Calculus

$$overrightarrow { ext{a}} ,,overrightarrow { ext{b}} ,,overrightarrow { ext{c}} $$ xa0 are three orthogonal vectors, Given that [overrightarrow {
m{a}} = {
m{hat i}} + 2{
m{hat j}} + 5{
m{hat k}}] xa0 xa0and [overrightarrow {
m{b}} = {
m{hat i}} + 2{
m{hat j}} - {
m{hat k}}] xa0 xa0, the vector $$,overrightarrow { ext{c}} $$ is parallel to

Given $$overrightarrow { ext{F}} = left( {{{ ext{x}}^2} - 2{ ext{y}}}
ight)overrightarrow { ext{i}} - 4{ ext{yz}}overrightarrow { ext{j}} + 4{ ext{x}}{{ ext{z}}^2}overrightarrow { ext{k}} ,$$ xa0 xa0 xa0 the value of the line integral $$intlimits_{ ext{c}} {overrightarrow { ext{F}} cdot doverrightarrow l } $$ xa0 along the straight line c from (0, 0, 0) to (1,1,1) is

At x = 0, the function f(x) = |x| has

A parabola x = y 2 with 0 ≤ x ≤ 1 is shown in the figure. The volume of the solid of rotation obtained by rotating the shaded area by 360° around the x-axis is

The line integral $$intlimits_{{{ ext{P}}_1}}^{{{ ext{P}}_2}} {left( {{ ext{ydx}} + { ext{xdy}}}
ight)} $$ xa0 for P 1 (x 1 , y 1 ) to P 2 (x 2 , y 2 ) along the semicircle P 1 , P 2 shown in the figure is

The angle between two unit-magnitude coplanar vectors P(0.866, 0.500, 0) and Q(0.259, 0.966, 0) will be

The following surface integral is to be evaluated over a sphere for the given steady velocity vector field F = xi + yj + zk defined with respect to a Cartesian coordinate system having i, j and k as unit base vectors. $$iintlimits_{ ext{S}} {frac{1}{4}left( {{ ext{F}} cdot { ext{n}}}
ight){ ext{dA}}}$$ xa0 xa0where S is the sphere, x 2 + y 2 + z 2 = 1 and n is the outward unit normal vector to the sphere. The value of the surface integral is

If a function is continuous at a point,

The vector that is normal to the surface 2xz 2 - 3xy - 4x = 7 at the point (1, -1, 2) is

The derivative of the symmetric function drawn in given figure will look like

The parabolic arc y = √x, 1 ≤ x ≤ 2 is revolved around the x-axis. The volume of the solid of revolution is

Which of the following functions would have only odd powers of x in its Taylor series expansion about the point x = 0?

A function f(x) = 1 - x 2 + x 3 is defined in the closed interval [-1, 1]. The value of x in the open interval (-1, 1) for which the mean value theorem is satisfied, is

For 0 ≤ t < $$infty $$, the maximum value of the function f(t) = e -t - 2e -2t occurs at

Let x be a continuous variable defined over the interval $$left( { - infty ,,infty }
ight)$$ xa0, and f(x) = e -x-e -x . The integral $${ ext{g}}left( { ext{x}}
ight) = int {{ ext{f}}left( { ext{x}}
ight){ ext{dx}}} $$ xa0 xa0is equal to

Divergence of vector field [overrightarrow {
m{V}} left( {{
m{x}},,{
m{y}},,{
m{z}}}
ight) = - left( {{
m{x}}cos {
m{xy}} + {
m{y}}}
ight){
m{hat i}} + left( {{
m{y}}cos {
m{xy}}}
ight){
m{hat j}} + left[ {left( {sin {{
m{z}}^2}}
ight) + {{
m{x}}^2} + {{
m{y}}^2}}
ight]{
m{hat k}}] xa0 xa0 xa0 xa0 xa0 xa0 is

One of the roots of the equation x 3 = j, where j is the positive square root of -1, is

The value of the integral $$intlimits_{ - infty }^infty {frac{{{ ext{dx}}}}{{1 + {x^2}}}} $$ xa0is

If $${ ext{f}}left( { ext{x}}
ight) = frac{{2{{ ext{x}}^2} - 7{ ext{x}} + 3}}{{5{{ ext{x}}^2} - 12{ ext{x}} - 9}},$$ xa0 xa0 then $$mathop {lim }limits_{{ ext{x}} o 3} { ext{f}}left( { ext{x}}
ight)$$ xa0will be

For two non-zero vectors $$overrightarrow { ext{A}} $$ and $$overrightarrow { ext{B}} $$, if $$overrightarrow { ext{A}} $$ + $$overrightarrow { ext{B}} $$ is perpendicular to $$overrightarrow { ext{A}} $$ - $$overrightarrow { ext{B}} $$ then,

What is curl of the vector field 2x 2 yi + 5z 2 j - 4uyzk?

A rail engine accelerates from its stationary position for 8 seconds and travels a distance of 280 m. According to the Mean Value Theorem, the speedometer at a certain time during acceleration must read exactly

Let f(x) = x e -x . The maximum value of the function in the interval (0, [infty ]) is

The contour on the x - y plane, where the partial derivative of x 2 + y 2 with respect to y is equal to the partial derivative of 6y + 4x with respect
to x, is

Consider the following equations [x08egin{gathered}
frac{{partial { ext{V}}left( {{ ext{x, y}}}
ight)}}{{partial { ext{x}}}} = { ext{p}}{{ ext{x}}^2} + {{ ext{y}}^2} + 2{ ext{xy}} hfill \
frac{{partial { ext{V}}left( {{ ext{x, y}}}
ight)}}{{partial { ext{y}}}} = {{ ext{x}}^2} + { ext{q}}{{ ext{y}}^2} + 2{ ext{xy}} hfill \
end{gathered} ] where p and q are constants. V(x, y) that satisfies the above equations is

$$mathop {lim }limits_{{ ext{x}} o 0} frac{{{{ ext{e}}^{ ext{x}}} - left( {1 + { ext{x}} + frac{{{{ ext{x}}^2}}}{2}}
ight)}}{{{{ ext{x}}^3}}} = ?$$

The quadratic approximation of f(x) = x 3 - 3x 2 - 5 a the point x = 0 is

The following inequality is true for all x close to
0. [2 - frac{{{{ ext{x}}^2}}}{3} < frac{{{ ext{x}}sin { ext{x}}}}{{1 - cos { ext{x}}}} < 2] What is the value of [mathop {lim }limits_{{ ext{x}} o 0} frac{{{ ext{x}}sin { ext{x}}}}{{1 - cos { ext{x}}}}?]

The series [sumlimits_{{ ext{n}} = 0}^infty {frac{1}{{{ ext{n}}!}}} ] xa0converges to

As x increased from [ - infty ] xa0to [infty ] , the function [{
m{f}}left( {
m{x}}
ight) = frac{{{{
m{e}}^{
m{x}}}}}{{1 + {{
m{e}}^{
m{x}}}}}]

The value of [mathop {lim }limits_{{ ext{x}} o 1} frac{{{{ ext{x}}^7} - 2{{ ext{x}}^5} + 1}}{{{{ ext{x}}^3} - 3{{ ext{x}}^2} + 2}}]

[mathop {{ ext{Lim}}}limits_{{ ext{x}} o 0} frac{{{ ext{x}} - sin { ext{x}}}}{{1 - cos { ext{x}}}}]

Let f = y x . What is [frac{{{partial ^2}{ ext{f}}}}{{partial { ext{x}}partial { ext{y}}}}] xa0at x = 2, y = 1?

Given y = x 2 + 2x + 10, the value of [{left. {frac{{{ ext{dy}}}}{{{ ext{dx}}}}}
ight|_{{ ext{x}} = 1}}] xa0is equal to

A sphere of unit radius is centered at the origin. The unit normal at a point (x, y, z) on the surface of the sphere is the vector

The total derivative of function xy is

[mathop {{ ext{Lim}}}limits_{{ ext{x}} o 0} left( {frac{{{{ ext{e}}^{2{ ext{x}}}} - 1}}{{sin left( {4{ ext{x}}}
ight)}}}
ight)] xa0 is equal to

The value of the function [{ ext{f}}left( { ext{x}}
ight) = mathop {lim }limits_{{ ext{x}} o 0} frac{{{{ ext{x}}^3} + {{ ext{x}}^2}}}{{2{{ ext{x}}^3} - 7{{ ext{x}}^2}}}] xa0 xa0is

Consider the function f(x) = |x| 3 , where x is real. Then the function f(x) at x = 0 is

[mathop {lim }limits_{{ ext{x}} o infty } {left( {1 + frac{1}{{ ext{x}}}}
ight)^{2{ ext{x}}}}] xa0 is equal to

[mathop {lim }limits_{{ ext{x}} o infty } frac{{{ ext{x}} - sin { ext{x}}}}{{{ ext{x}} + cos { ext{x}}}}] xa0 equals

For the spherical surface x 2 + y 2 + z 2 = 1, the unit outward normal vector at the point [left( {frac{1}{{sqrt 2 }},,frac{1}{{sqrt 2 }},,0}
ight)] xa0 is given by

Divergence of the vector field [{{
m{x}}^2}{
m{zhat i}} + {
m{xyhat j}} - {
m{y}}{{
m{z}}^2}{
m{hat k}}] xa0 xa0at (1, -1, 1) is

[mathop {lim }limits_{{ ext{x}} o 0} left( {frac{{1 - cos { ext{x}}}}{{{{ ext{x}}^2}}}}
ight)] xa0 is

As x varies from -1 to +3, which one of the following describes the behaviour of the function f(x) = x 3 - 3x 2 + 1?

The following plot shows a function y which varies linearly with x. The value of the integral [{ ext{I}} = intlimits_1^2 {{ ext{y dx}}} ] xa0 is

Given: [{ ext{x}}left( { ext{t}}
ight) = 3sin left( {1000pi { ext{t}}}
ight)] xa0 xa0and [{ ext{y}}left( { ext{t}}
ight) = 5cos left( {1000pi { ext{t}} + frac{pi }{4}}
ight)] The X-Y plot will be

The infinite series [1 + { ext{x}} + frac{{{{ ext{x}}^2}}}{{2!}} + frac{{{{ ext{x}}^3}}}{{3!}} + frac{{{{ ext{x}}^4}}}{{4!}} + ,...] xa0 xa0 xa0corresponds to

If [phi = 2{{ ext{x}}^3}{{ ext{y}}^2}{{ ext{z}}^4}] xa0 then [{
abla ^2}phi ] xa0is

The function f(x) = 2x 3 - 3x 2 - 36x + 2 has its maxima at

The minimum value of the function $${ ext{f}}left( { ext{x}}
ight) = left( {frac{{{{ ext{x}}^3}}}{3}}
ight) - { ext{x}}$$ xa0 xa0occurs at

The function f(x) = 2x - x 2 + 3 has

The function y = |2 - 3x|

Which of the following is correct?

The double integral $$intlimits_0^{ ext{a}} {intlimits_0^{ ext{y}} {{ ext{f}}left( {{ ext{x, y}}}
ight){ ext{dx dy}}} } $$ xa0 xa0is equivalent to

The values of x for which the function $${ ext{f}}left( { ext{x}}
ight) = frac{{{{ ext{x}}^2} - 3{ ext{x}} - 4}}{{{{ ext{x}}^2} + 3{ ext{x}} - 4}}$$ xa0 xa0is NOT continuous are

The expression $${ ext{V}} = int_0^{ ext{H}} {pi {{ ext{R}}^2}{{left( {1 - frac{{ ext{h}}}{{ ext{H}}}}
ight)}^2}} { ext{dh}}$$ xa0 xa0 xa0for the volume of a cone is equal to

Consider the function f(x) = |x| in the interval -1 < x ≤ 1. At the point x = 0, f(x) is

For a scalar function f(x, y, z) = x 2 + 3y 2 + 2z 2 , the gradient at the point P(1, 2, -1) is

The integral [intlimits_0^pi {{{sin }^3} heta ,{ ext{d}} heta } ] xa0 is given by

The volume of an object expressed in spherical co-ordinates is given by [{ ext{V}} = int_0^{2pi } {int_0^{frac{pi }{3}} {int_0^1 {{{ ext{r}}^2}sin phi ,{ ext{dr d}}phi ,{ ext{d}} heta .} } } ] The value of the integral is

The value of the integral [int_0^infty {int_0^infty {{{ ext{e}}^{ - {{ ext{x}}^2}}}{{ ext{e}}^{ - {{ ext{y}}^2}}}} } { ext{dx dy}}] xa0 xa0 is

The range of value of k for which the function f(x) = (k 2 - 4)x 2 + 6x 3 + 8x 4 has a local maxima at point x = 0 is

The area of the region bounded by the parabola y = x 2 + 1 and the straight line x + y = 3 is

The maximum value of f(x) = x 3 - 9x 2 + 24x + 5 in the interval [1, 6] is

The value of the integral [int_0^{2pi } {left( {frac{3}{{9 + {{sin }^2} heta }}}
ight){ ext{d}} heta } ] xa0 xa0 is

A velocity vector is given as [overrightarrow { ext{V}} = 5{ ext{xy}}overrightarrow { ext{i}} + 2{{ ext{y}}^2}overrightarrow { ext{j}} + 3{ ext{y}}{{ ext{z}}^2}overrightarrow { ext{k}} .] xa0 xa0 xa0 The divergence of this velocity vector at (1, 1, 1) is

Given a vector field [overrightarrow {
m{F}} = {{
m{y}}^2}{
m{x}}{{{
m{hat a}}}_{
m{x}}} - {
m{yz}}{{{
m{hat a}}}_{
m{y}}} - {{
m{x}}^2}{{{
m{hat a}}}_{
m{z}}},] xa0 xa0 the line integral [int {overrightarrow { ext{F}} cdot overrightarrow {{ ext{d}}l} } ] xa0evaluated along a segment on the x-axis from x = 1 to x = 2 is

At the point x = 0, the function f(x) = x 3 has

Curl of vector V(x, y, z) = 2x 2 i + 3z 2 j + y 3 k at x = y = z = 1 is

For a scalar function f(x, y, z) = x 2 + 3y 2 + 2z 2 , the directional derivative at the point P(1, 2, -1) in the direction of a vector [overrightarrow { ext{i}} - overrightarrow { ext{j}} + 2overrightarrow { ext{k}} ] xa0 is

The directional derivative of the function f(x, y) = x 2 + y 2 along a line directed from (0, 0) to (1, 1), evaluated at point x = 1, y = 1 is

Value of the integral [ointlimits_{ ext{c}} {left( {{ ext{xydy}} - {{ ext{y}}^2}{ ext{dx}}}
ight)} ] xa0 , where c is the square cut from the first quadrant by the lines x = 1 and y = 1 will be (Use Green's theorem to change the line integral into double integral)

The tangent to the curve represented by y = x $$l$$nx is required to have 45° inclination with the x-axis. The coordinates of the tangent point would be

Consider the function f(x) = sin (x) in the interval [{ ext{x}} in left[ {frac{pi }{4},,frac{{7pi }}{4}}
ight].] xa0 The number and location(s) of the local minima of this function are

Euclidean norm (length) of the vector [4 -2 -6] T is

∇ × ∇ × P where P is a vector is equal to

A parabolic cable is held between two supports at the same level. The horizontal span between the supports is L. The sag at the mid-span is h. The equation of the parabola is [{ ext{y}} = 4{ ext{h}}left( {frac{{{{ ext{x}}^2}}}{{{{ ext{L}}^2}}}}
ight)] xa0, where x is the horizontal coordinate and y is the vertical coordinate with the origin at the centre of the cable. The expression for the total length of the cable is

The value of the definite integral $$int_1^{ ext{e}} {sqrt { ext{x}} } ln left( { ext{x}}
ight){ ext{dx}}$$ xa0 xa0is

In the Taylor series expansion of e x about x = 2, the coefficient of (x - 2) 4 is

The value of the integral $$int_0^pi {{ ext{x}}{{cos }^2}{ ext{xdx}}} $$ xa0xa0is

The value of the line integral $$intlimits_{ ext{c}} {left( {2{ ext{x}}{{ ext{y}}^2}{ ext{dx}} + 2{{ ext{x}}^2}{ ext{ydy}} + { ext{dz}}}
ight)} $$ xa0 xa0 along a path joining the origin (0, 0, 0) and the point (1, 1, 1) is

Consider a vector field $$overrightarrow { ext{A}} left( {overrightarrow { ext{r}} }
ight).$$ xa0The closed loop line integral $$oint {overrightarrow { ext{A}} cdot overrightarrow {{ ext{d}}l} } $$ xa0can be expressed as

The area between the parabolas y 2 = 4ax and x 2 = 4ay is

$$mathop {lim }limits_{{ ext{x}} o infty } {{ ext{x}}^{frac{1}{{ ext{x}}}}}$$ xa0is

For a position vector [{
m{r}} = {
m{xhat i}} + {
m{yhat j}} + {
m{zhat k}}] xa0 xa0the norm of the vector can be defined as $$left| {overrightarrow { ext{r}} }
ight| = sqrt {{{ ext{x}}^2} + {{ ext{y}}^2} + {{ ext{z}}^2}} .$$ xa0 xa0 Given a function $$phi = ln left| {overrightarrow { ext{r}} }
ight|,$$ xa0 its gradient $$
abla phi $$ xa0is

According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value $$xi $$ in this interval such that $$intlimits_{ ext{a}}^{ ext{b}} {{ ext{f}}left( { ext{x}}
ight){ ext{dx}} = } $$

Let $$
abla cdot left( {{ ext{f}}overrightarrow { ext{v}} }
ight) = {{ ext{x}}^2}{ ext{y}} + {{ ext{y}}^2}{ ext{z}} + {{ ext{z}}^2}{ ext{x}},$$ xa0 xa0 xa0where f and v are scalar and vector fields respectively. If $$overrightarrow { ext{v}} = { ext{y}}overrightarrow { ext{i}} + { ext{z}}overrightarrow { ext{j}} + { ext{x}}overrightarrow { ext{k}} ,$$ xa0 xa0 then $$overrightarrow { ext{v}} cdot
abla { ext{f}}$$ xa0 is

The polynomial p(x) = x 5 + x + 2 has

A function y = 5x 2 + 10x is defined over an open interval x = (1, 2). At least at one point in this interval, $$frac{{{ ext{dy}}}}{{{ ext{dx}}}}$$xa0is exactly

Let x and y be two vectors in a 3 dimensional space and <x, y> denote their dot product. Then the determinant det [left[ {x08egin{array}{*{20}{c}}
{ < { ext{x}},{ ext{x}} > }&{ < { ext{x}},{ ext{y}} > } \
{ < { ext{y}},{ ext{x}} > }&{ < { ext{y}},{ ext{y}} > }
end{array}}
ight].]

$$mathop {{ ext{Lim}}}limits_{{ ext{x}} o infty } left( {frac{{{ ext{x}} + sin { ext{x}}}}{{ ext{x}}}}
ight)$$ xa0 xa0equal to

For a vector E, which one of the following statements is NOT TRUE ?

Which one of the following functions is continuous at x = 3?

If A(0, 4, 3), B(0, 0, 0) and C(3, 0, 4) are three points defined in x, y, zeo-ordinate system, then which of the following vector is perpendicular to both vectors [overrightarrow {{ ext{AB}}} ] xa0and [overrightarrow {{
m{BC}}} .]

A function f(x) is defined as [{ ext{f}}left( { ext{x}}
ight) = left{ {x08egin{array}{*{20}{c}}
{{{ ext{e}}^x}}&{{ ext{x}} < 1} \
{ln { ext{x}} + { ext{a}}{{ ext{x}}^2} + { ext{bx}},}&{{ ext{x}} geqslant 1}
end{array}}
ight.] where x[ in ] R which one of the following statements is TRUE?

The inner (dot) product of two non zero vectors [overrightarrow { ext{P}} ] and [overrightarrow { ext{Q}} ] is zero. The angle (degrees) between the two vectors is

Let f(x) = e x + x 2 for real x. From among the following, choose the Taylor series approximation of f(x) around x = 0, which includes all powers of x less than or equal to 3,

The value of the quantity P, where $${ ext{P}} = intlimits_0^1 {{ ext{x}}{{ ext{e}}^{ ext{x}}}{ ext{dx,}}} $$ xa0 is equal to

What is the value of $$mathop {lim }limits_{{ ext{x}} o frac{pi }{4}} frac{{cos { ext{x}} - sin { ext{x}}}}{{{ ext{x}} - frac{pi }{4}}}$$

The value of $$mathop {lim }limits_{{ ext{x}} o infty } {left( {1 + {{ ext{x}}^2}}
ight)^{{{ ext{e}}^{ - { ext{x}}}}}}$$ xa0 is

For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the angle between the hypotenuse and the side is

The value of $$mathop {lim }limits_{{ ext{x}} o 8} frac{{{{ ext{x}}^{frac{1}{3}}} - 2}}{{left( {{ ext{x}} - 8}
ight)}}$$

The vector that is NOT perpendicular to the vectors (i + j + k) and (i + 2j + 3k) is . . . . . . . .

Given $${ ext{i}} = sqrt { - 1} ,$$ xa0 what will be the evaluation of the definite integral $$int_0^{frac{pi }{2}} {frac{{cos { ext{x}} + { ext{i}}sin { ext{x}}}}{{cos { ext{x}} - { ext{i}}sin { ext{x}}}}} { ext{dx}},?$$

At x = 0, the function $${ ext{f}}left( { ext{x}}
ight) = left| {sin frac{{2pi { ext{x}}}}{{ ext{L}}}}
ight|$$ xa0 , xa0(-$$infty $$ < x < $$infty $$, L > 0) is

The value of $$int_0^infty {frac{1}{{1 + {{ ext{x}}^2}}}} { ext{dx}} + int_0^infty {frac{{sin { ext{x}}}}{{ ext{x}}}} { ext{dx}}$$ xa0 xa0 xa0is

The values of the integrals $$intlimits_0^1 {left( {intlimits_0^1 {frac{{{ ext{x}} - { ext{y}}}}{{{{left( {{ ext{x}} + { ext{y}}}
ight)}^3}}}{ ext{dy}}} }
ight){ ext{dx}}} $$ xa0 xa0 and $$intlimits_0^1 {left( {intlimits_0^1 {frac{{{ ext{x}} - { ext{y}}}}{{{{left( {{ ext{x}} + { ext{y}}}
ight)}^3}}}{ ext{dx}}} }
ight){ ext{dy}}} $$ xa0 xa0 are

What is the value of the definite integral, $$intlimits_0^{ ext{a}} {frac{{sqrt { ext{x}} }}{{sqrt { ext{x}} + sqrt {{ ext{a}} - { ext{x}}} }}} { ext{dx}},{ ext{?}}$$

The direction of vector A is radially outward from the origin, with |A| = kr n where r 2 = x 2 + y 2 + z 2 and k is a constant. The value of n for which $$
abla cdot { ext{A}} = 0$$ xa0 is

The angle of intersection of the curves x 2 = 4y and y 2 = 4x at point (0, 0) is

The expression e -$$l$$n x for x > 0 is equal to

While minimizing the function f(x), necessary and sufficient conditions for a point x 0 to be a minima are

The function f(x) = x 2 = x + x ...... x times, is defined

The position vector $$overrightarrow {{ ext{OP}}} $$xa0of P(20, 10) is rotated anti-clockwise in X-Y plane by an angle θ = 30° such that the point P occupies position Q, as shown in the figure. The coordinates (x, y) of Q are

If z = xy $$l$$n(xy), then

If $$overrightarrow { ext{a}} $$ and $$overrightarrow { ext{b}} $$ are two arbitrary vectors with magnitudes a and b, respectively, $${left| {overrightarrow { ext{a}} imes overrightarrow { ext{b}} }
ight|^2}$$ xa0will be equal to

The value of the integral given below is $$intlimits_0^pi {{{ ext{x}}^2}cos ,{ ext{x dx}}} $$

Let $${ ext{f}}left( {{ ext{x, y}}}
ight) = frac{{{ ext{a}}{{ ext{x}}^2} + { ext{b}}{{ ext{y}}^2}}}{{{ ext{xy}}}},$$ xa0 xa0 where a and b are constants. If $$frac{{partial { ext{f}}}}{{partial { ext{x}}}} = frac{{partial { ext{f}}}}{{partial { ext{y}}}}$$ xa0 at x = 1 and y = 2, then the relation between a and b is

The curve y = f(x) is such that the tangent to the curve at every point (x, y) has a Y-axis intercept c, given by c = -y. Then f(x) is proportional to

If f(x) is an even function and a is a positive real number, then $$int_{ - { ext{a}}}^{ ext{a}} {{ ext{f}}left( { ext{x}}
ight){ ext{dx}}} $$ xa0 equals

Curl of vector [overrightarrow {
m{F}} = {{
m{x}}^2}{{
m{z}}^2}{
m{hat i}} - 2{
m{x}}{{
m{y}}^2}{
m{zhat j}} + 2{{
m{y}}^2}{{
m{z}}^3}{
m{hat k}}] xa0 xa0 xa0is

The directional derivative of the scalar function f(x, y, z) = x 2 + 2y 2 + z at the point P = (1, 1, 2) in the direction of the vector [overrightarrow {
m{a}} = 3{
m{hat i}} - 4{
m{hat j}}] xa0 is

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

The value of $$mathop {lim }limits_{left( {{ ext{x,}},{ ext{y}}}
ight) o left( {0,,0}
ight)} frac{{{{ ext{x}}^2} - { ext{xy}}}}{{sqrt { ext{x}} - sqrt { ext{y}} }}$$ xa0 xa0is

If $${{ ext{e}}^{ ext{y}}} = {{ ext{x}}^{frac{1}{{ ext{x}}}}},$$ xa0then y has a

$$mathop {lim }limits_{{ ext{x}} o 0} frac{{{{sin }^2}{ ext{x}}}}{{ ext{x}}}$$ xa0 is equal to

Equation of the line normal to function $${ ext{f}}left( { ext{x}}
ight) = {left( {{ ext{x}} - 8}
ight)^{frac{2}{3}}} + 1$$ xa0 xa0at P(0, 5) is

For a small value of h, the Taylor series expansion for f(x + h) is

The divergence of the vector field [{
m{3xzhat i}} + 2{
m{xyhat j}} - {
m{y}}{{
m{z}}^2}{
m{hat k}}] xa0 xa0at a point (1, 1, 1) is equal to

The right circular cone of largest volume that can be enclosed by a sphere of 1 m radius has a height of

The area enclosed between the curves y 2 = 4x and x 2 = 4y is

It is known that two roots of the nonlinear equation x 3 - 6x 2 + 11x - 6 = are 1 and 3. The third root will be

The derivative of f(x) = cos x can be estimated using the approximation [{ ext{f}}'left( { ext{x}}
ight) = frac{{{ ext{f}}left( {{ ext{x}} + { ext{h}}}
ight) - { ext{f}}left( {{ ext{x}} - { ext{h}}}
ight)}}{{2{ ext{h}}}}.] The percentage error is calculated as [left( {frac{{{ ext{Exact value}} - { ext{Approx value}}}}{{{ ext{Exact value}}}} imes 100}
ight)] The percentage error in the derivative of f(x) at [{ ext{x}} = frac{pi }{6}] xa0radian choosing h = 0.1 radian is

If [overrightarrow {
m{A}} = {
m{xy}}{{{
m{hat a}}}_{
m{x}}} + {{
m{x}}^2}{{{
m{hat a}}}_{
m{y}}},,ointlimits_{
m{c}} {overrightarrow {
m{A}} cdot {
m{d}}overrightarrow l } ] xa0 xa0 over the path shown in the figure is

The expression [mathop {lim }limits_{alpha o 0} frac{{{{ ext{x}}^alpha } - 1}}{alpha }] xa0 is equal to

It is known that two roots of the non-linear equation x 3 - 6x 2 + 11x - 6 = 0 and 1 and 3. The third root will be

Consider the function y = x 2 - 6x + 9. The maximum value of y obtained when x varies over the interval 2 to 5 is

What is the value of [mathop {mathop {lim }limits_{{ ext{x}} o 0} }limits_{{ ext{y}} o 0} frac{{{ ext{xy}}}}{{{{ ext{x}}^2} + {{ ext{y}}^2}}}?]

The Taylor series expansion of 3 sinx + 2 cosx is . . . . . . . .

[mathop {lim }limits_{{ ext{x}} o infty } sqrt {{{ ext{x}}^2} + { ext{x}} - 1} - { ext{x is}}]

The Taylor series expansion of [frac{{sin { ext{x}}}}{{{ ext{x}} - pi }}] xa0at [{ ext{x}} = pi ] xa0is given by

If x = a (θ + sin θ) and y = a (1 - cos θ), then [frac{{{ ext{dy}}}}{{{ ext{dx}}}}]xa0will be equal to

Consider the line integral [intlimits_{ ext{C}} {left( {{ ext{xdy}} - { ext{ydx}}}
ight)} ] xa0 the integral being taken in a counterclockwise direction over the closed curve C that forms the boundary of the region R shown in the figure below. The region R is the area enclosed by the union of a 2 × 3 rectangle and a semi-circle of radius 1. The line integral evaluates to

Consider the shaded triangular region P shown in the figure. What is If $$iintlimits_{ ext{P}} {{ ext{xydxdy}},?}$$

Let [phi ] be an arbitrary smooth real valued scalar function and V be an arbitrary smooth vector valued function in a three-dimensional space. Which one of the following is an identity?

The magnitude of the directional derivative of the function f(x, y) = x 2 + 3y 2 in a direction normal to the circle x 2 + y 2 = 2, at the point (1, 1), is

If $${ ext{S}} = intlimits_1^infty {{{ ext{x}}^{ - 3}}{ ext{dx,}}} $$ xa0 then S has the value

By a change of variable x(u, v) = uv, y(u, v) = v/u is double integral, the integrand f(x, y) changes to f(uv, v/u) [phi ] (u, v). Then, [phi ] (u, v) is

Which one of the following describes the relationship among the three vectors, [{
m{hat i}} + {
m{hat j}} + {
m{hat k}},,2{
m{hat i}} + 3{
m{hat j}} + {
m{hat k}}] xa0 xa0 and [{
m{5hat i}} + 6{
m{hat j}} + 4{
m{hat k}},{
m{?}}]

Let the function [{ ext{f}}left( heta
ight) = left| {x08egin{array}{*{20}{c}}
{sin heta }&{cos heta }&{ an heta } \
{sin left( {frac{pi }{6}}
ight)}&{cos left( {frac{pi }{6}}
ight)}&{ an left( {frac{pi }{6}}
ight)} \
{sin left( {frac{pi }{3}}
ight)}&{cos left( {frac{pi }{3}}
ight)}&{ an left( {frac{pi }{3}}
ight)}
end{array}}
ight|] where [ heta in left[ {frac{pi }{6},,frac{pi }{3}}
ight]] xa0 and [{ ext{f'}}left( heta
ight)] xa0denote the derivative of f with respect to [ heta ]. Which of the following statements is/are TRUE? I. There exists [ heta in left( {frac{pi }{6},,frac{pi }{3}}
ight)] xa0 such that [{ ext{f'}}left( heta
ight) = 0.] II. There exists [ heta in left( {frac{pi }{6},,frac{pi }{3}}
ight)] xa0 such that
[{ ext{f'}}left( heta
ight)
e 0]

The value of the integral [intlimits_0^2 {intlimits_0^{ ext{x}} {{{ ext{e}}^{{ ext{x}} + { ext{y}}}}} } { ext{dy dx}}]

For the scalar field [{ ext{u}} = frac{{{{ ext{x}}^2}}}{2} + frac{{{{ ext{y}}^2}}}{3},] xa0 magnitude of the gradient at the point (1, 3) is

A series expansion for the function sin θ is

Directional derivative of [phi ] = 2xz - y 2 at the point (1, 3, 2) becomes maximum in the direction of:

The area enclosed between the straight line y = x and the parabola y = x 2 in the x - y plane is

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

The local minima of function f(x) = x 2 - x 4 in the range -0.8 ≤ x ≤ 0.8 is located at

Let f be a real-valued function of a real variable defined as f(x) = x 2 for x ≥ 0, and f(x) = -x 2 for x < 0. Which one of the following statements is true?

The value of [mathop {lim }limits_{{ ext{x}} o infty } {left( {1 + frac{1}{{ ext{x}}}}
ight)^{ ext{x}}}] xa0 is

The divergence of the vector field [overrightarrow {
m{U}} = {{
m{e}}^{
m{x}}}left( {cos ,{
m{yhat i}} + sin {
m{yhat j}}}
ight)] xa0 xa0 is

Given a function f(x, y) = 4x 2 + 6y 2 - 8x - 4y + 8. The optimal value of f(x, y)

To evaluate the double integral [int_0^8 {left( {int_{frac{{ ext{y}}}{2}}^{frac{{ ext{y}}}{2} + 1} {left( {frac{{2{ ext{x}} - { ext{y}}}}{2}}
ight){ ext{dx}}} }
ight){ ext{dy,}}} ] xa0 xa0 xa0we make the substitution [{ ext{u}} = frac{{2{ ext{x}} - { ext{y}}}}{2}] xa0 and [{ ext{v}} = frac{{ ext{y}}}{2}.] xa0The integral will reduce to

[iint {left( {
abla imes { ext{P}}}
ight) cdot { ext{ds,}}}] xa0 xa0where P is a vector, is equal to

[mathop {{ ext{Lim}}}limits_{{ ext{x}} o 0} frac{{sin { ext{x}}}}{{ ext{x}}}] xa0 is

At t = 0, the function [{ ext{f}}left( { ext{t}}
ight) = frac{{sin { ext{t}}}}{{ ext{t}}}] xa0 has

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve 3x 4 - 16x 3 - 24x 2 + 37 is

If P, Q and R are three points having coordinates (3, -2, -1), (1, 3, 4), (2, 1, -2) in XYZ space, then the distance from point P to plane OQR (O being the origin of the coordinate system) is given by

The value of the integral [intlimits_0^2 {frac{{{{left( {{ ext{x}} - 1}
ight)}^2}sin left( {{ ext{x}} - 1}
ight)}}{{{{left( {{ ext{x}} - 1}
ight)}^2} + cos left( {{ ext{x}} - 1}
ight)}}} { ext{dx}}] xa0 xa0 xa0is

Stokes theorem connects

Which of the following integrals is unbounded?

Changing the order of the integration in the double integral [{ ext{I}} = intlimits_0^8 {intlimits_{frac{{ ext{x}}}{4}}^2 {{ ext{f}}left( {{ ext{x,}},{ ext{y}}}
ight){ ext{dydx}}} } ] xa0 xa0 leads to [{ ext{I}} = intlimits_{ ext{r}}^{ ext{s}} {intlimits_{ ext{p}}^{ ext{q}} {{ ext{f}}left( {{ ext{x,}},{ ext{y}}}
ight){ ext{dx dy}}} .} ] xa0 xa0 What is q?

The divergence of the vector field [overrightarrow {
m{A}} = {
m{x}}{{{
m{hat a}}}_{
m{x}}} + {
m{y}}{{{
m{hat a}}}_{
m{y}}} + {
m{z}}{{{
m{hat a}}}_{
m{z}}}] xa0 xa0is

Compute [mathop {lim }limits_{{ ext{x}} o 3} frac{{{{ ext{x}}^4} - 81}}{{2{{ ext{x}}^2} - 5{ ext{x}} - 3}}]

The angle (in degree) between two planes vectors [overrightarrow {
m{a}} = frac{{sqrt 3 }}{2}{
m{hat i}} + frac{1}{2}{
m{hat j}}] xa0 xa0and [overrightarrow {
m{b}} = frac{{ - sqrt 3 }}{2}{
m{hat i}} + frac{1}{2}{
m{hat j}}] xa0 xa0is

The length of the curve [{ ext{y}} = frac{2}{3}{{ ext{x}}^{frac{3}{2}}}] xa0between x = 0 and x = 1 is

Consider the function f(x) = x 2 - x - 2. The maximum value of f(x) in the closed interval [-4, 4] is

[mathop {lim }limits_{{ ext{x}} o 0} frac{{{{log }_{ ext{e}}}left( {1 + 4{ ext{x}}}
ight)}}{{{{ ext{e}}^{3{ ext{x}}}} - 1}}] xa0 is equal to

Velocity vector of a flow field is given as [overrightarrow {
m{V}} = 2{
m{xyhat i}} - {{
m{x}}^2}{
m{zhat j}}{
m{.}}] xa0xa0The vorticity vector at (1, 1, 1) is

If $$overrightarrow { ext{r}} $$ is the position vector of any point on a closed surface S that encloses volume V then $$iintlimits_{ ext{S}} {overrightarrow { ext{r}} cdot { ext{d}}overrightarrow { ext{S}} }$$ xa0is equal to

[mathop {lim }limits_{ heta o 0} frac{{sin frac{ heta }{2}}}{ heta }] xa0is

At x = 0, the function f(x) = x 3 + 1 has

If T(x, y, z) = x 2 + y 2 + 2z 2 defines the temperatures at any location (x, y, z) then magnitude of temperature gradient at P(1, 1, 1) is

The value of [mathop {lim }limits_{{ ext{x}} o 0} frac{{{{ ext{x}}^3} - sin left( { ext{x}}
ight)}}{{ ext{x}}}] xa0 is

The divergence of the vector field [left( {{
m{x}} - {
m{y}}}
ight){
m{hat i}} + left( {{
m{y}} - {
m{x}}}
ight){
m{hat j}} + left( {{
m{x}} + {
m{y}} + {
m{z}}}
ight){
m{hat k}}] xa0 xa0 xa0 is

The solution of [intlimits_1^{ ext{a}} {intlimits_1^{ ext{b}} {frac{{{ ext{dxdy}}}}{{{ ext{xy}}}}} } ] xa0is

The line integral [int {overrightarrow { ext{V}} .{ ext{d}}overrightarrow { ext{r}} } ] xa0of the vector [overrightarrow {
m{V}} .left( {overrightarrow {
m{r}} }
ight) = 2{
m{xyzhat i}} + {{
m{x}}^2}{
m{zhat j}} + {{
m{x}}^2}{
m{yhat k}}] xa0 xa0 xa0from the origin to the point P(1, 1, 1)

The minimum value of function y = x 2 in the interval [1, 5] is

Let f(x) = [{{
m{x}}^{ - frac{1}{3}}}]xa0and A denote the area of the region bounded by f(x) and the X-axis, when x varies from -1 to 1. Which of the following statements is/are True? 1. f is continuous in [-1, 1] 2. f is not bounded in [-1, 1] 3. A is nonzero and finite

What is the area common to the circles r = a and r = 2a cos θ?

What should be the value of [lambda ] such that the function defined below is continuous at [{ ext{x}} = frac{pi }{2}?] [{ ext{f}}left( { ext{x}}
ight) = left{ {x08egin{array}{*{20}{c}}
{frac{{lambda cos { ext{x}}}}{{frac{pi }{2} - { ext{x}}}}}&{{ ext{if x}}
e frac{pi }{2}} \
1&{{ ext{if x}} = frac{pi }{2}}
end{array}}
ight.]

For the function f(x) = x 2 e -x , the maximum occurs when x is equal to

The value of integral [intlimits_{ - frac{pi }{2}}^{frac{pi }{2}} {left( {{ ext{x}}cos { ext{x}}}
ight){ ext{dx}}} ] xa0 is

For the two functions, f(x, y) = x 3 - 3xy 2 and g(x, y) = 3x 2 y - y 2 , which one of the following options is correct?

A function f(x) is continuous in the interval [0, 2]. It is known that f(0) = f(2) = -1 and f(1) = 1. Which one of the following statements must be true?

How many distinct values of x satisfy the equation sin(x) = [frac{{ ext{x}}}{2}], where x is in radians?

The [mathop {lim }limits_{{ ext{x}} o 0} frac{{sin left[ {frac{2}{3}{ ext{x}}}
ight]}}{{ ext{x}}}] xa0 is

The value expression [mathop {lim }limits_{{ ext{x}} o 0} frac{{sin { ext{x}}}}{{{{ ext{e}}^{ ext{x}}}{ ext{x}}}}] xa0is

Assuming [{ ext{i}} = sqrt { - 1} ] xa0 and t is a real number, [intlimits_0^{frac{pi }{3}} {{{ ext{e}}^{{ ext{it}}}}} { ext{dt}}] xa0 is

If [{ ext{f}}left( { ext{x}}
ight) = { ext{R}}sin left( {frac{{pi { ext{x}}}}{2}}
ight) + { ext{S}},{ ext{f}}'left( {frac{1}{2}}
ight) = sqrt 2 ] xa0 xa0 xa0 and [intlimits_0^1 {{ ext{f}}left( { ext{x}}
ight){ ext{dx}} = frac{{2{ ext{R}}}}{pi }} ,] xa0 xa0 then the constants R and S are, respectively

The vector function [overrightarrow { ext{A}} ] is given by [overrightarrow { ext{A}} = overrightarrow
abla { ext{u,}}] xa0where u(x, y) is a scalar function, Then [left| {overrightarrow
abla imes overrightarrow { ext{A}} }
ight|]

If [{ ext{u}} = log left( {frac{{{{ ext{x}}^2} + {{ ext{y}}^2}}}{{{ ext{x}} + { ext{y}}}}}
ight),] xa0 xa0what is the value of [{ ext{x}}frac{{partial { ext{u}}}}{{partial { ext{x}}}} + { ext{y}}frac{{partial { ext{u}}}}{{partial { ext{y}}}},?]

Consider the plot f(x) versus x as shown below Suppose [{ ext{F}}left( { ext{x}}
ight) = int_{ - 5}^x {{ ext{f}}left( { ext{y}}
ight)} { ext{dy}}{ ext{.}}] xa0 xa0Which one of the following is a graph of F(x)?

Consider the following definite integral: [{ ext{I}} = intlimits_0^1 {frac{{{{left( {{{sin }^{ - 1}}{ ext{x}}}
ight)}^2}}}{{sqrt {1 - {{ ext{x}}^2}} }}{ ext{dx}}} ] The value of the integral is

The value of the directional derivative of the function [phi ](x, y, z) = xy 2 + yz 2 + zx 2 at the point (2, -1, 1) in the direction of the vector p = i + 2j + 2k is

Which one of the following graphs describes the function f(x) = e -x (x 2 + x + 1)?

[intlimits_0^{frac{pi }{4}} {frac{{left( {1 - an { ext{x}}}
ight)}}{{left( {1 + an { ext{x}}}
ight)}}{ ext{dx}}} ] xa0 xa0evaluates to

Divergence of the three-dimensional radial vector field [overrightarrow { ext{r}} ] is

In the Taylor series expansion of exp(x) + sin(x) about the point x = π, the coefficient of (x - π) 2 is

The value of the integral of the function g(x, y) = 4x 3 + 10y 4 along the straight line segment from the point (0, 0) to the point (1, 2) in the x - y plane is

The value of [mathop {lim }limits_{{
m{x}} o 0} frac{{1 - cos left( {{{
m{x}}^2}}
ight)}}{{2{x^4}}}] xa0 is

Consider points P and Q in the x-y plane, with P = (1, 0) and Q = (0, 1). The line integral
[2intlimits_{
m{P}}^{
m{Q}} {left( {{
m{xdx}} + {
m{ydy}}}
ight)} ] xa0 xa0along the semicircle with the line segment PQ as its diameter

A surface S(x, y) = 2x + 5y - 3 is integrated once over a path consisting of the points that satisfy (x + 1) 2 + (y - 1) 2 = √2. The integral evaluates to

Let w = f(x, y), where x and y are functions of t. Then, according to the chain rule, [frac{{{
m{dw}}}}{{{
m{dt}}}}] is equal

For the parallelogram OPQR shown in the sketch, [overline {{
m{OP}}} = {
m{ahat t}} + {
m{bhat j}}] xa0 and [overline {{
m{OR}}} = {
m{chat t}} + {
m{dhat j}}{
m{.}}] xa0 The area of the parallelogram is

The distance between the origin and the point nearest to it on the surface z 2 = 1 + xy is

The value of [intlimits_0^3 {intlimits_0^{
m{x}} {left( {6 - {
m{x}} - {
m{y}}}
ight)} {
m{dx,dy}}} ] xa0 xa0 is

If for non-zero x, [{
m{af}}left( {
m{x}}
ight) + {
m{bf}}left( {frac{1}{{
m{x}}}}
ight) = frac{1}{{
m{x}}} - 25] xa0 xa0 xa0where a ≠ b then [intlimits_1^2 {{
m{f}}left( {
m{x}}
ight){
m{dx}}} ] xa0 is

The improper integral [intlimits_0^infty {{{
m{e}}^{ - 2{
m{t}}}}} {
m{dt}}] xa0converges to

Minimum of the real valued function [{
m{f}}left( {
m{x}}
ight) = {left( {{
m{x}} - 1}
ight)^{frac{2}{3}}}] xa0 occurs at x equal to

Given the following statements about a function f : R → R , select the right option: P: If f(x) is continuous at x = x 0 , then it is differential at x = x 0 . Q: If f(x) is continuous at x = x 0 , then it may not be differentiable at x = x 0 . R: If f(x) is differentiable at x = x 0 , then it is also continuous at x = x 0 .

The integral [frac{1}{{sqrt {2pi } }}intlimits_{ - infty }^infty {{{
m{e}}^{ - frac{{{x^2}}}{2}}}} {
m{dx}}] xa0 xa0is equal to

The area of a triangle formed by the tips of vectors [overline {
m{a}} {
m{,}},overline {
m{b}} ] xa0and [overline {
m{c}} ] is

For real x the maximum value of [frac{{{{ ext{e}}^{sin { ext{x}}}}}}{{{{ ext{e}}^{cos { ext{x}}}}}}]xa0is

Which of the following is not associated with vector calculus?

The series [sumlimits_{{ ext{m}} = 0}^infty {frac{1}{{{4^{ ext{m}}}}}{{left( {{ ext{x}} - 1}
ight)}^{2{ ext{m}}}}} ] xa0 converges for

f(x, y) is a continuous function defined over (x, y) [ in ] [0, 1] × [0, 1]. Given the two constraints, x > y 2 and y > x 2 , the volume under f(x, y) is

The value of [intlimits_0^{frac{pi }{6}} {{{cos }^4}3 heta ,{{sin }^3}6 heta ,{ ext{d}} heta } ] xa0 xa0is

Which one of the following is NOT a correct statement?

For the function e -x , the linear approximation around x = 2 is

The infinite series [{ ext{f}}left( { ext{x}}
ight) = { ext{x}} - frac{{{{ ext{x}}^3}}}{{3!}} + frac{{{{ ext{x}}^5}}}{{5!}} - frac{{{{ ext{x}}^7}}}{{7!}},...,infty ] xa0 xa0 xa0 converges to

If the vector function [overrightarrow {
m{F}} = {{
m{hat a}}_{
m{x}}}left( {3{
m{y}} - {{
m{k}}_1}{
m{z}}}
ight) + {{
m{hat a}}_{
m{y}}}left( {{{
m{k}}_2}{
m{x}} - 2{
m{z}}}
ight) - {{
m{hat a}}_{
m{z}}}left( {{{
m{k}}_3}{
m{y}} + {
m{z}}}
ight)] is irrotational, then the values of the constants k 1 , k 2 and k 3 respectively, are

Consider a differentiable function f(x) on the set of real numbers such that f(-1) = 0 and |f'(x)| ≤ 2. Given these conditions, which one of the following inequalities is necessarily true for all x[ in ] [-2, 2]?

The curve y = x 4 is

Which one of the following functions is strictly bounded?

A parametric curve defined by [{
m{x}} = cos left( {frac{{pi {
m{u}}}}{2}}
ight),,{
m{y}} = sin left( {frac{{pi {
m{u}}}}{2}}
ight)] xa0 xa0xa0in the range 0 ≤ u ≤ 1 is rotated about the x-axis by 360°. Area of the surface generated is

Let [{
m{g}}left( {
m{x}}
ight) = left{ {x08egin{array}{*{20}{c}}
{ - {
m{x,}}}&{{
m{x}} le 1}\
{{
m{x}} + 1,}&{{
m{x}} ge 1}
end{array}}
ight.] xa0 xa0 and [{
m{f}}left( {
m{x}}
ight) = left{ {x08egin{array}{*{20}{c}}
{1 - {
m{x,}}}&{{
m{x}} le 0}\
{{{
m{x}}^2},}&{{
m{x}} > 0}
end{array}}
ight..] Consider the composition of f and g i.e. (fog) (x) = f(g(x)). The number of discontinuities in (fog) (x) present in the interval ([ - infty ,] xa00) is:

What is [mathop {lim }limits_{ heta o 0} frac{{sin heta }}{ heta }] xa0equal to?

A cubic polynomial with real coefficients

If [{ ext{f}}left( { ext{x}}
ight) = sin left| { ext{x}}
ight|] xa0 then value of [frac{{{ ext{df}}}}{{{ ext{dx}}}}]xa0at [{ ext{x}} = frac{{ - pi }}{4}] xa0is

The value of integral [mathop{{int!!!!!int}mkern-21mu x08igcirc}limits_{ ext{S}}
{overrightarrow { ext{r}} cdot overrightarrow { ext{n}} { ext{ds}}} ] xa0 over the closed surface S bounding a volume, where [overrightarrow {
m{r}} = {
m{xhat i}} + {
m{yhat j}} + {
m{zhat k}}] xa0 xa0is the position vector and [{{
m{mathord{x08uildrel{lower3pthbox{$scriptscriptstyle
ightharpoonup$}}
over n} }}}] is the normal to the surface S, is

Which of the following statements are correct regarding dot product of vectors? 1. Dot product is less than or equal to the product of magnitudes of two vectors. 2. When two vectors are perpendicular to each other, then their dot product is non-zero. 3. Dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than [frac{pi }{2}]. 4. Dot product is equal to the product of one vector and the projection of the vector on the first one. Select the correct answer:

The volume of the solid surrounded by the surface [{left( {frac{{
m{x}}}{{
m{a}}}}
ight)^{frac{2}{3}}} + {left( {frac{{
m{y}}}{{
m{b}}}}
ight)^{frac{2}{3}}} + {left( {frac{{
m{z}}}{{
m{c}}}}
ight)^{frac{2}{3}}} = 1] xa0 xa0 xa0is

For the function, f(x, y) = x 2 - y 2 defined on R 2 , the point (0, 0) is

A path AB in the form of one quarter of a circle of unit radius is shown in the figure. Integration of (x + y) 2 on path AB traversed in a counterclockwise sense is

The minimum value of the function f(x) = x 3 - 3x 2 - 24x + 100 in the interval [-3, 3] is

The directional derivative of f(x, y, z) = 2x 2 + 3y 2 + z 2 at the point P(2, 1, 3) in the direction of the vector a = i - 2k is

What is the value of [mathop {lim }limits_{{
m{n}} o infty } {left( {1 - frac{1}{{
m{n}}}}
ight)^{2{
m{n}}}}?]

The line integral of function F = yzi, in the counter-clockwise direction, along the circle x 2 + y 2 = 1 at z = 1 is

Given a vector [overline {
m{u}} = frac{1}{3}left( { - {{
m{y}}^3}{
m{hat i}} + {{
m{x}}^3}{
m{hat j}} + {{
m{z}}^3}{
m{hat k}}}
ight)] xa0 xa0 and [{{
m{hat n}}}] as the unit normal vector to the surface of the hemisphere (x 2 + y 2 + z 2 = 1

z ≥ 0), the value of integral [int {left( {
abla imes overline {
m{u}} }
ight) cdot {
m{hat n}}} {
m{dS}}] xa0 xa0evaluated on the curved surface of the hemisphere S is

If Z = e ax + by F(ax - by)
the value of [{ ext{b}} cdot frac{{partial { ext{Z}}}}{{partial { ext{x}}}} + { ext{a}} cdot frac{{partial { ext{Z}}}}{{partial { ext{y}}}}] xa0 is