List of top Questions asked in GATE MA- 2026

For a transportation problem, let \(c_{ij}\) denote the unit cost of the cell \((i, j)\). The known unit costs are shown below (a dash marks a cell whose cost is not given).

	j = 1	j = 2	j = 3
i = 1	10	12	19
i = 2	11	13	-
i = 3	-	-	-

Let \(\alpha_i\) and \(\beta_j\), \(i, j = 1, 2, 3\), be the simplex multipliers associated with a basis corresponding to this unit cost table, so that \(\alpha_1 = x\), \(\alpha_2 = x + 1\), \(\beta_1 = y\) and \(\beta_2 = y + 2\). The relative cost coefficient \(d_{ij}\) is the difference between the current solution and the new improved solution.
If \(x = 4\), \(c_{13} = 19\) and \(\beta_3 = y + 5\), then which one of the following is TRUE?

Let \(S = \bigcup_{n=1}^{\infty} \left\{ (x, y) \in \mathbb{R}^2 : (x - n)^2 + y^2 = \dfrac{1}{n^2} \right\}\) with the usual topology. Which of the following statements is/are TRUE?

Circles \(C_1\), \(C_2\), and \(C_3\), with centers \(O_1\), \(O_2\), and \(O_3\), and radii \(r_1\), \(r_2\), and \(r_3\), respectively, touch each other as shown in the following figure.

Given \(r_1 = 2\) cm, \(r_2 = 1\) cm and the angle \(\angle O_1 O_3 O_2\) is \(90^{\circ}\), \(r_3 =\) _____ cm.

The number of zeros of the complex polynomial \( z^6 + 5z^3 + 4z + 11 \) in the annulus \( \{z \in \mathbb{C} : 1 < |z| < 3\} \) is equal to ______. (Answer in integer)

Let \( X = \{5, 6, 7, 8, 9, 10\} \) be equipped with the topology
\[ \tau = \{ \phi, X, \{5,6,7\}, \{8,9,10\} \} \]
Then the number of subsets of \( X \) which are neither open nor closed is ______.

If Jacobi method is used to solve the following system of linear equations
\[ \begin{pmatrix} 1 & 2 & 1 \\ 0 & 2 & 2 \\ 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix} \]
with the initial guess \( x^{(0)} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \) and \( x^{(i)} = \begin{pmatrix} x_1^{(i)} \\ x_2^{(i)} \\ x_3^{(i)} \end{pmatrix} \), \( i = 1,2,3,\ldots \), denotes the \( i^{th} \) iterate, then the value of \( \left| x_1^{(2)} + x_2^{(2)} + x_3^{(2)} \right| \) is equal to ______.

Let \( D = \{(x, y) \in \mathbb{R}^2 : 0 \leq x \leq 2,\ 0 \leq y \leq 2\} \) and let \( f(t) \) denote the smallest integer greater than or equal to \( t \). Then the value of the integral
\[ \iint_D f(x+y)\, dx\, dy \]
is ______.

Let \( \Omega = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 < 1\} \) be the open unit disc and \( \partial\Omega \) be its boundary. If \( u(x, y) \) is the solution of the following Dirichlet problem
\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \quad \text{in } \Omega \]
\[ u(x, y) = x^2 - y^2 \quad \text{on } \partial\Omega, \]
then the value of \( 4\left(u\left(\frac{1}{2}, 0\right) - u\left(0, \frac{1}{2}\right)\right) \) is ______.

Let \( \alpha, \beta \in \mathbb{R} \). If \( (4, 0, 2, \beta) \) is an optimal solution of the Linear Programming Problem:
\[ \text{minimize} \quad x_1 + 3x_2 + 2x_3 - \alpha x_4 \]
subject to
\[ 4x_1 + x_2 + x_3 = 18 \]
\[ -3x_1 + 2x_3 + x_4 = 2 \]
\[ x_1, x_2, x_3, x_4 \geq 0, \]
then the maximum value of \( 22(\alpha + \beta) \) is equal to ______.

Let \( L^2[0, \pi] \) denote the space of all real valued Lebesgue square integrable functions on \( [0, \pi] \). Let \( T: L^2[0, \pi] \to L^2[0, \pi] \) be defined as follows: \[ T(f(x)) = \sin x \int_0^{\pi} f(t)\cos t \, dt + \cos x \int_0^{\pi} f(t)\sin t \, dt \]

Then the value of \( \dfrac{4}{\pi}\|T\| \) is equal to ______. (Answer in integer)

Let \( P_3(\mathbb{R}) \) be the vector space of all polynomials of degree at most three with real coefficients under usual polynomial addition and scalar multiplication. Let \( T: P_3(\mathbb{R}) \to \mathbb{R}^2 \) be the linear transformation defined as \[ T(p) = \left(p(1), p'(1)\right) \] for all \( p \in P_3(\mathbb{R}) \), where \( p' \) is the derivative of \( p \).

Then the nullity of \( T \) is equal to ______. (Answer in integer)

Consider the problem of maximizing \[ z = \begin{pmatrix} x_1 & x_2 & x_3 \end{pmatrix} \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 1 & 2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \] subject to \[ \begin{pmatrix} x_1 & x_2 & x_3 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = 1, \] where \( \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \in \mathbb{R}^3 \).

Then the maximum value of \( z \) is ______. (Answer in integer)

Let \( \mathbb{Q}[x] \) be the ring of all polynomials with coefficients in \( \mathbb{Q} \) under the usual polynomial addition and multiplication. Let \( T: \mathbb{Q}[x] \to \mathbb{Q}[x] \) be defined by \[ T(p(x)) = p(x^2), \quad \text{for all } p(x) \in \mathbb{Q}[x]. \] Which of the following statements is/are TRUE?

Let \[ J = \begin{pmatrix} 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 0 & 3 \end{pmatrix}. \] Then the geometric multiplicity of the eigenvalue \(2\) of \(J\) is equal to ________. (Answer in integer)

Let \(X\) be any normed linear space and \(X'\) be the dual space of \(X\). Which of the following statements is/are TRUE?

Let \( (\mathbb{Z}_n, +) \) be the group of integers modulo \(n\). Let \( G = \mathbb{Z}_{30} \oplus \mathbb{Z}_{12} \) be the external direct product of \(\mathbb{Z}_{30}\) and \(\mathbb{Z}_{12}\). Then which of the following statements is/are TRUE?

Let \( \alpha = \iint_S \vec{F} \cdot \hat{n} \, dS \), where \( \vec{F} = (2x + 3z)\hat{i} + (xz - y)\hat{j} + (y^2 + 2z)\hat{k} \) and \( S \) is the sphere with centre at \( (3, -1, 2) \) and radius \( 9 \). Here \( \hat{n} \) is the unit normal drawn outward and \( \hat{i}, \hat{j}, \hat{k} \) are unit vectors.

Then the value of \( \dfrac{1}{36\pi}\alpha \) is equal to ______. (Answer in integer)

Let \( \alpha \) and \( \beta \) be the roots of the indicial equation obtained in the method of finding the Frobenius series solution to the differential equation: \[ 2x^2 \frac{d^2y}{dx^2} - x\frac{dy}{dx} + (1 - x^2) y = 0. \] Which of the following statements is/are TRUE?

Let \(P\) and \(Q\) be \(3 \times 3\) nonzero real matrices. Assume that there exists a \(3 \times 3\) real nonsingular matrix \(S\) such that \(S^{-1}PS\) and \(S^{-1}QS\) are both upper triangular. Which of the following statements is/are TRUE?

Let \(X\) and \(Y\) be topological spaces and \(f: X \to Y\) be a continuous and bijective mapping. Which one of the following statements is TRUE?

Let \(f(z) = \dfrac{z}{1-z}\) and \(g(z) = \dfrac{1+z}{1-z}\) be two Mobius transformations defined on the unit disc \(D = \{z \in \mathbb{C} : |z| < 1\}\). Consider the following statements:
\(S_1\): \(f(D) \subseteq g(D)\)
\(S_2\): \(g(D) \subseteq f(D)\)
Which of the following statements is/are CORRECT?

Let \(u(x,t)\) satisfy the wave equation
\[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \qquad -\infty < x < \infty, \; t > 0, \; c > 0 \] with
\[ u(x,0) = \begin{cases} 1 & \text{if } |x| < 1 \\ 0 & \text{otherwise} \end{cases} \] and
\[ \frac{\partial u}{\partial t}(x,0) = 0. \] By using D'Alembert's formula, the maximum value of \(u(0,t)\) for \(t > 0\) is

Let \(E_1\) and \(E_2\) be subsets of a normed linear space \(X\), and
\[ E_1 + E_2 = \{x+y : x \in E_1,\ y \in E_2\} \] \[ E_1 \times E_2 = \{(x,y) : x \in E_1,\ y \in E_2\}. \] Which one of the following is NOT TRUE?

Consider the Laplace equation
\[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0, \qquad 0 < x < 1,\; 0 < y < 1, \] with the boundary conditions
\[ T(x,0) = x, \qquad T(0,y) = y \] \[ T(x,1) = 1+x, \qquad T(1,y) = 1+y. \] Then the value of \(T\left(\dfrac{1}{2}, \dfrac{1}{3}\right)\) is equal to

Let \(\ell^{\infty} = \{x = (x_n)_{n \geq 1} \mid x_n \in \mathbb{R},\ \sup\{|x_n| : n = 1, 2, \ldots\} < \infty\}\) with the supremum norm. Let \(T : \ell^{\infty} \to \ell^{\infty}\) be given by \(T(x_1, x_2, x_3, \ldots) = \left(x_1, \dfrac{x_2}{2}, \dfrac{x_3}{3}, \ldots \right)\). Which one of the following is TRUE?