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 ________