CAIE P3 2021 June — Question 2 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2021
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicExponential Equations & Modelling
TypeQuadratic in exponential form
DifficultyStandard +0.3 This is a standard quadratic-in-exponential problem requiring substitution (let y = 4^x), solving a quadratic, then taking logarithms. It's slightly above average difficulty due to the manipulation required, but follows a well-established technique taught in P3 with no novel insight needed.
Spec1.06g Equations with exponentials: solve a^x = b
2 Solve the equation \(4 ^ { x } = 3 + 4 ^ { - x }\). Give your answer correct to 3 decimal places.
Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
State or imply \(u^2 - 3u - 1 = 0\), or equivalent in \(4^x\)B1
Solve for \(u\) or \(4^x\)M1
Obtain root \(\frac{1}{2}(3 + \sqrt{13})\), or decimal in \([3.30, 3.31]\)A1
Use correct method for finding \(x\) from a positive rootM1
Obtain answer \(x = 0.862\) and no otherA1
Total5 
**Question 2:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $u^2 - 3u - 1 = 0$, or equivalent in $4^x$ | B1 | |
| Solve for $u$ or $4^x$ | M1 | |
| Obtain root $\frac{1}{2}(3 + \sqrt{13})$, or decimal in $[3.30, 3.31]$ | A1 | |
| Use correct method for finding $x$ from a positive root | M1 | |
| Obtain answer $x = 0.862$ and no other | A1 | |
| **Total** | **5** | |
2 Solve the equation $4 ^ { x } = 3 + 4 ^ { - x }$. Give your answer correct to 3 decimal places.\\

\hfill \mbox{\textit{CAIE P3 2021 Q2 [5]}}
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