1. **Simplify the surds:** (a)(i) Simplify $\sqrt{2} + 3\sqrt{3} - 5\sqrt{2} + \sqrt{3}$. Group like terms: $$\sqrt{2} - 5\sqrt{2} + 3\sqrt{3} + \sqrt{3} = (1 - 5)\sqrt{2} + (3 + 1)\sqrt{3} = -4\sqrt{2} + 4\sqrt{3}$$ Answer: $$-4\sqrt{2} + 4\sqrt{3}$$ (a)(ii) Simplify $\sqrt{50} - \sqrt{75} + \sqrt{27} - \sqrt{18}$. First simplify each surd: $$\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$$ $$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$$ $$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$$ $$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$$ Rewrite expression: $$5\sqrt{2} - 5\sqrt{3} + 3\sqrt{3} - 3\sqrt{2} = (5 - 3)\sqrt{2} + (-5 + 3)\sqrt{3} = 2\sqrt{2} - 2\sqrt{3}$$ Answer: $$2\sqrt{2} - 2\sqrt{3}$$ 2. **Rationalise the denominator:** Rationalise $$\frac{2 + \sqrt{3}}{4 + \sqrt{2}}$$. Multiply numerator and denominator by the conjugate of the denominator $4 - \sqrt{2}$: $$\frac{2 + \sqrt{3}}{4 + \sqrt{2}} \times \frac{4 - \sqrt{2}}{4 - \sqrt{2}} = \frac{(2 + \sqrt{3})(4 - \sqrt{2})}{(4 + \sqrt{2})(4 - \sqrt{2})}$$ Calculate denominator: $$(4)^2 - (\sqrt{2})^2 = 16 - 2 = 14$$ Calculate numerator by expansion: $$2 \times 4 = 8$$ $$2 \times (-\sqrt{2}) = -2\sqrt{2}$$ $$\sqrt{3} \times 4 = 4\sqrt{3}$$ $$\sqrt{3} \times (-\sqrt{2}) = -\sqrt{6}$$ Sum numerator terms: $$8 - 2\sqrt{2} + 4\sqrt{3} - \sqrt{6}$$ Final expression: $$\frac{8 - 2\sqrt{2} + 4\sqrt{3} - \sqrt{6}}{14}$$ Simplify by factoring out 2 in numerator terms where possible if desired, but left as is is acceptable. Answer: $$\frac{8 - 2\sqrt{2} + 4\sqrt{3} - \sqrt{6}}{14}$$ or $$\frac{4 - \sqrt{2} + 2\sqrt{3} - \frac{1}{2}\sqrt{6}}{7}$$ 3. **Graph of the cubic function:** The function is given by: $$y = (x + 1)(x - 1)(x - 3)$$ It has roots at $$x = -1, 1, 3$$ which match the intercepts on the x-axis. The curve passes through the origin since: $$y(0) = (0 + 1)(0 - 1)(0 - 3) = 1 \times (-1) \times (-3) = 3$$ Actually, it passes through (0, 3), not origin; there might be a slight mistake in problem description but this is the function matching given roots and signs. The shape is cubic with three distinct roots and sign changes as described.