List of top Questions asked in GATE MA

If \( u(x,t) = A e^{-t} \sin x \) solves the following initial boundary value problem:
\[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, 0 < x < \pi, t > 0, \] \[ u(0,t) = u(\pi,t) = 0, t > 0, \] \[ u(x,0) = \begin{cases} 60, & 0 < x \leq \frac{\pi}{2}, \\ 40, & \frac{\pi}{2} < x < \pi, \end{cases} \] then \( \pi A = \underline{\hspace{1cm}} \).

Let \( V = \{ p : p(x) = a_0 + a_1 x + a_2 x^2, a_0, a_1, a_2 \in \mathbb{R} \} \) be the vector space of all polynomials of degree at most 2 over the real field \( \mathbb{R} \). Let \( T: V \to V \) be the linear operator given by
\[ T(p) = (p(0) - p(1)) + (p(0) + p(1)) x + p(0) x^2. \] Then the sum of the eigenvalues of \( T \) is \(\underline{\hspace{2cm}}\) .

The quadrature formula \[ \int_0^2 x f(x) \, dx \approx \alpha f(0) + \beta f(1) + \gamma f(2) \] is exact for all polynomials of degree \( \leq 2 \). Then \( 2 \beta - \gamma = \underline{\hspace{1cm}} \).

Let \( \tilde{x} = \begin{bmatrix} 11/3 \\ 2/3 \\ 0 \end{bmatrix} \) be an optimal solution of the following Linear Programming Problem P: 
Maximize \( 4x_1 + x_2 - 3x_3 \) 
subject to \[ 2x_1 + 4x_2 + ax_3 \leq 10, \] \[ x_1 - x_2 + bx_3 \leq 3, \] \[ 2x_1 + 3x_2 + 5x_3 \leq 11, \] \[ x_1 \geq 0, x_2 \geq 0, x_3 \geq 0, \text{where} a, b \text{ are real numbers.} \] If \( \tilde{y} = \begin{bmatrix} p \\ q \\ r \end{bmatrix} \) is an optimal solution of the dual of P, then \( p + q + r \)= \(\underline{\hspace{1cm}}\) (round off to 2 decimal places).

For each \( x \in (0, 1) \), consider the decimal representation \( x = d_1 d_2 d_3 \cdots d_n \cdots \). Define \( f: [0, 1] \to \mathbb{R} \) by \( f(x) = 0 \) if \( x \) is rational, and \( f(x) = 18n \) if \( x \) is irrational, where \( n \) is the number of zeroes immediately after the decimal point up to the first nonzero digit in the decimal representation of \( x \). Then the Lebesgue integral \[ \int_0^1 f(x) \, dx = \(\underline{\hspace{1cm}}\). \]
Let \( \langle \cdot, \cdot \rangle : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} \) be an inner product on the vector space \( \mathbb{R}^n \) over \( \mathbb{R} \). Consider the following statements: P: \( |\langle u, v \rangle| \leq \frac{1}{2} \left( \langle u, u \rangle + \langle v, v \rangle \right) \) for all \( u, v \in \mathbb{R}^n \). Q: If \( \langle u, v \rangle = \langle 2u, -v \rangle \) for all \( v \in \mathbb{R}^n \), then \( u = 0 \). Then, which of the following is correct?
Let \( G \) be a group of order \( 5^4 \) with center having \( 5^2 \) elements. Then the number of conjugacy classes in \( G \) is \(\underline{\hspace{1cm}}\) .
Let \( F \) be a finite field and \( F^{\times} \) be the group of all nonzero elements of \( F \) under multiplication. If \( F^{\times} \) has a subgroup of order 17, then the smallest possible order of the field \( F \) is \(\underline{\hspace{1cm}}\) .
Let \( R = \{ z = x + iy \in \mathbb{C} : 0 < x < 1 \text{ and } - 11 \pi < y < 11 \pi \} \) and \( r \) be the positively oriented boundary of \( R \). Then the value of the integral \[ \frac{1}{2 \pi i} \int_r \frac{e^z}{e^z - 2} \, dz \] \text{is \(\underline{\hspace{1cm}}\) .}
The number of zeros (counting multiplicity) of \( P(z) = 3z^5 + 2iz^2 + 7iz + 1 \) in the annular region \( \{ z \in \mathbb{C} : 1 < |z| < 7 \} \) is \(\underline{\hspace{2cm}}\) .
If \( y = \sum_{k=0}^{\infty} a_k x^k \), \( (a_0 \neq 0) \) is the power series solution of the differential equation
\[ \frac{d^2y}{dx^2} - 24x^2y = 0, \text{ then } \frac{a_4}{a_0} = \text{\(\underline{\hspace{2cm}}\).} \]
Let \( V \) be the solid region in \( \mathbb{R}^3 \) bounded by the paraboloid \( y = x^2 + z^2 \) and the plane \( y = 4 \). Then the value of \[ \iiint_V 15 \sqrt{x^2 + z^2} \, dV \] is
Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be given by \[ f(x, y) = 4xy - 2x^2 - y^4. \] Then \( f \) has
The equation \[ xy - z \log y + e^{xz} = 1 \] can be solved in a neighborhood of the point \( (0, 1, 1) \) as \( y = f(x, z) \) for some continuously differentiable function \( f \). Then
Consider the following topologies on the set \( \mathbb{R} \) of all real numbers. \[ T_1 \text{ is the upper limit topology having all sets } (a, b) \text{ as basis.} \] \[ T_2 = \{ U \subset \mathbb{R}: U \text{ is finite} \} \cup \{\emptyset\}. \] \[ T_3 \text{ is the standard topology having all sets } (a, b) \text{ as basis.} \] Then:
Let \( \mathbb{R} \) denote the set of all real numbers. Consider the following topological spaces. \[ X_1 = (\mathbb{R}, T_1), \text{where} \, T_1 \text{ is the upper limit topology having all sets } (a, b) \text{ as basis.} \] \[ X_2 = (\mathbb{R}, T_2), \text{where} \, T_2 = \{ U \subset \mathbb{R} : \mathbb{R} \setminus U \text{ is finite} \} \cup \{\emptyset\}. \] Then:
Let \( y(t) \) be the solution of the initial value problem \[ \frac{d^2 y}{dt^2} + a \frac{dy}{dt} + b y = f(t), a > 0, b > 0, a \neq b, a^2 - 4b = 0, \] with initial conditions \( y(0) = 0 \), \( \frac{dy}{dt}(0) = 0 \), obtained by the method of Laplace transform. Then
The critical point of the differential equation \[ \frac{d^2y}{dt^2} + 2 \alpha \frac{dy}{dt} + \beta^2 y = 0, \alpha > 0, \, \beta > 0, \] is a
The initial value problem \[ \frac{dy}{dt} = f(t, y), t > 0, y(0) = 1, \] where \( f(t, y) = -10 y \), is solved by the following Euler method: \[ y_{n+1} = y_n + h f(t_n, y_n), n \geq 0, \text{with step-size} \, h. \] Then \( y_n \to 0 \) as \( n \to \infty \), provided
Let \( L^2[-1, 1] \) be the Hilbert space of real-valued square integrable functions on [-1, 1] equipped with the norm \[ \|f\| = \left( \int_{-1}^{1} |f(x)|^2 \, dx \right)^{1/2}. \] Consider the subspace \[ M = \{ f \in L^2[-1, 1] : \int_{-1}^{1} f(x) \, dx = 0 \}. \] For \( f(x) = x^2 \), define \[ d = \inf \{ \|f - g\| : g \in M \}. \] Then
Let \( C[0, 1] \) be the Banach space of real valued continuous functions on [0, 1] equipped with the supremum norm. Define \( T: C[0, 1] \to C[0, 1] \) by \[ (Tf)(x) = \int_0^x t f(t) \, dt. \] Let \( R(T) \) denote the range space of \( T \). Consider the following statements: \[\begin{array}{rl} \bullet & \text{P: \( T \) is a bounded linear operator.} \\ \bullet & \text{Q: \( T^{-1}: R(T) \to C[0, 1] \) exists and is bounded.} \\ \end{array}\]
Let \( f(x) = x^4 + 2x^3 - 11x^2 - 12x + 36 \) for \( x \in \mathbb{R}. \) The order of convergence of the Newton-Raphson method \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}, n \geq 0, \] with \( x_0 = 2.1 \), for finding the root \( \alpha = 2 \) of the equation \( f(x) = 0 \) is \(\underline{\hspace{1cm}}\) .
Let \( \mathbb{Z} \) denote the ring of integers. Consider the subring \[ R = \{ a + b\sqrt{-17} : a, b \in \mathbb{Z} \} \] of the field \( \mathbb{C} \) of complex numbers. Consider the following statements: P: \( 2 + \sqrt{-17} \) is an irreducible element.
Q: \( 2 + \sqrt{-17} \) is a prime element. Then:
Consider the second-order partial differential equation (PDE) \[ \frac{\partial^2 u}{\partial x^2} + 4 \frac{\partial^2 u}{\partial x \partial y} + (x^2 + 4y^2) \frac{\partial^2 u}{\partial y^2} = \sin(x + y) \] Consider the following statements: P: The PDE is parabolic on the ellipse \( \frac{x^2}{4} + y^2 = 1 \).
Q: The PDE is hyperbolic inside the ellipse \( \frac{x^2}{4} + y^2 = 1 \). Then:
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