Standard +0.3 This is a straightforward logarithm equation requiring application of standard log laws (power law to bring down the 2, addition law to combine the right side) followed by solving a quadratic equation. It's slightly easier than average as it follows a predictable pattern with clear steps and no conceptual surprises, though students must remember to check validity of solutions given the domain restrictions.
3 It is given that \(k\) is a positive constant. Solve the equation \(2 \ln x = \ln ( 3 k + x ) + \ln ( 2 k - x )\), expressing \(x\) in terms of \(k\).
Obtain \(x^2 = (3+x)(2-x)\) or equivalent with no logarithms
A1
Solve 3-term quadratic equation
M1
Obtain \(x = \frac{3}{2}\) and no other solutions
A1
[5]
Use $2\ln x = \ln x^2$ | B1 |
Use law for addition or subtraction of logarithms | M1 |
Obtain $x^2 = (3+x)(2-x)$ or equivalent with no logarithms | A1 |
Solve 3-term quadratic equation | M1 |
Obtain $x = \frac{3}{2}$ and no other solutions | A1 | [5]
3 It is given that $k$ is a positive constant. Solve the equation $2 \ln x = \ln ( 3 k + x ) + \ln ( 2 k - x )$, expressing $x$ in terms of $k$.
\hfill \mbox{\textit{CAIE P2 2016 Q3 [5]}}