List of top Mathematics Questions asked in IIT JAM MA

Let \( f(x) = \frac{\sin \left( \frac{n\pi x}{\pi \sin x} \right)}{ \sin x} \), Let \( x_0 \in (0, \pi) \) and let \( f'(x_0) = 0 \). Then \[ (f(x_0))^2 \left( 1 + (\pi^2 - 1)\sin^2 x_0 \right) = \text{.........}. \]

Let

Let \( M \) be the matrix whose columns are \( v_1, v_2, 2v_1 - v_2, v_1 + 2v_2 \) in that order. Then the number of linearly independent solutions of the homogeneous system of linear equations \( Mx = 0 \) is ...........

Let \(G\) be a subgroup of \(GL_2(\mathbb{R})\) generated by

Then the order of \(G\) is ...........

Let

Then \[ \lim_{n \to \infty} M^n x \]

If

satisfies the assumptions of Rolle’s theorem in the interval \( [-1, 1] \), then the ordered pair \( (p, q) \) is

Let \( f_1(x), f_2(x), g_1(x), g_2(x) \) be differentiable functions on \( \mathbb{R} \). Let

Then \( F'(x) \) is equal to

If \[ y(x) = \int_{\sqrt{x}}^x e^t \, dt, \quad x>0 \] then \( y'(1) = \) .............

The radius of convergence of the power series \[ \sum_{n=0}^{\infty} n! x^{n^2} \] is ...........

For \( x>1 \), let \[ f(x) = \int_1^x \left( \sqrt{\log t - \frac{1}{2} \log \sqrt{t}} \right) dt. \] The number of tangents to the curve \( y = f(x) \) parallel to the line \( x + y = 0 \) is ..............

For a real number \( x \), define \( \left\lceil x \right\rceil \) to be the smallest integer greater than or equal to \( x \). Then \[ \int_0^1 \int_0^1 \left( \left\lceil x \right\rceil + \left\lceil y \right\rceil + \left| z \right| \right) dx \, dy \, dz = \text{.............}. \]

The maximum order of a permutation \( \sigma \) in the symmetric group \( S_{10} \) is ..........

Let \( a_n = \sqrt{n}, n \geq 1 \), and let \( s_n = a_1 + a_2 + \cdots + a_n \). Then \[ \lim_{n \to \infty} \left( \frac{a_n}{s_n} \Big/ \ln \left( 1 - \frac{a_n}{s_n} \right) \right) = \quad \_\_\_\_\_\_\_\_. \]

Let \( T \) be the smallest positive real number such that the tangent to the helix \[ \mathbf{r}(t) = \cos(t) \hat{i} + \sin(t) \hat{j} + \frac{t}{\sqrt{2}} \hat{k} \] at \( t = T \) is orthogonal to the tangent at \( t = 0 \). Then the line integral of \( \mathbf{F} = x \hat{i} - y \hat{j} \) along the section of the helix from \( t = 0 \) to \( t = T \) 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 ..........

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

Let \( y(x), x>0 \) be the solution of the differential equation \[ x^2 \frac{d^2y}{dx^2} + 5x \frac{dy}{dx} + 4y = 0 \] satisfying the conditions \( y(1) = 1 \) and \( y'(1) = 0 \). Then the value of \( e^2 y(e) \) is ...........

Let \( P \) be a \( 7 \times 7 \) matrix of rank 4 with real entries. Let \( a \in \mathbb{R}^7 \) be a column vector. Then the rank of \( P + aa^T \) is at least ..........

Let \( P \) be the point on the surface \( z = \sqrt{x^2 + y^2} \) closest to the point \( (4, 2, 0) \). Then the square of the distance between the origin and \( P \) is ..........

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

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

Consider the permutations \[ \sigma = (1 2 3 4 5 6 7 8), \quad \tau = (1 2 3 4 5 6 7 8) (4 5 3 7 8 6 1 2), \] in \( S_8 \). The number of \( \eta \in S_8 \) such that \( \eta^{-1} \sigma \eta = \tau \) is equal to ............

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

Let \( k, \ell \in \mathbb{R} \) be such that every solution of \[ \frac{d^2y}{dx^2} + 2k \frac{dy}{dx} + \ell y = 0 \] satisfies \( \lim_{x \to \infty} y(x) = 0 \). 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?

Let \( M \) be an \( n \times n \) matrix with real entries such that \( M^3 = I \). Suppose that \( Mv \neq v \) for any non-zero vector \( v \). Then which of the following statements is/are TRUE?