Simplication MCQs

The least fraction to be subtracted from the expression $$\frac{{3\frac{1}{4} - \frac{4}{5}{\text{ of }}\frac{5}{6}}}{{4\frac{1}{3} \div \frac{1}{5} - \left( {\frac{3}{{10}} + 21\frac{1}{5}} \right)}}$$     to make it an integer?

Find the value of the following expression.
$$\frac{{1\frac{2}{3} \div \frac{5}{6} \times 6 + \frac{4}{5} \times \frac{1}{2} + \frac{2}{3}}}{{2 - \left[ {1\frac{1}{3} \times \left( { - \frac{3}{5}} \right) - 6\left\{ {\frac{3}{5} - \left( {3 - \frac{3}{{10}}} \right)} \right\}} \right]}}$$

Which of the following statement is correct?
I. The value of 100² - 99² + 98² - 97² + 96² - 95² + 94² - 93² + . . . . + 2² - 1² is 5050.
II. If 8x + $$\frac{8}{{\text{x}}}$$ = -16 and x < 0, then the value of x¹⁹⁷ + x^-197 is 2.

If 56 × 75 × 60 × 84 × 210 = 2^p × 3^q × 5^r × 7^s, then what is the value of $$\left[ {\frac{{\left( {{\text{p}} + {\text{q}}} \right)}}{{\text{s}}}} \right] + {\text{r}}?$$

Simplify the following:
$$\frac{{{{\left( {0.1} \right)}^2} + {{\left( {0.01} \right)}^2} + {{\left( {0.009} \right)}^2} + 2\left[ {0.1 \times 0.01 + 0.01 \times 0.009 + 0.009 \times 0.1} \right]}}{{0.01 + 0.001 + 0.0009}}$$

$$\eqalign{ & {\text{If}} \cr & {\text{I = }}\frac{3}{4} \div \frac{5}{6}{\text{,}} \cr & {\text{II = 3}} \div \left[ {\left( {4 \div 5} \right) \div 6} \right]{\text{,}} \cr & {\text{III = }}\left[ {{\text{3}} \div \left( {4 \div 5} \right)} \right] \div {\text{6,}} \cr & {\text{IV = 3}} \div {\text{4}} \div \left( {5 \div 6} \right), \cr & {\text{Then - }} \cr} $$

Direction: In the question given below the given mathematical symbols are changed from '+' to '÷', '-' to '×', '÷' to '-' and from '×' to '+', then choose your answers from the following options.
67 × 119 + 17 - 27 × 259 = ?

The sum of the first 35 terms of the series $$\frac{1}{2}$$ $$ + $$ $$\frac{1}{3}$$ $$ - $$ $$\frac{1}{4}$$ $$ - $$ $$\frac{1}{2}$$ $$ - $$ $$\frac{1}{3}$$ $$ + $$ $$\frac{1}{4}$$ $$ + $$ $$\frac{1}{2}$$ $$ + $$ $$\frac{1}{3}$$ $$ - $$ $$\frac{1}{4}$$ . . . . . is = ?

The simplified value of $$\frac{{\sqrt {32} + \sqrt {48} }}{{\sqrt 8 + \sqrt {12} }}{\text{ is = ?}}$$