List of top Questions asked in IIT JAM MA

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

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

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

If \( x^2 + xy^2 = c \), where \( c \in \mathbb{R} \), is the general solution of the exact differential equation \[ M(x, y)\, dx + 2xy\, dy = 0, \] then \( M(1,1) \) is ............

Let \( C \) be the boundary of the square with vertices \( (0,0), (1,0), (1,1), (0,1) \) oriented counterclockwise. Then the value of the line integral \[ \oint_C x^2y^2 dx + (x^2 - y^2) dy \] is ............ (rounded off to two decimal places).

Let \( f(x) = \sqrt{x} + \alpha x, \; x > 0 \) and \[ g(x) = a_0 + a_1(x - 1) + a_2(x - 1)^2 \] be the sum of the first three terms of the Taylor series of \( f(x) \) around \( x = 1 \). If \( g(3) = 3 \), then \( \alpha \) is .............

Consider the expansion of the function \( f(x) = \dfrac{3}{(1 - x)(1 + 2x)} \) in powers of \( x \), valid in \( |x| < \dfrac{1}{2}. \) Then the coefficient of \( x^4 \) is ................

If \( \displaystyle \int_0^1 \int_{2y}^2 e^{x^2} \, dx \, dy = k(e^4 - 1), \) then \( k \) equals ...............

Let \( f(x, y) = e^x \sin y, \, x = t^3 + 1, \, y = t^4 + t. \) Then \( \dfrac{df}{dt} \) at \( t = 0 \) is ............. (rounded off to two decimal places).

Let \( S = \left\{ \frac{1}{n} : n \in \mathbb{N} \right\} \) and \( f : S \to \mathbb{R} \) be defined by \( f(x) = \frac{1}{x}. \) Then \[ \max \left\{ \delta : |x - \tfrac{1}{3}| < \delta \Rightarrow |f(x) - f(\tfrac{1}{3})| < 1 \right\} \] is ............. (rounded off to two decimal places).

Let \( G \) be a group with identity \( e \). Let \( H \) be an abelian non-trivial proper subgroup of \( G \) with the property that \( H \cap gHg^{-1} = \{ e \} \) for all \( g \notin H \). If \( K = \{ g \in G : gh = hg \text{ for all } h \in H \}, \) then

Let \( S \) be that part of the surface of the paraboloid \( z = 16 - x^2 - y^2 \) which is above the plane \( z = 0 \) and \( D \) be its projection on the xy-plane. Then the area of \( S \) equals

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?

Consider the following system of linear equations:
\[ \begin{cases} x + y + 5z = 3, \\ x + 2y + mz = 5, \\ x + 2y + 4z = k. \end{cases} \]

The system is consistent if

Let \( a, b \in \mathbb{R} \) and \( a < b. \) Which of the following statement(s) is/are true?

Let \( F = \{\omega \in \mathbb{C} : \omega^{2020} = 1\}. \)

Consider the groups \[ G = \left\{ \begin{pmatrix} \omega & z \\ 0 & 1 \end{pmatrix} : \omega \in F, z \in \mathbb{C} \right\} \text{and} H = \left\{ \begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix} : z \in \mathbb{C} \right\} \] under matrix multiplication.

Then the number of cosets of \( H \) in \( G \) is