For the following questions answer them individually
$$p^3 + q^3 + r^3 - 3pqr = 4$$. If $$a = q + r, b = r + p$$ and $$c = p + q$$, then what is the value of $$a^3 + b^3 + c^3 - 3abc$$?
If $$\alpha$$ and $$\beta$$ are the roots of the equation $$x^2 + x - 1 = 0$$, then what is the equation whose roots are $$\alpha^5$$ and $$\beta^5$$?
If $$x$$ and $$y$$ are natural numbers such that $$x + y = 2017$$, then what is the value of $$(- 1)^x + (- 1)^y$$?
If $$x + (\frac{1}{x}) = \frac{(\surd3 + 1)}{2}$$, then what is the value of $$x^4 + (\frac{1}{x^4})$$?
If $$a - \left(\frac{1}{a}\right) = b, b - \left(\frac{1}{b}\right) = c$$ and $$c - \left(\frac{1}{c}\right) = a$$, then what is the value of $$\left(\frac{1}{ab}\right) + \left(\frac{1}{bc}\right) + \left(\frac{1}{ca}\right)$$?
If the roots of the equation $$a(b - c)x^2 +b(c - a)x + c(a - b) = 0$$ are equal, then which of the following is true?
If $$[\surd(a^2 + b^2 + ab)] + [\surd(a^2 + b^2 - ab)] = 1$$, then what is the value of $$(1 - a^2)(1 - b^2)$$?
