IIT JAM Mathematics: Analysis, Algebra and Differential Equations
IIT JAM MA tests real analysis, calculus, linear algebra, differential equations, vector calculus and group theory. The exam favors precise definitions, counterexample awareness and short computations that reveal whether a condition is necessary, sufficient or both.
Core rules and concepts
- Real analysis: ε–N convergence; monotone bounded sequences converge; absolute convergence implies convergence, but conditional convergence does not.
- Continuity and differentiation: differentiability implies continuity, not conversely; Rolle’s and mean value theorems require hypotheses on a closed interval and differentiability inside.
- Integration: a bounded function with finitely many discontinuities is Riemann integrable; continuous functions on closed intervals are integrable.
- Linear algebra: rank–nullity theorem; det(A) = 0 means A is not invertible; real symmetric matrices are orthogonally diagonalizable.
- Differential equations: first-order linear equations use an integrating factor; second-order linear equations use characteristic roots.
- Vector calculus: gradient, divergence and curl; Green, Stokes and divergence theorems convert boundary and region integrals.
- Group theory: closure, associativity, identity and inverses; Lagrange’s theorem gives subgroup order dividing group order.
Worked example
For A = [[2, 1], [1, 2]], find the eigenvalues. det(A − λI) = (2 − λ)² − 1 = 0, so (2 − λ)² = 1 and λ = 1, 3. For λ = 3, (A − 3I)v = 0 gives −v₁ + v₂ = 0, so v = (1, 1). For λ = 1, v₁ + v₂ = 0, so v = (1, −1).
Traps that cost marks
Do not confuse pointwise and uniform convergence. A continuous function need not be differentiable. A singular matrix has determinant zero and no inverse. Groups need not be abelian. Check whether an eigenvector satisfies the original matrix equation, not merely a rearranged row.
Key points
- Differentiability implies continuity, not the reverse.
- Absolute convergence is stronger than ordinary convergence.
- Rank–nullity: rank + nullity = number of columns.
- Real symmetric matrices have real eigenvalues and orthogonal eigenvectors.
- Lagrange’s theorem constrains subgroup orders.