Answer: Option B. -> 8649
$$\eqalign{
& {\text{Let}}, \cr
& \sqrt {11881} \times \sqrt x = 10137 \cr
& {\text{Then}}, \cr
& 109 \times \sqrt x = 10137 \cr
& \Leftrightarrow \sqrt x = \frac{{10137}}{{109}} = 93 \cr
& \Leftrightarrow x = {\left( {93} \right)^2} \cr
& \Leftrightarrow x = 8649 \cr} $$

Answer: Option B. -> 2209
$$\eqalign{
& {\text{Let}}, \cr
& \sqrt x \times \sqrt {484} = 1034 \cr
& {\text{Then}}, \cr
& \sqrt x \times 22 = 1034 \cr
& \Leftrightarrow \sqrt x = \frac{{1034}}{{22}} = 47 \cr
& \Leftrightarrow x = {\left( {47} \right)^2} \cr
& \Leftrightarrow x = 2209 \cr} $$

Answer: Option C. -> 81
$$\eqalign{
& \frac{{4050}}{{\sqrt x }} = 450 \cr
& \Leftrightarrow \sqrt x = \frac{{4050}}{{450}} \cr
& \Leftrightarrow \sqrt x = 9 \cr
& \Leftrightarrow x = {\left( 9 \right)^2} \cr
& \Leftrightarrow x = 81 \cr} $$

Answer: Option D. -> 279841
$$\eqalign{
& {\text{Let, }} \cr
& {\left( {15} \right)^2} + {\left( {18} \right)^2} - 20 = \sqrt x \cr
& {\text{Then, }} \cr
& \sqrt x = 225 + 324 - 20 \cr
& \Leftrightarrow x = {\left( {529} \right)^2} \cr
& \Leftrightarrow x = 279841 \cr} $$

Answer: Option C. -> 16
$$\eqalign{
& {\text{Let}}, \cr
& \sqrt {\frac{{16}}{{25}}} \times \sqrt {\frac{x}{{25}}} \times \frac{{16}}{{25}} = \frac{{256}}{{625}} \cr
& {\text{Then}}, \cr
& \frac{4}{5} \times \frac{{\sqrt x }}{5} \times \frac{{16}}{{25}} = \frac{{256}}{{625}} \cr
& \Leftrightarrow \frac{{64\sqrt x }}{{625}} = \frac{{256}}{{625}} \cr
& \Leftrightarrow \sqrt x = \frac{{256}}{{625}} \times \frac{{625}}{{64}} \cr
& \Leftrightarrow x = {4^2} \cr
& \Leftrightarrow x = 16 \cr} $$

Answer: Option B. -> $$\frac{4}{9}$$
$$\eqalign{
& = \sqrt {\frac{{25}}{{81}} - \frac{1}{9}} \cr
& = \sqrt {\frac{{25 - 9}}{{81}}} \cr
& = \sqrt {\frac{{16}}{{81}}} \cr
& = \frac{{\sqrt {16} }}{{\sqrt {81} }} \cr
& = \frac{4}{9} \cr} $$

Answer: Option D. -> 500
$$\eqalign{
& = 10y\sqrt {{y^3} - {y^2}} \cr
& = 10 \times 5\sqrt {{5^3} - {5^2}} \cr
& = 50 \times \sqrt {125 - 25} \cr
& = 50 \times \sqrt {100} \cr
& = 50 \times 10 \cr
& = 500 \cr} $$

Answer: Option C. -> 240
$$\eqalign{
& = \sqrt {\left( {{{272}^2} - {{128}^2}} \right)} \cr
& = \sqrt {\left( {272 + 128} \right)\left( {272 - 128} \right)} \cr
& = \sqrt {400 \times 144} \cr
& = \sqrt {57600} \cr
& = 240 \cr} $$

Answer: Option E. -> None of these
$$\eqalign{
& {\text{Let,}} \cr
& {\left[ {{{\left( {\sqrt {81} } \right)}^2}} \right]^2} = {\left( x \right)^2} \cr
& {\text{Then,}} \cr
& \Leftrightarrow {x^2} = {\left( {81} \right)^2} \cr
& \Leftrightarrow x = 81 \cr} $$

Answer: Option C. -> 0.021
$$\eqalign{
& = \sqrt {0.000441} \cr
& = \sqrt {\frac{{441}}{{{{10}^6}}}} \cr
& = \frac{{\sqrt {441} }}{{\sqrt {{{10}^6}} }} \cr
& = \frac{{21}}{{{{10}^3}}} \cr
& = \frac{{21}}{{1000}} \cr
& = 0.021 \cr} $$