Moderate -0.3 This is a standard quadratic-in-exponential problem requiring substitution (let y = 3^x, so 9^x = y²), solving y² + y - 240 = 0, then applying logarithms. It's a routine multi-step question with well-practiced techniques, making it slightly easier than average but still requiring proper execution of multiple steps.
2 Given that \(9 ^ { x } + 3 ^ { x } = 240\), find the value of \(3 ^ { x }\) and hence, using logarithms, find the value of \(x\) correct to 4 significant figures.
Apply logarithms and use power law for \(3^x=k\) where \(k>0\)
M1
Dependent \*M, need to see \(x\ln 3=\ln k\), \(x=\log_3 k\) oe
Obtain \(2.465\)
A1
May be done using \(9^{\frac{x}{2}}\), same processes
## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| Recognise $9^x$ as $(3^x)^2$ or $3^{2x}$ | B1 | May be implied by $3^x(3^x+1)(=240)$ |
| Attempt solution of quadratic equation in $3^x$ | \*M1 | Perhaps using substitution $u=3^x$ |
| Obtain, finally, $3^x=15$ only | A1 | |
| Apply logarithms and use power law for $3^x=k$ where $k>0$ | M1 | Dependent \*M, need to see $x\ln 3=\ln k$, $x=\log_3 k$ oe |
| Obtain $2.465$ | A1 | May be done using $9^{\frac{x}{2}}$, same processes |
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2 Given that $9 ^ { x } + 3 ^ { x } = 240$, find the value of $3 ^ { x }$ and hence, using logarithms, find the value of $x$ correct to 4 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2018 Q2 [5]}}