CAIE P3 2019 November — Question 3 4 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2019
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicExponential Functions
TypeSolve rational exponential equation
DifficultyStandard +0.3 This is a straightforward exponential equation requiring substitution (let y = 3^x), algebraic manipulation to form a quadratic, and solving. While it involves multiple steps and careful algebra, the technique is standard for P3 level with no novel insight required—slightly easier than average.
Spec1.06g Equations with exponentials: solve a^x = b
3 Showing all necessary working, solve the equation \(\frac { 3 ^ { 2 x } + 3 ^ { - x } } { 3 ^ { 2 x } - 3 ^ { - x } } = 4\). Give your answer correct to 3 decimal places.
Question 3:
AnswerMarks Guidance
AnswerMarks Guidance
Reduce the equation to a homogeneous equation in \(3^{3x}\), \(3^{3x+1}\) or \(27^x\)M1
Simplify and reach \(3(3^{3x}) = 5\), \(3(27^x) = 5\), or equivalentA1
Use correct method for finding \(x\) from a positive value of \(3^{3x}\), \(3^{3x+1}\) or \(27^x\)M1
Obtain answer \(x = 0.155\)A1
Total4 
**Question 3:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| Reduce the equation to a homogeneous equation in $3^{3x}$, $3^{3x+1}$ or $27^x$ | M1 | |
| Simplify and reach $3(3^{3x}) = 5$, $3(27^x) = 5$, or equivalent | A1 | |
| Use correct method for finding $x$ from a positive value of $3^{3x}$, $3^{3x+1}$ or $27^x$ | M1 | |
| Obtain answer $x = 0.155$ | A1 | |
| **Total** | **4** | |
3 Showing all necessary working, solve the equation $\frac { 3 ^ { 2 x } + 3 ^ { - x } } { 3 ^ { 2 x } - 3 ^ { - x } } = 4$. Give your answer correct to 3 decimal places.\\

\hfill \mbox{\textit{CAIE P3 2019 Q3 [4]}}
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Q1 4 Q2 5 Q3 4 Q4 7 Q5 7 Q6 7 Q7 9 Q8 10 Q9 10 Q10 12