Here are the solutions to the questions: 1. (i) Given that the roots of the equation $3x^2 - 4x + 2 = 0$ are $\alpha$ and $\beta$. a) Show that $\alpha^3 + \beta^3 = -\frac{8}{27}$. Step 1: Identify the sum and product of roots from the given quadratic equation $3x^2 - 4x + 2 = 0$. For $ax^2 + bx + c = 0$, the sum of roots is $\alpha + \beta = -\frac{b}{a}$ and the product of roots is $\alpha\beta = \frac{c}{a}$. Here, $a=3$, $