Questions without answers in Engineering Mathematics

A two-dimensional temperature profile is given as $T=2 x^{2}+3 x y+y^{2}$. If $\hat{\boldsymbol{\imath}}$ and $\hat{\boldsymbol{\jmath}}$ are the unit vectors along $x$ a...

An experiment is performed by rolling an unbiased die once. Events A and B are defined as follows:Event A: The outcome of the rolled die is a prime number.Event B: The ou...

Using single step trapezoidal rule, the value of\[\int_{0}^{0.5}\left(1+16 x-20 x^{2}\right) d x\]is $\_\_\_\_$ (rounded off to two decimal places).

Consider a differential equation:\[\begin{array}{c}x^{2} \dfrac{d^{2} y}{d x^{2}}+3 x \dfrac{d y}{d x}-3 y=0 \\\left.\begin{array}{r}y=3 \\\dfrac{d y}{d x}=-5\end{array}\...

The data given in the table are fitted to the equation $y=m x$ using the method of least squares. The value of $m$ is $\_\_\_\_$ (rounded off to one decimal place).\[\beg...

​​​​​The sum of the following infinite series is:$$ \frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\cdots$$$\pi$$1+e$$e-1$$e$

​​​A circle with center at $(x, y)=(0.5,0)$ and radius $=0.5$ intersects with another circle with center at $(x, y)=(1,1)$ and radius $=1$ at two points. One of the point...

​​​​Consider a Cartesian coordinate system defined over a $3$-dimensional vector space with orthogonal unit basis vectors $\hat{\imath}, \hat{\jmath}$ and $\hat{k}$. Let ...

​​​​Consider two complex numbers $z_{1}=1-i$ and $z_{2}=i$. The argument of $z_{1} z_{2}$ is$0$$\frac{\pi}{4}$$\frac{\pi}{2}$$\pi$

​​​​A box contains $3$ identical green balls and $7$ identical blue balls. Two balls are randomly drawn without replacement from the box. The probability of drawing $1$ g...

​​​​Consider the differential equation$$ \frac{d y}{d x}+\frac{y}{x}=0$$Choose the CORRECT option(s) for the solution $y$.$y=x+c ; c$ is a constant$y=\frac{c}{x} ; c$ is ...

The Newton-Raphson method is used to find the root of$$f(x) \equiv x^{2}-x-1=0$$Starting with an initial guess $x_{(0)}=1$, the second iterate $x_{(2)}$ is $\_\_\_\_\_$ (...

The first non-zero term in the Taylor series expansion of $(1-x)-e^{-x}$ about $x=0$ is$1$$-1$$\frac{x^{2}}{2}$$-\frac{x^{2}}{2}$

If $z_{1}=-1+i$ and $z_{2}=2 i$, where $i=\sqrt{-1}$, then $\operatorname{Arg}\left(z_{1} / z_{2}\right)$ is$\frac{3 \pi}{4}$$\frac{\pi}{4}$$\frac{\pi}{2}$$\frac{\pi}{3}$

​​​​​​Consider a linear homogeneous system of equations $\text{Ax}=0$, where $\text{A}$ is an $n \times n$ matrix, $\text{x}$ is an $n \times 1$ vector and $0$ is an $n \...

Consider a matrix $\text{A}=\left[\begin{array}{cc}-5 & a \\ -2 & -2\end{array}\right]$, where $a$ is a constant. If the eigenvalues of $\text{A}$ are $-1$ and $-6$, then...

​​​​​​Let $r$ and $\theta$ be the polar coordinates defined by $x=r \cos \theta$ and $y=r \sin \theta$. The area of the cardioid $r=a(1-\cos \theta), 0 \leq \theta \leq 2...

The Newton-Raphson method is used to solve $f(x)=0$, where $f(x)=e^{x}-5 x$.If the initial guess $x^{(0)}=1.0$, the value of the next iterate, $x^{(1)}$, rounded off to $...

Consider the ordinary differential equation $x^{2} \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-3 y=0$, with the boundary conditions $y(x=1)=2$ and $y(x=2)=17 / 2$. The solu...

Consider the function $f(x, y, z)=x^{4}+2 y^{3}+z^{2}$. The directional derivative of the function at the point $P(-1,1,-1)$ along $(\hat{\boldsymbol{\imath}}+\hat{\bolds...

For a two-dimensional plane, the unit vectors, $\left(\hat{e}_r, \hat{e}_\theta\right)$ of the polar coordinate system and $(\hat{\imath}, \hat{\jmath})$ of the cartesian...

The position $x(t)$ of a particle, at constant $\omega$, is described by the equation$$\frac{d^2 x}{d t^2}=-\omega^2 x.$$The initial conditions are $x(t=0)=1$ and $\left....

An exhibition was held in a hall on $15$ August $2022$ between $\text{3 PM}$ and $\text{4 PM}$ during which any person was allowed to enter only once. Visitors who entere...

Simpson's one-third rule is used to estimate the definite integral$$I=\int_{-1}^1 \sqrt{\left(1-x^2\right)} d x$$with an interval length of $0.5.$ Which one of the follow...

The first derivative of the function$$U(r)=4\left[\left(\frac{1}{r}\right)^{12}-\left(\frac{1}{r}\right)^6\right]$$evaluated at $r=1$ is___________(in integer).

The value of $(1 + i)^{12}$, where $i = \sqrt{-1}$, is$-64i$$64i$$64$$-64$

Let $f\left ( x \right ) = e ^{-\left |x \right |}$, where $x$ is real. The value of $\dfrac{df}{dx}$ at $x = -1$ is$-e$$e$$\frac{1}{e}$$\frac{-1}{e}$

The value of the real variable $x \geq 0$, which maximizes the function $f\left ( x \right ) = x^{e} e ^{-x}$ is$e$$0$$\frac{1}{e}$$1$

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. ...