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### JKPSC/JKPCS Maths Optional Syllabus

JKPSC/JKPCS Maths Optional Syllabus Paper-1

**PAPER-I **

**(1) Linear
Algebra:** Vector spaces over R and C, linear
dependence and independence, subspaces, bases, dimensions, Linear
transformations, rank and nullity, matrix of a linear transformation. Algebra
of Matrices; Row and column reduction, Echelon form, congruence’s and
similarity; Rank of a matrix; Inverse of a matrix; Solution of system of linear
equations; Eigenvalues and eigenvectors, characteristic polynomial,
Cayley-Hamilton theorem, Symmetric, skew-symmetric, Hermitian, skew-Hermitian,
orthogonal and unitary matrices and their eigenvalues.

**(2) Calculus:** Real numbers, functions of a real variable,
limits, continuity, differentiability, mean-value theorem, Taylor’s theorem
with remainders, indeterminate forms, maxima and minima, asymptotes; Curve
tracing; Functions of two or three variables; Limits, continuity, partial
derivatives, maxima and minima, Lagrange’s method of multipliers, Jacobian.
Riemann’s definition of definite integrals; Indefinite integrals; Infinite and
improper integral; Double and triple integrals (evaluation techniques only);
Areas, surface and volumes.

**(3) Analytic
Geometry:** Cartesian and polar coordinates in
three dimensions, second degree equations in three variables, 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.

**(4) Ordinary
Differential Equations:** Formulation
of differential equations; Equations of first order and first degree, integrating
factor; Orthogonal trajectory; Equations of first order but not of first
degree, Clairaut’s equation, singular solution. Second and higher order liner
equations with constant coefficients, complementary function, particular
integral and general solution. Second order linear equations with variable
coefficients, Euler-Cauchy equation; Determination of complete solution when
one solution is known using method of variation of parameters. Laplace and
Inverse Laplace transforms and their properties, Laplace transforms of
elementary functions. Application to initial value problems for 2nd order
linear equations with constant coefficients.

**(5) Dynamics and
Statics:** Rectilinear motion, simple harmonic
motion, motion in a plane, projectiles; Constrained motion; Work and energy,
conservation of energy; Kepler’s laws, orbits under central forces. Equilibrium
of a system of particles; Work and potential energy, friction, Common catenary;
Principle of virtual work; Stability of equilibrium, equilibrium of forces in
three dimensions.

**(6) Vector
Analysis:** Scalar and vector fields,
differentiation of vector field of a scalar variable; Gradient, divergence and
curl in cartesian and cylindrical coordinates; Higher order derivatives; Vector
identities and vector equation. Application to geometry: Curves in space,
curvature and torsion; Serret-Frenet’s formulae. Gauss and Stokes’ theorems,
Green's identities.

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.