CAIE P3 Specimen — Question 2 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
SessionSpecimen
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicExponential Functions
TypeSolve exponential equation by substitution
DifficultyModerate -0.3 This is a straightforward substitution problem requiring students to recognize that 3^(2x) = u² and 3^(3x) = u³, leading to a cubic equation u³ - u² - u = 0 that factors easily. While it involves multiple steps (substitution, factorization, taking logarithms), each step is routine and the substitution is explicitly given, making it slightly easier than average.
Spec1.06g Equations with exponentials: solve a^x = b
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.
Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
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 rootM1
Obtain \(x = 0.438\) and no other answerA1
Total: 5 
## Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| 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 | |
| **Total: 5** | | |

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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.\\

\hfill \mbox{\textit{CAIE P3  Q2 [5]}}
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