List of top Questions asked in IIT JAM MA

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 \[ \int_0^1 \int_0^1 \left( \left\lceil x \right\rceil + \left\lceil y \right\rceil + \left| z \right| \right) dx \, dy \, dz = \text{.............}. \]

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

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

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 \( T \) be the smallest positive real number such that the tangent to the helix \[ x = \cos t, \quad y = \sin t, \quad z = \frac{t}{\sqrt{2}} \] at \( t = T \) is orthogonal to the tangent at \( t = 0 \). Then the line integral of \( \mathbf{F} = xj - y\mathbf{i} \) along the section of the helix from \( t = 0 \) to \( t = T \) 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 \( e^2y(e) \) is .............

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 ..........

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 ..........

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

Let \( v_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \) and \( v_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \). 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 \(Mx = 0\) is ...........

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

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 ................

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 ............

Let G be a subgroup of GL₂(ℝ) generated by

\[ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix} \] Then the order of G is ...........

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 \( \alpha = \) ................

If \( X \) and \( Y \) are \( n \times n \) matrices with real entries, then which of the following is/are TRUE?

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 \( \{x_n\} \) be a real sequence such that \( 7x_{n+1} = x_n^3 + 6 \) for \( n \geq 1 \). Then which of the following statements 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?

Let \( y(x) \) be the solution of the differential equation \[ \frac{dy}{dx} = (y - 1)(y - 3) \] satisfying the condition \( y(0) = 2 \). Then which of the following is/are TRUE?

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?

For \( a, \beta \in \mathbb{R} \), define the map \( \varphi_{a,\beta}: \mathbb{R} \to \mathbb{R} \) by \[ \varphi_{a,\beta}(x) = ax + \beta. \] Let \[ G = \{ \varphi_{a,\beta} \mid (a, \beta) \in \mathbb{R}^2 \}. \] For \( f, g \in G \), define \( g \circ f \in G \) by \[ (g \circ f)(x) = g(f(x)). \] Then which of the following statements is/are TRUE?