List of top Engineering Mathematics Questions asked in GATE Naval Architecture and Marine Engineering

Consider the function \( f(x) = |x| - 1 \). Which of the following statements is/are true in the interval \([-10, 10]\)?
Let \( M = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \) and \( K = \begin{bmatrix} 2 & -1 \\ -1 & 1 \end{bmatrix} \) satisfy the eigenvalue problem given by:
\[ (M - \alpha K)\phi = 0. \] The lowest eigenvalue \( \alpha \) is ________ (rounded off to two decimal places).
For all real values of \( x \) and \( y \), the partial differential equation in terms of \( \psi(x, y) \) given by \[ \frac{\partial^2 \psi}{\partial x^2} + 2 \frac{\partial^2 \psi}{\partial x \partial y} - 3 \frac{\partial^2 \psi}{\partial y^2} = 0 \] is ________
The Fourier series expansion of \( f(x) = \sin^2 x \) in the interval \((- \pi, \pi)\) is ______.
The value of \( \lim_{t \to 1} \frac{\ln t}{t^2 - 1} \) is ________.
The value of \( \int_1^3 \int_2^5 x^2 y \, dx \, dy \) is ________(answer in integer).
The gradient of \( y = 3x^2 \sin(2x) \) at \( (0.2, 1) \) is:
Consider \( f(t) = \cos(at) \), where \( a \) is a real constant. The Laplace transform of \( f(t) \) is _________.
A box contains 12 red and 8 blue balls. Two balls are drawn randomly from the box without replacement. The probability of drawing a pair of balls having the same color is ________
Consider the function \( z = \tan^{-1}\left(\frac{y}{x}\right) \), where \( x = u \sin v \) and \( y = u \cos v \). The partial derivative, \( \frac{\partial z}{\partial v} \) is
Consider the function \( z = x^3 - 2x^2y + xy^2 + 1 \). The directional derivative of z at the point (1, 2) along the direction \( 3\hat{i} + 4\hat{j} \) is
Let \(y(x)\) be the solution of the differential equation \(y'' - 4y' - 12y = 3e^{5x}\) satisfying \(y(0)=\frac{18{7}\) and \(y'(0)=-\frac{1}{7}\). Then \(y(1)\) is ______________ (rounded off to nearest integer).}
We have two coins. One is biased with probability of head = 1.0 and the other is a fair coin. One coin is chosen at random and tossed twice. If both outcomes are heads, the probability that the chosen coin is fair is ____________ (correct to one decimal place).
Let \[ M = \begin{pmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{pmatrix} \] Which of the following are TRUE?
The value of the line integral \[ \oint_{C} (-3y\,dx + 3x\,dy + z\,dz) \] along the circle \( C: x^{2} + y^{2} = 1,\ z = 1 \) oriented in the clockwise sense as seen from the origin, is
A simply supported beam is loaded with a uniformly distributed load (UDL). Which statements about strain energy are TRUE?
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Consider the complex functions: \[ f(z) = (x^2 - y^2) + i 2xy \] and \[ g(z) = 2xy + i(x^2 - y^2) \] Then on the complex plane,
Let \( A \) be a real non-zero square matrix of order \( n \). If the homogeneous system of linear equations \( Ax = 0 \) has only the trivial solution, then:
If a population has exponential distribution with mean 1, then its median is: