IIT JAM Mathematics Question Paper 2024
Given below are a few questions from each section of the IIT JAM Mathematics question paper 2024. By referring to these questions, the candidates will get to understand what they can expect from the actual IIT JAM 2025 Mathematics question papers.
Section A
Q.1 Let G be a group of order 39 such that it has exactly one subgroup of order 3 and exactly one subgroup of order 13. Then, which one of the following statements is TRUE?
(A) G is necessarily cyclic.
(B) G is abelian but need not be cyclic.
(C) G need not be abelian.
(D) G has 13 elements of order 13.
Q.2 For a positive integer n, let U(n) = {r ∈ ℤₙ : gcd(r, n) = 1} be the group under multiplication modulo n. Then, which one of the following statements is TRUE?
(A) U(5) is isomorphic to U(8).
(B) U(10) is isomorphic to U(12).
(C) U(8) is isomorphic to U(10).
(D) U(8) is isomorphic to U(12).
Q.3 Let y(x) be the solution of the differential equation
dy/dx = 1 + y sec x, for x ∈ (-π/2, π/2)
that satisfies y(0) = 0. Then, the value of y(π/6) equals:
(A) √3 log (3/2)
(B) (√3/2) log (3/2)
(C) (√3/2) log 3
(D) √3 log 3
Q.4 Let g: R → R be a continuous function. Which one of the following is the solution of the differential equation
d²y/dx² + y = g(x), for x ∈ R,
satisfying the conditions y(0) = 0, y'(0) = 1?
(A) y(x) = sin(x) - ∫₀ˣ sin(x - t) g(t) dt
(B) y(x) = sin(x) + ∫₀ˣ sin(x - t) g(t) dt
(C) y(x) = sin(x) - ∫₀ˣ cos(x - t) g(t) dt
(D) y(x) = sin(x) + ∫₀ˣ cos(x - t) g(t) dt
Section B
Q.1 Which of the following statements is/are TRUE?
(A) The additive group of real numbers is isomorphic to the multiplicative group of positive real numbers.
(B) The multiplicative group of nonzero real numbers is isomorphic to the multiplicative group of nonzero complex numbers.
(C) The additive group of real numbers is isomorphic to the multiplicative group of nonzero complex numbers.
(D) The additive group of real numbers is isomorphic to the additive group of rational numbers.
Q.2 The center Z(G) of a group G is defined as
Z(G) = {x ∈ G : xg = gx for all g ∈ G}.
Let |G| denote the order of G. Then, which of the following statements is/are TRUE for any group G?
(A) If G is non-abelian and Z(G) contains more than one element, then the center of the quotient group G/Z(G) contains only one element.
(B) If |G| ≥ 2, then there exists a non-trivial homomorphism from Z to G.
(C) If |G| ≥ 2 and G is non-abelian, then there exists a non-identity isomorphism from G to itself.
(D) If |G| = p³, where p is a prime number, then G is necessarily abelian.
Q.3 For a matrix M, let Rowspace(M) denote the linear span of the rows of M and Colspace(M) denote the linear span of the columns of M. Which of the following hold(s) for all A, B, C ∈ M₁₀(R) satisfying A = BC?
(A) Rowspace(A) ⊆ Rowspace(B)
(B) Rowspace(A) ⊆ Rowspace(C)
(C) Colspace(A) ⊆ Colspace(B)
(D) Colspace(A) ⊆ Colspace(C)
Q.4 Let
S = {(x, y, z) ∈ R³ : x² + y² + z² = 4, (x − 1)² + y² ≤ 1, z ≥ 0}.
Then, the surface area of S equals _______ (rounded off to two decimal places).
Section C
Q.1 For α ∈ (−2π, 0), consider the differential equation
x²(d²y/dx²) + αx(dy/dx) + y = 0 for x > 0.
Let D be the set of all α ∈ (−2π, 0) for which all corresponding real solutions to the above differential equation approach zero as x → 0⁺. Then, the number of elements in D ∩ Z equals _______.
Q.2 For n ∈ N, if
aₙ = (1 / (n³ + 1)) + (2² / (n³ + 2)) + ... + (n² / (n³ + n))
then the sequence {aₙ} (n=1 to ∞) converges to _______ (rounded off to two decimal places).
Q.3 For k ∈ N, let 0 = t₀ < t₁ < ⋯ < tₖ < tₖ₊₁ = 1. A function f : [0,1] → R is said to be piecewise linear with nodes t₁, ⋯, tₖ if for each j = 1, 2, ⋯, k + 1, there exist aⱼ ∈ R and bⱼ ∈ R such that
f(t) = aⱼ + bⱼt for tⱼ₋₁ < t < tⱼ.
Let V be the real vector space of all real-valued continuous piecewise linear functions on [0,1] with nodes 1/4, 1/2, and 3/4. Then, the dimension of V equals _______.
