## Mathematics for Competitive Examinations |

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### Contents

Preface ChapterI ABSTRACT ALGEBRA 129 | 1 |

2 Groups and Subgroups | 5 |

3 Normal Subgroups Quotient Groups Homomorphisms and Isomorphisms | 8 |

4 Cyclic Groups | 10 |

5 Permutation Groups | 12 |

6 Rings Ideals Integral Domains Fields | 14 |

7 Polynomial Rings | 18 |

ChapterH LINEAR ALGEBRA 3063 | 30 |

2 Sequence and Series | 67 |

3 Functions Limit and Continuity | 73 |

4 Differentiation | 78 |

5 Riemann Integration | 81 |

6 Sequence of functions and Series of functions | 82 |

7 Power series | 83 |

8 Functions of bounded variation and compact set | 85 |

ChapterV DIFFERENTIAL EQUATION 118137 | 118 |

2 Determinants | 34 |

3 Vector Spaces Subspaces Quotient Spaces Linear Independence L I Bases and Dimension | 36 |

4 Linear Transformation | 40 |

5 Eigen Values Eigen Vectors and Canonical Forms | 43 |

6 Inner Product Spaces | 45 |

7 Miscellaneous | 46 |

ChapterIll REAL ANALYSIS 64101 | 64 |

ChapterVI LINEAR PROGRAMMING PROBLEM 138147 | 138 |

ChapterVII PROBABILITY 148162 | 148 |

ChapterVHI ADVANCED ANALYSIS 163176 | 163 |

AppendixI GENERAL MATHEMATICS 177188 | 177 |

AppendixII COMPUTER SCIENCE 189198 | 189 |

AppendixHI PREVIOUS YEARS QUESTIONS OF DIFFERENT | 199 |

### Common terms and phrases

abelian analytic B3pi binary Binomial distribution bounded circle closed commutative ring compact complex numbers constant continuous function convex countable cyclic group defined denotes diagonisable divergent dnojS eigen value eigen vector equal exist field F finite dimensional following is true given differential equation group of order hence Hints homeomorphism homomorphism identity infinite number initial value problem inner product space integral domain isomorphic Let G linear mapping linear transformation linearly independent maximal ideal monotone non-commutative non-singular non-zero normal subgroup Note number of elements one-one orthogonal P(AB pole positive integer prime ideal probability Prove radius of convergence random variable rank rational numbers real numbers real valued sequence Show singular solve square matrix subgroup of G subgroup of order subset subspace theorem total number unbounded uncountable uniformly unique vector space zero divisors