**Lalit Narayan Mithila University, Darbhanga B.Sc. part-1 Mathematics Honours Syllabus –**

__Paper-I and Paper-II__B.Sc. part-1 Mathematics honours syllabus |

__Paper-I__

__Paper-I__

**Twelve questions to be set, and six questions you will have to answer –**

Schoeller Partial and total order relations, countability, Cardinality.

Bernstein theorem, Cardinal and ordinal numbers and their arithmetic, Axiom choice and its various forms, Zor well-ordering theorem (3 questions)

properties, Definition of a group with examples and simple of groups, Cyclic groups, Coset decomposition, Lagrange’s theorem and its consequences, Fer and Euler’s theorems, Homomorphism and Isomorphism, Normal subgroups. Quotient groups, Fundamental theorem of homomorphism.

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Permutation groups. Even and odd permutations, the alternating groups, Cayley’s theorem, Introduction of rings, subrings. Integral domains and fields, characteristic of a ring. (3 questions) Symmetric, Skew symmetric, Hermitian and Skew Hermitian matrices. Elementary operations on matrices, Inverse of a matrix.

Linear independence of row and column matrices. The rank of a matrix, Eigenvalues, Eigenvectors and the characteristic equation of matrices to a system of linear (both homogenous and non-homogeneous) equations Theorems on the consistency of a system of a linear equation. (3questions)

General properties of polynomials and equations. The fundamental theorem of algebra, Descarte’s rule of sing, Relation between roots and coefficients. Evaluations of symmetric functions of roots of cubic and biquadratic, of equations, Reciprocal equations Transformation of cubic and biquadratic.

__Paper-II__

__Paper-II__

**Twelve questions to be set, and six you will have to answer.**

**Analytical Geometer of Two dimensions: –**General equation of second degree, tracing of conics, the system of conics, confocal conics, polar equation of conic. Equation of chord. Tangent, normal. Asymptote and director circle. (4 questions)

**Analytical Geometry of Three Dimensions: –**Equations of a plane and straight lines, Coplanarity. Shortest distance. The volume of the tetrahedron Sphere, Radical plane. The tangent plane, cone, generating line, condition for three mutually perpendicular generators. Central conoids, normal and conjugate diameters of ellipsoids and its properties questions)

**Higher Trigonometry: –**De movers theorem and its applications circular, inverse circular and hyperbolic functions. Logarithm o complex quantity. Expansion of trigonometrical functions, Gregory’s series, the summation of series, Resolution into factors.

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__LNMU B.Sc. part-1 Mathematics subsidiary syllabus__

__LNMU B.Sc. part-1 Mathematics subsidiary syllabus__

**Lalit Narayan Mithila University, Darbhanga B.Sc. part-1mathematics (subsidiary) Syllabus**

Time- 3 hours

Full marks- 100

**Sixteen questions to be set and Eight to be answered. You will have select at least one from each Group.**

__Group-A__

__Group-A__

Set theory, Modern Algebra and Linear Algebra

**(7 questions)****Set Theory: –**De Morgan’s laws, Cartesian product of sets. Equivalence relation, Partition of a set Fundamental theorem on equivalence relation. Countability of rational, real and algebraic numbers.

**(2 questions)**

**Abstract Algebra: –**Nations of the group, abelian group with simple examples, The uniqueness of identity and inverse, Cancellation laws in a group. Reversal rule. Definitions and examples of subgroup and cyclic group

**(1 question)**

Definitions, simple examples and simple properties of Ring, integral domains and field

**(1 question)**Matrices, Transpose, Orthogonal, adjoint and in of the matrix Rank of Matrix

**(2 questions)**Definition, examples, and properties of vector space, vector subspace.

**(1 question)**

__Group-B__**Real Analysis (4 questions) –**

Sequence and their convergence, Cauchy’s general principle of convergence Bounded sequence, Monotonic sequences. (1 question)

Convergent and divergent Series. The convergence of the positive term series. comparison test.

Cauchy’s root test. D’ Alembert’s ratio test. Raabe’s test and Liebnitz’s test questions)

Continuity and differentiability of a real function of a single real variable with examples (1 question)

__Group-C__**Analytic Geometry of Two and Three Dimensions (5 questions)**

**Analytical Geometry of Two dimensions: –**System of circles. Coaxial circles. Radical axis, General equation of second degree and its reduction to standard forms of a parabola. Ellipse and Hyperbola. Equation of tangents and normal for general equation.

**(3 questions)**

**Analytical Geometry of three dimensions: –**Rectangular Cartesian co-ordinates. Direction cosines. The angle between two straight lines. Equations of a plane and straight line. Condition for coplanarity of straight lines. Equation of shortest distance.

**(2 questions)**