Moderate -0.3 This question requires integrating two exponential functions using standard results, equating them, and solving for k. While it involves multiple steps (integrate, apply limits, equate, solve), each step is routine: the integrals are direct applications of ∫e^(ax)dx = (1/a)e^(ax), and the algebra to solve for k is straightforward. It's slightly easier than average because it's a standard technique question with no conceptual surprises, though the need to manipulate the resulting equation prevents it from being purely mechanical.
Use limits correctly to obtain an equation in \(e^{2x}, e^{4k}\)
M1
Carry out recognisable solution method for quadratic in \(e^{2x}\)
M1
Obtain \(e^{2x} = 1\) and \(e^{4k} = 3\)
A1
Use logarithmic method to solve an equation of the form \(e^{2a} = b\), where \(b > 0\)
M1
Obtain answer \(k = \frac{1}{2}\ln 3\)
A1
[6]
State at least one correct integral | B1 |
Use limits correctly to obtain an equation in $e^{2x}, e^{4k}$ | M1 |
Carry out recognisable solution method for quadratic in $e^{2x}$ | M1 |
Obtain $e^{2x} = 1$ and $e^{4k} = 3$ | A1 |
Use logarithmic method to solve an equation of the form $e^{2a} = b$, where $b > 0$ | M1 |
Obtain answer $k = \frac{1}{2}\ln 3$ | A1 | [6] |
4 Find the exact value of the positive constant $k$ for which
$$\int _ { 0 } ^ { k } e ^ { 4 x } d x = \int _ { 0 } ^ { 2 k } e ^ { x } d x$$
\hfill \mbox{\textit{CAIE P2 2011 Q4 [6]}}