Answer: Option D. -> 1290
$$\eqalign{
& TD = \frac{{BG{\text{ }} \times 100}}{{TR}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{540 \times 100}}{{6 \times 12}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{90 \times 100}}{{12}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{15 \times 100}}{2} \cr
& \,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,750 \cr
& BG = BD - TD \cr
& \Rightarrow 540 = BD - 750 \cr
& \Rightarrow BD = 540 + 750 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 1290 \cr} $$

Answer: Option A. -> Rs. 60
F = Rs. 3000
R = 10%
Date on which the bill is drawn = 14th July at 5 months
Nominally Due Date = 14th December
Legally Due Date = 14th December + 3 days = 17th December
Date on which the bill is discounted = 5th October
Unexpired Time
= [6th to 31th of October] + [30 Days in November] + [1th to 17th of December]
= 26 + 30 + 17
= 73 Days
$$\eqalign{
& = \frac{{73}}{{365}}\,{\text{year}} \cr
& {\text{ = }}\frac{1}{5}\,{\text{year}} \cr} $$
Banker's Discount = Simple Interest on the face value of the bill for unexpired time = $$\frac{{{\text{FTR}}}}{{100}}$$
$$\eqalign{
& = \frac{{3000 \times \frac{1}{5} \times 10}}{{100}} \cr
& = 30 \times \frac{1}{5} \times 10 \cr
& = {\text{Rs}}{\text{.}}\,60 \cr} $$

Answer: Option D. -> Rs. 21
$$\eqalign{
& BG = \frac{{{{(TD)}^2}}}{{PW}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{{20}^2}}}{{400}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,1 \cr
& BG = BD - TD \cr
& \Rightarrow 1 = BD - 20 \cr
& \Rightarrow BD = 1 + 20 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,21 \cr} $$

Answer: Option A. -> Rs. 1600
Since the compound interest is taken here,
$$\eqalign{
& PW{\text{ }}{\left( {1 + \frac{5}{{100}}} \right)^2} = 1764 \cr
& PW\,{\left( {1 + \frac{1}{{20}}} \right)^2} = 1764 \cr
& PW\,{\left( {\frac{{21}}{{20}}} \right)^2} = 1764 \cr
& PW \times \frac{{441}}{{400}} = 1764 \cr
& \Rightarrow PW = \frac{{1764 \times 400}}{{441}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4 \times 400 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,1600 \cr} $$

Answer: Option B. -> Rs. 1.25
$$\eqalign{
& F = {\text{Rs}}{\text{.}}\,8100 \cr
& R = 5\% \cr
& T = 3\,{\text{months}} = \frac{1}{4}\,{\text{years}} \cr
& BD = \frac{{FTR}}{{100}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\, = \frac{{8100 \times \frac{1}{4} \times 5}}{{100}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2025}}{{20}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\, = \frac{{405}}{4} \cr
& \,\,\,\,\,\,\,\,\,\,\,\, = {\text{ Rs}}{\text{.}}\,101.25 \cr
& TD = \frac{{FTR}}{{100 + (TR)}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{8100 \times \frac{1}{4} \times 5}}{{100 + \left( {\frac{1}{4} \times 5} \right)}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{2025 \times 5}}{{100 + \left( {\frac{5}{4}} \right)}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{2025 \times 5 \times 4}}{{400 + 5}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{2025 \times 5 \times 4}}{{405}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{405 \times 5 \times 4}}{{81}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{45 \times 5 \times 4}}{9} \cr
& \,\,\,\,\,\,\,\,\,\,\, = 5 \times 5 \times 4 \cr
& \,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,100 \cr
& BD - TD = \,101.25 - 100 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,1.25 \cr} $$

Answer: Option D. -> 6%
$$\eqalign{
& BD\,{\text{for}}\,3\,{\text{years}} = {\text{Rs}}{\text{.}}\,1116 \cr
& BD\,{\text{for}}\,4\,{\text{years}} = \frac{{1116}}{3} \times 4 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}1488 \cr
& TD\,{\text{for}}\,4\,{\text{years}} = {\text{Rs}}{\text{.}}\,1200 \cr
& F = \frac{{BD \times TD}}{{BD - TD}} \cr
& \,\,\,\,\,\,\, = \frac{{1488 \times 1200}}{{1488 - 1200}} \cr
& \,\,\,\,\,\,\, = \frac{{1488 \times 1200}}{{288}} \cr
& \,\,\,\,\,\,\, = \frac{{124 \times 1200}}{{24}} \cr
& \,\,\,\,\,\,\, = \frac{{124 \times 100}}{2} \cr
& \,\,\,\,\,\,\, = 62 \times 100 \cr
& \,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,6200 \cr} $$
⇒ Rs. 1488 is the simple interest on Rs. 6200 for 4 years
$$\eqalign{
& \Rightarrow 1488 = \frac{{6200 \times 4 \times R}}{{100}} \cr
& \Rightarrow R = \frac{{1488 \times 100}}{{6200 \times 4}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{372 \times 100}}{{6200}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{372}}{{62}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 6\% \cr} $$

Answer: Option D. -> Rs. 18
$$\eqalign{
& T = 6{\text{ months}} = \frac{1}{2}{\text{year}} \cr
& R = 6\% \cr
& TD = \frac{{BD \times 100}}{{100 + TR}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{18.54 \times 100}}{{100 + \left( {\frac{1}{2} \times 6} \right)}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{18.54 \times 100}}{{103}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{1854}}{{103}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,18 \cr} $$

Answer: Option D. -> Rs. 21
$$\eqalign{
& BG = \frac{{{{(TD)}^2}}}{{PW}} = \frac{{{{20}^2}}}{{400}} = {\text{Rs}}{\text{. }}1 \cr
& BG = BD - TD \cr
& \Rightarrow 1 = BD - 20 \cr
& \Rightarrow BD = 1 + 21 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,21 \cr} $$

Answer: Option D. -> Rs. 240
$$\eqalign{
& TD = \sqrt {PW \times BG} \cr
& \,\,\,\,\,\,\,\,\,\, = \sqrt {5760 \times 10} \cr
& \,\,\,\,\,\,\,\,\,\, = \sqrt {57600} \cr
& \,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}240 \cr} $$

Answer: Option D. -> $$\frac{{150}}{{11}}\,\% $$
$$\eqalign{
& {\text{Sum}} = \frac{{B.D. \times T.D.}}{{B.D. - T.D.}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,\frac{{120 \times 110}}{{120 - 110}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,1320 \cr} $$
Since B.D. is S.I. on sum due, so S.I. on Rs. 1320 for 8 months is Rs. 120.
$$\eqalign{
& {\text{Rate}} = \frac{{100 \times 120}}{{1320 \times \frac{2}{3}}}\,\% \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{150}}{{11}}\,\% \cr} $$