A curve $C$ has equation $y=\frac{{{x}^{2}}-2x+7}{x-3}$. Using an algebraic method, find the ran

Question

A curve $C$ has equation $y=\frac{{{x}^{2}}-2x+7}{x-3}$. Using an algebraic method, find the range of values that $y$ cannot take. Leave your answer in exact form.[4] Sketch the curve $C$, stating the equations of any asymptotes and the exact coordinates of any points where $C$ crosses the axes and of any turning points.[4] By considering a suitable graph, find the range of values of $k$ such that the equation ${{x}^{4}}-2{{x}^{3}}+7{{x}^{2}}=kx-3k$ has exactly one positive root and one negative root.[2]

Accepted Answer

(i) $4-2\sqrt{10}<y<4+2\sqrt{10}$ (ii) Asymptotes $x=3$ and $y=x+1$, no $x$-intercepts, $y$-intercept $\left(0,-\dfrac73\right)$, turning points $(3-\sqrt{10},4-2\sqrt{10})$ and $(3+\sqrt{10},4+2\sqrt{10})$ (iii) $k<0$

2022 VJC Promo Q8

2022 VJC Promo Q8

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