2023 JPJC P1 Q4
The curve $C$ has equation $y=\frac{{{x}^{2}}+x+3}{x-2}$.
(i)
Without using a calculator, find the set of values that $y$ can take.
[4]
(ii)
Sketch the graph of $C$, stating clearly the equations of any asymptotes and the coordinates of any turning points and axial intercepts.
[3]
(iii)
By drawing a suitable graph on the same diagram in (ii), find the range of values of $k$, where $k>0$, such that the equation ${{\left( x+1 \right)}^{2}}{{\left( x-2 \right)}^{2}}+{{\left( {{x}^{2}}+x+3 \right)}^{2}}={{k}^{2}}{{\left( x-2 \right)}^{2}}$ has at least one positive real root, giving your answer correct to $3$ decimal places.
[3]
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