Recent questions in Engineering Mathematics

The area of a circular field is $25\: m^{2}$. The radius, $r$, is to be determined using the Newton-Raphson iterative method. For an initial guess of $r = 2.500\: m$, the...

The partial differential equation $$\frac{\partial u}{\partial t} = \frac{1}{\pi^{2}} \frac{\partial ^{2} u}{\partial x^{2}}$$ where, $t \geq 0$ and $x \in [0,1]$, is sub...

The directional derivative of $f\left ( x, y, z \right ) = 4x^{2}+2y^{2}+z^{2}$ at the point $(1, 1, 1)$ in the direction of the vector $\vec{v} = \hat{i} - \hat{k}$ is _...

The equation $\dfrac{dy}{dx} = xy^{2}+ 2y+ x- 4.5$ with the initial condition $y(x= 0) = 1$ is to be solved using a predictor-corrector approach. Use a predictor based on...

An ordinary differential equation $\text{(ODE)}, \: \dfrac{dy}{dx}=2y$, with an initial condition $y(0)=1$, has the analytical solution $y=e^{2x}$.Using Runge-Kutta secon...

The function $\cos(x)$ is approximated using Taylor series around $x=0$ as $\cos(x) \approx 1 + ax + bx^2 + cx^3 + dx^4$. The values of $a,b,c$ and $d$ are$a=1, \: b=-0....

A three-dimensional velocity field is given by $V=5x^2y \: i + Cy \: j-10xyz \: k$, where $i,j,k$ are the unit vectors in $x,y,z$ directions, respectively, describing a c...

For the function $f(x) = \begin{cases} -x, & x<0 \\ x^2, & x \geq 0 \end{cases}$ the $\text{CORRECT}$ statement(s) is/are$f(x)$ is continuous at $x=1$$f(x) $ is different...

A source placed at the origin of a circular sample holder (radius $r=1 \: m$) emits particles uniformly in all directions. A detector of length $l=1 \: cm$ has been place...

Let $A$ be a square matrix of size $n \times n (n>1)$. The elements of $A=\{a_{ij}\}$ are given by $$a_{ij} = \begin{cases} i \times j, & \text{if } i \geq j \\ 0, & \tex...

To solve an algebraic equation $f(x)=0$, an iterative scehme of the type $x_{n+1} = g(x_n)$ is proposed, where $g(x)=x-\dfrac{f(x)}{f’(x)}$.At the solution $x=s,\: g’(s)=...

The probability distribution function of a random variable $X$ is shown in the following figure.From this distribution, random samples with sample size $n=68$ are taken. ...

For the ordinary differential equation $$\dfrac{d^3y}{dt^3}+6 \dfrac{d^2y}{dt^2}+11 \dfrac{dy}{dt}+6y=1$$ with initial conditions $y(0)=y’(0) = y’’(0)=y’’’(0)=0$, the val...

Sum of the eigenvalues of the matrix $\begin{bmatrix} 2 & 4 & 6 \\ 3 & 5 & 9 \\ 12 & 1 & 7 \end{bmatrix}$ is ______ (round off to nearest integer).

In a box, there are $5$ green balls and $10$ blue balls. A person picks $6$ balls randomly. The probability that the person has picked $4$ green balls and $2$ blue balls ...

The maximum value of the function $f(x)=-\dfrac{5}{3} x^3 +10x^2-15x+16$ in the interval $(0.5,3.5)$ is$0$$8$$16$$48$

Given $\dfrac{dy}{dx}=y-20$, and $y \mid_{x=0}=40$, the value of $y$ at $x=2$ is _________ (round off to nearest integer)

Consider the following dataset.$$\begin{array}{|c|c|c|c|c|c|} \hline x & 1 & 3 & 5 & 15 & 25 \\ \hline f(x) & 6 & 8 & 10 & 12 & 5 \\ \hline \end{array}$$The value of the ...

Which one of the following methods requires specifying an initial interval containing the root (i.e., bracketing) to obtain the solution of $f(x) =0$, where $f(x)$ is a c...

Consider the hyperbolic functions in $\textbf{Group-1}$ and their definitions in $\textbf{Group-2}$.$\begin{array}{llll} & \textbf{Group-1} & & \textbf{Group-2} \\ \textb...

Consider the following continuously differentiable function $$\textbf{v}(x,y,z)=3x^2y \textbf{ i} + 8y^2z \textbf{ j} + 5xyz \textbf{ k}$$ where $\textbf{i, j,}$ and $\te...

Consider the following unit step function.The Laplace transform of this function is$\dfrac{e^{-3s}}{s} \\ $$\dfrac{e^{-3s}}{s^2} \\$$\dfrac{e^{-3s}}{3s} \\$$\dfrac{e^{-6s...

The value of the expression $\underset{x\rightarrow \frac{\pi }{2}}{\lim}\: \mid \frac{\tan\:x}{x} \mid $ is $\infty$$0$$1$$-1$

The product of the eigenvalues of the matrix $\begin{pmatrix} 2 &3 \\ 0& 7 \end{pmatrix}$ is ____________ (rounded off to one decimal place).

The solution of the ordinary differential equation$\frac{dy}{dx}+3y=1$, subjects to the initial condition $y=1$ at $x=0$, is$\frac{1}{3}\left ( 1+2e^{-x/3} \right )$$\fra...

The value of the complex number $i^{-1/2}$ (where $i=\sqrt{-1}$) is $\frac{1}{\sqrt{2}}\left ( 1-i \right )$$-\frac{1}{\sqrt{2}}i$$\frac{1}{\sqrt{2}}i$$\frac{1}{\sqrt{2}}...

If $x,y$ and $z$ are directions in a Cartesian coordinate system and $i$, $j$ and $k$ are the respective unit vectors, the directional derivative of the function $u\left ...

Two unbiased dice are thrown. Each dice can show any number between $1$ and $6$. The probability that the sum of the outcomes of the two dice is divisible by $4$ is _____...

The Newton-Raphson method is used to determine the root of the equation $f(x)=e^{-x}-x$. If the initial guess for the root is 0, the estimate of the root after two iterat...

The real part of $6e^{i\pi /3}$ is _______________.

The number of positive roots of the function $f(x)$ shown below in the range $0<x<6$ is ___________________.

Let $i$ and $j$ be the unit vectors in the $x$ and $y$ directions, respectively. For the function $F\left ( x,y \right )=x^{3}+y^{2}$ the gradient of the function. i.e.. ...

The marks obtained by a set of students are: $38,\:84,\:45,\:70,\:75,\:60,\;48$.The mean and median marks, respectively, are$45$ and $75$$55$ and $48$$60$ and $60$$60$ an...

For the initial value problem$$\frac{dx}{dt}=\sin\left ( t \right ),\:\:\:\:x\left ( 0 \right )=0$$the value of $x$ at $t=\pi /3$ , is _____________.

The Laplace transform of a function is $\frac{s+1}{s\left ( s+2 \right )}$The initial and final values, respectively, of the function are $0$ and $1$$1$ and $\frac{1}{2}$...

Match the problem type in Group-$1$ with the numerical method in Group-$2$.$\begin{array}{cc}\&\text{Group-$1$} & \text{Group-$2$} \\ &\text{P. System of linear algebraic...

A box has $6$ red balls and $4$ white balls. A ball is picked at random and replaced in the box, after which a second ball is picked.The probability of both the balls bei...