List of top Questions asked in IIT JAM MA

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

Let \( a, b, c \in \mathbb{R} \) such that \( a < b < c. \) Which of the following is/are true for any continuous function \( f : \mathbb{R} \to \mathbb{R} \) satisfying \( f(a) = b, f(b) = c \) and \( f(c) = a? \)

Let \( f : [0, 1] \to \mathbb{R} \) be a continuous function such that \( f\left(\dfrac{1}{2}\right) = -\dfrac{1}{2} \) and \[ |f(x) - f(y) - (x - y)| \le \sin(|x - y|^2) \] for all \( x, y \in [0, 1]. \) Then \( \int_0^1 f(x) \, dx \) is

The value of the triple integral \( \iiint_V (x^2y + 1) \, dx\,dy\,dz \), where \( V \) is the region given by \( x^2 + y^2 \le 1, \, 0 \le z \le 2, \) is

Let \( M \) be a \( 4 \times 3 \) real matrix and let \( \{e_1, e_2, e_3\} \) be the standard basis of \( \mathbb{R}^3. \) Which of the following is true?

A solution of the differential equation \( 2x^2\dfrac{d^2y}{dx^2} + 3x\dfrac{dy}{dx} - y = 0, \, x > 0 \) that passes through the point (1, 1) is

Consider the differential equation \( L[y] = (y - y^2)dx + xdy = 0. \) The function \( f(x, y) \) is said to be an integrating factor of the equation if \( f(x, y)L[y] = 0 \) becomes exact. If \( f(x, y) = \dfrac{1}{x^2 y^2}, \) then

Let \( M \) be an \( n \times n \) ( \( n \ge 2 \) ) non-zero real matrix with \( M^2 = 0 \) and let \( \alpha \in \mathbb{R} \setminus \{0\}. \) Then

Let \( M \) be a real \( 6 \times 6 \) matrix. Let 2 and -1 be two eigenvalues of \( M. \) If \( M^5 = aI + bM \), where \( a, b \in \mathbb{R}, \) then