<p>The curve $C$ has equation $y=\frac{{{x}^{2}}+2x+q}{x-p}$, $x\ne p$, where $p$ and $q$ are constants. It is given that the line $y=x+3$ is an asymptote of $C$.</p>
<p>Show that $p=1$.<span class="inline-marks">[2]</span></p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">If $C$ has $2$ stationary points, show that $q>-3$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Sketch $C$ for $q=1$. By adding another graph, deduce that for all positive $\beta $, the equation $\left( \beta +1 \right){{x}^{2}}+2x+1+\beta {{x}^{3}}$ has exactly one real root.<span class="inline-marks">[5]</span></div></li></ol>

The curve $C$ has equation $y=\frac{{{x}^{2}}+2x+q}{x-p}$, $x\ne p$, where $p$ and $q$ are constants. It is given that the line $y=x+3$ is an asymptote of $C$.

If $C$ has $2$ stationary points, show that $q>-3$.

Sketch $C$ for $q=1$. By adding another graph, deduce that for all positive $\beta $, the equation $\left( \beta +1 \right){{x}^{2}}+2x+1+\beta {{x}^{3}}$ has exactly one real root.

2018 DHS Promo Q4

2018 DHS Promo Q4

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