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

The value of the integral \( \int_{0}^{\infty} \frac{dx}{e^x + e^{-x}} \) is:

The approximate value of the integral \( \int_{2}^{3} \frac{dx}{x} \) using Simpson's rule with \( h = 0.5 \) is

Consider the following function:
\[ f(x) = \begin{cases} k, & \text{if } x = 1 \\\frac{\sqrt{3x + 1} - \sqrt{2x + 2}}{x - 1}, & \text{if } x > -\frac{1}{3}, x \neq 1 \end{cases} \]
If \( f(x) \) is continuous at \( x = 1 \), the value of \( k \) (correct up to 2 decimal places) is

A textile company decides to find the coefficient of correlation (\( r \)) between fibre quality (\( X \)) and yarn quality (\( Y \)). The company randomly selects 10 samples and observes the following:\[\sum X = 50, \quad \sum Y = 40, \quad \sum X^2 = 260, \quad \sum Y^2 = 228, \quad \sum XY = 222, \quad \text{and} \quad r(X, Y) = 0.84.\]If the correct pairs \((X = 4, Y = 11)\) and \((X = 6, Y = 9)\) are taken wrongly as \((X = 6, Y = 15)\) and \((X = 4, Y = 5)\), respectively, then the correct value of \( r(X, Y) \) (rounded off to 2 decimal places) is

The three eigenvalues of the matrix \[ \begin{pmatrix} -2 & X & -3 \\ 2 & 1 & -6 \\ -1 & -2 & y \end{pmatrix} \] are \(\lambda_1 = 5\) and \(\lambda_2 = \lambda_3 = -3\). The value of \(X\) is _______.

A scientist wants to find the root of the equation \(2x^3 + x^2 - 1 = 0\) lying in \((0,1)\). He applies the Secant method only once by taking two initial guesses, 0.5 and 0.7. The value of the root is approximately:

If \( g(x) \) is a continuous function such that\[ \int_{a}^{b} g(x) \, dx = \beta \]then the correct statement(s), amongst the following, is/are:

A continuous random variable \(X\) has the following probability density function:\[ f(x) = \begin{cases} kx^2(1 - x^3), & 0 \leq x \leq 1 \\0, & \text{elsewhere}\end{cases} \]If the mean of \(X\) is 0.64, then the value of \(k\) (rounded off to 2 decimal places) is ________________.

The differential equation \( \frac{dy}{dx} = \frac{M(x,y)}{N(x,y)} \) is exact if _______. (