of integrable functions.
• The fundamental theorem of calculus, first and second mean value theorems of calculus.
• Improper integrals and their convergence, comparison tests, Abel’s and Dirichlet’s tests.

Sequences and series

• Definition of a sequence, theorems on limits of sequences, bounded and monotonic sequences and their convergence.
• Cauchy’s convergence criterion, algebra of sequences, main theorems, monotonic sequences.
• Series of non-negative terms, comparison test, Cauchy’s Integral test, Ratio test, Raabe’s test, logarithmic test, Gauss’s test.
• Alternating series, Leibnitz’s test. Absolute and conditional convergence.

Metric Spaces

• Definition and examples of metric spaces. Limits in metric spaces.
• Functions continuous on metric spaces. Open sets. Closed sets.
• Connected sets. Complete metric spaces. Compact metric spaces.
• Continuous functions on compact metric spaces, uniform continuity.

Complex Analysis

• Complex numbers, Geometric representation of Complex numbers.
• Analytic function, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula.
• Conformal mapping, Bilinear Transformation (Mobius transformation).

SECTION- B

Partial Differential Equations

• First order partial differential equations: Partial differential equations of the first order in two independent variables, formulation of first order partial differential equation.
• Solution of linear first order partial differential equations (Lagrange’s Method), integral surfaces passing through a given curve, surfaces orthogonal to a given system of surfaces.
• Solution of non-linear partial differential equations of first order by Charpit’s method.
• Second order partial differential equations: Origin and classification of second order partial differential equation.
• Solution of linear partial differential equation with constant coefficients.
• Monge’s method to solve the non-linear partial differential equation Rr+Ss+Tt = V.

Laplace Transforms

• Introduction, basic theory of Laplace transforms, solution of initial value problem using Laplace transforms.
• Shifting theorems, unit step function, Dirac-delta function.
• Differentiation and integration of Laplace transforms. Convolution theorem.

Calculus of Variations

• Variation problems with fixed boundaries-Euler’s equation for functionals containing first order derivative and one independent variable. Extremals.
• Functionals dependent on higher order derivatives. Functionals dependent on more than one independent variable.
• Variational problems in parametric form. Invariance of Euler’s equation under coordinates transformation.
• Variational problems with moving boundaries-functionals dependent on one and two functions.
• Sufficient conditions for an Extremum-Jacobi and Legendre conditions.

Numerical Analysis and computer programming

• Numerical Methods: Solution of algebraic and transcendental equations of one variable by Bisection, Secant, Regula Falsi, Newton-Raphson Method, Roots of Polynomials.
• Linear Equations: Solution of system of linear equations by Gaussian elimination method, Gauss-Siedel iterative method.
• Interpolation: Lagrange and Newton interpolation, divided differences, difference schemes, interpolation formulas using differences.
• Numerical Differentiation: Solution of ordinary differential equations by Euler’s method, Runge-Kutta’s II and IV order method.
• Numerical Integration: Simpson’s 1/3 rule, Simpson’s 3/8 rule, Trapezodial rule, Gaussian quadrature formula.
• Programming in C: Algorithms and flow-charts for solving numerical problems. Developing simple programs in C language for problems involving techniques covered in the numerical analysis.