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# (IAS Planner) Optional Subjects Syllabus: (Mathematics)

## Syllabus of Examination

## Mathematics (Optional Subjects)

## Paper-I

**(1) Linear Algebra: **Vector spaces over R and C, linear
dependence and independence, subspaces, bases, dimension; 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 integrals; 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 linear 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 & 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 equations. Application to geometry: Curves in space,
Curvature and torsion; Serret-Frenet’s formulae. Gauss and Stokes’ theorems,
Green’s identities.

## 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 functions,
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; 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 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-Jordan (direct),
Gauss-Seidel(iterative) methods. Newton's (forward and backward) interpolation,
Lagrange's interpolation.

Numerical integration: Trapezoidal rule, Simpson's rules, Gaussian quadrature
formula.

Numerical solution of ordinary differential equations: Euler and Runga 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: **Generalized
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.

## Suggested Reading:

**Paper-I - Part- A**

__1. Linear Algebra__

a) Linear Algebra by Sharma and Vashishta

b) Theory of Matrices by Sharma and Vashishta, Krishna Series.

c) B.Sc Text Books by S. Chand Publications

__2. Calculus__

a) Differential Calculus by Sharma and Vashishta, Krishna Series.

b) Integral Calculus by Sharma and Vashishta, Krishna Series

__3. Analytical Geometry__

a) Two Dimensional Geometry by S.Chand Publications, Simplified Series

b) Analytical Geometry by P.N.Chatterjee

**Part – B**

__1. Ordinary Differential Equations__

a) Ordinary Differential Equations by Rai Singhania

__2. Statics, Dynamics and Hydrostatics__

a) Statics by S. Chand Publications

b) Dynamics by P.N.Chatterjee.

c) Hydrostatics by S. Chand Publications.

__3. Vector Calculus__

a) Vector Algebra by M.L.Khanna

b) Vector Analysis by M.L.Khanna

c) B.Sc. 1st year Mathematics Textbook by S.Chand Publications

**Paper – II - Part – A**

__1. Algebra__

a) Modern Algebra by Sharma and Vashishta. Krishna Series

b) Algebra by Herstein, John Wiley Publications

__2. Real Analysis__

a) B.Sc. 2nd & 3rd year Mathematics Books by S. Chand Publications

b) Real Analysis by Sharma and Vashishta, Krishna Series.

c) Real Analysis by M.L. Khanna

__3. Linear Programming__

a) Linear Programming by S.D.Sharma

__4. Complex Analysis__

a) Complex Analysis by S.P.Tyagi

b) Complex Analysis byM.L.Khanna

**Part – B**

__1. Partial Differential Equations__

a) Boundary Value Problems by Rai Singhania

b) Partial Differential Equations by Ion Sneddon

__2. Numerical Methods__

a) Numerical Analysis by S S Sastry, TMH Publications

b) Numerical Methods by V.Rajaramn

c) Progrmming in Basic by E.Balguruswamy, TMH Publications

d) Brilliant Tutorials Material set No.10.

__3. Mechanics and Hydrodynamics__

a) Mechanics by S.Chand Publications

b) Mechanics by M.L.Khanna

c) Hydrodynamics by Rai Singhania.