Simplification - Aptitude MCQ Questions and Solutions with Explanations

xyz = 256 ⇒ (2z) (2z) z = 256 ⇒ 4z³
= 256 ⇒ z³ = 64 ⇒ z = 4
∴ x = 2z = (2 × 4) = 8

  $$\left( {1 - \frac{1}{{{3^2}}}} \right)$$ $$\left( {1 - \frac{1}{{{4^2}}}} \right)$$ $$\left( {1 - \frac{1}{{{5^2}}}} \right)$$   . . . . . $$\left( {1 - \frac{1}{{{{11}^2}}}} \right)$$ $$\left( {1 - \frac{1}{{{{12}^2}}}} \right)$$
  $$ = \left( {\frac{{{3^2} - 1}}{{{3^2}}}} \right)\left( {\frac{{{4^2} - 1}}{{{4^2}}}} \right)\left( {\frac{{{5^2} - 1}}{{{5^2}}}} \right)....$$       $$\left( {\frac{{{{11}^2} - 1}}{{{{11}^2}}}} \right)$$ $$\left( {\frac{{{{12}^2} - 1}}{{{{12}^2}}}} \right)$$
  $$ = \left[ {\frac{{\left( {3 + 1} \right)\left( {3 - 1} \right)}}{{\left( {2 + 1} \right)\left( {4 - 1} \right)}}} \right]$$   $$\left[ {\frac{{\left( {4 + 1} \right)\left( {4 - 1} \right)}}{{\left( {3 + 1} \right)\left( {5 - 1} \right)}}} \right]$$   $$\left[ {\frac{{\left( {5 + 1} \right)\left( {5 - 1} \right)}}{{\left( {4 + 1} \right)\left( {6 - 1} \right)}}} \right]$$   . . . . . $$\left[ {\frac{{\left( {11 + 1} \right)\left( {11 - 1} \right)}}{{\left( {10 + 1} \right)\left( {12 - 1} \right)}}} \right]$$   $$\left[ {\frac{{\left( {12 + 1} \right)\left( {12 - 1} \right)}}{{\left( {11 + 1} \right)\left( {13 - 1} \right)}}} \right]$$
$$\eqalign{ & = \frac{{\left( {3 - 1} \right)}}{{\left( {2 + 1} \right)}} \times \frac{{\left( {12 + 1} \right)}}{{\left( {13 - 1} \right)}} \cr & = \frac{2}{3} \times \frac{{13}}{{12}} \cr & = \frac{{13}}{{18}} \cr} $$

$$\eqalign{ & {\text{According to question,}} \cr & \Rightarrow \frac{{{{\left( {7.5} \right)}^3} + 1}}{{{{\left( {7.5} \right)}^2} - 6.5}}\,\,\,\,\,\,\, \cr & \,\,\,\,\,\,\,\,\,\,\,\left[ {\therefore {a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)} \right] \cr & \Rightarrow \frac{{\left( {7.5 + 1} \right)\left[ {{{\left( {7.5} \right)}^2} + 1 - 7.5 \times 1} \right]}}{{{{\left( {7.5} \right)}^2} - 7.5 \times 1 + {1^2}}} \cr & \Rightarrow 8.5 \cr} $$

$$\eqalign{ & {\text{According to question}} \cr & \sqrt {13} {\text{ + }}\sqrt {1300} {\text{ + }}\sqrt {0.013} \cr & = 3.6 + 10\sqrt {13} + \sqrt {\frac{{130}}{{10000}}} \cr & = 3.6 + 10 \times 3.6 + \frac{{11.4}}{{100}} \cr & = 39.714 \cr} $$

Given expression,
 $${\text{ = }}\left( {1 - \frac{1}{2}} \right)$$   $$ + $$ $$\left( {\frac{1}{2} - \frac{1}{3}} \right)$$   $$ + $$ $$\left( {\frac{1}{3} - \frac{1}{4}} \right)$$   $$ + $$ $$\left( {\frac{1}{4} - \frac{1}{5}} \right)$$   $$ + $$ . . . . . $$ + $$ $$\left( {\frac{1}{{11}} - \frac{1}{{12}}} \right)$$
$$\eqalign{ & = \left( {1 - \frac{1}{{12}}} \right) \cr & = \frac{{11}}{{12}} \cr} $$

Clearly , sum of first 6 terms is zero , So sum of first 30 terms = 0
∴ Required sum
$$\eqalign{ & {\text{ = }}\left( {\frac{1}{2} + \frac{1}{3} - \frac{1}{4} - \frac{1}{2} - \frac{1}{3}} \right) \cr & = - \frac{1}{4} \cr} $$

$$\eqalign{ & {\text{Given expression,}} \cr & \left( {1 + 11 + 111 + 1111} \right) + \left( {\frac{1}{2} \times 4} \right) \cr & = 1234 + 2 = 1236 \cr} $$

$$\eqalign{ & {\text{According to question,}} \cr & \frac{{3\frac{1}{4} - \frac{4}{5}{\text{ of }}\frac{5}{6}}}{{4\frac{1}{3} \div \frac{1}{5} - \left( {\frac{3}{{10}} + 21\frac{1}{5}} \right)}}{\text{ }} \cr & \Rightarrow \frac{{\frac{{13}}{4} - \frac{4}{5} \times \frac{5}{6}}}{{\frac{{13}}{3} \times 5 - \left( {\frac{3}{{10}} + \frac{{106}}{5}} \right)}}{\text{ }} \cr & \Rightarrow \frac{{\frac{{13}}{4} - \frac{2}{3}}}{{\frac{{65}}{3} - \frac{3}{{10}} - \frac{{106}}{5}}} \cr & \Rightarrow \frac{{\frac{{39 - 8}}{{12}}}}{{\frac{{650 - 9 - 636}}{{30}}}} \cr & \Rightarrow \frac{{31}}{{12}} \times \frac{{30}}{5} \cr & \Rightarrow \frac{{31}}{2} \cr & \Rightarrow 15\frac{1}{2} \cr} $$
∴ Least fraction number should be subtracted is:
$${\text{15}}\frac{1}{2} - 15 = \frac{1}{2}$$     (least fraction number)

$$\eqalign{ & {\text{According to question,}} \cr & \,\,\,\, \sqrt {5 + \sqrt {11 + \sqrt {19 + \sqrt {29 + \sqrt {49} } } } } \cr & \Rightarrow \sqrt {5 + \sqrt {11 + \sqrt {19 + \sqrt {29 + 7} } } } \cr & \Rightarrow \sqrt {5 + \sqrt {11 + \sqrt {19 + \sqrt {36} } } } \cr & \Rightarrow \sqrt {5 + \sqrt {11 + \sqrt {19 + 6} } } \cr & \Rightarrow \sqrt {5 + \sqrt {11 + \sqrt {25} } } \cr & \Rightarrow \sqrt {5 + \sqrt {11 + 5} } \cr & \Rightarrow \sqrt {5 + \sqrt {16} } \cr & \Rightarrow \sqrt {5 + 4} \cr & \Rightarrow \sqrt 9 \cr & \Rightarrow 3 \cr} $$

As we know that square of
'31' is = 961
∴ 1000 - 961 =39