List of top Engineering Mathematics Questions asked in GATE Textile Engineering and Fibre Science

If \[ \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix} \quad \text{is the eigenvector of the matrix} \quad A = \begin{bmatrix} 8 & 11 & 3 \\ 4 & -1 & 3 \\ -4 & 10 & 6 \end{bmatrix}, \quad \text{then the corresponding eigenvalue is:} \]

The solution of the following differential equation represents \[ \frac{dy}{dx} = \frac{y + 1}{x} \]

Let \( X \) be a Poisson distributed random variable with parameter \( \lambda (> 0) \) such that it satisfies the equation \[ P(X = 1) = 3P(X = 3) - P(X = 2) {. Then, the value of } \lambda { is:} \]

If \( u(x, y, z) = x^2y + y^2z + z^2x \), the value of \[ \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z} { at the point } (1,1,1) { is:} \]

The diameter of a fibre is assumed to be a continuous random variable \( X \) with probability density function \[ f(x) = 6x(1 - x), \quad 0<x \leq 1 \] If \( P(X<\beta) = P(X>\beta) \), then the value of \( \beta \) (rounded off to 1 decimal place) is _________.

Consider the partial differential equation: \[ \frac{\partial^2 u}{\partial x^2} = \frac{1}{k} \frac{\partial u}{\partial t} + \sin x, \quad k>0 \] Amongst the following, the correct statement(s) for the above equation is/are:

The value of \( \alpha \) for which the Euler method with step size \( h = 0.1 \) provides \( y(1.1) = 1.2 \) for the following initial value problem is \[ \frac{dy}{dx} = -2xy^\alpha, \quad y(1) = 2 \]