The curve \(C\) has equation \(y = \mathrm{e}^{2x} - 2 + \frac{10}{x}\).
Sketch the graph of \(C\), stating the coordinates of any turning points and the equations of any asymptotes clearly.[3]
Determine the equation of a curve which, when used with the equation of \(C\), will enable you to solve the equation \(\frac{\mathrm{e}^{2x}-2}{x} + \frac{6}{x^2} - 1 = 0.\)[1]
Find the range of values of \(k\) for which the curve \(C\) intersects the curve \(y = \mathrm{e}^{2x} - \frac{1}{2}x + k\) at two distinct points.[4]