List of top Questions asked in IIT JAM MA

Let \( S^1 = \{z \in \mathbb{C} : |z| = 1\} \) be the circle group under multiplication and \( i = \sqrt{-1}. \) Then the set \( \{\theta \in \mathbb{R} : (e^{i2\pi\theta}) \text{ is infinite}\} \) is

Let \( D = \{(x, y) \in \mathbb{R}^2 : |x| + |y| \le 1\} \) and \( f : D \to \mathbb{R} \) be a non-constant continuous function. Which of the following is TRUE?

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} \) and \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k}, \, x, y, z \in \mathbb{R}. \) Which of the following is FALSE?

Let \( S \) be the part of the cone \( z^2 = x^2 + y^2 \) between the planes \( z = 0 \) and \( z = 1. \) Then the value of the surface integral \( \iint_S (x^2 + y^2) \, dS \) is

Let \[ M = \begin{bmatrix} 9 & 2 & 7 & 1 \\ 0 & 7 & 2 & 1 \\ 0 & 0 & 11 & 6 \\ 0 & 0 & -5 & 0 \end{bmatrix}. \] Then, the value of \( \det((8I - M)^3) \) is ................

Suppose that \( G \) is a group of order 57 which is not cyclic. If \( G \) contains a unique subgroup \( H \) of order 19, then for any \( g \notin H \), the order of \( g \) is ................

Consider the real vector space \( P_{2020} = \left\{ \sum_{i=0}^n a_i x^i : a_i \in \mathbb{R}, \, 0 \le n \le 2020 \right\}. \) Let \( W \) be the subspace given by \[ W = \left\{ \sum_{i=0}^n a_i x^i \in P_{2020} : a_i = 0 \text{ for all odd } i \right\}. \] Then the dimension of \( W \) is ................

Let \( \phi : S_3 \to S_1 \) be a non-trivial non-injective group homomorphism. Then the number of elements in the kernel of \( \phi \) is .............

The sum of the series \[ \frac{1}{2(2^2 - 1)} + \frac{1}{3(3^2 - 1)} + \frac{1}{4(4^2 - 1)} + \cdots \] is ...........

Let \( x_n = n^{1/n} \) and \( y_n = e^{1 - x_n}, \, n \in \mathbb{N}. \) Then the value of \( \lim_{n \to \infty} y_n \) is ..............

Let \( \vec{F} = x\hat{i} + y\hat{j} + z\hat{k} \) and \( S \) be the sphere given by \( (x - 2)^2 + (y - 2)^2 + (z - 2)^2 = 4. \) If \( \hat{n} \) is the unit outward normal to \( S \), then \( \dfrac{1}{\pi} \iint_S \vec{F} \cdot \hat{n} \, dS \) is .............

Consider the differential equation \[ \frac{dy}{dx} + 10y = f(x), x > 0, \] where \( f(x) \) is a continuous function such that \( \lim_{x \to \infty} f(x) = 1. \) Then the value of \( \lim_{x \to \infty} y(x) \) is .................

Let \( V \) be a non-zero vector space over a field \( F \). Let \( S \subset V \) be a non-empty set. Consider the following properties of \( S \):
(I) For any vector space \( W \) over \( F \), any map \( f : S \to W \) extends to a linear map from \( V \) to \( W \).
(II) For any vector space \( W \) over \( F \) and any two linear maps \( f, g : V \to W \) satisfying \( f(s) = g(s) \) for all \( s \in S \), we have \( f(v) = g(v) \) for all \( v \in V \).
(III) \( S \) is linearly independent.
(IV) The span of \( S \) is \( V \).
Which of the following statement(s) is/are true?

Which of the following is FALSE?

If \( y(x) = 2e^{2x} + e^{\beta x}, \beta \neq 2 \), is a solution of the differential equation \[ \frac{d^2 y}{dx^2} + \frac{dy}{dx} - 6y = 0, \] satisfying \( \frac{dy}{dx} (0) = 5 \), then \( y(0) \) is equal to

Let \( M \) and \( N \) be any two \( 4 \times 4 \) matrices with integer entries satisfying

\[ MN = \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} = 2 \]

Then the maximum value of \( \det(M) + \det(N) \) is ...............

Let \( M \) be a \( 3 \times 3 \) matrix with real entries such that

\[ M^2 = M + 2I, \quad \text{where} \quad I \text{ denotes the } 3 \times 3 \text{ identity matrix.} \]

If \( \alpha, \beta, \gamma \) are eigenvalues of \( M \) such that \( \alpha\beta\gamma = -4, \) then \( \alpha + \beta + \gamma \) is equal to .............

Let \( y(x) = x v(x) \) be a solution of the differential equation \[ x^2 \frac{d^2y}{dx^2} - 3x \frac{dy}{dx} + 3y = 0. \] If \( v(0) = 0 \) and \( v(1) = 1, \) then \( v(-2) \) is equal to .................

If \( y(x) \) is the solution of the initial value problem \[ \frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + 4y = 0, \quad y(0) = 2, \quad \frac{dy}{dx}(0) = 0, \] then \( y(\ln 2) \) is (round off to 2 decimal places) equal to ...............

If

\[ \begin{pmatrix} z \\ y \end{pmatrix} \]

is an eigenvector corresponding to a real eigenvalue of the matrix

\[ \begin{pmatrix} 0 & 0 & 2 \\ 1 & 0 & -4 \\ 0 & 1 & 3 \end{pmatrix} \]

then \( z - y \) is equal to .................

Let

\[ f(x, y) = \begin{cases} \frac{x^3 + y^3}{x^2 - y^2}, & x^2 - y^2 \neq 0 \\ 0, & x^2 - y^2 = 0 \end{cases} \]

Then the directional derivative of \( f \) at \( (0, 0) \) in the direction of \( \frac{4}{5} \hat{i} + \frac{3}{5} \hat{j} \) is ............

The coefficient of \( (x - \frac{\pi}{2}) \) in the Taylor series expansion of the function

\[ f(x) = \begin{cases} \frac{4(1 - \sin(x))}{2x - \pi}, & x \neq \frac{\pi}{2} \\ 0, & x = \frac{\pi}{2} \end{cases} \]

about \( x = \frac{\pi}{2} \) is ............

Let \( \vec{F}(x, y) = -y \hat{i} + x \hat{j} \) and let \( C \) be the ellipse

\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]

with counterclockwise orientation. Then the value of \( \oint_C \vec{F} \cdot d\vec{r} \) (rounded to 2 decimal places) is .............

Let

\[ f(x, y) = \begin{cases} \frac{|x|}{|x| + |y|} \sqrt{x^4 + y^2}, & (x, y) \neq (0, 0), \\ 0, & (x, y) = (0, 0). \end{cases} \]

Then at \( (0, 0) \),

Let \( G \) be a noncyclic group of order 4. Consider the statements I and II:
I. There is NO injective (one-one) homomorphism from \( G \) to \( \mathbb{Z}_4 \)
II. There is NO surjective (onto) homomorphism from \( \mathbb{Z}_4 \) to \( G \)