Recent questions in Engineering Mathematics

The total cost $(C_{T})$ of an equipment in terms of the operating variables $x$ and $y$ is$$C_{T}=2x+\frac{12000}{xy}+y+5$$The optimal value of $C_{T}$, rounded to $1$ d...

Which one of the following is an iterative technique for solving a system of simultaneous linear algebraic equations?Gauss eliminationGauss-JordanGauss-Seidel$LU$ decompo...

The Laplace transform of $e^{at}\sin\left ( bt \right )$ is$\frac{b}{\left ( s-a \right )^{2}+b^{2}}$$\frac{s-a}{\left ( s-a \right )^{2}+b^{2}}$$\frac{s-a}{\left ( s-a \...

What are the modulus $\left ( r \right )$ and argument $\left ( \theta \right )$ of the complex number $3+4i$ ?$r=\sqrt{7}, \theta =tan^{-1}\left ( \frac{4}{3} \right )$$...

A set of simultaneous linear algebraic equations is represented in a matrix form as shown below.$\begin{bmatrix} 0 &0 &0 &4 &13 \\ 2& 5& 5& 2& 10\\ 0&0 &2 &5 &3 \\ 0&0 &0...

What is the solution for the second order differential equation $\frac{\mathrm{d} ^{2}y}{\mathrm{d} x}^{2}+y=0$, with the initial conditions $y\left | _{x-0}=5\:and\:\fra...

The model $y=mx^{2}$ is to be fit to the data given below.$$\begin{array}{|cl|cI|}\hline&{x} & {1} & {\sqrt{2}} & {\sqrt{3}} \\ \hline &{y} & {2} & {5} & {8} \\ \hline \e...

The Lagrange mean-value theorem is satisfied for $f\left ( x \right )=x^{3}+5$, in the interval $\left ( 1,4 \right )$ at a value (rounded off to the second decimal place...

Values of $f\left ( x \right )$ in the interval $\left [ 0,4 \right ]$ are given below.$\begin{array}{|cl|cI|cI|cI|cI|cI|}\hline&{x} & \text{0} & \text{1} & \text{2} & \t...

The following set of the three vectors$$\begin{pmatrix} 1\\2\\1 \end{pmatrix}, \begin{pmatrix} x\\6\\x \end{pmatrix}\:and \:\begin{pmatrix} 3\\4 \\2 \end{pmatrix},$$is li...

Consider a linear ordinary differential equation: $\frac{dy}{dx}+p\left ( x \right )y=r\left ( x \right )$. Functions $p\left ( x \right )$ and $r\left ( x \right )$ are...

A complex-valued function, $f(z)$, given below is analytic in domain $D$:$$f\left ( z \right )=u\left ( x,y\right )+i\nu \left ( x,y\right )\:\:\:\:\:\:\:z=x+iy$$Which of...

A scalar function in the $xy$-plane is given by $\phi \left ( x,y \right )=x^{2}+y^{2}$. If $\hat{i}$ and $\hat{j}$ are unit vectors in the $x$ and $y$ directions, the d...

The diameters of sand particles in a sample range from $50$ to $150$ microns. The number of particles of diameter x in the sample is proportional to $\frac{1}{50+x}$. The...

A vector $u=-2y\hat{i}+2x\hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors in $x$ and $y$ directions, respectively. Evaluate the line integral$$I=\oint _{C}u.dr$$w...

The solution of the non-linear equation$$x^{3}-x=0$$is to be obtained using Newton-Raphson method. If the initial guess is $x=0.5$, the method converges to which one of t...

For complex variable $z$, the value of the contour integral $\frac{1}{2\pi i} \underset{C}{\int }\frac{e^{-2z}}{z(z-3)}dz$ along the clockwise contour $C:\left | z \right...

Consider the following two normal distributions$f_{1}\left ( x \right )=exp\left ( -\pi x^{2} \right )$$f_{2}\left ( x \right )=\frac{1}{2 \pi}exp\left \{ -\frac{1}{4\pi ...

In rolling of two fair dice, the outcome of an experiments is considered to be the sum of the numbers appearing on the dice. The probability is highest for the outcome of...

Consider the following differential equation$\frac{dy}{dx}=x+In\:\left ( y \right ); y=2 \: at \: x=0$The solution of this equation at $x=0.4$ using Euler method with a s...

The integrating factor for the differential equation$\frac{dy}{dx}-\frac{y}{1+x}=\left ( 1+x \right )$ is$\frac{1}{1+x}$$(1+x)$$x(1+x)$$\frac{x}{1+x}$

The differential equation $\frac{d^{2}y}{dx^{2}}+x^{2}\frac{dy}{dx}+x^{3}y=e^{x}$ is a non-linear differential equation of first degreelinear differential equation of fir...

Integral of the time-weighted absolute error $(ITAE)$ is expressed as$\int _{0}^{\infty }\frac{\left | \varepsilon \left ( t \right ) \right |}{t^{2}}dt$$\int _{0}^{\inft...

For the time domain function $f(t)=t^{2}$, which $ONE$ of the following is the Laplace transform of $\int _{0}^{t}f\left ( t \right )dt$?$\frac{3}{s^{4}}$$\frac{1}{4s^{2}...

If $f*(x)$ is the complex conjugate of $f(x)=\cos(x) + i\: \sin(x)$, then for real $a$ and $b$, $\int _{a}^{b}f*\left ( x \right )f\left ( x \right )$ is $ALWAYS$positive...

If $f(x)$ is a real and continuous function of $x$, the Taylor series expansion of $f(x)$ about its minima will $NEVER$ have a term containingfirst derivativesecond deriv...

Gradient of a scalar variable is alwaysa vectora scalara dot productzero

For the functionf(z) = $\frac{1}{(2-z)(z+2)}$the residue at z = 2 is $\_\_\_\_\_$

The solution of the differential equation$\frac{dy}{dx}$ – y$^{2}$ = 0, given y=1 at x=0 is$\frac{1}{1+x}$$\frac{1}{1-x}$$\frac{1}{(1-x)^2}$$\frac{x^3}{3}$+1

The solution of the differential equation$\frac{d^2y}{dx^2}$ – $\frac{dy}{dx}$ + 0.25y = 0, given y = 0 at x = 0 and $\frac{dy}{dx}$ = 1 at x = 0 isxe$^{0.5x}$ – xe$^{-0...

The value of the integral${\scriptstyle \int^{0.5} _{0.1}}$ e$^{-x^3}$dxevaluated by Simpson’s rule using 4 subintervals (up to 3 digits after the decimal point) is $\_\_...

Evaluate ${\displaystyle \int \frac{dx}{e^x – 1}}$(Note: C is a constant of integration)$\frac{e^x}{e^x -1}$ + C$\frac {In(e^x -1)}{e^x}$ + CIn$(\frac {e^x}{e^x -1})$ + ...

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$\frac{\p...

The number of emails received on six consecutive days is 11, 9, 18, 18, 4 and 15,respectively.what are the median and the mode for these data?18 and 11, respectively13 an...

For two rolls of a fair die, the probability of getting a 4 in the first roll and a number less than 4 in the second roll, up to 3 digits after the decimal point, is ____...

Which of the following statements are TRUE?P.The eigenvalues of a symmetric matrix are realQ.The value of the determinant of an orthogonal matrix can only be +1R.The tr...

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$

The Newton – Raphson method is used to find the roots of the equation $f(x) = x- \cos\pi x$ $0\le...

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...