<p>The curve \(C\) has equation</p>
<p style="text-align: center">\[y = x - 2 + \frac{3k(x - 2)}{x^2 - 2k}, \quad \text{where}\hspace{0.5em} k &gt; 0, \ k \ne 2.\]</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Without the use of a calculator, find the coordinates of any points of intersection of \(C\) with the axes.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Write down the equations of the asymptotes of \(C\) in terms of \(k\).<span class="inline-marks">[2]</span></div></li></ol>
<p>It is now given that&nbsp;\(k = 1\).</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Show that the \(x\)-coordinates of the turning points of \(C\) satisfy the equation \[x^4 - 7x^2 + 12x - 2 = 0.\] Hence find these \(x\)-coordinates.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Sketch \(C\), labelling all essential features.<span class="inline-marks">[3]</span></div></li></ol>

\[y = x - 2 + \frac{3k(x - 2)}{x^2 - 2k}, \quad \text{where } k > 0, \ k \ne 2.\]

Without the use of a calculator, find the coordinates of any points of intersection of \(C\) with the axes.

Write down the equations of the asymptotes of \(C\) in terms of \(k\).

Show that the \(x\)-coordinates of the turning points of \(C\) satisfy the equation \[x^4 - 7x^2 + 12x - 2 = 0.\] Hence find these \(x\)-coordinates.

Sketch \(C\), labelling all essential features.

2024 EJC Promo Q12

2024 EJC Promo Q12