The diagram depicts part of the curve $y = x(9 - x^2)$ together with the line $y = 5x$, which meet at the origin $O$ and at the point $R$. Point $P$ is on the line $y = 5x$ between $O$ and $R$, and its $x$-coordinate is $t$. Point $Q$ is on the curve, and $PQ$ is parallel to the $y$-axis.
(a(i))[2]
Express the length of $PQ$ in terms of $t$, and simplify your answer.
(a(ii))[3]
Given that $t$ may vary, find the greatest value of the length of $PQ$.
(c(i))[2]
Express the length of $PQ$ in terms of $t$, and simplify your answer.
(c(ii))[3]
As $t$ is allowed to vary, find the greatest value of the length of $PQ$.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “The coordinates of $P$ and $Q$ are $P(t,5t)$ and $Q(t,t(9-t^2))$.” …