Answer: Option B. -> 1320
$$\eqalign{
& {\text{Sum}} = \frac{{B.D. \times T.D.}}{{B.D. - T.D.}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{120 \times 110}}{{120 - 110}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\, = 1320 \cr} $$

Answer: Option A. -> 4 months
S.I. on Rs. 1800 = T.D. on Rs. 1872
P.W. of Rs. 1872 is Rs. 1800
Rs. 72 is S.I. on Rs. 1800 at 12%
$$\eqalign{
& {\text{Time}} = \frac{{100 \times 72}}{{12 \times 1800}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{3}\,{\text{years}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4\,{\text{months}} \cr} $$

Answer: Option A. -> Rs. 432
F = Rs. 2160
TD = Rs. 360
PW = F - TD = 2160 - 360 = Rs. 1800
True Discount is the Simple Interest on the present value for unexpired time
⇒ Simple Interest on Rs. 1800 for unexpired time = Rs. 360
Banker's Discount is the Simple Interest on the face value of the bill for unexpired time
= Simple Interest on Rs. 2160 for unexpired time
$$\eqalign{
& = \frac{{360}}{{1800}} \times 2160 \cr
& = \frac{1}{5} \times 2160 \cr
& = {\text{Rs}}{\text{.}}\,432 \cr} $$

Answer: Option A. -> 15
$$\eqalign{
& {\text{Sum}} = \frac{{B.D. \times T.D.}}{{B.D. - T.D.}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{B.D. \times T.D.}}{{B.G.}} \cr
& \frac{{T.D.}}{{B.G.}} = \frac{{{\text{Sum}}}}{{B.D.}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{1650}}{{165}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 10 \cr} $$
Thus, if B.G. is Rs. 1, T.D. = Rs. 10
If B.D. is Rs. 11, T.D. = Rs. 10
If B.D. is Rs. 165,
$$\eqalign{
& T.D. = {\text{Rs}}{\text{.}}\,\frac{{10}}{{11}} \times 165 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,150 \cr} $$
And, B.G. = Rs. (165 - 150) = Rs. 15

Answer: Option C. -> 5880
Face value of the bill = Rs. 6000
Date on which the bill was drawn = July 14 at 5 months
Nominally due date = December 14
Legally due date = December 17
Date on which the bill was discounted = October 5
Unexpired time :
Oct.   Nov.   Dec.
26   +   30   +   17 = 73 days = $$\frac{1}{5}$$ Years
B.D. = S.I. on Rs. 6000 for $$\frac{1}{5}$$ year
$$\eqalign{
& = {\text{Rs}}{\text{.}}\,\left( {6000 \times 10 \times \frac{1}{5} \times \frac{1}{{100}}} \right) \cr
& = {\text{Rs}}{\text{.}}\,120 \cr
& {\text{T}}{\text{.D}}{\text{.}} = {\text{Rs}}{\text{.}}\,\left[ {\frac{{6000 \times 10 \times \frac{1}{5}}}{{100 + \left( {10 \times \frac{1}{5}} \right)}}} \right] \cr
& = {\text{Rs}}{\text{.}}\,\frac{{12000}}{{102}} \cr
& = {\text{Rs}}{\text{.}}\,117.64 \cr} $$
B.G. = B.D. - T.D. = Rs. 120 - 117.64 = Rs. 2.36
Money received by the holder of the bill = Rs. 6000 - 120 = Rs. 5880

Answer: Option B. -> $$11\frac{1}{9}\,\% $$
Let the amount = Rs. 100
Then BD = Rs.10 (∵ banker's discount, BD is the simple Interest on the face value of the bill for unexpired time and bill is discounted at 10% per annum)
Proceeds = Rs. 100 - Rs. 10 = Rs. 90
Hence we should get Rs. 10 as the interest of Rs. 90 for 1 year so that nothing will be lost
$$\eqalign{
& \Rightarrow 10 = \frac{{90 \times 1 \times R}}{{100}} \cr
& \Rightarrow R = \frac{{10 \times 100}}{{90}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{100}}{9} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 11\frac{1}{9}\,\% \cr} $$

Answer: Option A. -> Rs. 1360
BG = Rs. 360
T = 3 years
R = 12%
$$\eqalign{
& TD = \frac{{BG \times 100}}{{TR}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = \frac{{360 \times 100}}{{3 \times 12}} \cr
& \,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}1000 \cr} $$
BG = BD - TD
⇒ BD = BG + TD = 360 + 1000 = Rs. 1360

Answer: Option D. -> 9 months
F = Rs. 498
TD = Rs. 18
PW = F - TD = 498 - 18 = Rs. 480
R = 5%
$$\eqalign{
& TD = \frac{{PW \times TR}}{{100}} \cr
& \Rightarrow 18 = \frac{{480 \times T \times 5}}{{100}} \cr
& \Rightarrow 18 = 24 \times T{\text{ }} \cr
& \Rightarrow {\text{ }}T{\text{ }} = \frac{{18}}{{24}} = \frac{3}{4}{\text{ years}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{12 \times 3}}{4}{\text{ months}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = 9 months}} \cr} $$

Answer: Option A. -> Rs. 180
$$\eqalign{
& F = \frac{{BD \times TD}}{{(BD - TD)}} \cr
& \,\,\,\,\,\,\, = \frac{{36 \times 30}}{{(36 - 30)}} \cr
& \,\,\,\,\,\,\, = \frac{{36 \times 30}}{6} \cr
& \,\,\,\,\,\,\, = 36 \times 5 \cr
& \,\,\,\,\,\,\, = {\text{ Rs}}{\text{. }}180 \cr} $$

Answer: Option C. -> Rs. 5000
T = 4 years
R = 5%
Banker's Gain, BG = Rs. 200
$$\eqalign{
& TD = \frac{{BG \times 100}}{{TR}} \cr
& \,\,\,\,\,\,\,\,\,\, = \frac{{200 \times 100}}{{4 \times 5}} \cr
& \,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,1000 \cr
& \cr
& TD = \sqrt {PW \times BG} \cr
& 1000 = \sqrt {PW \times 200} \cr
& 1000000 = PW \times 200 \cr
& PW = \frac{{1000000}}{{200}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}5000 \cr} $$