List of top Mathematics Questions asked in GATE Mathematics

Let \(u(x, t)\) be the solution of the non-homogeneous wave equation \[ \frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial t^2} = \sin x \sin(2t), \quad 0<x<\pi, \ t>0 \] \[ u(x, 0) = 0, \text{ and } \frac{\partial u}{\partial t}(x, 0) = 0, \quad \text{for } 0 \le x \le \pi, \] \[ u(0, t) = 0, \quad u(\pi, t) = 0, \quad \text{for } t \ge 0. \] Then the value of \( u \left( \frac{\pi}{2}, \frac{3\pi}{2} \right) \) is ................. (rounded off to 2 decimal places).

Let \(\sigma \in S_8\), where \(S_8\) is the permutation group on 8 elements. Suppose \(\sigma\) is the product of \(\sigma_1\) and \(\sigma_2\), where \(\sigma_1\) is a 4-cycle and \(\sigma_2\) is a 3-cycle in \(S_8\). If \(\sigma_1\) and \(\sigma_2\) are disjoint cycles, then the number of elements in \(S_8\) which are conjugate to \(\sigma\) is ..................

Let A be a \(3 \times 3\) real matrix with det(\(A + i I\)) = 0, where \(i = \sqrt{-1}\) and I is the \(3 \times 3\) identity matrix. If det(A) = 3, then the trace of \(A^2\) is ..................

Let \(\Omega\) be the disk \(x^2 + y^2<4\) in \(\mathbb{R}^2\) with boundary \(\partial\Omega\). If \(u(x, y)\) is the solution of the Dirichlet problem \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad (x, y) \in \Omega, \] \[ u(x, y) = 1 + 2x^2, \quad (x, y) \in \partial\Omega, \] then the value of \(u(0,1)\) is ..................

For every \(k \in \mathbb{N} \cup \{0\}\), let \(y_k(x)\) be a polynomial of degree k with \(y_k(1) = 5\). Further, let \(y_k(x)\) satisfy the Legendre equation \[ (1-x^2)y'' - 2xy' + k(k+1)y = 0. \] If \[ \frac{1}{2} \sum_{k=1}^{n} \int_{-1}^{1} (y_k(x) - y_{k-1}(x))^2 dx - \sum_{k=1}^{n} \int_{-1}^{1} (y_k(x))^2 dx = 24, \] for some positive integer n, then the value of n is ..................

Consider the transportation problem between five sources and four destinations as given in the cost table below. The supply and demand at each of the source and destination are also provided:
\begin{tabular}{|c|c|c|c|c|c|c|} \cline{3-7} \multicolumn{2}{c|}{} & \multicolumn{4}{c|}{DESTINATIONS} & Supply
\cline{3-7} \multicolumn{2}{c|}{} & P & Q & R & S &
\hline \multirow{5}{*}{\rotatebox[origin=c]{90}{SOURCES}} & 1 & 13 & 8 & 12 & 9 & 20
\cline{2-7} & 2 & 10 & 7 & 5 & 20 & 10
\cline{2-7} & 3 & 3 & 19 & 5 & 12 & 50
\cline{2-7} & 4 & 4 & 9 & 7 & 15 & 30
\cline{2-7} & 5 & 14 & 0 & 1 & 7 & 40
\hline \multicolumn{2}{|c|}{Demand} & 60 & 10 & 20 & 60 &
\hline \end{tabular}
Let \(C_N\) and \(C_L\) be the total cost of the initial basic feasible solution obtained from the North-West corner method and the Least-Cost method, respectively. Then \(C_N - C_L\) equals ..................

Let C be the curve of intersection of the cylinder \( x^2 + y^2 = 4 \) and the plane \( z - 2 = 0 \). Suppose C is oriented in the counterclockwise direction around the z-axis, when viewed from above. If \[ \int_C ( \sin x + e^x ) dx + 4x dy + e^z \cos^2 z dz = \alpha\pi, \] then the value of \( \alpha \) equals ..................

For \(h>0\), and \(\alpha, \beta, \gamma \in \mathbb{R}\), let \[ D_h f(a) = \frac{\alpha f(a-h) + \beta f(a) + \gamma f(a+2h)}{6h} \] be a three-point formula to approximate \(f'(a)\) for any differentiable function \(f: \mathbb{R} \to \mathbb{R}\) and \(a \in \mathbb{R}\).
If \(D_h f(a) = f'(a)\) for every polynomial f of degree less than or equal to 2 and for all \(a \in \mathbb{R}\), then

Let \( f \) be a twice continuously differentiable function on \( [a, b] \) such that \( f'(x)<0 \) and \( f''(x)<0 \) for all \( x \in (a, b) \). Let \( f(\zeta) = 0 \) for some \( \zeta \in (a, b) \). The Newton-Raphson method to compute \( \zeta \) is given by \[ x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}, \quad k = 0, 1, 2, ...... \] for an initial guess \( x_0 \). If \( x_k \in (\zeta, b) \) for some \( k \geq 0 \), then which of the following statements is/are correct?

Let \(\phi\) and \(\psi\) be two linearly independent solutions of the ordinary differential equation \[ y'' + (2 - \cos x) y = 0, \quad x \in \mathbb{R}. \] Let \(\alpha, \beta \in \mathbb{R}\) be such that \(\alpha<\beta\), \(\phi(\alpha) = \phi(\beta) = 0\) and \(\phi(x) \neq 0\) for all \(x \in (\alpha, \beta)\).
Consider the following statements:
P: \(\phi'(\alpha)\phi'(\beta)>0\).
Q: \(\psi(x) \neq 0\) for all \(x \in (\alpha, \beta)\).
Then

Let \((\mathbb{R}, \tau)\) be a topological space, where the topology \(\tau\) is defined as \[ \tau = \{U \subseteq \mathbb{R} : U = \emptyset \text{ or } 1 \in U\}. \] Which of the following statements is/are correct?

Consider the Cauchy problem \[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = u; \] \[ u = f(t) \text{ on the initial curve } \Gamma = (t, t); t>0. \] Consider the following statements:
P: If \(f(t) = 2t + 1\), then there exists a unique solution to the Cauchy problem in a neighbourhood of \(\Gamma\).
Q: If \(f(t) = 2t - 1\), then there exist infinitely many solutions to the Cauchy problem in a neighbourhood of \(\Gamma\).
Then

For a fixed \(c \in \mathbb{R}\), let \(\alpha = \int_0^c (9x^2 - 5cx^4)dx\).
If the value of \(\int_0^c (9x^2 - 5cx^4)dx\) obtained by using the Trapezoidal rule is equal to \(\alpha\), then the value of c is .................. (rounded off to 2 decimal places).

If for some \(a \in \mathbb{R}\), \[ \int_1^4 \int_{-x}^x \frac{1}{x^2 + y^2} dydx = \int_{-\pi/4}^{\pi/4} \int_{\sec\theta}^{a \sec\theta} \frac{1}{r} drd\theta, \] then the value of \(a\) equals ................

Let S be the portion of the plane \(z = 2x + 2y - 100\) which lies inside the cylinder \(x^2 + y^2 = 1\). If the surface area of S is \(a\pi\), then the value of a is equal to ................

Consider the following Linear Programming Problem P:
Minimize \(3x_1 + 4x_2\)
subject to \[ x_1 - x_2 \le 1, \] \[ x_1 + x_2 \ge 3, \] \[ x_1 \ge 0, x_2 \ge 0. \] The optimal value of the problem P is ..................

Let \[ \frac{z}{1 - z - z^2} = \sum_{n=0}^{\infty} a_n z^n, \quad a_n \in \mathbb{R} \] for all z in some neighbourhood of 0 in \(\mathbb{C}\).
Then the value of \(a_6 + a_5\) is equal to ................

Let \(u(x, t)\) be the solution of \[ \frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} = 0, \quad x \in (-\infty, \infty), t>0, \] \[ u(x, 0) = \sin x, \quad x \in (-\infty, \infty), \] \[ \frac{\partial u}{\partial t}(x, 0) = \cos x, \quad x \in (-\infty, \infty), \] for some positive real number c.
Let the domain of dependence of the solution u at the point P(3,2) be the line segment on the x-axis with end points Q and R.
If the area of the triangle PQR is 8 square units, then the value of \(c^2\) is ................

Let \(T: \mathbb{R}^4 \to \mathbb{R}^4\) be a linear transformation and the null space of \(T\) be the subspace of \(\mathbb{R}^4\) given by \[ \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : 4x_1 + 3x_2 + 2x_3 + x_4 = 0 \}. \] If \(\text{Rank}(T - 3I) = 3\), where \(I\) is the identity map on \(\mathbb{R}^4\), then the minimal polynomial of \(T\) is

Let \(D = \{z \in \mathbb{C} : |z|<1\}\) and \(f: D \to \mathbb{C}\) be defined by \[ f(z) = z - 25z^3 + \frac{z^5}{5!} - \frac{z^7}{7!} + \frac{z^9}{9!} - \frac{z^{11}}{11!} \] Consider the following statements:
P: \(f\) has three zeros (counting multiplicity) in D.
Q: \(f\) has one zero in \(U = \{z \in \mathbb{C} : \frac{1}{2}<|z|<1\}\).
Then

Let \(N \subseteq \mathbb{R}\) be a non-measurable set with respect to the Lebesgue measure on \(\mathbb{R}\).
Consider the following statements:
P: If \(M = \{x \in N : x \text{ is irrational}\}\), then \(M\) is Lebesgue measurable.
Q: The boundary of \(N\) has positive Lebesgue outer measure.
Then

For \(k \in \mathbb{N}\), let \(E_k\) be a measurable subset of \([0,1]\) with Lebesgue measure \(\frac{1}{k^2}\).
Define \[ E = \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} E_k \quad \text{and} \quad F = \bigcup_{n=1}^{\infty} \bigcap_{k=n}^{\infty} E_k \] Consider the following statements:
P: Lebesgue measure of \(E\) is equal to zero.
Q: Lebesgue measure of \(F\) is equal to zero.
Then