List of top Engineering Mathematics Questions

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?

Consider the differential equation $\frac{dy}{dx} + \frac{y}{x} = 0$. Choose the CORRECT option(s) for the solution $y$.

Let the difference equation $ y[n] = \alpha y[n-1] + x[n] $, where $ \alpha>1 $ and $ \alpha $ is real, represent a causal discrete-time linear time invariant system. The system is initially at rest. If $ x[n] = -\delta[n-p] $, where $ p>10 $, the value of $ y[p+1] $ is:

Let $A$ and $B$ be real symmetric matrices of the same size. Which one of the following options is correct?

If \[ P = \begin{pmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{pmatrix} \] and $P + P^T = I$, the value of $\alpha \; (0 \leq \alpha \leq \pi/2)$ is

The length of the edge of a variable cube is increasing at the rate of 25 cm s$^{-1}$. If the initial length of the edge of the cube is 10 cm, the rate of increase of the surface area of the cube is .............. cm$^2$ s$^{-1}$. (answer in integer)

The area bounded by the curve $ y = \sin x $ and the x-axis between $ x = 0 $ and $ x = \frac{3\pi}{2} $ is ............. sq. units. (answer in integer)

Let $X$ and $Y$ be two random variables with mean $0$, variance $1$, and correlation coefficient $\tfrac{1}{3}$. Then the value of $\mathrm{Var}(X+3Y)$ is equal to:

Let $f(z)$ be an analytic function such that \[ Re(f'(z)) = 3x^2 - 4y - 3y^2, f(i)=0, f'(0)=0, \] where $i = \sqrt{-1}$. Then the value of $f(1)$ is:

For $a,b \in \mathbb{R}$, consider the system of linear equations: \[ x + y + az = 2, \] \[ 2y + 2z = 1, \] \[ ax + 2z = b \] If the system has infinitely many solutions, then which of the following statements is/are correct?

Let $u(x,t)$ be the solution of the initial boundary value problem: \[ \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0, 0<x<\pi, \; t>0, \] \[ u(x,0) = 2 \sin\!\left(\tfrac{3x}{2}\right)\cos\!\left(\tfrac{x}{2}\right), 0<x<\pi, \] \[ u(0,t) = u(\pi,t) = 0, t>0. \] Then the value of $\lim_{t\to\infty u\!\left(\tfrac{3\pi}{4},t\right)$ is equal to (rounded off to two decimal places).}

Suppose that 2 is an eigenvalue of the matrix \[ A = \begin{bmatrix} 0 & 3 & -\alpha\\ 0 & 1 & 0 \\1 & -1 & 3 \end{bmatrix}. \] Then the value of $\alpha$ is equal to (Answer in integer):

Consider the function \[ f(x,y) = x^2y + 2xy^2 - 2x^2y^2. \] Then which one of the following statements is correct?

Let $C$ be the positively oriented boundary of the domain bounded by the curves $y=2x^{2}$ and $y^{2}=4x$. Then the value of the line integral \[ \oint_{C} (2y^{2}+2xy+4y)\,dx + (x^{2}+4xy+8x)\,dy \] is equal to:

Consider the infinite series \[ (P):\ \sum_{n=2}^{\infty}\frac{1}{(n\log n)^{1/n}}, \qquad (Q):\ \sum_{n=1}^{\infty}\frac{n^n}{(2n)!}. \] Then which one of the following statements is correct?

Suppose the polynomial $p(x)=a+bx+cx^{2}+dx^{3}$ interpolates the data $(-1,1),\ (0,3),\ (1,2),\ (2,4)$. Then which one of the following statements is correct?

Let $y(x)$ be the solution of the initial value problem, \[ x^{2}y'' + xy' - y = 0, x>0, \] with $y(1)=0$, $y'(1)=2$. Then the value of $y'\!\left(\tfrac{1}{2}\right)$ is equal to (Answer in integer):

Consider the second-order PDE: \[ 4x^2 \frac{\partial^2 u}{\partial x^2} + 4(x+y)\frac{\partial^2 u}{\partial x \partial y} + (x^2+y^2)\frac{\partial^2 u}{\partial y^2} - u=0. \] Which one of the following statements is correct?

The eigenvalues of the matrix \[ A = \begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{bmatrix} \] are $\lambda_1, \lambda_2, \lambda_3$. The value of $\lambda_1 \lambda_2 \lambda_3 (\lambda_1 + \lambda_2 + \lambda_3)$ is:

A stationary tank is cylindrical in shape with two hemispherical ends and is horizontal, as shown in the figure. $R$ is the radius of the cylinder as well as of the hemispherical ends. The tank is half filled with an oil of density $\rho$ and the rest of the space in the tank is occupied by air. The air pressure, inside the tank as well as outside it, is atmospheric. The acceleration due to gravity ($g$) acts vertically downward. The net horizontal force applied by the oil on the right hemispherical end (shown by the bold outline in the figure) is:

Let $f(x) = \ln x$. The first derivative $f'(x)$ is to be calculated at $x=1$ using numerical differentiation. $f'(1)$ is calculated using: - First-order forward difference ($f'_{FD}$), - First-order backward difference ($f'_{BD}$), - Second-order central difference ($f'_{CD}$), with interval width $h = 0.1$. The correct order of the values of $f'_{FD}, f'_{BD}, f'_{CD}$ is:

If $\dfrac{dy}{dx} + y = x$, and $y(0) = 0$, then the value of $y(1)$ is:

The product of the roots of the equation $x^4 + 1 = 0$ is ............ (answer in integer).

Which of the following statement(s) is/are CORRECT?

Let $\vec{A} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{B} = \hat{i} + \hat{j}$, where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors. The projection of $\vec{B}$ on $\vec{A}$ is: