2020 SAJC P2 Q1

Sketch the graph of \(y=\frac{{{x}^{2}}+2}{2x+1}\), stating the equation(s) of the asymptote(s), the coordinates of the turning point(s) and intersection(s) with the axes.[4]

Let \(P\) be the point of intersection of the oblique asymptote of the graph \(y=\frac{{{x}^{2}}+2}{2x+1}\) and the \(y\)-axis. Show that \(P\) lies on the line \(4y=mx-1\) for all real values of \(m\).[2]

Deduce the number of real root(s) for the equation \(4{{x}^{2}}+8=\left( mx-1 \right)\left( 2x+1 \right)\), where \(0 < m < 2\).[2]

Hence, what can be said about the two roots of the equation \(4{{x}^{2}}+8=\left( mx-1 \right)\left( 2x+1 \right)\), where \(0 < m < 2\)?[1]