<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=3{{t}^{2}}$, $y=2{{t}^{3}}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the equation of the tangent to $C$ at the point with parameter $t$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Points $P$ and $Q$ on $C$ have parameters $p$ and $q$ respectively. The tangent at $P$ meets the tangent at $Q$ and some point $R$. Show that the $x$-coordinates of $R$ is ${{p}^{2}}+pq+{{q}^{2}}$, and find the $y$-coordinates of $R$ in terms of $p$ and $q$. Given that $pq=-1$, show that $R$ lies on the curve with equation $x={{y}^{2}}+1$.</div><p style="text-align: right;">[5]</p></li></ol>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1760242144093-gsk7u74l81p.png" width="400" class="aligncenter" /></figure>
<p>A curve $L$ has equation $x={{y}^{2}}+1$. The diagram shows parts of $C$ and $L$ for which $y\ge 0$. The curves $C$ and $L$ touch at the point $M$.</p>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">Show that $4{{t}^{6}}-3{{t}^{2}}+1=0$ at $M$. Hence, or otherwise, find the exact coordinates of $M$.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Find the exact value of the area of the shaded region bounded by $C$ and $L$ for which $y\ge 0$.</div><p style="text-align: right;">[6]</p></li></ol>

Find the equation of the tangent to $C$ at the point with parameter $t$.

Points $P$ and $Q$ on $C$ have parameters $p$ and $q$ respectively. The tangent at $P$ meets the tangent at $Q$ and some point $R$. Show that the $x$-coordinates of $R$ is ${{p}^{2}}+pq+{{q}^{2}}$, and find the $y$-coordinates of $R$ in terms of $p$ and $q$. Given that $pq=-1$, show that $R$ lies on the curve with equation $x={{y}^{2}}+1$.

A curve $L$ has equation $x={{y}^{2}}+1$. The diagram shows parts of $C$ and $L$ for which $y\ge 0$. The curves $C$ and $L$ touch at the point $M$.

Show that $4{{t}^{6}}-3{{t}^{2}}+1=0$ at $M$. Hence, or otherwise, find the exact coordinates of $M$.

Find the exact value of the area of the shaded region bounded by $C$ and $L$ for which $y\ge 0$.

N2013 P1 Q11

N2013 P1 Q11

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