Krishna's - B.Sc. Mathematics PDF
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SYLLABUS- SOLUTION SET: B.SC. MATHEMATICS,
Linear Algebra & Linear Programming
THIRD YEAR"FIRST PAPER
Note: Five questions are to be attempted selecting at least one from section (B).
(A) Linear Algebra (Eight questions)
Vector spaces-vector space, Sub spaces, Linear combinations, Linear spans, Sums and
direct sums, linear dependence, Bases and dimensions, Dimensions and subspaces,
coordinates and change of bases. Linear transformations-Linear transformations,
rank and nullity, Operations with linear transformations, Linear Operators, algebra of
linear operators, Invertible linear operators, Matrix of linear transformation,
Matrices and linear trainsformation, Matrix of a linear operator, Change of basis,
similarity.
Linear functionals-linear functional, Dual space and dual basis, Double dual space,
Annihilators, Transpose of linear transformation. Bilinear, Quadratic and Hermitian
forms -bilinear forms, Quadratic forms.
(B) Linear Programming (Two questions)
Linear programming Graphical method, Simplex method, The dual of linear
programming problem.
Analysis
THIRD YEAR"SECOND PAPER
Note: Five questions are to be attempted selecting at least one from each section.
(A) Complex analysis (Four questions)
Complex number and its representation in the Argand plane, Function of a complex
variables, Analytic functions, Complex integration, Cauchy's theorem, Cauchy's
integral formula, Taylor's theorem, Laurent's theorem, Liovilles theorem, poles and
essential singularities, Residues, Cauchy's integral theoem and its application in the
evaluation of integrals.
(B) Real analysis (Six questions)
Continuity of functions, Properties of continuous functions, Types of discontinuities,
Uniform continuity, Differentiability Taylor's theorem with various forms of
remainders, Riemann integral- definition and properties, Condition of integrability,
Convergence and uniform convergence of integrals.
Points wise convergence, Uniform convergence, Tests of uniform convergence
continuity and uniform convergence.
Numerical Analysis
THIRD YEAR"THIRD PAPER
Note: Attempt any five questions.
(A) Numerical Analysis (Ten questions)
Fintie difference, Difference operators, Newton's interpolation formula, Divided
differences, Interpolation with unequal interval of arguments, Lagrange's formula,
Sarling and Bassel formula (application only).
Integration, Simpson's rule, Trapezoidal rule and men academy, Numerucal solution
of algebraic equations in two unknown quantities, Regula Falsi, Newton Raphson,
Graffi's root squaring method.
Numeral method of matrix, inversion, determination of eigen values and eigen
vectors.
Linear Algebra & Linear Programming
THIRD YEAR"FIRST PAPER
Note: Five questions are to be attempted selecting at least one from section (B).
(A) Linear Algebra (Eight questions)
Vector spaces-vector space, Sub spaces, Linear combinations, Linear spans, Sums and
direct sums, linear dependence, Bases and dimensions, Dimensions and subspaces,
coordinates and change of bases. Linear transformations-Linear transformations,
rank and nullity, Operations with linear transformations, Linear Operators, algebra of
linear operators, Invertible linear operators, Matrix of linear transformation,
Matrices and linear trainsformation, Matrix of a linear operator, Change of basis,
similarity.
Linear functionals-linear functional, Dual space and dual basis, Double dual space,
Annihilators, Transpose of linear transformation. Bilinear, Quadratic and Hermitian
forms -bilinear forms, Quadratic forms.
(B) Linear Programming (Two questions)
Linear programming Graphical method, Simplex method, The dual of linear
programming problem.
Analysis
THIRD YEAR"SECOND PAPER
Note: Five questions are to be attempted selecting at least one from each section.
(A) Complex analysis (Four questions)
Complex number and its representation in the Argand plane, Function of a complex
variables, Analytic functions, Complex integration, Cauchy's theorem, Cauchy's
integral formula, Taylor's theorem, Laurent's theorem, Liovilles theorem, poles and
essential singularities, Residues, Cauchy's integral theoem and its application in the
evaluation of integrals.
(B) Real analysis (Six questions)
Continuity of functions, Properties of continuous functions, Types of discontinuities,
Uniform continuity, Differentiability Taylor's theorem with various forms of
remainders, Riemann integral- definition and properties, Condition of integrability,
Convergence and uniform convergence of integrals.
Points wise convergence, Uniform convergence, Tests of uniform convergence
continuity and uniform convergence.
Numerical Analysis
THIRD YEAR"THIRD PAPER
Note: Attempt any five questions.
(A) Numerical Analysis (Ten questions)
Fintie difference, Difference operators, Newton's interpolation formula, Divided
differences, Interpolation with unequal interval of arguments, Lagrange's formula,
Sarling and Bassel formula (application only).
Integration, Simpson's rule, Trapezoidal rule and men academy, Numerucal solution
of algebraic equations in two unknown quantities, Regula Falsi, Newton Raphson,
Graffi's root squaring method.
Numeral method of matrix, inversion, determination of eigen values and eigen
vectors.
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