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[#106] 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

(A) continuous and differentiable

(B) Not continuous and not differentiable

(C) Not continuous but differentiable

(D) Continuous but not differentiable

Correct Answer

(D) Continuous but not differentiable

[#107] 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

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

(B) $$pi $$

(C) $$frac{{3pi }}{2}$$

(D) 1

Correct Answer

(B) $$pi $$

[#108] 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

(A) same and equal to 0.5

(B) same and equal to -0.5

(C) 0.5 and -0.5 respectively

(D) -0.5 and 0.5 respectively

Correct Answer

(C) 0.5 and -0.5 respectively

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

(A) 0

(B) $$frac{{ ext{a}}}{2}$$

(C) a

(D) 2a

Correct Answer

(B) $$frac{{ ext{a}}}{2}$$

[#110] 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

(A) -2

(B) 2

(C) 1

(D) 0

Correct Answer

(A) -2