1998 TYS

A sketch of the curve \(y=\frac{a{{x}^{2}}+bx+c}{x+d}\), where \(a\), \(b\), \(c\) and \(d\) are constants, is shown, not to scale, in the diagram. The equations of the asymptotes, also shown in the diagram, are \(x=2\) and \(y=3-2x\).

1. Write the value of \(d\).
2. State the value of \(a\) and show that \(b=7\).
3. Given that the curve has a stationary point where \(x=1\), find the value of \(c\) and the \(x\)-coordinate of the other stationary point.
4. Copy the above sketch and, by drawing a sketch of another suitable curve on the same diagram, show that the equation \(2{{x}^{4}}-7{{x}^{3}}+8{{x}^{2}}+x-2=0\) has exactly two real roots.