List of top Questions asked in GATE MA

Consider the balanced transportation problem with three sources $ S_1, S_2, S_3 $, and four destinations $ D_1, D_2, D_3, D_4 $, for minimizing the total transportation cost whose cost matrix is as follows:

where $ \alpha, \lambda>0 $. If the associated cost to the starting basic feasible solution obtained by using the North-West corner rule is 290, then which of the following is/are correct?

Let $ p_A(x) $ denote the characteristic polynomial of a square matrix $ A $. Then, for which of the following invertible matrices $ M $, the polynomial $ p_M(x) - p_{M^{-1}}(x) $ is constant?

Let the functions $ f: \mathbb{R} \to \mathbb{R} $ and $ g: \mathbb{R}^2 \to \mathbb{R} $ be given by \[ f(x_1, x_2) = x_1^2 + x_2^2 - 2x_1x_2, \quad g(x_1, x_2) = 2x_1^2 + 2x_2^2 - x_1x_2. \] Consider the following statements:
S1: For every compact subset $ K $ of $ \mathbb{R} $, $ f^{-1}(K) $ is compact.
S2: For every compact subset $ K $ of $ \mathbb{R} $, $ g^{-1}(K) $ is compact. Then, which one of the following is correct?

Consider the function $ F: \mathbb{R}^2 \to \mathbb{R}^2 $ given by \[ F(x, y) = (x^3 - 3xy^2 - 3x, 3x^2y - y^3 - 3y). \] Then, for the function $ F $, the inverse function theorem is:

Let $ u(x, t) $ be the solution of the following initial-boundary value problem: \[ \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0, \quad x \in (0, \pi), \quad t>0, \] with the boundary conditions: \[ u(0, t) = u(\pi, t) = 0, \quad u(x, 0) = \sin 4x \cos 3x. \] Then, for each $ t>0 $, the value of $ u\left( \frac{\pi}{4}, t \right) $ is

The partial differential equation \[ (1 + x^2) \frac{\partial^2 u}{\partial x^2} + 2x(1 - y^2) \frac{\partial^2 u}{\partial x \partial y} + (1 - y^2) \frac{\partial^2 u}{\partial y^2} + x \frac{\partial u}{\partial x} + (1 - y^2) \frac{\partial u}{\partial y} = 0 \] is:

Let $ g(x, y) = f(x, y)e^{2x + 3y} $ be defined in $ \mathbb{R}^2 $, where $ f(x, y) $ is a continuously differentiable non-zero homogeneous function of degree 4. Then,

\[ x \frac{\partial g}{\partial x} + y \frac{\partial g}{\partial y} = 0 \text{ holds for} \]

Let $ X = \{ f \in C[0,1] : f(0) = 0 = f(1) \} $ with the norm $ \|f\|_\infty = \sup_{0 \leq t \leq 1} |f(t)| $, where $ C[0,1] $ is the space of all real-valued continuous functions on $ [0,1] $.

Let $ Y = C[0,1] $ with the norm $ \|f\|_2 = \left( \int_0^1 |f(t)|^2 \, dt \right)^{\frac{1}{2}} $. Let $ U_X $ and $ U_Y $ be the closed unit balls in $ X $ and $ Y $ centered at the origin, respectively. Consider $ T: X \to \mathbb{R} $ and $ S: Y \to \mathbb{R} $ given by

\[ T(f) = \int_0^1 f(t) \, dt \quad \text{and} \quad S(f) = \int_0^1 f(t) \, dt. \]

Consider the following statements:
S1: $ \sup |T(f)| $ is attained at a point of $ U_X $.
S2: $ \sup |S(f)| $ is attained at a point of $ U_Y $.

Then, which one of the following is correct?

Consider the system of ordinary differential equations \[ \frac{dX}{dt} = MX, \] where $ M $ is a $ 6 \times 6 $ skew-symmetric matrix with entries in $ \mathbb{R} $. Then, for this system, the origin is a stable critical point for

Consider the following subsets of the Euclidean space $ \mathbb{R}^4 $:
$ S = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \} $,
$ T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \} $,
$ U = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \} $.
Then, which one of the following is TRUE?

For the linear programming problem: \[ {Maximize} \quad Z = 2x_1 + 4x_2 + 4x_3 - 3x_4 \] subject to \[ \alpha x_1 + x_2 + x_3 = 4, \quad x_1 + \beta x_2 + x_4 = 8, \quad x_1, x_2, x_3, x_4 \geq 0, \] consider the following two statements:
S1: If $ \alpha = 2 $ and $ \beta = 1 $, then $ (x_1, x_2)^T $ forms an optimal basis.
S2: If $ \alpha = 1 $ and $ \beta = 4 $, then $ (x_3, x_2)^T $ forms an optimal basis. Then, which one of the following is correct?

To find a real root of the equation $ x^3 + 4x^2 - 10 = 0 $ in the interval $ \left( 1, \frac{3}{2} \right) $ using the fixed-point iteration scheme, consider the following two statements:
Statement 1 S1: The iteration scheme $ x_{k+1} = \sqrt{\frac{10}{4 + x_k}}, \, k = 0, 1, 2, \ldots $ converges for any initial guess $ x_0 \in \left( 1, \frac{3}{2} \right) $.
Statement 2 S2: The iteration scheme $ x_{k+1} = \frac{1}{2} \sqrt{10 - x_k^3}, \, k = 0, 1, 2, \ldots $ diverges for some initial guess $ x_0 \in \left( 1, \frac{3}{2} \right) $.

Given that the Laplace transforms of $ J_0(x), J_0'(x), $ and $ J_0''(x) $ exist, where $ J_0(x) $ is the Bessel function. Let $ Y = Y(s) $ be the Laplace transform of the Bessel function $ J_0(x) $. Then, which one of the following is TRUE?

Then, which one of the following is TRUE?

Let $ y = P_n(x) $ be the unique polynomial of degree $ n $ satisfying the Legendre differential equation \[ (1 - x^2)y'' - 2xy' + n(n + 1)y = 0 \quad {and} \quad y(1) = 1. \] Then, the value of $ P_{11}'(1) $ is equal to _________ (in integer).

Consider the following Linear Programming Problem $ P $: Minimize $ x_1 + 2x_2 $, subject to
$ 2x_1 + x_2 \leq 2 $,
$ x_1 + x_2 = 1 $,
$ x_1, x_2 \geq 0 $.
The optimal value of the problem $ P $ is equal to:

Let $D = \{(x, y) \in \mathbb{R}^2 : x > 0 \text{ and } y > 0\}$. If the following second-order linear partial differential equation
$y^2 \frac{\partial^2 u}{\partial x^2} - x^2 \frac{\partial^2 u}{\partial y^2} + y \frac{\partial u}{\partial y} = 0$ on $D$
is transformed to
$\left( \frac{\partial^2 u}{\partial \eta^2} - \frac{\partial^2 u}{\partial \xi^2} \right) + \left( \frac{\partial u}{\partial \eta} + \frac{\partial u}{\partial \xi} \right) \frac{1}{2\eta} + \left( \frac{\partial u}{\partial \eta} - \frac{\partial u}{\partial \xi} \right) \frac{1}{2\xi} = 0$ on $D$,
for some $a, b \in \mathbb{R}$, via the coordinate transform $\eta = \frac{x^2}{2}$ and $\xi = \frac{y^2}{2}$, then which one of the following is correct?

Consider the following limit: $ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{0}^{\infty} e^{-x / \epsilon} \left( \cos(3x) + x^2 + \sqrt{x + 4} \right) dx. $
Which one of the following is correct?

Let $ \{(a, b) : a, b \in {R, a<b \} }$ be a basis for a topology $ \tau $ on $ {R} $. Which of the following is/are correct?

Let $ T, S : {R}^4 \to {R}^4 $ be two non-zero, non-identity $ {R} $-linear transformations. Assume $ T^2 = T $. Which of the following is/are true?

Let $ p_1<p_2 $ be the two fixed points of the function $ g(x) = e^x - 2 $, where $ x \in {R} $. For $ x_0 \in {R} $, let the sequence $ (x_n)_{n \geq 1} $ be generated by the fixed-point iteration \[ x_n = g(x_{n-1}), \quad n \geq 1. \] Which one of the following is/are correct?

Which of the following is/are eigenvalue(s) of the Sturm–Liouville problem \[ y'' + \lambda y = 0, \quad 0 \leq x \leq \pi, \] with the boundary conditions \[ y(0) = y'(0), \quad y(\pi) = y'(\pi)? \]

The problem: Let $ f : \mathbb{R}^2 \to \mathbb{R} $ be a function such that \[ f(x, y) = \begin{cases} \left( 1 - \cos\left(\frac{x^2}{y^2}\right) \right) \sqrt{x^2 + y^2}, & \text{if } y \neq 0, x \in \mathbb{R}, \\ 0, & \text{otherwise.} \end{cases} \] Which of the following is/are correct?

For an integer $ n $, let $ f_n(x) = xe^{-nx }$, where $ x \in [0, 1] $. Let $ S := \{f_n : n \geq 1\} $. Consider the metric space $ (C([0, 1]), d) $, where \[ d(f, g) = \sup_{x \in [0, 1]} |f(x) - g(x)|, \quad f, g \in C([0, 1]). \] Which of the following statement(s) is/are true?}

Let $ T : {R}^4 \to {R}^4 $ be an $ {R} $-linear transformation such that 1 and 2 are the only eigenvalues of $ T $. Suppose the dimensions of ${Kernel}(T - I_4)$ and ${Range}(T - 2I_4)$ are 1 and 2, respectively. Which of the following is/are possible (upper triangular) Jordan canonical form(s) of $ T $?