Algebra ( बीजगणित ) B.Sc First Semester Bihar

MJC-01: Algebra (Theory: 06 credits)

¼chtxf.kr ¼fl)kar& 06 ØsfMV½½

Unit

Subject ¼fo"k;½

No. of Lectures

1

Polar representation of complex numbers, De –Moivre’s theorem and its applications, Logarithms of complex quantities, Hyperbolic functions, Gregory series, Summation of series, Resolution into factors.

lfEeJ la[;kvksa dk /kzqoh; fu:i.k] Mh&ekW;oj çes; vkSj mlds vuqç;ksx] lfEeJ jkf’k;ksa ds y?kqx.kd] vfrijoyf;d Qyu] xzsxjh Js.kh] Js.kh dk ;ksx] xq.ku[kaMksa esa [kaMuA

10

2

Cartesian product of sets, Equivalence relations, partition, partial and total order relation Functions, Composition of functions, Invertible functions, Cardinality of a set, Countable and Uncountable sets, Cantor’s theorem.

leqPp;ksa dk dkrhZ; xq.kuQy] rqY;rk laca/k] foHkktu] vkaf'kd vkSj dqy Øe laca/k Qyu] Qyuksa dh lajpuk] O;qRØe.kh; Qyu] leqPp; dh x.kukad] x.kuh; vkSj vx.kuh;  leqPp;] dSaVj dk çes;A

12

3

Well-ordering property of positive integers, Division algorithm, Euclidean algorithm, Fundamental Theorem of Arithmetic, Modular arithmetic and basic properties of congruence’s, Principle of mathematical induction.

/kukRed iw.kkaZdksa dk lqO;ofLFkr xq.k] foHkktu ,YxksfjFe] ;wfDyfM;u ,YxksfjFe] vadxf.kr dk ewyHkwr çes;] e‚Mîwyj vadxf.kr vkSj lokaZxlerk ds ewy xq.k] xf.krh; çsj.k dk fl)karA

12

4

Matrices, Operation on Matrices, Kinds of matrices, Transpose, symmetric & skew symmetric Matrices, Hermitian, skew Hermitian Matrices, Adjoint and Inverse of a matrix, orthogonal matrix, Solution of a system of linear equations by matrix methods. Echelon forms, Rank of a matrix.

vkO;wg] vkO;wg ij lafØ;k] vkO;wg ds çdkj] :ikarj.k] lefer vkSj fo"ke lefer vkO;wg] gfeZfV;u] fo"ke gfeZVh vkO;wg] ,d vkO;wg dk lg[kaMt vkSj O;qRØe] v‚FkksZxksuy vkO;wg] vkO;wg fof/k;ksa }kjk jSf[kd lehdj.kksa dh ,d ç.kkyh dk gy] lksikud :i] ,d vkO;wg dh jSadA  

12

5

Fundamental theorem of algebra, Relation between roots and coefficients of a polynomial equation, Symmetric Function of roots, Transformation of equation, Descartes rule of signs, Solution of Cubic equation (Cardon’s method) and bi quadratic equation (Elder’s method).

chtxf.kr dk vk/kkjHkwr çes;] cgqin lehdj.k ds ewyksa vkSj xq.kkad ds chp laca/k] ewyksa dk lefer Qyu] lehdj.k dk :ikarj.k] ladsrksa dk MsldkVsZl fu;e] ?ku lehdj.k dk gy ¼dkMZu dh fof/k½ vkSj f}?kkr lehdj.k ¼,YMlZ fof/k½A

14

MIC-01: Algebra (Theory: 03 credits)

¼chtxf.kr ¼fl)kar& 03 ØsfMV½½

Unit

Subject ¼fo"k;½

No. of Lectures

1

Polar representation of complex numbers, De –Moivre’s theorem and its applications, Logarithms of complex quantities, Hyperbolic functions, Gregory series, Summation of series,.

lfEeJ la[;kvksa dk /kzqoh; fu:i.k] Mh&ekW;oj çes; vkSj mlds vuqç;ksx] lfEeJ jkf’k;ksa ds y?kqx.kd] vfrijoyf;d Qyu] xzsxjh Js.kh] Js.kh dk ;ksxA

08

2

Cartesian product of sets, Equivalence relations, Functions, Composition of functions, Invertible functions, Partial and Total order relation, Countable and Uncountable sets,

leqPp;ksa dk dkrhZ; xq.kuQy] rqY;rk laca/k] Qyu] Qyuksa dh lajpuk] O;qRØe.kh; Qyu] vkaf'kd vkSj dqy Øe laca/k] x.kuh; vkSj vx.kuh; leqPp;]

07

3

Matrices, Operation on Matrices, Kinds of matrices, Transpose, symmetric & skew symmetric matrices, Hermitian and skew Hermitian matrices, Adjoint and Inverse of a matrix, solution of a system of linear equations by matrix methods.

vkO;wg] vkO;wg ij lafØ;k] vkO;wg ds çdkj] :ikarj.k] lefer vkSj fo"ke lefer vkO;wg] gfeZVh vkSj fo"ke gfeZVh vkO;wg] vkO;wg ds lg[kaMt vkSj O;qRØe] vkO;wg fof/k;ksa }kjk jSf[kd lehdj.kksa dh ,d ç.kkyh dk gyA  

08

4

Fundamental theorem of algebra, Relation between roots and coefficients of a polynomial equation, Evaluation of symmetric functions of roots, Transformation of equation, Solution of Cubic equation (Cardon’s method).

chtxf.kr dk vk/kkjHkwr çes;] cgqin lehdj.k dh ewyksa vkSj xq.kkadksa ds chp laca/k] ewyksa ds lefer Qyuksa dh x.uk] lehdj.k dk :ikarj.k] f=?kkr lehdj.k dk gy ¼dkMZu dh fof/k½A

07