List of top Mathematics Questions asked in IIT JAM MA

If \( x^h y^k \) is an integrating factor of the differential equation \[ y(1 + xy) \, dx + x(1 - xy) \, dy = 0, \] then the ordered pair \( (h, k) \) is equal to

Let \( \{a_n\}_{n=0}^{\infty} \) and \( \{b_n\}_{n=0}^{\infty} \) be sequences of positive real numbers such that \( n a_n<b_n<n^2 a_n \), for all \( n \geq 2 \). If the radius of convergence of the power series \[ \sum_{n=0}^{\infty} a_n x^n \] is 4, then the power series \[ \sum_{n=0}^{\infty} b_n x^n \] is

Let \( g: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function. Define \( f: \mathbb{R}^3 \to \mathbb{R} \) by \[ f(x, y, z) = g(x^2 + y^2 - 2z^2). \] Then \[ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \] is equal to

Let \( S \) be the set of all limit points of the set \[ \left\{ \frac{n}{\sqrt{2}} + \frac{\sqrt{2}}{n} : n \in \mathbb{N} \right\}. \] Let \( \mathbb{Q}_+ \) be the set of all positive rational numbers. Then

Let \( M_{n \times p}(\mathbb{R}) \) be the subspace of \( M_{n \times p}(\mathbb{R}) \) defined by \[ V = \{ X \in M_{n \times p}(\mathbb{R}) : TX = 0 \}. \] Then the dimension of \( V \) is

Let \( a_1 = b_1 = 0 \), and for each \( n \geq 2 \), let \( a_n \) and \( b_n \) be real numbers given by \[ a_n = \sum_{m=2}^{n} (-1)^{m} m (\log(m))^m \] \[ b_n = \sum_{m=2}^{n} \frac{1}{(\log(m))^m}. \] Then which one of the following is TRUE about the sequences \( \{a_n\} \) and \( \{b_n\} \)?

Let \( \mathbb{C}^* = \mathbb{C} \setminus \{0\} \) denote the group of non-zero complex numbers under multiplication. Suppose \[ Y_n = \{ z \in \mathbb{C} \mid z^n = 1 \}, n \in \mathbb{N}. \] Which of the following is (are) subgroup(s) of \( \mathbb{C}^* \)?

Let \( P \) and \( Q \) be two non-empty disjoint subsets of \( \mathbb{R} \). Which of the following is (are) FALSE?

Suppose \( f, g, h \) are permutations of the set \( \{ \alpha, \beta, \gamma, \delta \} \), where
f interchanges \( \alpha \) and \( \beta \) but fixes \( \gamma \) and \( \delta \),
g interchanges \( \beta \) and \( \gamma \) but fixes \( \alpha \) and \( \delta \),
h interchanges \( \gamma \) and \( \delta \) but fixes \( \alpha \) and \( \beta \).
Which of the following permutations interchange(s) \( \alpha \) and \( \delta \) but fix(es) \( \beta \) and \( \gamma \)?

Consider the group \( \mathbb{Z}^2 = \{ (a, b) | a, b \in \mathbb{Z} \} \) under component-wise addition. Then which of the following is a subgroup of \( \mathbb{Z}^2 \)?

Let \( G \) be a group satisfying the property that \( f: G \to \mathbb{Z}_{221} \) is a homomorphism implies \( f(g) = 0 \), \( \forall g \in G \). Then a possible group \( G \) is

Suppose that \( f, g : \mathbb{R} \to \mathbb{R} \) are differentiable functions such that \( f \) is strictly increasing and \( g \) is strictly decreasing. Define \( p(x) = f(g(x)) \) and \( q(x) = g(f(x)) \), \( \forall x \in \mathbb{R} \). Then, for \( t > 0 \), the sign of \( \int_0^t p'(x) (q'(x)-3) \, dx \) is

Let \( f : \mathbb{R}^3 \to \mathbb{R} \) be a scalar field, \( \vec{v} : \mathbb{R}^3 \to \mathbb{R}^3 \) be a vector field and let \( \vec{a} \in \mathbb{R}^3 \) be a constant vector. If \( \vec{r} \) represents the position vector \( x\hat{i} + y\hat{j} + z\hat{k} \), then which one of the following is FALSE?

Let \( a \) be a positive real number. If \( f \) is a continuous and even function defined on the interval \( [-a, a] \), then \( \int_{-a}^{a} \dfrac{f(x)}{1+e^x} dx \) is equal to

Which one of the following is TRUE?

Suppose \( \alpha, \beta, \gamma \in \mathbb{R} \). Consider the following system of linear equations.
\[ x + y + z = \alpha, x + \beta y + z = \gamma, x + y + \alpha z = \beta. \] If this system has at least one solution, then which of the following statements is (are) TRUE?

If \( \vec{\mathbf{F}} (x, y, z) = (2x + 3yz) \hat{i} + (3xz + 2y) \hat{j} + (3xy + 2z) \hat{k} \) for \( (x, y, z) \in \mathbb{R}^3 \), then which among the following is (are) TRUE?

Which of the following subsets of \( \mathbb{R} \) is (are) connected?

Let \( S \) be a subset of \( \mathbb{R} \) such that 2018 is an interior point of \( S \). Which of the following is (are) TRUE?

The order of the element \( (1 \ 2 \ 3)(2 \ 4 \ 5)(4 \ 5 \ 6) \) in the group \( S_6 \) is ..............

Let \( A_6 \) be the group of even permutations of 6 distinct symbols. Then the number of elements of order 6 in \( A_6 \) is ..................

Suppose \( x, y, z \) are positive real numbers such that \( x + 2y + 3z = 1 \). If \( M \) is the maximum value of \( xyz^2 \), then the value of \( \frac{1}{M} \) is ..................

If \( a = \int_{\pi/3}^{\pi/6} \dfrac{\sin t + \cos t}{\sqrt{\sin 2t}} \, dt, \) then the value of \( \left( 2 \sin \frac{\alpha}{2} + 1 \right)^2 \text{ is ............}. \)

Let \( H \) be the quotient group \( \mathbb{Q} / \mathbb{Z} \). Consider the following statements.
I. Every cyclic subgroup of \( H \) is finite.
II. Every finite cyclic group is isomorphic to a subgroup of \( H \).
Which one of the following holds?

Let \( \alpha, \beta, \gamma, \delta \) be the eigenvalues of the matrix

Then \( \alpha^2 + \beta^2 + \gamma^2 + \delta^2 = \) ..........