Calculus (EE)

Let $a_R$ be the unit radial vector in the spherical co-ordinate system. For which of the following value(s) of $n$, the divergence of the radial vector field $f(R)=a_R \frac{1}{R^n}$ is independent of $R$ ?

Let $(x, y) \in \Re^2$. The rate of change of the real valued function, $V(x, y)=x^2+x+y^2+1$ at the origin in the direction of the point $(1,2)$ is _________ (round off to the nearest integer)

What is the value of $\left(\frac{3^{81}}{27^4}\right)^{1 / 3}$ ?

Spheres of unit diameter are centered at ( $l, m, n$ ) where $l, m$ and $n$ take every possible integer values. The distance between two spheres is computed from the center of one sphere to the center of the other sphere...

Consider the set $S$ of points $(x, y) \in R^2$ which minimize the real valued function $$ f(x, y)=(x+y-1)^2+(x+y)^2 $$ Which of the following statements is true about the set $S$ ?

Consider a vector $\vec{u} = 2\hat{x} + \hat{y} + 2\hat{z}$, where $\hat{x}$, $\hat{y}$, $\hat{z}$ represent unit vectors along the coordinate axes $x$, $y$, $z$ respectively. The directional derivative of the function $...

Consider the function $f(t) = (\text{max}(0,t))^2$ for $- \infty

Let $f(t)$ be a real-valued function whose second derivative is positive for $- \infty

A surveyor has to measure the horizontal distance from her position to a distant reference point C. Using her position as the center, a 200 m horizontal line segment is drawn with the two endpoints A and B. Points A, B,...

For a given vector $${[\matrix{ 1 & 2 & 3 \cr } ]^T}$$, the vector normal to the plane defined by $${w^T}x = 1$$ is

Consider the following equation in a 2-D real-space. $$|{x_1}{|^p} + |{x_2}{|^p} = 1$$ for $$p > 0$$ Which of the following statement(s) is/are true.

A quadratic function of two variables is given as $$f({x_1},{x_2}) = x_1^2 + 2x_2^2 + 3{x_1} + 3{x_2} + {x_1}{x_2} + 1$$ The magnitude of the maximum rate of change of the function at the point (1, 1) is ___________ (Rou...

Let $$\overrightarrow E (x,y,z) = 2{x^2}\widehat i + 5y\widehat j + 3z\widehat k$$. The value of $$\mathop{\int\!\!\!\int\!\!\!\int}\limits_{\kern-5.5pt V} {(\overrightarrow \nabla \,.\,\overrightarrow E )dV} $$, where V...

Let $$f(x) = \int\limits_0^x {{e^t}(t - 1)(t - 2)dt} $$. Then f(x) decreases in the interval.

Let $$f(x,y,z) = 4{x^2} + 7xy + 3x{z^2}$$. The direction in which the function f(x, y, z) increases most rapidly at point P = (1, 0, 2) is

Suppose the circles $x^2+y^2=1$ and $(x-1)^2+(y-1)^2=r^2$ intersect each other orthogonally at the point $(u, v)$. Then $u+v=$ $\_\_\_\_$ .

In the open interval $(0,1)$, the polynomial $p(x)=x^4-4 x^3+2$ has

Let $f(x)$ be a real-valued function such that $f^{\prime}\left(x_0\right)=0$ for some $x_0 \in(0,1)$ and $f^{\prime \prime}\left(x_0\right)>0$ for all $x \in(0,1)$. Then $f(x)$ has

For a regular polygon having 10 sides, the interior angle between the sides of the polygon, in degrees, is

For what values of k given below is $${{{{\left( {k + 2} \right)}^2}} \over {k - 3}}$$ an integer?

The three roots of the equation f(x) = 0 are x = {-2, 0, 3}. What are the three values of x for which f(x -3) = 0?

Functions F(a, b) and G(a, b) are defined as follows: F(a, b) = (a - b) 2 and G(a, b) = |a - b| , where |x| represents the absolute value of x. What would be the value of G(F(1, 3), G(1, 3))?

Consider a function $$f\left( {x,y,z} \right)$$ given by $$f\left( {x,y,z} \right) = \left( {{x^2} + {y^2} - 2{z^2}} \right)\left( {{y^2} + {z^2}} \right).$$ The partial derivative of this function with respect to $$x$$...

Let $$g\left( x \right) = \left\{ {\matrix{ { - x} & {x \le 1} \cr {x + 1} & {x \ge 1} \cr } } \right.$$ and $$f\left( x \right) = \left\{ {\matrix{ {1 - x,} & {x \le 0} \cr {{x^{2,}}} & {x > 0} \cr } } \right..$$ Consid...

For a complex number $$z,$$ $$\mathop {Lim}\limits_{z \to i} {{{z^2} + 1} \over {{z^3} + 2z - i\left( {{z^2} + 2} \right)}}$$ is

A function $$f(x)$$ is defined as $$f\left( x \right) = \left\{ {\matrix{ {{e^x},x < 1} \cr {\ln x + a{x^2} + bx,x \ge 1} \cr } \,\,,\,\,} \right.$$ where $$x \in R.$$ Which one of the following statements is TRUE?

The line integral of the vector field $$\,\,F = 5xz\widehat i + \left( {3{x^2} + 2y} \right)\widehat j + {x^2}z\widehat k\,\,$$ along a path from $$(0, 0, 0)$$ to $$(1,1,1)$$ parameterized by $$\left( {t,{t^2},t} \right)...

Let $$\,\,S = \sum\limits_{n = 0}^\infty {n{\alpha ^n}} \,\,$$ where $$\,\,\left| \alpha \right| < 1.\,\,$$ The value of $$\alpha $$ in the range $$\,\,0 < \alpha < 1,\,\,$$ such that $$\,\,S = 2\alpha \,\,$$ is ________...

The maximum value attained by the function $$f(x)=x(x-1) (x-2)$$ in the interval $$\left[ {1,2} \right]$$ is _________.

The value of line integral $$\,\,\int {\left( {2x{y^2}dx + 2{x^2}ydy + dz} \right)\,\,} $$ along a path joining the origin $$(0, 0, 0)$$ and the point $$(1, 1, 1)$$ is

The value of the integral $$\,\,2\int_{ - \infty }^\infty {\left( {{{\sin \,2\pi t} \over {\pi t}}} \right)} dt\,\,$$ is equal to

The volume enclosed by the surface $$f\left( {x,y} \right) = {e^x}$$ over the triangle bounded by the lines $$x=y;$$ $$x=0;$$ $$y=1$$ in the $$xy$$ plane is ________.

If a continuous function $$f(x)$$ does not have a root in the interval $$\left[ {a,b} \right],\,\,$$ then which one of the following statements is TRUE?

The line integral of function $$F=yzi,$$ in the counterclockwise direction, along the circle $${x^2} + {y^2} = 1$$ at $$z=1$$ is

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

Let $$\,\,\nabla .\left( {fV} \right) = {x^2}y + {y^2}z + {z^2}x,\,\,$$ where $$f$$ and $$V$$ are scalar and vector fields respectively. If $$V=yi+zj+xk,$$ then $$\,V.\left( {\nabla f} \right)$$ is

A particle, starting from origin at $$t=0$$ $$s,$$ is traveling along $$x$$-axis with velocity $$v = {\pi \over 2}\cos \left( {{\pi \over 2}t} \right)m/s$$ At $$t=3$$ $$s,$$ the difference between the distance covered by...

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

To evaluate the double integral $$\int\limits_0^8 {\left( {\int\limits_{y/2}^{\left( {y/2} \right) + 1} {\left( {{{2x - y} \over 2}} \right)dx} } \right)dy,\,\,} $$ we make the substitution $$u = \left( {{{2x - y} \over...

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

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

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

Given a vector field $$\overrightarrow F = {y^2}x\widehat a{}_x - yz\widehat a{}_y - {x^2}\widehat a{}_z,$$ the line integral $$\int {F.dl} $$ evaluated along a segment on the $$x-$$axis from $$x=1$$ to $$x=2$$ is

The direction of vector $$A$$ is radially outward from the origin, with $$\left| A \right| = K\,{r^n}$$ where $${r^2} = {x^2} + {y^2} + {z^2}$$ and $$K$$ is constant. The value of $$n$$ for which $$\nabla .A = 0\,\,$$ is

The maximum value of $$f\left( x \right) = {x^3} - 9{x^2} + 24x + 5$$ in the interval $$\left[ {1,6} \right]$$ is

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

The value of the quantity, where $$P = \int\limits_0^1 {x{e^x}\,dx\,\,\,} $$ is

Divergence of the $$3$$ $$-$$ dimensional radial vector field $$\overrightarrow r $$ is

Divergence of the three-dimentional radial vector field $$\overrightarrow F$$ is

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

If $$(x, y)$$ is continuous function defined over $$\left( {x,y} \right) \in \left[ {0,1} \right] \times \left[ {0,1} \right].\,\,\,$$ Given two constants, $$\,x > {y^2}$$ and $$\,y > {x^2},$$ the volume under $$f(x, y)$...

$$$F\left(x,y\right)=\left(x^2\;+\;xy\right)\;{\widehat a}_x\;+\;\left(y^2\;+\;xy\right)\;{\widehat a}_y$$$. Its line integral over the straight line from (x, y)=(0,2) to (2,0) evaluates to

$$F\left( {x,y} \right) = \left( {{x^2} + xy} \right)\,\widehat a{}_x + \left( {{y^2} + xy} \right)\,\widehat a{}_y.\,\,$$ Its line integral over the straight line from $$(x, y)=(0,2)$$ to $$(x,y)=(2,0)$$ evaluates to

Consider the function $$f\left( x \right) = {\left( {{x^2} - 4} \right)^2}$$ where $$x$$ is a real number. Then the function has

Divergence of the vector field $$v\left( {x,y,z} \right) = - \left( {x\,\cos xy + y} \right)\widehat i + \left( {y\,\cos xy} \right)\widehat j + \left[ {\left( {\sin {z^2}} \right) + {x^2} + {y^2}} \right]\widehat k\,\,$...

The integral $$\,\,{1 \over {2\pi }}\int\limits_0^{2\Pi } {Sin\left( {t - \tau } \right)\cos \tau \,d\tau \,\,\,} $$ equals

Divergence of the vector field $$$\overrightarrow V\left(x,y,z\right)=-\left(x\cos xy\;+\;y\right)\;\widehat i\;+\;\left(y\cos xy\right)\;\widehat j\;+\;\left(\sin\;z^2\;+\;x^2\;+\;y^2\right)\widehat k$$$ is

The expression $$V = \int\limits_0^H {\pi {R^2}{{\left( {1 - {h \over H}} \right)}^2}dh\,\,\,} $$ for the volume of a cone is equal to _________.

If $$S = \int\limits_1^\infty {{x^{ - 3}}dx} $$ then $$S$$ has the value

for the scalar field $$u = {{{x^2}} \over 2} + {{{y^2}} \over 3},\,\,$$ the magnitude of the gradient at the point $$(1,3)$$ is

For the function $$f\left( x \right) = {x^2}{e^{ - x}},$$ the maximum occurs when $$x$$ is equal to

The area enclosed between the parabola $$y = {x^2}$$ and the straight line $$y=x$$ is _______.

Given a vector field $${\overrightarrow F ,}$$ the divergence theorem states that

$$\mathop {Lim}\limits_{\theta \to 0} \,{{\sin \,m\,\theta } \over \theta },$$ where $$m$$ is an integer, is one of the following :

$$\mathop {Lim}\limits_{x \to \infty } \,x\sin {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle x$}} = \_\_\_\_\_.$$

If $$f(0)=2$$ and $$f'\left( x \right) = {1 \over {5 - {x^2}}},$$ then the lower and upper bounds of $$f(1)$$ estimated by the mean value theorem are ______.

The directional derivative of $$f\left( {x,y} \right) = 2{x^2} + 3{y^2} + {z^2}\,\,$$ at point $$P\left( {2,1,3} \right)\,\,$$ in the direction of the vector $$\,\,a = \overrightarrow i - 2\overrightarrow k \,\,$$ is

The integration of $$\int {{\mathop{\rm logx}\nolimits} \,dx} $$ has the value

The volume generated by revolving the area bounded by the parabola $${y^2} = 8x$$ and the line $$x=2$$ about $$y$$-axis is