Answer: Option B. -> $$3\frac{1}{2}$$ years
$$\eqalign{
& {\text{Rate }}\% = {\text{12}}\% \cr
& {\text{Principal = Rs}}{\text{. 1860}} \cr
& {\text{Amount = Rs}}{\text{. 2641}}{\text{.20}} \cr
& {\text{Interest}} \cr
& {\text{ = Rs}}{\text{. }}\left( {2641.20 - 1860} \right) \cr
& = {\text{Rs}}{\text{. 781}}{\text{.20}} \cr
& {\text{By using formula,}} \cr
& {\text{Required time }} \cr
& = \frac{{781.20 \times 100}}{{1860 \times 12}} \cr
& = 3\frac{1}{2}{\text{ years}} \cr} $$

Answer: Option B. -> 3 year
According to the question,
Principal    
    Interest
10  
  3
$$\eqalign{
& {\text{Rate }}\% {\text{ = 10}}\% \cr
& {\text{Time = }}\frac{3}{{10}} \times \frac{{100}}{{10}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\, = 3\,{\text{years}} \cr} $$

Answer: Option B. -> II alone is correct
$$\eqalign{
& {\text{Let sum be x}}{\text{.}} \cr
& {\text{Then,}} \cr
& {\text{S}}{\text{.I}}{\text{.}} = x \cr
& {\text{I - Time}} \cr
& = \frac{{100 \times x}}{{x \times \frac{{50}}{3}}} \cr
& = 6\,{\text{years(false)}} \cr
& {\text{II}} - {\text{Time}} \cr
& = \frac{{100 \times x}}{{x \times 20}} \cr
& = 5\,{\text{years(True)}} \cr
& {\text{III}} - {\text{Suppose sum}} = x. \cr
& {\text{Then, S}}{\text{.I}}{\text{. }} = x \cr
& {\text{Time }} = {\text{5 }}{\text{years}}{\text{.}} \cr
& {\text{Rate}} = \left( {\frac{{100 \times x}}{{x \times 5}}} \right)\% \cr
& \,\,\,\,\,\,\,\,\,\,\,\, = 20\% . \cr
& {\text{Now, sum}} = x,\,{\text{S}}{\text{.I}}{\text{.}} = 3x\,{\text{and}}\,{\text{Rate}} = 20\% \cr & \therefore {\text{Time}} = \left( {\frac{{100 \times 3x}}{{x \times 20}}} \right){\text{years}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 15\,{\text{years}}(\text{false}) \cr
& {\text{So, B alone is correct}}{\text{.}} \cr} $$

Answer: Option C. -> Rs. 550
Let the amount invested = Rs. P
According to the question,
$$\eqalign{
& {\text{P}} + \frac{{{\text{P}} \times 10 \times 4}}{{100}} = 770 \cr
& \Rightarrow {\text{P}} + \frac{{4{\text{P}}}}{{10}} = 770 \cr
& \Rightarrow \frac{{14{\text{P}}}}{{10}} = 770 \cr
& \Rightarrow {\text{P}} = \frac{{770 \times 10}}{{14}} \cr
& \Rightarrow {\text{P}} = {\text{Rs 550}} \cr} $$
Hence, required invested amount = Rs. 550
Alternate
$$\eqalign{
& {\text{10}}\% {\text{ = }}\frac{{1 \to {\text{Interest}}}}{{10 \to {\text{Principal}}}} \cr
& {\text{Interest in 4 years}} \cr
& {\text{ = 1}} \times {\text{4}} \cr
& = {\text{4}} \cr
& {\text{Amount = }} \cr
& = \left( {{\text{Interest + Principal}}} \right) \cr
& = 4 + 10 \cr
& = 14 \cr
& {\text{According to the question,}} \cr
& {\text{14 units = 770}} \cr
& {\text{1 unit = }}\frac{{770}}{{14}} \cr
& {\text{10 units = }}\frac{{770}}{{14}} \times {\text{10}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = Rs}}{\text{. 550 }} \cr
& {\text{The amount invested}} \cr
& {\text{ = Rs}}{\text{. 550}} \cr} $$

Answer: Option D. -> Date inadequate
$$\eqalign{
& {\text{Let sum}} = x{\text{.}} \cr
& {\text{Then,}} \cr
& {\text{S}}{\text{.I}}{\text{.}} = \frac{x}{2} \cr
& \therefore \frac{x}{2} = \frac{{x \times 8 \times 6}}{{100}} \cr} $$
Clearly, data is inadequate.

Answer: Option B. -> 6400
$$\eqalign{
& {\text{Present Population}} \cr
& = {\text{P}}{\left( {\frac{{1 - {\text{R}}}}{{100}}} \right)^n} \cr
& = 10000{\left( {\frac{{1 - 20}}{{100}}} \right)^2} \cr
& = 10000{\left( {\frac{{100 - 20}}{{100}}} \right)^2} \cr
& = 10000{\left( {\frac{{80}}{{100}}} \right)^2} \cr
& = 10000{\left( {\frac{4}{5}} \right)^2} \cr
& = 10000 \times \frac{{16}}{{25}} \cr
& = 400 \times 16 \cr
& = 6400 \cr} $$

Answer: Option C. -> Rs. 20000
Capital after paying income tax
$$\eqalign{
& \Rightarrow {\text{4}}\% - {\text{3}}{\text{.75}}\% = {\text{48}} \cr
& \Rightarrow {\text{0}}{\text{.25}}\% {\text{ = 48}} \cr
& {\text{100}}\% {\text{ = }}\frac{{48}}{{0.25}} \times 100 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 19200 \cr} $$
⇒ Capital without paying income tax
⇒ 19200 = Capital × 96%
Net capital = 20000

Answer: Option A. -> 2 : 1
Let two parts are P1 and P2 respectively
According to the question
$$\eqalign{
& \frac{{{{\text{P}}_1} \times 3 \times 12}}{{100}} = \frac{{{{\text{P}}_2} \times 9 \times 16}}{{2 \times 100}} \cr
& 36{{\text{P}}_1} = 72{{\text{P}}_2} \cr
& \frac{{{{\text{P}}_1}}}{{{{\text{P}}_2}}} = \frac{{72}}{{36}} = \frac{2}{1} \cr
& {{\text{P}}_1}{\text{:}}{{\text{P}}_2} = 2:1 \cr
& {\text{Hence,}} \cr
& {\text{Required ratio = 2 : 1}} \cr} $$

Answer: Option A. -> $$1\frac{1}{4}$$ years
$$\eqalign{
& {\text{Let sum}} = x. \cr
& {\text{Then,}} \cr
& {\text{S}}{\text{.I}}{\text{.}} = 0.125x = \frac{1}{8}x \cr
& {\text{R}} = 10\% \cr
& \therefore \text{Rate} \cr
& = \left( {\frac{{100 \times x}}{{x \times 8 \times 10}}} \right){\text{years}} \cr
& = \frac{5}{4}{\text{years}} \cr
& = {\text{1}}\frac{1}{4}{\text{years}} \cr} $$

Answer: Option A. -> Rs. 5000
$$\eqalign{
& {\text{Let principal is P}} \cr
& {\text{then,}} \cr
& {\text{300 = }}\frac{{{\text{P}} \times 3 \times 2}}{{100}} \cr
& {\text{P = 5000}} \cr} $$