List of top Engineering Mathematics Questions asked in GATE Aerospace Engineering

Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]
An approximate solution of the equation \( x^3 - 17 = 0 \) is to be obtained using the Newton-Raphson method. If the initial guess is \( x_0 = 2 \), the value at the end of the first iteration is \( x_1 = \) \_\_\_\_\_\ (rounded off to two decimal places).
\( \hat{i} \) and \( \hat{j} \) denote unit vectors in the \( x \) and \( y \) directions, respectively. The outward flux of the two-dimensional vector field \( \vec{v} = x \hat{i} + y \hat{j} \) over the unit circle centered at the origin is \_\_\_\_\_\_ (rounded off to two decimal places).
Consider the ordinary differential equation: \[ \frac{1}{2} \frac{dy}{dx} + \frac{y}{x} = 1. { If } y = \frac{2}{3} { at } x = 1, { then the value of } y { at } x = 3 { is } \_\_\_\_\_\_ { (rounded off to the nearest integer).} \]
The equation of a closed curve in two-dimensional polar coordinates is given by \( r = \frac{2}{\sqrt{\pi}} (1 - \sin \theta) \). The area enclosed by the curve is \_\_\_\_\_\_\_\_ (answer in integer).
The maximum value of the function \( f(x) = (x - 1)(x - 2)(x - 3) \) in the domain [0, 3] occurs at \( x = \) \_\_\_\_\_\_ (rounded off to two decimal places).
Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]
If \( i = \sqrt{-1} \), \[ \frac{(i+1)^3}{i-1} = \_\_\_\_\_\_\_\_. \]