Practice MCQ Questions and Answer on Simplification

Solution:

 $$\eqalign{ & {\text{According to question,}} \cr & \,\,\,\left[ {4 - \frac{5}{{1 + \frac{1}{{3 + \frac{1}{{2 + \frac{1}{4}}}}}}}} \right] \cr & = 4 - \frac{5}{{1 + \frac{1}{{3 + \frac{1}{{\frac{9}{4}}}}}}} \cr & = 4 - \frac{5}{{1 + \frac{1}{{3 + \frac{4}{9}}}}} \cr & = 4 - \frac{5}{{1 + \frac{1}{{\frac{{31}}{9}}}}} \cr & = 4 - \frac{5}{{1 + \frac{9}{{31}}}} \cr & = 4 - \frac{5}{{\frac{{40}}{{31}}}} \cr & = 4 - \frac{{5 \times 31}}{{40}} \cr & = 4 - \frac{{31}}{8} \cr & = \frac{1}{8} \cr & {\text{According to question}} \cr & \frac{1}{8}{\text{part = 10 minutes}} \cr & {\text{1 part = 10 minutes}} \cr & \frac{3}{5}{\text{ part = 80}} \times \frac{3}{5} \cr & {\text{ = 48 minutes}} \cr} $$

Solution:

 $$\eqalign{ & {\text{Given,}} \cr & \left( {x + \frac{1}{x}} \right) = 3{\text{ }} \cr & {\text{On squaring both sides, we get}} \cr & {\left( {x + \frac{1}{x}} \right)^2} = {3^2} \cr & \Rightarrow {\left( {x + \frac{1}{x}} \right)^2} = 9 \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} + 2 = 9 \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} = 7 \cr} $$

Solution:

 $$\eqalign{ & {\text{A}} = \left[ {\frac{3}{7}{\text{ of }}4\frac{1}{5} \div \frac{{18}}{{25}} + \frac{{17}}{{24}}} \right]{\text{ of }}\left[ {\frac{{289}}{{16}} \div {{\left( {\frac{3}{4} + \frac{2}{3}} \right)}^2}} \right] \cr & {\text{A}} = \left[ {\frac{3}{7}{\text{ of }}\frac{{21}}{5} \div \frac{{18}}{{25}} + \frac{{17}}{{24}}} \right]{\text{ of }}\left[ {\frac{{289}}{{16}} \div \frac{{{{17}^2}}}{{{{12}^2}}}} \right] \cr & {\text{A}} = \left[ {\frac{9}{5} \times \frac{{25}}{{18}} + \frac{{17}}{{24}}} \right]{\text{ of }}\left[ {\frac{{289}}{{16}} \times \frac{{144}}{{289}}} \right] \cr & {\text{A}} = \left[ {\frac{5}{2} + \frac{{17}}{{24}}} \right]{\text{ of 9}} \cr & {\text{A}} = \frac{{77}}{{24}} \times 9 \cr & {\text{8A}} = 8 \times \frac{{77}}{{24}} \times 9 \cr & {\text{8A}} = 231 \cr} $$

Solution:

 $$\eqalign{ & {\text{According to question,}} \cr & \frac{5}{{3 + \frac{3}{{1 - \frac{2}{3}}}}}\,\, \cr & \Rightarrow \frac{5}{{3 + \frac{3}{{\frac{1}{3}}}}}\, \cr & \Rightarrow \frac{5}{{3 + 9}} \cr & \Rightarrow \frac{5}{{12}} \cr} $$

Solution:

 $$\eqalign{ & \left( {x + \frac{1}{x}} \right){\text{ = 2}} \cr & \Rightarrow {\left( {x + \frac{1}{x}} \right)^2}{\text{ = }}{{\text{2}}^2} \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} + 2 = 4 \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} = 2 \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} - 2.x.\frac{1}{x} \cr & \,\,\,\,\,\,\,\, = 2 - 2 = 0 \cr & \Rightarrow {\left( {x - \frac{1}{x}} \right)^2}{\text{ = 0}} \cr & \Rightarrow x - \frac{1}{x} = 0 \cr} $$

Solution:

 $$\eqalign{ & {\text{According to question,}} \cr & \sqrt {\frac{{0.009 \times 0.036 \times 0.016 \times 0.08}}{{0.002 \times 0.0008 \times 0.0002}}} \, \cr & \Rightarrow \sqrt {\frac{{9 \times 36 \times 16 \times 8}}{{2 \times 8 \times 2}}} \cr & \Rightarrow 3 \times 6 \times 2 \cr & \Rightarrow 36 \cr} $$

Solution:

 $$\eqalign{ & 1 + \frac{1}{{1 + \frac{2}{{2 + \frac{3}{{1 + \frac{4}{5}}}}}}} \cr & = 1 + \frac{1}{{1 + \frac{2}{{2 + \frac{3}{{\frac{9}{5}}}}}}} \cr & = 1 + \frac{1}{{1 + \frac{2}{{2 + \frac{5}{3}}}}} \cr & = 1 + \frac{1}{{1 + \frac{2}{{\frac{{11}}{3}}}}} \cr & = 1 + \frac{1}{{1 + \frac{6}{{11}}}} \cr & = 1 + \frac{1}{{\frac{{17}}{{11}}}} \cr & = 1 + \frac{{11}}{{17}} \cr & = 1\frac{{11}}{{17}} \cr} $$

Solution:

 $$\eqalign{ & \frac{{{{\left( {469 + 174} \right)}^2} - {{\left( {469 - 174} \right)}^2}}}{{\left( {469 \times 174} \right)}} = ? \cr & {\text{We}}\,{\text{ know that}}, \cr & 4ab = {\left( {a + b} \right)^2} - {\left( {a - b} \right)^2} \cr & \therefore \frac{{{{\left( {469 + 174} \right)}^2} - {{\left( {469 - 174} \right)}^2}}}{{\left( {469 \times 174} \right)}} \cr & = \frac{{4 \times 469 \times 174}}{{469 \times 174}} \cr & = 4 \cr} $$

Solution:

 $$\eqalign{ & {\text{According to question,}} \cr & \frac{{\frac{5}{3} \times \frac{7}{{51}}{\text{ of }}\frac{17}{5} - \frac{1}{3}}}{{\frac{2}{9} \times \frac{5}{7}{\text{ of }}\frac{{28}}{5} - \frac{2}{3}}}\, \cr & \Rightarrow \frac{{\frac{5}{3} \times \frac{7}{{51}} \times \frac{17}{5} - \frac{1}{3}}}{{\frac{2}{9} \times \frac{5}{7} \times \frac{{28}}{5} - \frac{2}{3}}}\, \cr & \Rightarrow \frac{{\frac{5}{3} \times \frac{7}{{15}} - \frac{1}{3}}}{{\frac{2}{9} \times 4 - \frac{2}{3}}} \cr & \, \Rightarrow \frac{{\frac{7}{9} - \frac{1}{3}}}{{\frac{8}{9} - \frac{2}{3}}} \cr & \, \Rightarrow \frac{{\frac{{7 - 3}}{9}}}{{\frac{{8 - 6}}{9}}} \cr & \Rightarrow \frac{4}{2} \cr & \Rightarrow 2 \cr} $$

Solution:

 $$\eqalign{ & \frac{{0.75}}{{10}} + \frac{{0.31}}{{100}} + \frac{{0.09}}{{1000}} \cr & = \frac{{75}}{{1000}} + \frac{{3.1}}{{1000}} + \frac{{0.09}}{{1000}} \cr & = \frac{{78.19}}{{10000}} \cr & = 0.07819 \cr} $$