Recent activity in Engineering Mathematics

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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...
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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....
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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...
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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$$f(x) $ is different...
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$\textbf{A, B, C}$ and $\textbf{D}$ are vectors of length $4$. $$\textbf{A} = \begin{bmatrix} a_1 & a_2 & a_3 & a_4 \end{bmatrix} \\ \textbf{B} = \begin{bmatrix}b_1 & b_2...
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Let $A$ be a square matrix of size $n \times n (n>1)$. The elements of $A=\{a_{ij}\}$ are given by $$a_{ij} = \begin{cases} i \times j, & \text{if } i \geq j \\ 0, & \tex...
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To solve an algebraic equation $f(x)=0$, an iterative scehme of the type $x_{n+1} = g(x_n)$ is proposed, where $g(x)=x-\dfrac{f(x)}{f’(x)}$.At the solution $x=s,\: g’...
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The probability distribution function of a random variable $X$ is shown in the following figure.From this distribution, random samples with sample size $n=68$ are taken. ...
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For the ordinary differential equation $$\dfrac{d^3y}{dt^3}+6 \dfrac{d^2y}{dt^2}+11 \dfrac{dy}{dt}+6y=1$$ with initial conditions $y(0)=y’(0) = y’’(0)=y’’’(0)...
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An equation of state is explicit in pressure $p$ and cubic in the specific volume $v$. At the critical point $‘c’$ , the isotherm passing through $‘c’$ satisfies$...
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If $i = \sqrt{-1}$, the value of the integral $$\oint_{c}\dfrac{7z + i}{z(z^2 + 1)}dz \quad\quad\quad\quad|z|<2$$,u...
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The Newton – Raphson method is used to find the roots of the equation $f(x) = x- \cos\pi x$ $0\...
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If $a$ is a constant, then the value of the integral $a^{2}\int^\infty_0 xe^{-ax}dx$ is$1/a$$a$$1$$0$
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Consider the following $(2\times2)$ matrix $\left( \begin{array}{c} 4 & 0 \\ 0 & 4 \end{array} \right)$Whi...
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For the function $f(t) = e^{-t}/\tau$,the Taylor series approximation for $t\ll$$\tau$ is$1+\frac{t}{\tau}$$1-\frac{t}{\tau}$$1-\frac{t^2}{2\tau^2}$$1+t$
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If $a$ and $b$ are arbitrary constants, then the solutions to the ordinary differential equation $\dfrac{d^...
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Consider the following set of linear algebraic equations $x{_1} + 2x{_2} + 3x{_3} = 2$ ...
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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 ...
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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)
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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$$48$
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In a box, there are $5$ green balls and $10$ blue balls. A person picks $6$ balls randomly. The probability that the person has picked $4$ green balls and $2$ blue balls ...
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Sum of the eigenvalues of the matrix $\begin{bmatrix} 2 & 4 & 6 \\ 3 & 5 & 9 \\ 12 & 1 & 7 \end{bmatrix}$ is ______ (round off to nearest integer).
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Consider the following unit step function.The Laplace transform of this function is$\dfrac{e^{-3s}}{s} \\ $$\dfrac{e^{-3s}}{s^2} \\$$\dfrac{e^{-3s}}{3s} \\$$\dfrac{e^{-6s...
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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...
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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...
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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...
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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 _____...
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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 ...
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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$$\frac{1}{\sqrt{2}}...
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The solution of the ordinary differential equation$\frac{dy}{dx}+3y=1$, subjects to the initial condition $y=1$ at $x=0$, is$\frac{1}{3}\left ( 1+2e^{-x/3} \right )$$\fra...
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The product of the eigenvalues of the matrix $\begin{pmatrix} 2 &3 \\ 0& 7 \end{pmatrix}$ is ____________ (rounded off to one decimal place).
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The value of the expression $\underset{x\rightarrow \frac{\pi }{2}}{\lim}\: \mid \frac{\tan\:x}{x} \mid $ is $\infty$$0$$1$$-1$
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A system of $n$ homogeneous linear equations containing $n$ unknowns will have non-trivial solutions if and only if the determinant of the coefficient matrix is$1$$-1$$0$...
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For the matrix $A=\begin{bmatrix} \cos\theta &-\sin\theta \\ \sin \theta & \cos\theta \end{bmatrix}$ if $\det$ stands for the determinant and $A^{T}$ is the transpose of ...
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If $y=e^{-x^{2}}$ then the value of $\underset{x\rightarrow \infty }{\lim}\frac{1}{x}\frac{dy}{dx}$ is __________
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The decay ratio for a system having complex conjugate poles as $\left ( -\frac{1}{10}+j\frac{2}{15} \right )$ and $\left ( -\frac{1}{10}-j\frac{2}{15} \right )$ is $7\ti...
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The figure which represents for $y=\frac{sin \:x}{x}$ for $x>0$ ($x$ in radians) is
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The fourth order Runge-Kutta ($RK4$) method to solve an ordinary differential equation $\frac{dy}{dx}=f\left ( x,y \right )$ is given as $$y\left ( x+h \right )=y\left ( ...
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Consider the following two equation:$\frac{dx}{dt}+x+y=0$$\frac{dy}{dt}-x=0$The above set of equations is represented by$\frac{d^{2}y}{dt^{2}}-\frac{dy}{dt}-y=0$$\frac{d^...
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The value of $\underset{x\rightarrow 0}{\lim}\:\frac{\tan\left ( x \right )}{x}$ is ________________.
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