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PMS Syllabus for The Subject of Mathematics Papers

 

Paper I
T otal Marks: 100
Candidates  will   be  asked  to   attempt  three   questions  from   Section A  and  two  questions
from section B.
Section  A
Limits,   Continuity ,   Differentiability   and   its   Applications,   General   theorems   (Rolle’s
Theorem,   Mean   value   theorem),   Asymptotes,   Applications   of   Maxima   and   Minima.
Definite   and   Indefinite   integrals   and   their   Application,   Quadrature,   Rectification,
Numerical methods of Integration (Trapezoidal and Simpson rule),Multiple integrals and
their   Applications.   Areas   and   Volumes,   Centre   of   Mass,   Reimann-Stijles   Integral,
Ordinary   Differential   Equations   (O.D.Eqs)   and   their   Applications   in   Rectilinear   motion
and Growth/Decay problems. 2nd Order Differential Equations with  Applications  (Spring
Mass and Simple Harmonic Oscillator Problems).
SectionB
Sequences   and   Series,   Convergence   tests,   Power   Series,   Radius   and   Interval   of
Convergence.   Complex  Analysis,   Function   of   Complex   V ariable,   Demoivre’s   Theorem
and   its  Application.  Analytic   Function,   Singularities,   Cauchy   theorem,   Cauchy   Integral
formula.
Conic   Sections   in   Cartesian   coordinates,   Plane   Polar   Coordinates   and   their   use   to
represent   the   straight   line   and   Conic   section.   Vector   equation   for   plane   and   space
curves.   T angents   and   Normals   and   Binormals,   Curvature   and   torsion,   Serre   Frenet’s
Formula.

Paper II
T otalMarks: 100
Section  A
Groups:  Definition  and examples   of  Groups,  Order  of  a  Group,  Order  of  an element of
a   Group,  Abelian   and   non-Abelian   Groups   and   Cyclic   groups.   Lagrange   theorem  and
applications,   Normal   subgroups,   Characteristic   Subgroups   of  a   group,   Normalizer   in   a
group,   Centralizer   in  a  group.   Fundamental   Theorem   of  Homomorphism,   Isomorphism
theorems of groups,  Automorphisms
Rings,   Fields   and   Vector   Spaces:  Examples   of   Rings,   Subrings,   Integral   domains,
Fields,   Vector   spaces,   Linear   independence/   dependence,   Basis   and   dimension   of
finitely   generated   spaces,   Examples   of   Field   extension   and   finite   fields,   Examples   of
finite and infinite dimensional vectorspaces.
Section B
Metric   Spaces   and   T opological   Spaces:  Definition   and   Examples   of   Metric   spaces
and   topological   spaces,   Closed   and   Open  Spheres,   Interior ,   Exterior   and  Frontier  of  a
Set, Sequences in Metric Spaces, Convergence of Sequences. Definition and examples
of Normed Spaces.Inner product spaces, Gram-Schmidt Process of Orthonormalization
Matrices   and   Linear   Algebra:  Linear   transformations,   Matrices   and   their   algebra,
Reduction  of  matrices   to   Echelon  and  Reduced  Echelon  form.   Solution   of  a  system   of
homogenous   and   Non-Homogenous   equations,   Numerical   methods   of   solving
equations   (Gauss-Siedal   method,   Jaccobi   method)   Properties   of   Determinants,
Eigenvalues and Eigenvectors and the Diagonalization of the Symmetric Matrices.

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