2026 RI Promo Q7

2026 RI Promo Q7

Junior College 1
8 marks

A curve \(C\) has equation \(y=\dfrac{x^2+k}{x+k}\), where \(k\in\mathbb R\setminus\{-1,0\}\).

  1. Find the range of values of \(k\) such that \(C\) has no stationary points.[3]

Let \(k=-4\) for the rest of the question.

  1. Sketch \(C\), stating the equations of any asymptotes, coordinates of any turning points and the points where \(C\) crosses the axes.[4]
  2. State the range of values of \(m\) for which the graph of \(y+2k=m(x+k)\) does not intersect with \(C\).[1]

Solution:

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Answer:(a) \(-1<k<0\). (b) Asymptotes \(x=4,\ y=x+4\); turns \((4\pm2\sqrt3,8\pm4\sqrt3)\); intercepts \((-2,0),(2,0),(0,1)\). (c) \(m\le1\).

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