List of top Mathematics Questions on Mathematics asked in IIT JAM MA

The number of subgroups of \( \mathbb{Z}_7 \times \mathbb{Z}_7 \) of order 7 is ............

For \( x>0 \), let \( \lfloor x \rfloor \) denote the greatest integer less than or equal to \( x \). Then \[ \lim_{x \to \infty} x \left( \lfloor x \rfloor + \lfloor x / 2 \rfloor + \lfloor x / 3 \rfloor + \cdots + \lfloor x / 10 \rfloor \right) \] is ..........

Evaluate \[ \frac{1}{2\pi} \left( \frac{\pi^3}{1 \cdot 3} - \frac{\pi^5}{3 \cdot 5} + \frac{\pi^7}{5 \cdot 7} - \cdots + (-1)^{n-1} \frac{\pi^{2n+1}}{(2n-1)!} \left( 2n+1 \right) \right). \]

Evaluate \[ \left( \int_0^1 x^4 (1 - x)^5 \, dx \right)^{-1}. \]

If the orthogonal trajectories of the family of ellipses \[ x^2 + 2y^2 = c_1, \quad c_1>0, \] are given by \[ y = c_2 x^\alpha, \quad c_2 \in \mathbb{R}, \] then \( \alpha = \) ................

Let \( G \) be a group of order 20 in which the conjugacy classes have sizes 1, 4, 5, 5, 5. Then which of the following is/are TRUE?

Let \( S \) be the set of all rational numbers in \( (0,1) \). Then which of the following statements is/are TRUE?