The curve $C$ has equation$y=k{{x}^{3}}$ . The tangent at the point $P$ on $C$ meets the curve again

Question

The curve $C$ has equation$y=k{{x}^{3}}$ . The tangent at the point $P$ on $C$ meets the curve again at point $Q$. The tangent at point $Q$ meets the curve again at point $R$. If the $x$ coordinates of $P$, $Q$ and $R$ are $p$ , $q$, and $r$ respectively where $p
e 0$ . Show that $p$ and $q$ satisfy the equation ${{\left( \frac{q}{p} ight)}^{2}}+\left( \frac{q}{p} ight)-2=0$ . [4] Show that $p$ , $q$, and $r$ are three consecutive terms of a geometric progression. Hence determine if this geometric series is convergent. [4] [You may use the identity ${{a}^{3}}-{{b}^{3}}=\left( a-b ight)\left( {{a}^{2}}+ab+{{b}^{2}} ight)$ for $a,b\in \mathbb{R}$.]

Accepted Answer

(i) $q/p = -2$; (ii) GP with ratio $-2$, not convergent

2017 TJC P1 Q5

2017 TJC P1 Q5

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