List of top Questions asked in IIT JAM MA- 2025

Let \( \mathbb{R}/\mathbb{Z} \) denote the quotient group, where \( \mathbb{Z} \) is considered as a subgroup of the additive group of real numbers \( \mathbb{R} \). Let \( m \) denote the number of injective (one-one) group homomorphisms from \( \mathbb{Z}_3 \) to \( \mathbb{R}/\mathbb{Z} \) and \( n \) denote the number of group homomorphisms from \( \mathbb{R}/\mathbb{Z} \) to \( \mathbb{Z}_3 \). Then, which one of the following is TRUE?

Let \( f_1, f_2, f_3 \) be nonzero linear transformations from \( \mathbb{R}^4 \) to \( \mathbb{R} \) and \[ \ker(f_1) \subset \ker(f_2) \cap \ker(f_3). \] Let \( T : \mathbb{R}^4 \to \mathbb{R}^3 \) be the linear transformation defined by \[ T(v) = (f_1(v), f_2(v), f_3(v)) \quad \text{for all } v \in \mathbb{R}^4. \] Then, the nullity of \( T \) is equal to:

Let \( \varphi : \mathbb{R} \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = (y - 1)(y - 3), \] satisfying \( \varphi(0) = 2 \). Then, which one of the following is TRUE?

Let \( C \) denote the family of curves described by \( yx^2 = \lambda \), for \( \lambda \in (0, \infty) \) and lying in the first quadrant of the \( xy \)-plane. Let \( O \) denote the family of orthogonal trajectories of \( C \). Which one of the following curves is a member of \( O \), and passes through the point \( (2, 1) \)?

Let \( \varphi : (0, \infty) \to \mathbb{R} \) be the solution of the differential equation \[ x \frac{dy}{dx} = \left( \ln y - \ln x \right) y, \] satisfying \( \varphi(1) = e^2 \). Then, the value of \( \varphi(2) \) is equal to:

Let \( T : P_2(\mathbb{R}) \to P_2(\mathbb{R}) \) be the linear transformation defined by \[ T(p(x)) = p(x + 1), \quad \text{for all } p(x) \in P_2(\mathbb{R}) \] If \( M \) is the matrix representation of \( T \) with respect to the ordered basis \( \{1, x, x^2\} \) of \( P_2(\mathbb{R}) \), then which one of the following is TRUE?

Let \( G \) be a finite abelian group of order 10. Let \( x_0 \) be an element of order 2 in \( G \). If \( X = \{ x \in G : x^3 = x_0 \} \), then which one of the following is TRUE?

Which one of the following is the general solution of the differential equation \[ \frac{d^2 y}{dx^2} - 8 \frac{dy}{dx} + 16y = 2e^{4x} ? \]

Define \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) by \[ T(x, y, z) = (x + z, 2x + 3y + 5z, 2y + 2z), \quad \text{for all } (x, y, z) \in \mathbb{R}^3 \] Then, which one of the following is TRUE?

For which one of the following choices of \( N(x, y) \), is the equation \[ (e^x \sin y - 2y \sin x) \, dx + N(x, y) \, dy = 0 \] an exact differential equation?