List of top Engineering Mathematics Questions

\( \hat{i} \) 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).
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).
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).
Consider the ordinary differential equation: \[ \frac{1}{2} \frac{dy}{dx} + \frac{y}{x} = 1. { If } y = \frac{2}{3} { at } x = 1, { then the value of } y { at } x = 3 { is } \_\_\_\_\_\_ { (rounded off to the nearest integer).} \]
Find the limit: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]
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).
If \( i = \sqrt{-1} \), \[ \frac{(i+1)^3}{i-1} = \_\_\_\_\_\_\_\_. \]
The partial differential equation \[ \frac{\partial^2 u}{\partial x^2} + 4 \frac{\partial^2 u}{\partial x \partial y} + 2 \frac{\partial^2 u}{\partial y^2} = 0 \] is \_\_\_\_\_\_.
Let the function \( f(x) \) be defined as: \[ f(x) = \begin{cases} A + x, & {if } x<2
1 + x^2, & {if } x \geq 2 \end{cases} \] If the function \( f(x) \) is continuous at \( x = 2 \), the value of \( A \) is \_\_\_\_\_.
\( \hat{i} \) 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).
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).
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 \( i = \sqrt{-1} \), \[ \frac{(i+1)^3}{i-1} = \_\_\_\_\_\_\_\_. \]
Find the maximum value of the function \( f(x) = -x^3 + 2x^2 \) in the interval \( [-1, 1.5] \).
Given the random variable \( X \) which takes the values 0, 1, 2, 7, 11, and 12 with the following probabilities: \[ P(X = 0) = 0.4, \quad P(X = 1) = 0.3, \quad P(X = 2) = 0.1, \quad P(X = 7) = 0.1, \quad P(X = 11) = ? \]
Given that \( \lambda \) is an eigenvalue of matrix \( A \) with the corresponding eigenvector \( x \), and \( x \) is also an eigenvector of \( B = A - 2I \), find the relationship between \( \lambda \) and the eigenvalue of \( B \).
Find the sum of the series: \[ 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots \]

If \(u = \sin^{-1}\left(\frac{x}{y}\right) + \tan^{-1}\left(\frac{y}{x}\right)\), then the value of \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} \) is:

If \( A = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} \), then eigen value and its corresponding eigen vector
Let E and F be the events of a sample space S of an experiment, if $ P(S/F) = P(F/F) $, then $ P(S/F) $ is equal to:

A constant force \(\vec{F} = (4\hat{i} + \hat{j} - 3\hat{k}) \, \text{N} \) moves a particle from \( A: (1, 2, 3) \, \text{m}  \text{to}  B: (5, 4, 1) \, \text{m}. \)
Find the work done by the force (in joules). Answer as an integer.