<p>The curve \(C\) is defined by the parametric equations \(x = 6t - 5\) and \(y = 2t^2 + 1\), where \(t \ge \frac{5}{6}\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the curve \(C\), labelling the exact coordinates of any axial intercept(s).</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the equation of the normal to \(C\) at the point \(P\) where \(t = 1\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the acute angle between the normal to \(C\) at the point \(P\) where \(t = 1\) and the tangent to \(C\) at the point \(M\) where \(t = 3\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The curve \(C\) is translated \(5\) units in the positive \(x\)-direction and translated \(3\) units in the negative \(y\)-direction, to form the curve \(D\). Find the equation of \(D\) in parametric form.</span><span class="inline-marks">[2]</span></div></li></ol>
<p>The curve \(E\) is defined by the parametric equations \(x = 7u\) and \(y = \frac{9}{u}\), where \(u \neq 0\).</p>
<ol class="abc-list" start="5"><li><div class="list-item-content"><span class="list-item-text">Show that at the point of intersection of the curves \(C\) and \(E\), \(6t^3 - 5t^2 + 3t - 34 = 0\). Deduce that there is only one point of intersection and find the coordinates of this point.</span><span class="inline-marks">[4]</span></div></li></ol>

The curve \(C\) is defined by the parametric equations \(x = 6t - 5\) and \(y = 2t^2 + 1\), where \(t \ge \frac{5}{6}\).

Sketch the curve \(C\), labelling the exact coordinates of any axial intercept(s).

Find the equation of the normal to \(C\) at the point \(P\) where \(t = 1\).

Find the acute angle between the normal to \(C\) at the point \(P\) where \(t = 1\) and the tangent to \(C\) at the point \(M\) where \(t = 3\).

The curve \(C\) is translated \(5\) units in the positive \(x\)-direction and translated \(3\) units in the negative \(y\)-direction, to form the curve \(D\). Find the equation of \(D\) in parametric form.

The curve \(E\) is defined by the parametric equations \(x = 7u\) and \(y = \frac{9}{u}\), where \(u \neq 0\).

Show that at the point of intersection of the curves \(C\) and \(E\), \(6t^3 - 5t^2 + 3t - 34 = 0\). Deduce that there is only one point of intersection and find the coordinates of this point.

2025 RVHS P1 Q12

2025 RVHS P1 Q12