<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">On a single diagram, sketch the graphs of \(y = \frac{3x}{x - 1}\) and \(y = 3x\), stating the equation(s) of any asymptote(s) and the coordinates of points of intersection.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the area bounded by the curve \(y = \frac{3x}{x - 1}\) and the lines \(y = 3x\) and \(y = 8\).</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">A curve has the equation \(y = \sqrt{x^2 + 1}\) where \(x\) is non-negative.</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">With the aid of a sketch of the above curve, explain why the value of \(S = \lim_{n \to \infty} \frac{1}{n} \left[ 1 + \sqrt{\frac{1}{n^2} + 1} + \sqrt{\frac{4}{n^2} + 1} + \dots + \sqrt{\frac{(n - 1)^2}{n^2} + 1} \right]\) is \(\int_0^1 \sqrt{x^2 + 1} \, \mathrm{d}x\).</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the value of \(S\).</span><span class="inline-marks">[1]</span></div></li></ol></li></ol>

On a single diagram, sketch the graphs of \(y = \frac{3x}{x - 1}\) and \(y = 3x\), stating the equation(s) of any asymptote(s) and the coordinates of points of intersection.

Hence find the area bounded by the curve \(y = \frac{3x}{x - 1}\) and the lines \(y = 3x\) and \(y = 8\).

A curve has the equation \(y = \sqrt{x^2 + 1}\) where \(x\) is non-negative.

With the aid of a sketch of the above curve, explain why the value of \(S = \lim_{n \to \infty} \frac{1}{n} \left[ 1 + \sqrt{\frac{1}{n^2} + 1} + \sqrt{\frac{4}{n^2} + 1} + \dots + \sqrt{\frac{(n - 1)^2}{n^2} + 1} \right]\) is \(\displaystyle \int_0^1 \sqrt{x^2 + 1} \, \mathrm{d}x\).

2025 SAJC P1 Q7

2025 SAJC P1 Q7