Hindi - Himachal Pradesh

Forces in three dimensions, Poinsot’s central axis, Wrenches, Null lines and planes, Stableand 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 andtransverse directions, central orbits.Kepler’s laws of motion, motion of a particle in three dimensions, acceleration in terms ofPolar and Cartesian co-ordinate systems.PAPER-II(Note: Use of Scientific non-programmable calculators will be allowed in this paper fornumerical analysis part.)SECTION –AAbstract Algebra:Mappings, elementary properties of integers. Definition of a Group and Subgroup theirexamples and properties. Normal subgroups, Quotient Groups. Homomorphism, Groupautomorphisms,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 ofintegrable functions. The fundamental theorem of calculus, first and second mean valuetheorems 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 monotonicsequences 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, alternatingseries, Leibnitz’s test. Absolute and conditional convergence.Metric Spaces:Definition and examples of metric spaces. Limits in metric spaces. Functions continuouson metric spaces. Open sets. Closed sets. Connected sets. Complete metric spaces.Compact metric spaces. Continuous functions on compact metric spaces, uniformcontinuity.Complex Analysis:Complex numbers, Geometric representation of Complex numbers. Analytic function,Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula. Conformalmapping, Bilinear Transformation (Mobius transformation).SECTION- BPartial Differential Equations:First order partial differential equations: Partial differential equations of the first order intwo independent variables, formulation of first order partial differential equation, solutionof linear first order partial differential equations (Lagrange’s Method), integral surfacespassing through a given curve, surfaces orthogonal to a given system of surfaces, solutionof non-linear partial differential equations of first order by Charpit’s method.