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FM Nov 2024 p11 q06
4141
The curve C has equation \(y = \frac{4x^2 + x + 1}{2x^2 - 7x + 3}\).
(a) Find the equations of the asymptotes of C.
(b) Find the coordinates of any stationary points on C.
(c) Sketch C, stating the coordinates of any intersections with the axes.
(d) Sketch the curve with equation \(y = \left| \frac{4x^2 + x + 1}{2x^2 - 7x + 3} \right|\) and state the set of values of \(k\) for which \(\left| \frac{4x^2 + x + 1}{2x^2 - 7x + 3} \right| = k\) has 4 distinct real solutions.
Solution
(a) To find the vertical asymptotes, set the denominator equal to zero: \(2x^2 - 7x + 3 = 0\). Solving gives \(x = \frac{1}{2}\) and \(x = 3\). The horizontal asymptote is found by considering the degrees of the polynomials: \(y = \frac{4}{2} = 2\).
(b) To find stationary points, differentiate \(y\) with respect to \(x\):
Set \(\frac{dy}{dx} = 0\) and solve \(-3x^2 + 2x + 1 = 0\) to find \(x = -\frac{1}{3}\) and \(x = 1\). Substitute back to find \(y\) values: \(\left(-\frac{1}{3}, \frac{1}{3}\right)\) and \((1, -3)\).
(c) Sketch the curve using the asymptotes and stationary points. The curve intersects the y-axis at \((0, \frac{1}{3})\).
(d) For \(\left| \frac{4x^2 + x + 1}{2x^2 - 7x + 3} \right| = k\) to have 4 distinct real solutions, \(k > 3\).
