Mathematics for IIT JAM MS
Mathematics
The Mathematics half of IIT JAM MS rewards fluency with the tools that statistics leans on: convergence of sequences and series, one-variable calculus, improper integrals, and linear algebra. Most items are computational with one conceptual twist, so command of standard forms (p-series, Taylor expansions, rank conditions) matters far more than clever tricks.
Convergence toolkit
- A monotone bounded sequence converges; a series of positive terms is settled by comparison, limit comparison, ratio (L<1 converges), root (L<1 converges) or the integral test.
- An alternating series converges when |a_n| decreases to 0, but it is only absolutely convergent when Σ|a_n| converges.
- Σ n^(−p) converges iff p>1; ∫₁^∞ x^(−p)dx behaves the same way, whereas ∫₀¹ x^(−p)dx converges iff p<1.
Calculus and linear algebra
- Rolle, mean value theorem, Taylor with remainder, and L'Hôpital applied only to genuine 0/0 or ∞/∞ forms after rewriting.
- For a square matrix A: invertible ⟺ det A ≠ 0 ⟺ full rank ⟺ no zero eigenvalue. The system Ax=b is consistent iff rank(A)=rank([A|b]), and the solution is unique iff that common rank equals n.
Worked example
For which λ does x+y+z=1, x+2y+3z=4, x+3y+λz=7 have infinitely many solutions? Expanding the determinant along the first row: (2λ−9) − (λ−3) + (3−2) = λ−5, so λ≠5 gives a unique solution. At λ=5, row2−row1 is y+2z=3 and row3−row1 is 2y+4z=6, the same equation. The system is consistent with one free variable, hence infinitely many solutions. Answer: λ=5.
Traps
Concluding convergence when the ratio test gives L=1 (the test is inconclusive - switch to comparison); assuming A²=0 forces A=0; confusing diagonalizable with invertible; and, above all, forgetting that a zero determinant still requires a separate consistency check.
Key points
- Determine whether a series converges absolutely, conditionally, or not at all.
- Use p-test thresholds, with p>1 at infinity and p<1 at zero.
- Check consistency of a linear system with rank, not just the determinant.
- Apply the mean value theorem only when its hypotheses hold on the stated interval.