List of top Questions asked in GATE MA- 2021

If \( u(x, y) \) is the solution of the Cauchy problem \[ x \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 1, u(x, 0) = -x^2, x > 0, \] then \( u(2, 1) \) is equal to

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,