List of top Questions asked in GATE MA

Let \( f(x, y) = (x^2 - y^2, 2xy) \), where \( x>0, y>0 \). Let \( g \) be the inverse of \( f \) in a neighborhood of \( f(2, 1) \). Then the determinant of the Jacobian matrix of \( g \) at \( f(2, 1) \) is equal to (round off to TWO decimal places):

Given a real subspace \( W \) of \( {R}^4 \), let \( W^\perp \) denote its orthogonal complement with respect to the standard inner product on \( {R}^4 \). Let \( W_1 = {Span}\{(1, 0, 0, -1)\ \) and \( W_2 = {Span}\{(2, 1, 0, -1)\} \). The dimension of \( W_1^\perp \cap W_2^\perp \) over \( {R} \) is equal to (answer in integer):}

Let \( p = (1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}) \in {R^4 \) and \( f : {R}^4 \to {R} \) be a differentiable function such that \( f(p) = 6 \) and \( f(Ax) = A^3 f(x) \), for every \( A \in (0, \infty) \) and \( x \in {R}^4 \). The value of \[ 12 \frac{\partial f}{\partial x_1}(p) + 6 \frac{\partial f}{\partial x_2}(p) + 4 \frac{\partial f}{\partial x_3}(p) + 3 \frac{\partial f}{\partial x_4}(p) \] is equal to (answer in integer):}

The number of group homomorphisms from \( {Z}/47 \) to \( S_4 \) is equal to (answer in integer):

The number of non-isomorphic finite groups with exactly 3 conjugacy classes is equal to (answer in integer):

Let \( {F}_3 \) be the field with exactly 3 elements. The number of elements in \( GL_2({F}_3) \) is equal to (answer in integer):

If \( y_1 \) and \( y_2 \) are two different solutions of the ordinary differential equation \[ y'' + \sin(e^x)y = \cos(e^x), \quad 0<x<1, \] then which one of the following is its general solution on \( [0, 1] \)?

Let \( u = u(x, t) \) be the solution of \[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \quad 0<x<1, \, t>0, \] with boundary conditions \( u(0, t) = u(1, t) = 0 \) and initial condition \( u(x, 0) = \sin(\pi x) \). Define \[ g(t) = \int_0^1 u^2(x, t) \, dx. \] Which one of the following is correct?

Let \( A \) be a \( 3 \times 3 \) real matrix and \( b \) be a \( 3 \times 1 \) real column vector. Consider the statements:
1. The Jacobi iteration method for the system \( (A + \epsilon I_3)x = b \) converges for any initial approximation and \( \epsilon>0 \).
2. The Gauss-Seidel iteration method for the system \( (A + \epsilon I_3)x = b \) converges for any initial approximation and \( \epsilon>0 \).
Which one of the following is correct?

Let \( A \in M_n({C}) \) be a normal matrix. Consider the following statements:
1. If all the eigenvalues of \( A \) are real, then \( A \) is Hermitian.
2. If all the eigenvalues of \( A \) have absolute value 1, then \( A \) is unitary.
Which one of the following is correct?

For the initial value problem \[ \frac{dy}{dx} = f(x, y), \quad y(x_0) = y_0, \] generate approximations \( y_n \) to \( y(x_n) \) using the recursion formula \[ y_n = y_{n-1} + a k_1 + b k_2, \] where \[ k_1 = h f(x_{n-1}, y_{n-1}), \quad k_2 = h f(x_{n-1} + \beta h, y_{n-1} + \beta k_1). \] Which one of the following choices of \( a, b, \alpha, \beta \) gives the Runge-Kutta method of order 2?

Given a prime number \( p \), let \( n_p(G) \) denote the number of \( p \)-Sylow subgroups of a finite group \( G \). Which one of the following is TRUE for every group \( G \) of order \( 2024 \)?

Let \( {R}[X^2, X^3] \) be the subring of \( {R}[X] \) generated by \( X^2 \) and \( X^3 \). Consider the following statements: 1. The ring \( {R}[X^2, X^3] \) is a unique factorization domain.
2. The ring \( {R}[X^2, X^3] \) is a principal ideal domain.
Which one of the following is correct?

Let \( A \in M_2({C}) \) be given by \[ A = \begin{bmatrix} 0 & 2
2 & 0 \end{bmatrix}. \] Let \( T: M_2({C}) \to M_2({C}) \) be the linear transformation given by \( T(B) = AB \). The characteristic polynomial of \( T \) is:

Consider the following statements:
1. Every compact Hausdorff space is normal.
2. Every metric space is normal.
Which one of the following is correct?

Consider the topology on \( {Z} \) with basis \( S(a,b) = \{an + b : n \in {Z\} \), where \( a, b \in {Z} \) and \( a \neq 0 \). Consider the following statements:} 1. \( S(a, b) \) is both open and closed for each \( a, b \in {Z} \) with \( a \neq 0 \).
2. The only connected set containing \( x \in {Z} \) is \( \{x\} \).
Which one of the following is correct?

Let \( X \) be a topological space and \( A \subseteq X \). Given a subset \( S \) of \( X \), let \( {int}(S), \partial S, \) and \( \overline{S} \) denote the interior, boundary, and closure, respectively, of the set \( S \). Which one of the following is NOT necessarily true?

Let \( C \) be the ellipse \( \{ z \in \mathbb{C} : |z - 2| + |z + 2| = 8 \} \) traversed counter-clockwise. The value of the contour integral \[ \int_C \frac{z^2 \, dz}{z^2 - 2z + 2} \] is equal to:

Consider the following condition on a function \( f : {C} \to {C} \): \[ |f(z)| = 1 \quad {for all } z \in {C} { such that } \operatorname{Im}(z) = 0. \] Which one of the following is correct?

In the given text, the blanks are numbered (i)-(iv). Select the best match for all the blanks. Yoko Roi stands \_\_\_ (i) \_\_\_ as an author for standing \_\_\_ (ii) \_\_\_ as an honorary fellow, after she stood \_\_\_ (iii) \_\_\_ her writings that stand \_\_\_ (iv) \_\_\_ the freedom of speech.

Seven identical cylindrical chalk-sticks are fitted tightly in a cylindrical container. The figure below shows the arrangement of the chalk-sticks inside the cylinder. \includegraphics[width=0.4\linewidth]{q7 MA.PNG} The length of the container is equal to the length of the chalk-sticks. The ratio of the occupied space to the empty space of the container is:

Visualize a cube that is held with one of the four body diagonals aligned to the vertical axis. Rotate the cube about this axis such that its view remains unchanged. The magnitude of the minimum angle of rotation is:

Five cubes of identical size and another smaller cube are assembled as shown in Figure A. If viewed from direction \( X \), the planar image of the assembly appears as Figure B. \includegraphics[width=0.5\linewidth]{q9 MA.PNG} If viewed from direction \( Y \), the planar image of the assembly (Figure A) will appear as:

The plot below shows the relationship between the mortality risk of cardiovascular disease and the number of steps a person walks per day. Based on the data, which one of the following options is true? \includegraphics[width=0.6\linewidth]{q8 MA.PNG}

A student was supposed to multiply a positive real number \( p \) with another positive real number \( q \). Instead, the student divided \( p \) by \( q \). If the percentage error in the student’s answer is 80\%, the value of \( q \) is: