IISC Integrated Ph.D Maths entrance exam syllabus

I am looking for Indian Institute of Science Integrated Ph.D Maths entrance exam syllabus, please provide here???

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Mathematical Sciences question paper will consist of multiple choice questions for 100 marks from mathematics. There will be negative marking for wrong answer.

SYLLABUS

1. Algebra

Theory of Equations Relations between roots and Coefficients. Newton’s identities. Rolle’s theorem, Reciprocal Equations, Des Cartes, Rule of Signs, Cubic and quark equations, Complex numbers and De Moivre’s Theorem. Determinants Cofactors, Properties of determinants, Solution of a Linear System, Cramer’s Rule. Inequalities AM-GM inequality, Cauchy-Schwarz inequality. Set Theory : Relations, Functions, Cardinality. Algebraic Structures Binary Operations, Groups, Rings: Definitions, Examples and Elementary Theorems. Vector Spaces Subspaces, Linear Independence, Bases, Dimension Linear Transformations, Matrices, Rank Nullity, Eigen values and Eigen vectors.

2. Geometry :

Two-dimensional Coordinate Geometry

Conics and their equations in Cartesian and Polar Coordinates, Ellipse, Parabola and Hyperbola.

Three-dimensional Co-ordinate Geometry

Planes, Lines, Spheres and Cones.

3. Vector Algebra and Vector Calculus : Vectors, addition, Scalar multiplication. Dot Product, Cross Product, Triple Product, Equations to the Line and the Plane. Grad, Divergence and Curl, Vector Integration, Green’s Gauss’ and Stokes’ Theorems.

4. Calculus and Analysis

Real Number System, Sequence and Series. Continuity, Differentiability. Mean Value Theorems, Indeterminate Value Theorem, L’Hospital Rule, Tangents and Normals, Maxima and Minima. Riemann Integration, Multiple Integrals, Partial differentiations, Lengths, areas and volumes by integration.

5. Differential Equations

First Order ODE Method of Separation of Variables: Exact equations: Euler’s equation: Orthogonal Family of curves, Second Order Linear ODE : Variation of Parameters.