# GATE Questions & Answers of Calculus Electrical Engineering

#### Calculus 23 Question(s)

Let $f\left(x\right)=x{e}^{-x}$. The maximum value of the function in the interval $\left(0,\infty \right)$ is

Minimum of the real valued function $f\left(x\right)={\left(x-1\right)}^{2}{3}}$ occurs at x equal to

To evaluate the double integral $\int\limits_0^8\left(\int\limits_\frac y2^{\left(\frac y2\right)+1}\left(\frac{2x-y}2\right)dx\right)dy$,we make the substitution $u=\left(\frac{2x-y}{2}\right)$ and $v=\frac{y}{2}$ The integral will reduce to

The minimum value of the function  $f\left(x\right)={x}^{3}-3{x}^{2}-24x+100$ in the interval [-3, 3] is

A particle, starting from origin at t = 0 s, is traveling along x-axis with velocity

$\mathrm{v}=\frac{\mathrm{\pi }}{2}\mathrm{cos}\left(\frac{\mathrm{\pi }}{2}t\right)\mathrm{m}/\mathrm{s}$

At t = 3 s, the difference between the distance covered by the particle and the magnitude of displacement from the origin is _________.

Let $\nabla .\left(f\mathbf{v}\right)={x}^{2}y+{y}^{2}z+{z}^{2}x$,where f and v are scalar and vector fields respectively. If $\mathbf{v}\mathbf{=}y\mathbit{i}+z\mathbit{j}+x\mathbit{k}$ then $\mathbf{v}\mathbf{.}\nabla f$is

The mean thickness and variance of silicon steel laminations are 0.2 mm and 0.02 respectively. The varnish insulation is applied on both the sides of the laminations. The mean thickness of one side insulation and its variance are 0.1 mm and 0.01 respectively. If the transformer core is made using 100 such varnish coated laminations, the mean thickness and variance of the core respectively are

The curl of the gradient of the scalar field defined by $V=2{x}^{2}y+3{y}^{2}z+4{z}^{2}x$ is

Given a vector field F= y2xax - yzay -x2az, the line integral evaluated along a segment on the x-axis from x =1 to x = 2 is

A function $y=5{x}^{2}+10x$ is defined over an open interval x=(1,2)  . At least at one point in this interval,$\frac{dy}{dx}$ is exactly

The maximum value of f(x) = x3 - 9x2 + 24x + 5 in the interval [1, 6] is

The direction of vector A is radially outward from the origin, with $\left|A\right|=k{r}^{n}$ where r2 = x2 +y2 +z2 and k is a constant. The value of n for which $\nabla ·A=0$ is

Roots of the algebraic equation x3+ x2+ x + 1 = 0 are

The function $f\left(x\right)=2x-{x}^{2}+3$ has

The two vectors [1,1,1] and [1,a,a2] where $a=\left(-\frac{1}{2}+\mathrm{j}\frac{\sqrt{3}}{2}\right)$ are

The value of the quantity P, where P=$\int\limits_0^1xe^x\;dx$ is equal to

Divergence of the three-dimensional radial vector field $\stackrel{\to }{r}$ is

At t = 0, the function $f\left(t\right)=\frac{\mathrm{sin}t}{t}$ has

f(x,y) is a continuous function defined over (x,y)$\in$ [0,1] × [0,1]. Given the two constraints, x > y2 and y > x2, the volume under f(x,y) is

A cubic polynomial with real coefficients

$\mathbf{F}\left(x,y\right)=\left({x}^{2}+xy\right){\stackrel{\mathbf{^}}{\mathbf{a}}}_{\mathbf{x}}+\left({y}^{2}+xy\right){\stackrel{\mathbf{^}}{\mathbf{a}}}_{y}$. It's line integral over the straight line from $\left(x,y\right)=\left(0,2\right)$ to $\left(x,y\right)=\left(2,0\right)$ evaluates to

The integral $\frac1{2\pi}\int\limits_0^{2\pi}\sin\left(t-\tau\right)\cos\tau\operatorname{d}\tau$ equals