Answer: Option B. -> 4R
$$\eqalign{
& \Rightarrow {\text{Q}} = \sqrt {\frac{{{{\left( {8{\text{R}}} \right)}^2}}}{4}} \cr
& \Rightarrow {\text{Q}} = \frac{{\sqrt {{{\left( {8{\text{R}}} \right)}^2}} }}{{\sqrt 4 }} \cr
& \Rightarrow {\text{Q}} = \frac{{8{\text{R}}}}{2} \cr
& \Rightarrow {\text{Q}} = 4{\text{R}} \cr} $$

Answer: Option A. -> 1000 and 2000
The smallest such number is 1444 $$\left[ {1444 = {{\left( {38} \right)}^2}} \right]$$
It lies between 1000 and 2000.

Answer: Option A. -> 1
$$\eqalign{
& {\text{We have,}} \cr
& 1 \times 2 \times 3 \times 4 = 24 \cr
& {\text{And }}24 + 1 = 25\left[ {25 = {5^2}} \right] \cr
& 2 \times 3 \times 4 \times 5 = 120{\text{ }} \cr
& {\text{And 1}}20 + 1 = 121\left[ {121 = {{11}^2}} \right] \cr
& 3 \times 4 \times 5 \times 6 = 360{\text{ }} \cr
& {\text{And }}360 + 1 = 361\left[ {361 = {{19}^2}} \right] \cr
& 4 \times 5 \times 6 \times 7 = 840{\text{ }} \cr
& {\text{And }}840 + 1 = 841\left[ {841 = {{29}^2}} \right] \cr
& \therefore p = 1 \cr} $$

Answer: Option A. -> 1806
Clearly, the man was born between 1800 and 1850 is 1849.
$${\text{And, }}1849 = {\left( {43} \right)^2}$$
So, the man was 43 years old in 1849
$$\eqalign{
& {\text{Thus, he was born in }} \cr
& = \left( {1849 - 43} \right) \cr
& = 1806 \cr} $$

Answer: Option C. -> 1.6
$$\eqalign{
& = \frac{{1 + \sqrt {0.01} }}{{1 - \sqrt {0.1} }} \cr
& = \frac{{1 + 0.1}}{{1 - 0.316}} \cr
& = \frac{{1.1}}{{0.684}} \cr
& = \frac{{1100}}{{684}} \cr
& = 1.6 \cr
& \,\,\,\,\,\,\,3|\overline 0 \,.\,\overline {10} \,\,\overline {00} \,\,\overline {00} \,(0.316 \cr
& \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,9 \cr
& \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr
& \,\,\,61|\,\,\,\,\,\,\,\,\,\,100 \cr
& \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,61 \cr
& \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr
& 626\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,3900 \cr
& \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,3756 \cr
& \,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr} $$

Answer: Option A. -> 23.15
$$\eqalign{
& = \sqrt {{\text{535}}{\text{.9225}}} \cr
& = \sqrt {\frac{{{\text{5359225}}}}{{10000}}} \cr
& = \frac{{2315}}{{100}} \cr
& = 23.15 \cr
& \,\,\,\,\,\,\,\,\,2|\overline 5 \,\,\overline {35} \,\,\overline {92} \,\,\overline {25} \,\,(2315 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,|4 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr
& \,\,\,\,\,\,\,43|\,\,1\,35 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,1\,2\,9 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr
& \,\,\,461\,|\,\,\,\,\,\,\,\,\,6\,92 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,4\,61 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr
& 4625\,|\,\,\,\,\,\,\,\,\,\,\,\,231\,25 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,231\,25 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{x}} \cr} $$

Answer: Option D. -> 6
$$\eqalign{
& {\text{Let the missing digit be x}} \cr
& \,\,\,\,\,\,\,1|\overline 1 \,\,\overline {53} \,\,\overline {7{\text{x}}} \,\,(124 \cr
& \,\,\,\,\,\,\,\,\,\,|\,\,1 \cr
& \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr
& \,\,\,22|\,\,\,\,\,\,\,53 \cr
& \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,44 \cr
& \,\,\,\,\,\,\,\,\,\,| - - - - - - - - \cr
& 244\,|\,\,\,\,\,\,\,\,\,\,\,\,\,97{\text{x}} \cr
& \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,976 \cr
& \,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr
& \,\,\,\,\,\,\,\,\,\,|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{x}} \cr
& \,\,\,\,\,\,\,\,\,\,| - - - - - - - \cr
& {\text{Then, x}} = 6 \cr} $$

Answer: Option C. -> 1.633
$$\eqalign{
& = \sqrt {\frac{8}{3}} \cr
& = \sqrt {\frac{{8 \times 3}}{{3 \times 3}}} \cr
& = \frac{{\sqrt {24} }}{3} \cr
& = \frac{{4.899}}{3} \cr
& = 1.633 \cr} $$

Answer: Option B. -> 0.447
$$\eqalign{
& = \frac{1}{{\sqrt 5 }} \cr
& = \frac{1}{{\sqrt 5 }} \times \frac{{\sqrt 5 }}{{\sqrt 5 }} \cr
& = \frac{{\sqrt 5 }}{5} \cr
& = \frac{{2.236}}{5} \cr
& = 0.447 \cr} $$

Answer: Option A. -> 0.000002
$$\eqalign{
& 0.000326 = \frac{{326}}{{{{10}^6}}} \cr
& \,\,\,\,1|\overline 3 \,\,\overline {26} \,\,(18 \cr
& \,\,\,\,\,\,\,|\,\,1 \cr
& \,\,\,\,\,\,\,| - - - - - - \cr
& 28|\,\,\,2\,26 \cr
& \,\,\,\,\,\,\,|\,\,\,2\,24 \cr
& \,\,\,\,\,\,\,| - - - - - - \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2 \cr} $$
∴ Required number to be subtracted
$$\eqalign{
& = \frac{2}{{{{10}^6}}} \cr
& = 0.000002 \cr} $$