JKPSC/JKPCS Maths Optional Syllabus Paper-2

**PAPER-II **

**(1) Algebra:** Groups, subgroups, cyclic groups, cosets,
Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of groups,
basic isomorphism theorems, permutation groups, Cayley’s theorem. Rings,
subrings and ideals, homomorphisms of rings; Integral domains, principal ideal
domains, Euclidean domains and unique factorization domains; Fields, quotient
fields.

**(2) Real
Analysis:** Real number system as an ordered
field with least upper bound property; Sequences, limit of a sequence, Cauchy
sequence, completeness of real line; Series and its convergence, absolute and
conditional convergence of series of real and complex terms, rearrangement of
series. Continuity and uniform continuity of functions, properties of
continuous functions on compact sets. Riemann integral, improper integrals;
Fundamental theorems of integral calculus. Uniform convergence, continuity,
differentiability and integrability for sequences and series of functions;
Partial derivatives of functions of several (two or three) variables, maxima
and minima.

**(3) Complex
Analysis:** Analytic function, Cauchy-Riemann
equations, Cauchy's theorem, Cauchy's integral formula, power series,
representation of an analytic function, Taylor’s series; Singularities;
Laurent’s series; Cauchy’s residue theorem; Contour integration.

**(4) Linear
Programming:** Linear
programming problems, basic solution, basic feasible solution and optimal
solution; Graphical method and simplex method of solutions; Duality.
Transportation and assignment problems.

**(5) Partial
Differential Equations:** Family of
surfaces in three dimensions and formulation of partial differential equations;
69 Solution of quasilinear partial differential equations of the first order,
Cauchy’s method of characteristics; Linear partial differential equations of
the second order with constant coefficients, canonical form; Equation of a
vibrating string, heat equation, Laplace equation and their solutions.

**(6) Numerical
Analysis and Computer Programming:**

Numerical Analysis: Numerical methods: Solution of algebraic and transcendental equations of one
variable by bisection, Regula-Falsi and Newton-Raphson methods, solution of
system of linear equations by Gaussian Elimination and Gauss-Jorden (direct),
Gauss-Seidel (iterative) methods. Newton’s (forward and backward) and
interpolation, Lagrange’s interpolation. Numerical integration: Trapezoidal
rule, Simpson’s rule, Gaussian quadrature formula. Numerical solution of
ordinary differential equations: Euler and Runge Kutta methods.

**Computer
Programming****: **Binary system;
Arithmetic and logical operations on numbers; Octal and Hexadecimal Systems;
Conversion to and from decimal Systems; Algebra of binary numbers. Elements of
computer systems and concept of memory; Basic logic gates and truth tables,
Boolean algebra, normal forms. Representation of unsigned integers, signed
integers and reals, double precision reals and long integers. Algorithms and
flow charts for solving numerical analysis problems.

**(7) Mechanics and
Fluid Dynamics:** Generalised
coordinates; D’Alembert’s principle and Lagrange’s equations; Hamilton
equations; Moment of inertia; Motion of rigid bodies in two dimensions.
Equation of continuity; Euler’s equation of motion for inviscid flow;
Stream-lines, path of a particle; Potential flow; Two-dimensional and
axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation
for a viscous fluid.