## PhD in Applied Mathematics

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**Minimum Eligibility**

A minimum of 17 years of regular education (12 years of regular schooling + a 3-year Bachelor’s degree and a 2-year Master’s degree or 4-year Bachelor’s degree + a 1 year Master’s degree leading to an MA/MSc/MTech degree in Mathematics/Computer Science/Statistics/Operations Research/Physics or 12 years of regular schooling + an integrated Bachelor's and Master's degree of at least 5 years duration in Mathematics/Computer Science/Statistics/Operations Research/Physics) from an institution recognized by the government of any of the SAARC countries, with a minimum of 55% marks or an equivalent grade.

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**Admission Procedure**: Entrance Test and Interview. A minimum of 50% marks will have to be secured separately, in both the Entrance Test and the Interview, in order to be eligible for admission. The weightage of the Entrance Test will be 70 marks and the weightage of the Interview will be 30 marks.

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**Format of the Entrance Test Paper **

The duration of the Entrance Test will be 3 hours and the question paper will consist of 70 multiple choice questions in two parts: Part A and Part B.

**PART A:** will have 30 questions on undergraduate level Mathematics

**PART B:** will have 40 questions on Master’s level Mathematics

Calculators will not be allowed. However, Log Tables may be used.

The areas from which questions may be asked will include the following:

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**Analysis:** Real functions; limit, continuity, differentiability; sequences; series; uniform convergence; functions of complex variables; analytic functions, complex integration; singularities, power and Laurent series; metric spaces; stereographic projection; topology, compactness, connectedness; normed linear spaces, inner product spaces; dual spaces, linear operators; Lebesgue measure and integration; convergence theorems.

**Algebra: **Basic theory of matrices and determinants; eigen values and eigen vectors; Groups and their elementary properties; subgroups, normal subgroups, cyclic groups, permutation groups; Lagrange's theorem; quotient groups, homomorphism of groups; Cauchy Theorem and p-groups; the structure of groups; Sylow's theorems and their applications; rings, integral domains and fields; ring homomorphism and ideals; polynomial rings and irreducibility criteria; vector space, vector subspace, linear independence of vectors, basis and dimensions of a vector space, inner product spaces, orthonormal basis; Gram-Schmidt process, linear transformations.

**Differential Equations:** First order ordinary differential equations (ODEs); solution of first order initial value problems; singular solution of first order ODEs; system of linear first order ODEs; method of solution of dx/P=dy/Q=dz/R; orthogonal trajectory; solution of Pfaffian differential equations in three variables; linear second order ODEs; Sturm-Liouville problems; Laplace transformation of ODEs; series solutions; Cauchy problem for first order partial differential equations (PDEs); method of characteristics; second order linear PDEs in two variables and their classification; separation of variables; solution of Laplace, wave and diffusion equations; Fourier transform and Laplace transform of PDEs.

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**Numerical Analysis: **Numerical solution of algebraic and transcendental equations; direct and iterative methods for system of linear equations; matrix eigenvalue problems; interpolation and approximations; numerical differentiation and integration; composite numerical integration; double numerical integration; numerical solution for initial value problems; finite difference and finite element methods for boundary value problems.

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**Probability and Statistics: **Axiomatic approach of probability; random variables; expectation, moments generating functions, density and distribution functions; conditional expectation.

**Linear Programming:** Linear programming problem and its formulation; graphical method, simplex method; artificial starting solution; sensitivity analysis; duality and post-optimality analysis.

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**Negative Marks for Wrong Answers: **If the answer given to any of the Multiple Choice Questions is wrong, ¼ of the marks assigned to that question will be deducted.

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**Interview:** Candidates up to five times the number of seats available will be short-listed for interview on the basis of their performance in the entrance test subject to a minimum cut off.

A final merit list will be drawn by adding together the marks of the entrance test and the interview. Separate merit lists will be made for (a) candidates from all SAARC countries other than India, and (b) candidates from India. Equal number of candidates will be offered admission from these two lists, according to merit and availability of seats.

PhD Applied Mathematics Test Paper for the year 2017