List of top Questions asked in GATE MA

Consider the following topologies on the set \( \mathbb{R} \) of all real numbers:
\( T_1 = \{ U \subset \mathbb{R} : 0 \notin U \text{ or } U = \mathbb{R} \} \),
\( T_2 = \{ U \subset \mathbb{R} : 0 \in U \text{ or } U = \emptyset \} \),
\( T_3 = T_1 \cap T_2 \).
Then the closure of the set \( \{1\} \) in \( (\mathbb{R}, T_3) \) is

Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be differentiable. Let \( D_u f(0,0) \) and \( D_v f(0,0) \) be the directional derivatives of \( f \) at \( (0,0) \) in the directions of the unit vectors \( u = \left( \frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}} \right) \) and \( v = \left( \frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}} \right) \), respectively. If \( D_u f(0,0) = \sqrt{5} \) and \( D_v f(0,0) = \sqrt{5} \), then \[ \frac{\partial f}{\partial x} (0,0) + \frac{\partial f}{\partial y} (0,0) = \(\underline{\hspace{1cm}}\). \]

Let \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) be a linear transformation such that \[ T \left( \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \right) = \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}, T^2 \left( \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \right) = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, T^2 \left( \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} \right) = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}. \] Then the rank of \( T \) is \(\underline{\hspace{2cm}}\) .

Let \( y(x) \) be the solution of the following initial value problem \[ x^2 \frac{d^2y}{dx^2} - 4x \frac{dy}{dx} + 6y = 0, x > 0, \] \[ y(2) = 0, \frac{dy}{dx}(2) = 4. \] Then \( y(4) = \(\underline{\hspace{1cm}}\) \).

Consider the fixed-point iteration \[ x_{n+1} = \varphi(x_n), n \geq 0, \] with \[ \varphi(x) = 3 + (x - 3)^3, x \in (2.5, 3.5), \] and the initial approximation \( x_0 = 3.25 \). Then, the order of convergence of the fixed-point iteration method is

Let \( \{ e_n : n = 1, 2, 3, \dots \} \) be an orthonormal basis of a complex Hilbert space \( H \). Consider the following statements: P: There exists a bounded linear functional \( f: H \to \mathbb{C} \) such that \( f(e_n) = \frac{1}{n} \) for \( n = 1, 2, 3, \dots \)
Q: There exists a bounded linear functional \( g: H \to \mathbb{C} \) such that \( g(e_n) = \frac{1}{\sqrt{n}} \) for \( n = 1, 2, 3, \dots \)

Let \( f: \mathbb{R}^3 \to \mathbb{R} \) be a twice continuously differentiable scalar field such that \( \text{div}(\nabla f) = 6 \). Let \( S \) be the surface \( x^2 + y^2 + z^2 = 1 \) and \( \hat{n} \) be the unit outward normal to \( S \). Then the value of \[ \iint_S (\nabla f \cdot \hat{n}) \, dS \] is

Consider the following statements:
P: Every compact metrizable topological space is separable.
Q: Every Hausdorff topology on a finite set is metrizable.
Then,

Let \( T(z) = \frac{az + b}{cz + d}, ad - bc \neq 0 \), be the Möbius transformation which maps the points \( z_1 = 0, z_2 = -i, z_3 = \infty \) in the z-plane onto the points \( w_1 = 10, w_2 = 5 - 5i, w_3 = 5 + 5i \) in the w-plane, respectively. Then the image of the set \( S = \{ z \in \mathbb{C} : \text{Re}(z) < 0 \ \) under the map \( w = T(z) \) is}

Let \( R \) be the row reduced echelon form of a \( 4 \times 4 \) real matrix \( A \) and let the third column of \( R \) be \[ \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}. \] Consider the following statements: \[ P: \text{If} \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix} \text{ is a solution of } A x = 0, \text{ then } \gamma = 0. \] \[ Q: \text{For all } b \in \mathbb{R}^4, \text{rank}[A | b] = \text{rank}[R | b]. \] Then: \\

The eigenvalues of the boundary value problem \[ \frac{d^2y}{dx^2} + \lambda y = 0, x \in (0, \pi), \lambda > 0, \] \[ y(0) = 0, y(\pi) - \frac{dy}{dx}(\pi) = 0 \] are given by:

The family of surfaces given by \[ u = xy + f(x^2 - y^2), \text{where} \, f : \mathbb{R} \to \mathbb{R} \, \text{is a differentiable function, satisfies:} \]

The function \( u(x, t) \) satisfies the initial value problem \[ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2}, \, x \in \mathbb{R}, \, t > 0, \] \[ u(x, 0) = 0, \, \frac{\partial u}{\partial t} (x, 0) = 4xe^{-x^2}. \] Then \( u(5, 5) \) is:

Let \( f(z) = u(x, y) + i v(x, y) \) for \( z = x + i y \in \mathbb{C} \), where \( x \) and \( y \) are real numbers, be a non-constant analytic function on the complex plane \( \mathbb{C} \). Let \( u_x, v_x \) and \( u_y, v_y \) denote the first order partial derivatives of \( u(x, y) = \text{Re(f(z)) \) and \( v(x, y) = \text{Im}(f(z)) \) with respect to real variables \( x \) and \( y \), respectively. Consider the following two functions defined on \( \mathbb{C} \):}
\[ g_1(z) = u_x(x, y) - i u_y(x, y) \text{for} z = x + i y \in \mathbb{C}, g_2(z) = v_x(x, y) + i v_y(x, y) \text{for} z = x + i y \in \mathbb{C}. \] Then,

Details of prices of two items P and Q are presented in the above table. The ratio of cost of item P to cost of item Q is 3:4. Discount is calculated as the difference between the marked price and the selling price. The profit percentage is calculated as the ratio of the difference between selling price and cost, to the cost.
The formula for Profit Percentage is:
\[ \text{Profit \%} = \frac{\text{Selling Price} - \text{Cost}}{\text{Cost}} \times 100 \] The discount on item Q, as a percentage of its marked price, is:

We have 2 rectangular sheets of paper, M and N, of dimensions 6 cm × 1 cm each. Sheet M is rolled to form an open cylinder by bringing the short edges of the sheet together. Sheet N is cut into equal square patches and assembled to form the largest possible closed cube. Assuming the ends of the cylinder are closed, the ratio of the volume of the cylinder to that of the cube is:

A circular sheet of paper is folded along the lines in the directions shown. The paper, after being punched in the final folded state as shown and unfolded in the reverse order of folding, will look like \(\underline{\hspace{2cm}}\)

A polygon is convex if, for every pair of points inside the polygon, the line segment joining them lies completely inside or on the polygon. Which one of the following is NOT a convex polygon?

The ratio of boys to girls in a class is 7 to 3. Among the options below, an acceptable value for the total number of students in the class is:

Some people suggest anti-obesity measures (AOM) such as displaying calorie information in restaurant menus. Such measures sidestep addressing the core problems that cause obesity: poverty and income inequality. Which one of the following statements summarizes the passage?

\(\underline{\hspace{1cm}}\) is to surgery as writer is to \(\underline{\hspace{1cm}}\)
Which one of the following options maintains a similar logical relation in the above sentence?

There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag. The probability that at least two chocolates are identical is:

Given below are two statements 1 and 2, and two conclusions I ans II.
Statement 1: All bacteria are microorganisms.
Statement 2: All pathogens are microorganisms.
Conclusion I: Some pathogens are bacteria.
Conclusion II: All pathogens are not bacteria.
Based on the given statements and conclusions, which option is logically correct?

Consider the following sentences:
(i) Everybody in the class is prepared for the exam.
(ii) Babu invited Danish to his home because he enjoys playing chess.
Which of the following is the CORRECT observation about the above two sentences?

f(x,y)=\(\left\{\begin{matrix} 1-cos(\frac{x^2}{y^2}) \\ \sqrt{x^2+y^2} \end{matrix}\right.\) ;if y≠0 & n∈R, then