The curve $C$ has equation $y=\frac{{{x}^{2}}+5x+1}{x-3}$, $x\in \mathbb{R}$, $x\ne 3$. With

Question

The curve $C$ has equation $y=\frac{{{x}^{2}}+5x+1}{x-3}$, $x\in \mathbb{R}$, $x\ne 3$. Without using a calculator, find the set of values that $y$ can take.[3] Sketch $C$, stating the equations of the asymptotes and the coordinates of any turning points.[3] By considering a suitable graph on the same diagram as $C$, find the range of positive values for $k$ such that $k{{\left( x-3 \right)}^{2}}-{{\left( \frac{{{x}^{2}}+5x+1}{x-3}-11 \right)}^{2}}=1$ has no solution.[2]

Accepted Answer

(i) $y\in( -\infty,1]\cup[21,\infty)$ (ii) Asymptotes $x=3$ and $y=x+8$, turning points $(-2,1)$ and $(8,21)$ (iii) $0<k\leq1$

2017 ACJC Promo Q8

2017 ACJC Promo Q8

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