Calculus - Study Mode

Show All Answers

[#1] $$overrightarrow { ext{a}} ,,overrightarrow { ext{b}} ,,overrightarrow { ext{c}} $$ 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

(A) [{ m{hat i}} + 2{ m{hat j}} + 3{ m{hat k}}]

(B) [{ m{2hat i}} + { m{hat j}}]

(C) [{ m{2hat i}} - { m{hat j}}]

(D) [4{ m{hat k}}]

Correct Answer

(C) [{ m{2hat i}} - { m{hat j}}]

[#2] 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

(A) $$frac{3}{{16}}$$

(B) 0

(C) $$frac{{ - 5}}{{12}}$$

(D) -1

Correct Answer

(D) -1

[#3] At x = 0, the function f(x) = |x| has

(A) a minimum

(B) a maximum

(C) a point of inflection

(D) neither a maximum nor minimum

Correct Answer

(A) a minimum

[#4] 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

(A) $$frac{pi }{4}$$

(B) $$pi $$

(C) $$frac{pi }{2}$$

(D) $$2pi $$

Correct Answer

(C) $$frac{pi }{2}$$

[#5] 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

(A) x 2 y 2 - x 1 y 1

(B) $$left( {{ ext{y}}_2^2 - { ext{y}}_1^2} ight) + left( {{ ext{x}}_2^2 - { ext{x}}_1^2} ight)$$

(C) (x 2 - x 1 ) (y 2 - y 1 )

(D) (y 2 - y 1 ) 2 + (x 2 - x 1 ) 2

Correct Answer

(A) x 2 y 2 - x 1 y 1