1. Problem 6: Given that $y$ varies directly as the square of $(x+1)$, we write the variation as: $$y = k(x+1)^2$$ where $k$ is the constant of proportionality. 2. Use the given values $y=200$ when $x=4$ to find $k$: $$200 = k(4+1)^2 = k(5)^2 = 25k$$ So, $$k = \frac{200}{25} = 8$$ 3. Therefore, the equation connecting $y$ and $x$ is: $$y = 8(x+1)^2$$ --- 4. Problem 7(a): Construct the locus of point $P$ such that $\angle MPN = 50^\circ$. - The locus of points $P$ forming a fixed angle $\theta$ with segment $MN$ is an arc of a circle passing through $M$ and $N$. - Using a ruler and compass, draw the circle segment where $\angle MPN = 50^\circ$. 5. Problem 7(b): Locate point $Q$ such that the area of $\triangle MQN$ is half that of $\triangle MLN$ and $\angle MQN = 50^\circ$. - Since $\angle MQN = 50^\circ$, point $Q$ lies on the same locus circle from part (a). - The area of $\triangle MLN$ is given by: $$\text{Area}_{MLN} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times MN \times 5$$ - To find $Q$ such that $\text{Area}_{MQN} = \frac{1}{2} \text{Area}_{MLN}$, adjust the position of $Q$ on the locus so that the base $MN$ is divided accordingly. - Use compass and ruler to mark $Q$ on the locus circle satisfying these conditions. This completes the construction steps for the problem.