IIT JAM MS study plan

Derive and state from memory the limit, continuity and differentiability criteria for functions of one real variable, including the epsilon-delta definition of a limit

Compute first and second derivatives using product, quotient and chain rules and apply them to locate maxima, minima and points of inflection

Evaluate standard limits and indeterminate forms using L'Hopital's rule and Taylor expansions to second order

Set up and solve first-order separable and linear ordinary differential equations, including integrating-factor method

Compute definite and indefinite integrals using substitution, integration by parts and partial fractions

Apply the fundamental theorem of calculus and evaluate improper integrals for convergence

Determine convergence or divergence of sequences and series using comparison, ratio and root tests

State and apply the mean value theorems (Rolle, Lagrange) and Taylor's theorem with remainder

Perform matrix operations, compute determinants and inverses, and identify rank and linear independence of row/column vectors

Solve systems of linear equations using Gaussian elimination and classify solutions by consistency

Find eigenvalues and eigenvectors of 2x2 and 3x3 matrices and diagonalize where possible

Verify the Cayley-Hamilton theorem on a 3x3 matrix and use it to compute a matrix inverse

Compute partial derivatives, gradients and directional derivatives for functions of two variables and locate unconstrained critical points

Use Lagrange multipliers to solve constrained optimization problems with one equality constraint

Define random variables, probability mass and density functions, and verify that a given function is a valid distribution

Compute expectation and variance from a given pmf or pdf using the definition and the shortcut formula

Derive and apply the binomial, Poisson, geometric, uniform, exponential and normal distributions, including their means and variances

Compute cumulative probabilities using standardization and the standard normal table for the normal distribution

Apply conditional probability, Bayes' theorem and the law of total probability to multi-stage problems

Determine independence and compute covariance and correlation for jointly distributed discrete variables

Construct the sampling distribution of the sample mean and apply the central limit theorem to approximate probabilities

Derive point estimators by the method of moments and by maximum likelihood for common distributions

Check unbiasedness and compute mean squared error of an estimator as variance plus squared bias

Construct confidence intervals for a population mean with known and unknown variance

Formulate null and alternative hypotheses and carry out one-sample and two-sample z and t tests at a stated significance level

Interpret p-values correctly and explain Type I and Type II errors in a given scenario

Compute the size and power of a simple test and identify the most powerful test in a simple-vs-simple case using the Neyman-Pearson lemma

Apply the chi-square goodness-of-fit test to a given frequency table

Complete three full-length timed mock papers under exam conditions and record the score for each

Rank recurring error types from the mock papers and re-solve every missed question without notes

Re-derive the ten formulas most often missed, covering both mathematics and statistics

Build a final one-page revision sheet of definitions, theorems and standard results for both domains