IIT JAM MA case studies
IIT JAM MA · Put your knowledge to work in context.
JAM Mathematics (MA) is one of the subject papers of the Joint Admission Test for Masters, an examination described by the JAM Office as testing "the comprehensive understanding of the candidates in various undergraduate subjects" - Mathematics being listed alongside Biotechnology, Chemistry, Economics, Geology, Mathematical Statistics and Physics. JAM has been conducted from 2004 to provide admissions to M.Sc. The IIT JAM MA case studies are 5 cases and about 3,571 words.
1Passing a limit through an integral: the moving peak
While revising sequences of functions, a candidate writes this solution for f n(x) = n x (1 − x²)^n on [0, 1]: 'For each fixed x in (0, 1], (1 − x²)^n decays exponentially, so f n(x) → 0; the pointwise limit is the zero function, hence ∫₀¹ f n(x) dx → 0.' A numerical check contradicts the conclusion: the substitution u = 1 − x² gives ∫₀¹ f n = n/(2(n + 1)), so the integrals are 0.4545 for n = 10, 0.4950 for n = 100, and they tend to 1/2. The candidate also notices that f n(1/√(2n+1)) ≈ 4.3 when n = 100, so the functions are not small everywhere on the interval.
745 words, about 4 min2A parameterised 3×3 system: when singularity means a line, not emptiness
A consulting exercise models three outputs (x, y, z) by x + y + z = 1, x + 2y + 3z = 4, x + 4y + αz = β, with tuning constants α and β. A candidate writes: 'three equations in three unknowns, so the coefficient matrix is invertible and the solution is unique for every α and β.' A colleague substitutes α = 7, β = 10 and finds that x = t − 2, y = 3 − 2t, z = t satisfies all three equations for every real t.
718 words, about 4 min3Extreme values on a triangle: throwing away the interior critical point
A candidate must find the extreme values of f(x, y) = x² + y² − 2x − 2y on the closed triangular region T with vertices O(0,0), A(3,0) and B(0,3). The submitted report reads: 'f has positive definite Hessian, so it is convex; a convex function on a polygon attains its maximum at a vertex, and a convex function cannot have an interior minimum unless the gradient vanishes there.
716 words, about 4 min4A logarithmic borderline: the series Σ 1/(n^p ln n)
A worksheet asks for which real p the series S(p) = Σ {n=2}^∞ 1/(n^p ln n) converges. The candidate submits three steps.
658 words, about 3 min5A curl-free field that is not conservative: the whole story
The vector field F(x, y) = (−y/(x² + y²), x/(x² + y²)) is defined on the plane with the origin removed. A candidate computes ∂/∂x [x/(x² + y²)] = (y² − x²)/(x² + y²)² and ∂/∂y [−y/(x² + y²)] = (y² − x²)/(x² + y²)², observes that the two are equal, and concludes that ∮ C F·dr = 0 for every closed curve C.
734 words, about 4 min