List of top Mathematics Questions asked in IIT JAM MA

For an integer \(k \ge 0\), let \(P_k\) denote the vector space of all real polynomials in one variable of degree less than or equal to \(k\). Define a linear transformation \(T : P_2 \to P_3\) by \[ T(f(x)) = f''(x) + x f(x). \] Which one of the following polynomials is not in the range of \(T\)?

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

Which one of the following subsets of \(\mathbb{R}\) has a non-empty interior?

For every \( n \in \mathbb{N} \), let \( f_n : \mathbb{R} \to \mathbb{R} \) be a function. From the given choices, pick the statement that is the negation of \[ \text{“For every } x \in \mathbb{R} \text{ and for every real number } \varepsilon>0, \text{ there exists an integer } N>0 \text{ such that } \sum_{i=1}^p |f_{N+i}(x)|<\varepsilon \text{ for every integer } p>0.” \]

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 \( f : \mathbb{R} \to \mathbb{R} \) 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

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

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

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 \( 0<\alpha<1 \) 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

Let \( a = \lim_{n \to \infty} \left( \frac{1}{n^2} + \frac{2}{n^2} + \cdots + \frac{n-1}{n^2} \right) \) and \( b = \lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{n+n} \right). \) Which of the following is/are true?

Which of the following is FALSE?

If \( y(x) = 2e^{2x} + e^{\beta x}, \beta \neq 2 \), is a solution of the differential equation \[ \frac{d^2 y}{dx^2} + \frac{dy}{dx} - 6y = 0, \] satisfying \( \frac{dy}{dx} (0) = 5 \), then \( y(0) \) is equal to

Let \( M \) and \( N \) be any two \( 4 \times 4 \) matrices with integer entries satisfying

\[ MN = \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} = 2 \]

Then the maximum value of \( \det(M) + \det(N) \) is ...............

Let \( M \) be a \( 3 \times 3 \) matrix with real entries such that

\[ M^2 = M + 2I, \quad \text{where} \quad I \text{ denotes the } 3 \times 3 \text{ identity matrix.} \]

If \( \alpha, \beta, \gamma \) are eigenvalues of \( M \) such that \( \alpha\beta\gamma = -4, \) then \( \alpha + \beta + \gamma \) is equal to .............

If

\[ \begin{pmatrix} z \\ y \end{pmatrix} \]

is an eigenvector corresponding to a real eigenvalue of the matrix

\[ \begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & -4 \\ 0 & 1 & 3 \end{pmatrix} \]

then \( z - y \) is equal to .................

Let

\[ f(x, y) = \begin{cases} \frac{x^3 + y^3}{x^2 - y^2}, & x^2 - y^2 \neq 0 \\ 0, & x^2 - y^2 = 0 \end{cases} \]

Then the directional derivative of \( f \) at \( (0, 0) \) in the direction of \( \frac{4}{5} \hat{i} + \frac{3}{5} \hat{j} \) is ............

The coefficient of \( (x - \frac{\pi}{2}) \) in the Taylor series expansion of the function

\[ f(x) = \begin{cases} \frac{4(1 - \sin(x))}{2x - \pi}, & x \neq \frac{\pi}{2} \\ 0, & x = \frac{\pi}{2} \end{cases} \]

about \( x = \frac{\pi}{2} \) is ............

Let \( \vec{F}(x, y) = -y \hat{i} + x \hat{j} \) and let \( C \) be the ellipse

\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]

with counterclockwise orientation. Then the value of \( \oint_C \vec{F} \cdot d\vec{r} \) (rounded to 2 decimal places) is .............

Let

\[ f(x, y) = \begin{cases} \frac{|x|}{|x| + |y|} \sqrt{x^4 + y^2}, & (x, y) \neq (0, 0), \\ 0, & (x, y) = (0, 0). \end{cases} \]

Then at \( (0, 0) \),