# BPSC Maths Optional Syllabus

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### BPSC Maths Optional Syllabus

#### SECTION- I

Linear Algebra, Calculus, Analytic Geometry of two and three dimensions and Differential Equations

Linear Algebra:

Vector space bases, dimension of finitely generated space. Linear transformations, Rank and nullity of a linear transformation, Cayley Hamilton theorem. Eigenvalues and Eigenvectors. Matrix of a linear transformation. Row and Column reduction. Echelon form. Equivalence. Congruence and similarity. Reduction to canonical forms. Orthogonal, symmetrical, skew-symmetrical, unitary, Hermitian and Skew-Hermitian matrices

#### SECTION- II

Algebra, Real Analysis, Complex Analysis, Partial Differential equations.  Mechanics, Hydrodynamics, Numerical Analysis. Statistics including probability operational research.

Algebra:

Groups, sub-groups, normal sub-groups, homomorphism of groups, quotient groups. Basic isomorphism theorems. Sylow theorems. Permutation Groups, Cayley's theorem. Rings and Ideals, Principal Ideal domains, Unique factorization domains and Euclidean domains. Field Extensions. Finite fields.

Real Analysis:

Metric spaces, their topology with special reference to Rsequence in a metric space, Cauchy sequence, Completeness, Completion Continuous functions, Uniform Continuity, Properties of Continuous function on Compact sets. Riemann-Stieltjes integral, Improper integrals and their conditions of existence. Differentiation of functions of several variable, Implicit function theorem, maxima and minima, Absolute and conditional convergence series of real and complex terms, Re-arrangement of series. Uniform convergence, Infinite products, Continuity, differentiability and integrability for series, Multiple integrals.

Complex Analysis:

Analytic functions, Cauchy's theorem, Cauchy's integral formula, Power series, Taylor's, Singularities, Cauchy's Residue theorem and Contour integration.

Partial Differential Equations:

Formations of partial differential equations. Types of integrals of partial differential Equations of first order Charpit

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