2024 EJC H1 Q3
2024 EJC H1 Q3
8 marks
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]
Solution:
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Answer:(a) Minimum approximately \((0.905,15.2)\); asymptotes \(x=0\), \(y=-2\). (b) \(y=x+\frac{4}{x}\), \(x\ne0\). (c) \(k<-2-2\sqrt5\) or \(k>-2+2\sqrt5\).
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