For the following questions answer them individually
Which of the following statement(s) is/are true?
$$I. (65)^{\frac{1}{6}} > (17)^{\frac{1}{4}} > (12)^{\frac{1}{3}}$$
$$II. (17)^{\frac{1}{4}} > (65)^{\frac{1}{6}} > (12)^{\frac{1}{3}}$$
$$III. (12)^{\frac{1}{3}} > (17)^{\frac{1}{4}} > (65)^{\frac{1}{6}}$$
If $$P = 7 + 4\surd3$$ and $$PQ = 1$$, then what is the value of $$\left( \frac{1}{P^2} \right) + \left(\frac{1}{Q^2}\right)$$?
If $$x = (\surd5) + 1$$ and $$y = (\surd5) - 1$$, then what is the value of $$\left(\frac{x^2}{y^2}\right) + \left(\frac{y^2}{x^2}\right) + 4[\left(\frac{x}{y}\right) + \left(\frac{y}{x}\right)] + 6$$?
If $$x = 2 +\surd3, y = 2 - \surd3$$ and $$z = 1$$, then what is the value of $$\left(\frac{x}{yz}\right) + \left(\frac{y}{xz}\right) + \left(\frac{z}{xy}\right) + 2 \left[\left(\frac{1}{x}\right) + \left(\frac{1}{y}\right) + \left(\frac{1}{z}\right)\right]$$?
A root of equation $$ax^2 + bx + c = 0$$ (where $$a, b$$ and $$c$$ are rational numbers) is $$5 + 3\surd3$$. What is the value of $$\frac{(a^2 + b^2 + c^2)}{(a + b + c)}$$?
If $$x = (\frac{a}{b}) + (\frac{b}{a}), y = (\frac{b}{c}) + (\frac{c}{b})$$ and $$z = (\frac{c}{a}) + (\frac{a}{c})$$, then what is the value of $$xyz - x^2 - y^2 - z^2$$?
If $$\left[a + (\frac{1}{a})\right]^2 - 2\left[a - (\frac{1}{a})\right] = 12$$, then which of the following is a value of '$$a$$'?
$$A = \frac{(x^8 - 1)}{(x^4 + 1)}$$ and $$B = \frac{(y^4 - 1)}{(y^2 + 1)}$$. If $$x = 2$$ and $$y = 9$$, then what is the value of $$A^2 + 2AB + A(A+B)^2$$?
