Answer: Option B. -> 0.3375
$$\eqalign{
& = {1.5^2} \times \sqrt {0.0225} \cr
& = {1.5^2} \times \sqrt {\frac{{225}}{{10000}}} \cr
& = 2.25 \times \frac{{15}}{{100}} \cr
& = 2.25 \times 0.15 \cr
& = 0.3375 \cr} $$

Answer: Option D. -> 2.13
Given expression,
$$ = \sqrt {\frac{1}{{100}}} + \sqrt {\frac{{81}}{{100}}} + \sqrt {\frac{{121}}{{100}}} + $$      $$\sqrt {\frac{9}{{10000}}} $$
$$\eqalign{
& = \frac{1}{{10}} + \frac{9}{{10}} + \frac{{11}}{{10}} + \frac{3}{{100}} \cr
& = 0.1 + 0.9 + 1.1 + 0.03 \cr
& = 2.13 \cr} $$

Answer: Option B. -> 1.25
$$\eqalign{
& \,\,\,\,\,\,\,1|\overline 1 \,.\,\overline {56} \,\,\overline {25} \,(1.25 \cr
& \,\,\,\,\,\,\,\,\,\,|1 \cr
& \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr
& \,\,\,22|\,\,\,\,\,\,56 \cr
& \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,44 \cr
& \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr
& 245\,|\,\,\,\,\,\,\,12\,25 \cr
& \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,12\,25 \cr
& \,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr
& \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\text{x} \cr
& \,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr
& \therefore \sqrt {1.5625} = 1.25 \cr} $$

Answer: Option D. -> 37.304
$$\eqalign{
& {\text{Given expression,}} \cr
& \sqrt {1.30} + \sqrt {1300} + \sqrt {0.0130} \cr
& = \sqrt {\frac{{130}}{{100}}} + \sqrt {13 \times 100} + \sqrt {\frac{{130}}{{10000}}} \cr
& = \frac{{\sqrt {130} }}{{10}} + \sqrt {13} \times 10 + \frac{{\sqrt {130} }}{{100}} \cr
& = \frac{{11.40}}{{10}} + 3.605 \times 10 + \frac{{11.40}}{{100}} \cr
& = 1.14 + 36.05 + 0.114 \cr
& = 37.304 \cr} $$

Answer: Option D. -> $$\sqrt 3 $$
$$\eqalign{
& a = \frac{{\sqrt 3 }}{2}{\text{ (given)}} \cr
& \therefore \sqrt {1 + a} + \sqrt {1 - a} \cr
& = \sqrt {1 + \frac{{\sqrt 3 }}{2}} + \sqrt {1 - \frac{{\sqrt 3 }}{2}} \cr
& = \sqrt {\frac{{2 + \sqrt 3 }}{2}} + \sqrt {\frac{{2 - \sqrt 3 }}{2}} \cr
& = \sqrt {\frac{{2\left( {2 + \sqrt 3 } \right)}}{4}} + \sqrt {\frac{{2\left( {2 - \sqrt 3 } \right)}}{4}} \cr
& = \sqrt {\frac{{4 + 2\sqrt 3 }}{4}} + \sqrt {\frac{{4 - 2\sqrt 3 }}{4}} \cr} $$
$$ = \sqrt {\frac{{3 + 1 + 2 \times \sqrt 3 \times 1}}{2}} + $$     $$\sqrt {\frac{{3 + 1 - 2 \times \sqrt 3 \times 1}}{2}} $$     \[\because \left\{ \begin{gathered}
{\left( {\sqrt 3 } \right)^2} + {\left( 1 \right)^2} - 2.\sqrt 3 .1 = {\left( {\sqrt 3 - 1} \right)^2} \hfill \\
{\left( {\sqrt 3 } \right)^2} + {\left( 1 \right)^2} + 2.\sqrt 3 .1 = {\left( {\sqrt 3 + 1} \right)^2} \hfill \\
{a^2} + {b^2} - 2ab = {\left( {a - b} \right)^2} \hfill \\
{a^2} + {b^2} - 2ab = {\left( {a + b} \right)^2} \hfill \\
\end{gathered} \right\}\]
$$\eqalign{
& = \sqrt {\frac{{{{\left( {\sqrt 3 + 1} \right)}^2}}}{2}} + \sqrt {\frac{{{{\left( {\sqrt 3 - 1} \right)}^2}}}{2}} \cr
& = \frac{{\sqrt 3 + 1 + \sqrt 3 - 1}}{2} \cr
& = \frac{{2\sqrt 3 }}{2} \cr
& = \sqrt 3 \cr} $$

Answer: Option B. -> 2
$$\frac{{{{\left( {0.75} \right)}^3}}}{{1 - 0.75}}$$ $${\text{ + }}$$$$\left[ {0.75 + {{\left( {0.75} \right)}^2} + 1} \right]$$
$$ = \frac{{{{\left( {0.75} \right)}^2} \times 0.75}}{{0.25}}$$   $${\text{ + }}$$ $$\left[ {0.75 + 0.5625 + 1} \right]$$
$$ = 0.5625 \times 3 \,\, + $$   $$\left[ {0.75 + 0.5625 + 1} \right]$$
$$\eqalign{
& = 1.6875 + 2.3125 \cr
& = 4 \cr} $$
Square root of 4 = 2

Answer: Option D. -> $$\sqrt 5 $$
$$\eqalign{
& {\text{Given,}} \cr
& \frac{{5 + \sqrt {10} }}{{5\sqrt 5 - 2\sqrt {20} - \sqrt {32} + \sqrt {50} }}{\text{ }} \cr
& = \frac{{5 + \sqrt {10} }}{{5\sqrt 5 - 2 \times 2\sqrt 5 - 2 \times 2\sqrt 2 + 5\sqrt 2 }} \cr
& = \frac{{5 + \sqrt {10} }}{{5\sqrt 5 - 4\sqrt 5 - 4\sqrt 2 + 5\sqrt 2 }} \cr
& = \frac{{5 + \sqrt {10} }}{{\sqrt 5 + \sqrt 2 }} \cr
& = \frac{{\sqrt 5 \left( {\sqrt 5 + \sqrt 2 } \right)}}{{\sqrt 5 + \sqrt 2 }} \cr
& = \sqrt 5 \cr} $$

Answer: Option D. -> 1.2245
$$\eqalign{
& = \frac{{3\sqrt 2 }}{{2\sqrt 3 }} \cr
& = \frac{{3\sqrt 2 }}{{2\sqrt 3 }} \times \frac{{\sqrt 3 }}{{\sqrt 3 }} \cr
& = \frac{{3\sqrt 6 }}{{2 \times 3}} \cr
& = \frac{{\sqrt 6 }}{2} \cr
& = \frac{{2.449}}{2} \cr
& = 1.2245 \cr} $$

Answer: Option B. -> 7.826
$$\eqalign{
& = \frac{{\sqrt 5 }}{2} - \frac{{10}}{{\sqrt 5 }} + \sqrt {125} \cr
& = \frac{{{{\left( {\sqrt 5 } \right)}^2} - 20 + 2\sqrt 5 \times 5\sqrt 5 }}{{2\sqrt 5 }} \cr
& = \frac{{5 - 20 + 50}}{{2\sqrt 5 }} \cr
& = \frac{{35}}{{2\sqrt 5 }} \times \frac{{\sqrt 5 }}{{\sqrt 5 }} \cr
& = \frac{{35\sqrt 5 }}{{10}} \cr
& = \frac{7}{2} \times 2.236 \cr
& = 7 \times 1.118 \cr
& = 7.826 \cr} $$

Answer: Option C. -> 6
294 = 7 × 7 × 2 × 3
To make it a perfect square, it must be multiplied by 2 × 3 i.e.,6
∴ Required number = 6