Algebra MCQs

If $$a = \frac{{\sqrt 3 - \sqrt 2 }}{{\sqrt 3 + \sqrt 2 }}$$    and $$b = \frac{{\sqrt 3 + \sqrt 2 }}{{\sqrt 3 - \sqrt 2 }}{\text{,}}$$    then the value of $$\frac{{{a^2}}}{b}$$  + $$\frac{{{b^2}}}{a}$$  = ?

If $$2 + x\sqrt 3 $$   = $$\frac{1}{{2 + \sqrt 3 }}{\text{,}}$$   then the simplest value of x is?

If $$a - \frac{1}{{a - 3}} = 5{\text{,}}$$   then the value of $${\left( {a - 3} \right)^3}$$   - $$\frac{1}{{{{\left( {a - 3} \right)}^3}}} = ?$$

If $$x + \frac{1}{x} = 5{\text{,}}$$   then the value of $$\frac{{5x}}{{{x^2} + 5x + 1}}$$   is?

If $${\left( {x + \frac{1}{x}} \right)^2} = 3{\text{,}}$$    then the value of (x⁷² + x⁶⁶ + x⁵⁴ + x²⁴ + x⁶ + 1) is?

If $$x + \frac{1}{x} = 1{\text{,}}$$   then the value of $$\frac{{{x^2} + 3x + 1}}{{{x^2} + 7x + 1}}$$   is?

If x² - 3x + 1 = 0, then the value of $$\frac{{\left( {{x^4} + \frac{1}{{{x^2}}}} \right)}}{{\left( {{x^2} + 5x + 1} \right)}}$$   is:

Simplified value of $$\left[ {\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\, - \,\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,} \right]$$              $$ ÷ $$  $$\left[ {\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\, + \,\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,} \right]$$       = ?

Find the value of a and b if (x - 1) and (x + 1) are factors of x⁴ + ax³ - 3x² + 2x + b = ?