Answer: Option A. -> 2
$$\eqalign{
& {\text{Amount}} = Rs.\,\left( {30000 + 4347} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,34347 \cr
& {\text{Let}}\,{\text{the}}\,{\text{time}}\,{\text{be}}\,n\,{\text{years}} \cr
& {\text{Then}},30000\,{\left( {1 + \frac{7}{{100}}} \right)^n} = 34347 \cr
& \Rightarrow {\left( {\frac{{107}}{{100}}} \right)^n} = \frac{{34347}}{{30000}} = \frac{{11449}}{{10000}} = {\left( {\frac{{107}}{{100}}} \right)^2} \cr
& \therefore n = 2\,{\text{years}} \cr} $$

Answer: Option A. -> Rs. 2.04
$$\eqalign{
& {\text{C}}{\text{.I}}{\text{.}}\,{\text{when}}\,{\text{interest}}\,{\text{compounded}}\,{\text{yearly}} \cr
& = Rs.\left[ {5000 \times \left( {1 + \frac{4}{{100}}} \right) \times \left( {1 + \frac{{\frac{1}{2} \times 4}}{{100}}} \right)} \right] \cr
& = Rs.\left( {5000 \times \frac{{26}}{{25}} \times \frac{{51}}{{50}}} \right) \cr
& = Rs.5304 \cr
& {\text{C}}{\text{.I}}{\text{.}}\,{\text{when}}\,{\text{interest}}\,{\text{in}}\,{\text{compounded}}\,{\text{half - yearly}} \cr
& = Rs.\,\left[ {5000 \times {{\left( {1 + \frac{2}{{100}}} \right)}^3}} \right] \cr
& = Rs.\,\left( {5000 \times \frac{{51}}{{50}} \times \frac{{51}}{{50}} \times \frac{{51}}{{50}}} \right) \cr
& = Rs.\,5306.04 \cr
& \therefore {\text{Difference}} = Rs.\,\left( {5306.04 - 5304} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,2.04 \cr} $$

Answer: Option C. -> 6.25%
$$\eqalign{
& {\text{S}}{\text{.I}}{\text{. on Rs}}{\text{. 672 for 1 year}} \cr
& {\text{ = Rs}}{\text{. }}\left( {714 - 672} \right) \cr
& {\text{ = Rs}}{\text{. 42}} \cr
& \therefore {\text{Rate = }}\left( {\frac{{100 \times 42}}{{672 \times 1}}} \right){\text{% }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = 6}}{\text{.25% }} \cr} $$

Answer: Option A. -> Rs. 8000
Let the share of the younger and elder sons be Rs. x and Rs. (16400 - x)
Then, amount of Rs. x after 3 years = Amount of Rs. (16400 - x) after 2 years
$$\eqalign{
& \Rightarrow x{\left( {1 + \frac{5}{{100}}} \right)^3} = \left( {16400 - x} \right){\left( {1 + \frac{5}{{100}}} \right)^2} \cr
& \Rightarrow x\left( {1 + \frac{5}{{100}}} \right) = \left( {16400 - x} \right) \cr
& \Rightarrow \frac{{21x}}{{20}} + x = 16400 \cr
& \Rightarrow \frac{{41x}}{{20}} = 16400 \cr
& \Rightarrow x = \left( {\frac{{16400 \times 20}}{{41}}} \right) \cr
& \Rightarrow x = 8000 \cr} $$

Answer: Option D. -> 5%
$$\eqalign{
& {\text{Principal = Rs 8000}} \cr
& {\text{Amount = Rs 8820}} \cr
& {\text{Let Rate = }}R \cr
& {\text{Time = 2 years}} \cr
& {\text{By using formula, }} \cr
& \Rightarrow 8820 = 8000{\left( {1 + \frac{R}{{100}}} \right)^2} \cr
& \Rightarrow \frac{{8820}}{{8000}} = {\left( {1 + \frac{R}{{100}}} \right)^2} \cr
& \Rightarrow \frac{{441}}{{400}} = {\left( {1 + \frac{R}{{100}}} \right)^2} \cr
& {\text{Taking square root of both sides,}} \cr
& \Rightarrow \frac{{21}}{{20}} = \left( {1 + \frac{R}{{100}}} \right) \cr
& \Rightarrow R = 5\% \cr} $$

Answer: Option C. -> 6 years
x becomes 3x in 3 years
Therefore 3x also becomes 9x in 3 years
Required years = 3 + 3 = 6

Answer: Option D. -> Rs. 400
$$\eqalign{
& {\text{Rate = 10}}\% \cr
& {\text{Time = 2 years}} \cr
& {\text{Effective rate of CI for 2 years}} \cr
& {\text{ = 10 + 10 + }}\frac{{10 \times 10}}{{100}} \cr
& = 21\% \cr
& {\text{Effective rate of SI for 2 years}} \cr
& {\text{ = 2}} \times {\text{10 = 20}}\% \cr
& {\text{Required SI}} \cr
& {\text{ = }}\frac{{420}}{{21}} \times {\text{20 = Rs. 400}} \cr} $$

Answer: Option C. -> Rs. 1750
$$\eqalign{
& {\text{C}}{\text{.I}}{\text{.}}\, = Rs.\,\left[ {4000 \times {{\left( {1 + \frac{{10}}{{100}}} \right)}^2} - 4000} \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {4000 \times \frac{{11}}{{10}} \times \frac{{11}}{{10}} - 4000} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,840 \cr
& \therefore {\text{Sum}} = Rs.\,\left( {\frac{{420 \times 100}}{{3 \times 8}}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,1750 \cr} $$

Answer: Option A. -> 8
$$\left[ {15000 \times {{\left( {1 + \frac{R}{{100}}} \right)}^2} - 15000} \right]$$       $$ - $$ $$\left( {\frac{{15000 \times R \times 2}}{{100}}} \right)$$    $$ = 96$$
$$ \Rightarrow 15000\left[ {{{\left( {1 + \frac{R}{{100}}} \right)}^2} - 1 - \frac{{2R}}{{100}}} \right] = 96$$
$$ \Rightarrow 15000$$  $$\left[ {\frac{{{{\left( {100 + R} \right)}^2} - 10000 - \left( {200 \times R} \right)}}{{10000}}} \right]$$       $$ = 96$$
$$\eqalign{
& \Rightarrow {R^2} = {\frac{{96 \times 2}}{3}} = 64 \cr
& \Rightarrow R = 8 \cr
& \therefore {\text{Rate}} = 8\% \cr} $$

Answer: Option B. -> Rs. 500
$$\eqalign{
& \text{Let the sum be Rs. P} \cr
& \text{Then, }\, {P{{\left( {1 + \frac{{10}}{{100}}} \right)}^2} - P} = 525 \cr
& \Rightarrow P\left[ {{{\left( {\frac{{11}}{{10}}} \right)}^2} - 1} \right] = 525 \cr
& \Rightarrow P = {\frac{{525 \times 100}}{{21}}} = 2500 \cr
& \therefore \text{Sum} = Rs.\,2500 \cr
& \text{So, S.I.} = Rs.\left( {\frac{{2500 \times 5 \times 4}}{{100}}} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,500 \cr} $$