# 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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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 _________.

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

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

##### Show Answer

A cubic polynomial with real coefficients

##### Show Answer

$\mathbf{F}\left(x,y\right)=\left({x}^{2}+xy\right){\stackrel{\mathbf{^}}{\mathbf{a}}}_{\mathbf{x}}+\left({y}^{2}+xy\right){\stackrel{\mathbf{^}}{\mathbf{a}}}_{y}$. 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

##### Show Answer

Consider function f(x)=(x2-4)2 where x is a real number. Then the function has

##### Show Answer

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