2023 VJC Promo Q10

The curve \(C\) has equation \(y = \frac{3x^2 + ax + 3}{x + 1}\), where \(a\) is a constant, \(a \neq 6\).

1. Find the set of values of \(a\) such that \(C\) does not cut the \(x\)-axis.[2]
2. It is given that the equation of \(C\) is \(y = \frac{3x^2 - 6x + 3}{x + 1}\). Sketch \(C\) and give the equations of any asymptotes. Also, state the coordinates of any points where \(C\) crosses the axes and of any turning points.[4]
3. Let \(k\) be a positive constant. By sketching a suitable graph in the same diagram in part (b), find an inequality satisfied by \(k\) such that the equation
   \[k^2(x - 1)^2 + \left(\frac{3x^2 - 6x + 3}{x + 1} + 12\right)^2 = k^2\]
   has 2 real and distinct roots.[2]