A curve $C$ has parametric equations $x=1-2{{t}^{2}}$, $\qquad$ $y={{t}^{3}}+1$, $\qquad$

Question

A curve $C$ has parametric equations $x=1-2{{t}^{2}}$, $\qquad$ $y={{t}^{3}}+1$, $\qquad$ for $t\in \mathbb{R}$. Sketch $C$, indicating the coordinates of the points of intersection with the axes.[3] The line $x+y=2$ cuts $C$ at the points $A$ and $B$ where the $x$-coordinate of $A$ is greater than the $x$-coordinate of $B$. Find the exact length of $AB$ in the form $k\sqrt{2}$ where $k$ is an integer to be determined.[3] The point $M$ is the midpoint of $AP$ where $P$ is the point $\left( 1-2{{p}^{2}},{{p}^{3}}+1 \right)$. Find a cartesian equation of the curve traced by $M$ as $p$ varies. Leave your answer in the form ${{\left( ax+b \right)}^{3}}={{\left( cy+d \right)}^{2}}$, where $a$, $b$, $c$ and $d$ are integers to be determined.[3] $y=\left[ \mathrm{f}\left( p \right) \right]x+{{p}^{3}}+\frac{8p}{3}+1-\frac{4}{3p}$ Where $\mathrm{f}\left( p \right)$ is an expression in $p$ to be determined.[3] Show that the equation of the normal to $C$ at the point $P$ is Find the acute angle between the normals to the curve at $x=-7$.[3]

Accepted Answer

(b) $8\sqrt{2}$ (b) ${{\left( 1-x \right)}^{3}}={{\left( 2y-2 \right)}^{2}}$ (e) ${{67.4}^{\circ }}$ or $1.18$ rad

2023 CJC Promo Q11

2023 CJC Promo Q11

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