## JAM MATHEMATICS

**ABOUT JAM**

The Indian Institute of Science Bangalore (IISc) and Indian Institutes of Technology (IITs) conduct the Joint Admission Test for M.Sc. (JAM) for admission to Integrated Ph.D. Degree Programmes at IISc Bangalore and M.Sc. (Two Years), Joint M.Sc.-Ph.D., M.Sc.-Ph.D. Dual Degree, M.Sc.-M.Tech., M.Sc.-M.S.(Research)/Ph.D. Dual Degree and other Post-Bachelors Degree Programmes at IITs. The main objective of JAM is to consolidate Science as a career option for bright students across the country. It is expected that the research infrastructure, the inter-disciplinary interaction, and the vibrant academic environment of IISc Bangalore and IITs will motivate these students to pursue R&D careers in frontier areas of basic sciences as well as inter-disciplinary areas of science and technology.

**WHAT IS NEW?**

**1.**Multiple Choice Questions (MCQ): Each MCQ type question has four choices out of which only one choice is the correct answer.

**2.**Multiple Select Questions (MSQ): Each MSQ type question is similar to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the four given choices.

**3.**Numerical Answer Type (NAT) Questions: For each NAT type question, the answer is a signed real number which needs to be entered using the virtual keypad on the monitor. No choices will be shown for these types of questions.

**IMPORTANT DATES**

September 02, 2016 | Commencement of ONLINE Registration and Application Process |

October 10, 2016 | Last Date for Payment of Fee through Challan |

October 14, 2016 | Closure of ONLINE Application Procedure |

February 07, 2017 | JAM 2017 Examination |

March 23, 2017 | Announcement of JAM 2017 Results |

**QUESTION PATTERN - JAM 2017**

JAM 2017 examination will be conducted for the seven test papers, namely: Biological Sciences (BL), Biotechnology (BT), Chemistry (CY), Geology (GG), Mathematics (MA), Mathematical Statistics (MS), and Physics (PH). All seven test papers will be of fully objective type, with three different patterns of questions as follows:

**Multiple Choice Questions (MCQ):**Each MCQ-type question has four choices out of which only one choice is the correct answer.

**Multiple Select Questions (MSQ):**Each MSQ type question is similar to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the four given choices.

**Numerical Answer Type (NAT) Questions:**For each NAT type question, the answer is a signed real number that needs to be entered using the virtual keypad on the monitor. No choices will be shown for these types of questions.

**PATTERN OF TEST PAPERS**

**SECTION A:**contains a total of 30 Multiple Choice Questions (MCQ) carrying one or two marks each. Each MCQ type question has four choices out of which only one choice is the correct answer.

Candidates can mark the answer by clicking the choice.

**SECTION B:**contains a total of 10 Multiple Select Questions (MSQ) carrying two marks each. Each MSQ type question is similar to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the four given choices. The candidate gets full credit if he/she selects all the correct answers only and no wrong answers. Candidates can mark the answer(s) by clicking the choice(s).

**SECTION C:**contains a total of 20 Numerical Answer Type (NAT) questions carrying one or two marks each. For these NAT-type questions, the answer is a signed real number that needs to be entered using the virtual keyboard on the monitor. No choices will be shown for these types of questions.

Candidates have to enter the answer by using a virtual numeric keypad.

**MARKING SCHEME**

**ONLINE TEST RULES**

## Regular Classroom Programs

**Basic Building Measures**(BBM) which has two components

**1. BBM- Tests:**After 2-3 lectures on any topic the academy offers a test to all the students. This test (Called BBM Test) contains very basic and fundamental questions about that topic. The performances over this test are analyzed comprehensively and on the basis of their performance, those students are identified who could not cross minimum thresh hold and requires extra care.

**2. BBM Classes:**The students Identified by BBM tests are clubbed into various groups and dedicated faculty are assigned to each group. These groups are given extra classes (BBM Classes). In these classes, this faculty helps the students clear their doubts and concept so that they too can equally participate in the regular classes along with the so-called average plus students.

## Syllabus / Books

**JAM: MATHEMATICS**

**FUNCTIONS OF ONE VARIABLE:**

**Functions of One Variable:**limit, continuity, differentiation, Rolles Theorem, Mean value theorem. Taylors theorem, Maxima and minima.

**Functions of Two Real Variables:**limit, continuity, partial derivatives, differentiability, maxima, and minima. Method of Lagrange multipliers, Homogeneous functions including Eulers theorem.

**Integral Calculus:**Integration as the inverse process of differentiation, definite integrals, and their properties, Fundamental theorem of integral calculus. Double and triple integrals, change of order of integration. Calculating surface areas and volumes using double integrals and applications. Calculating volumes using triple integrals and applications.

**Differential Equations:**Ordinary differential equations of the first order of the form y=f(x,y). Bernoullis equation, exact differential equations, integrating factor, Orthogonal trajectories, Homogeneous differential equations-separable solutions, Linear differential equations of second and higher-order with constant coefficients, Method of variation of parameters. Cauchy-Euler equation.

**Vector Calculus:**Scalar and vector fields, gradient, divergence, curl, and Laplacian. Scalar line integrals and vector line integrals, scalar surface integrals and vector surface integrals, Greens, Stokes, and Gauss theorems and their applications.

**Group Theory:**Groups, subgroups, Abelian groups, non-abelian groups, cyclic groups, permutation groups; Normal subgroups, Lagranges Theorem for finite groups, group homomorphisms, and basic concepts of quotient groups (only group theory).

**Linear Algebra:**Vector spaces, Linear dependence of vectors, basis, dimension, linear transformations, matrix representation with respect to an ordered basis, Range space, and null space, rank-nullity theorem; Rank and inverse of a matrix, determinant, solutions of systems of linear equations, consistency conditions. Eigenvalues and eigenvectors. Cayley-Hamilton theorem. Symmetric, skew-symmetric, hermitian, skew-hermitian, orthogonal, and unitary matrices.

**Real Analysis:**Interior points, limit points, open sets, closed sets, bounded sets, connected sets, compact sets; completeness of R, Power series (of real variable) including Taylors and Maclaurins, domain of convergence, term-wise differentiation, and integration of power series.

**JAM: MATHEMATICAL STATISTICS (MS)**

**MATHEMATICS Sequences and Series**: Convergence of sequences of real numbers, Comparison, root, and ratio tests for convergence of series of real numbers.

**Differential Calculus:**Limits, continuity, and differentiability of functions of one and two variables. Rolles theorem, mean value theorems, Taylors theorem, indeterminate forms, maxima, and minima of functions of one and two variables.

**Integral Calculus:**Fundamental theorems of integral calculus. Double and triple integrals, applications of definite integrals, arc lengths, areas, and volumes.

**Matrices:**Rank, the inverse of a matrix. Systems of linear equations. Linear transformations, eigenvalues, and eigenvectors. Cayley-Hamilton theorem, symmetric, skew-symmetric, and orthogonal matrices.

**Differential Equations:**: Ordinary differential equations of the first order of the form y = f(x,y). Linear differential equations of the second order with constant coefficients.

**STATISTICS**

**Probability:**Axiomatic definition of probability and properties, conditional probability, multiplication rule. The theorem of total probability. Bayes theorem and independence of events.

**Random Variables:**Probability mass function, probability density function, and cumulative distribution functions, distribution of a function of a random variable. Mathematical expectation, moments, and moment generating function. Chebyshevs inequality.

**Joint Distributions:**Joint, marginal, and conditional distributions. Distribution of functions of random variables. Product moments, correlation, simple linear regression. Independence of random variables.

**Sampling distributions:**Chi-square, t, and F distributions, and their properties.

**Limit Theorems:**Weak law of large numbers. Central limit theorem (i.i.d. with finite variance case only).

**Estimation:**Unbiasedness, consistency, and efficiency of estimators, method of moments, and method of maximum likelihood. Sufficiency, factorization theorem. Completeness, Rao-Blackwell, and Lehmann- Scheffe theorems, uniformly minimum variance unbiased estimators. Rao-Cramer inequality. Confidence intervals for the parameters of univariate normal, two independent normals, and one parameter exponential distributions.

**Testing of Hypotheses:**: Basic concepts, applications of Neyman-Pearson Lemma for testing simple and composite hypotheses. Likelihood ratio tests for parameters of univariate normal distribution.