Moderate -0.3 This is a straightforward substitution problem requiring students to recognize that 3^(2x) = u² and 3^(3x) = u³, leading to the quadratic u + u² = u³. The algebraic manipulation is routine (factoring to get u(u² - u - 1) = 0), and solving the resulting quadratic is standard. While it requires multiple steps and logarithms for the final answer, the technique is well-practiced and involves no conceptual difficulty beyond recognizing the substitution pattern, making it slightly easier than average.
2 Using the substitution \(u = 3 ^ { x }\), solve the equation \(3 ^ { x } + 3 ^ { 2 x } = 3 ^ { 3 x }\) giving your answer correct to 3 significant figures.
Obtain root \(\frac{1}{2}(1 + \sqrt{5})\), or decimal in \([1.61, 1.62]\)
A1
Use correct method for finding \(x\) from a positive root
M1
Obtain \(x = 0.438\) and no other answer
A1
[5]
State or imply $1 + u = u^2$ | B1 |
Solve for $u$ | M1 |
Obtain root $\frac{1}{2}(1 + \sqrt{5})$, or decimal in $[1.61, 1.62]$ | A1 |
Use correct method for finding $x$ from a positive root | M1 |
Obtain $x = 0.438$ and no other answer | A1 |
| [5] |