List of top Engineering Mathematics Questions

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 \_\_\_\_\_.
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: \[ \lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} \]
\( \hat{i} \) 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).
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 \( i = \sqrt{-1} \), \[ \frac{(i+1)^3}{i-1} = \_\_\_\_\_\_\_\_. \]

Consider designing a linear binary classifier \( f(x) = \text{sign}(w^T x + b), x \in \mathbb{R}^2 \) on the following training data: 



Class-2: \( \left\{ \left( \begin{array}{c} 0 \\ 0 \end{array} \right) \right\} \) 

Hard-margin support vector machine (SVM) formulation is solved to obtain \( w \) and \( b \). Which of the following options is/are correct?

Eigenvalue question: \[\begin{pmatrix} 6 & 8 \\ 4 & 2 \end{pmatrix} \quad \text{Eigenvalue options:} \quad \text{(i) 1, (ii) 2, (iii) 3, (iv) 4}\]
Given the matrix equation: \[A = \begin{pmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{pmatrix} \quad A \cdot X = B\]
Where \( A \) is the matrix, \( X \) is the unknown vector, and \( B \) is a constant vector. Solve for \( X \).
Find the sum of the series: \[ 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots \]
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 \( B \).
Solve the second-order differential equation: \[ y'' + 0.8 y' + 0.16 y = 0, \quad y(0) = 3, \quad y'(0) = 4.5 \]
Solve the cubic equation: \[ x^3 - \frac{15}{2} x^2 + 18x + 20 = 0 \]
Find the order and degree of the following differential equation: \[ \frac{d^3 y}{dx^3} + \frac{d^2 y}{dx^2} + \frac{dy}{dx} + y = 0 \]
Find the next term in the sequence: 3, 9, 19, 33, _
Given the matrix: \[A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix}\]

 If rank(A) is at least 3, then what are the possible values of \( \alpha, \beta, \gamma \)?

Two fair dice are rolled, and the random variable \( X \) denotes the sum of the outcomes. What is the expected value of \( X \)?
If a complex function \( f(z) \) is analytic everywhere inside a closed contour \( C \) with anti-clockwise direction, then which of the following statements are correct?
Consider three Boolean variables \( x, y, z \). A majority function outputs 1 if the majority of its inputs are 1, otherwise, it outputs 0. Derive the Boolean expression for the majority function and simplify it using Boolean algebra.

The variance for continuous probability function \(f(x) = x^2 e^{-x}\) when \(x \ge 0\) is