List of top Questions asked in IIT JAM MA- 2021

Let \(n>1\) be an integer. Consider the following two statements for an arbitrary \(n \times n\) matrix \(A\) with complex entries. I. If \(A^k = I_n\) for some integer \(k \ge 1\), then all the eigenvalues of \(A\) are \(k^{\text{th}}\) roots of unity.
II. If, for some integer \(k \ge 1\), all the eigenvalues of \(A\) are \(k^{\text{th}}\) roots of unity, then \(A^k = I_n.\)
Then

Let \( M_n(\mathbb{R}) \) be the real vector space of all \( n \times n \) matrices with real entries, \( n \ge 2 \). Let \( A \in M_n(\mathbb{R}) \). Consider the subspace \( W \) of \( M_n(\mathbb{R}) \) spanned by \(\{I_n, A, A^2, A^3, \ldots\}\). Then the dimension of \( W \) over \(\mathbb{R}\) is necessarily

Let \(y\) be the solution of \[ (1+x)y''(x) + y'(x) - \frac{1}{1+x} y(x) = 0, \quad x \in (-1,\infty), \] with initial conditions \(y(0) = 1, \ y'(0) = 0.\) Then

Consider the surface \[ S = \{(x,y,xy) \in \mathbb{R}^3 : x^2 + y^2 \le 1\}. \] Let \(\vec{F} = y\hat{i} + x\hat{j} + \hat{k}\). If \(\hat{n}\) is the continuous unit normal field to the surface \(S\) with positive \(z\)-component, then \[ \iint_S \vec{F} \cdot \hat{n} \, dS \] equals

Let \( P : \mathbb{R} \to \mathbb{R} \) be a continuous function such that \( P(x)>0 \) for all \(x \in \mathbb{R}\). Let \(y\) be a twice differentiable function on \(\mathbb{R}\) satisfying \[ y''(x) + P(x)y'(x) - y(x) = 0 \] for all \(x \in \mathbb{R}\). Suppose that there exist two real numbers \(a,b\) (\(a<b\)) such that \(y(a) = y(b) = 0\). Then

Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function satisfying \( f(x) = f(x+1) \) for all \(x \in \mathbb{R}\). Then

Let \( f : \mathbb{R} \to \mathbb{R} \) be a continuous function such that for all \(x \in \mathbb{R}\), \[ \int_0^1 f(xt) \, dt = 0. \quad \text{(*)} \] Then

Let \(p\) and \(t\) be positive real numbers. Let \(D_t\) be the closed disc of radius \(t\) centered at \((0,0)\), i.e., \[ D_t = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 \le t^2 \}. \] Define \[ I(p,t) = \iint_{D_t} \frac{dx\,dy}{(p^2 + x^2 + y^2)^2}. \] Then \(\lim_{t \to \infty} I(p,t)\) is finite

How many elements of the group \(\mathbb{Z}_{50}\) have order 10?

Let \( 0<\alpha<1 \) be a real number. The number of differentiable functions \( y : [0,1] \to [0,\infty) \), having continuous derivative on \([0,1]\) and satisfying \[ y'(t) = (y(t))^{\alpha}, \ t \in [0,1], \quad y(0) = 0, \] is