2023 JPJC P1 Q4

The curve \(C\) has equation \(y=\frac{{{x}^{2}}+x+3}{x-2}\).

1. Without using a calculator, find the set of values that \(y\) can take.[4]
2. Sketch the graph of \(C\), stating clearly the equations of any asymptotes and the coordinates of any turning points and axial intercepts.[3]
3. 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]