CAIE P3 2021 June — Question 2 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2021
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicFixed Point Iteration
TypeSolve exponential equation via iteration
DifficultyStandard +0.3 This is a straightforward numerical methods question requiring rearrangement of an exponential equation, showing sign change to prove uniqueness of root, and iterative solution to 3 decimal places. While it involves exponential functions, the technique is standard A-level fare with no novel insight required—slightly easier than average due to being a routine application of interval/sign-change methods.
Spec1.06g Equations with exponentials: solve a^x = b
2 Find the real root of the equation \(\frac { 2 \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { 2 + \mathrm { e } ^ { x } } = 3\), giving your answer correct to 3 decimal places. Your working should show clearly that the equation has only one real root.
Question 2:
AnswerMarks Guidance
AnswerMarks Guidance
Reduce to a 3-term quadratic \(u^2 + 6u - 1 = 0\) OEB1 Allow '= 0' implied
Solve a 3-term quadratic for \(u\)M1
Obtain root \(\sqrt{10} - 3\)A1
Obtain answer \(x = -1.818\) onlyA1 The question asks for 3 d.p.
Reject \(-\sqrt{10} - 3\) correctlyB1 e.g. by stating that \(e^x > 0\) or \(\ln(-10 - \sqrt{3})\) is impossible. Not "math error".
Alternative method:
AnswerMarks Guidance
AnswerMarks Guidance
Rearrange to obtain a correct iterative formulaB1 e.g. \(x_{n+1} = -\ln(6 + e^{x_n})\)
Use the iterative process at least twiceM1
Obtain answer \(x = -1.818\)A1
Show sufficient iterations to at least 4 d.p. to justify \(x = -1.818\)A1 \(1, -2.165..., -1.811..., -1.819..., -1.818..., -1.818...\)
Clear explanation of why there is only one real rootB1
Total5 
## Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Reduce to a 3-term quadratic $u^2 + 6u - 1 = 0$ OE | B1 | Allow '= 0' implied |
| Solve a 3-term quadratic for $u$ | M1 | |
| Obtain root $\sqrt{10} - 3$ | A1 | |
| Obtain answer $x = -1.818$ only | A1 | The question asks for 3 d.p. |
| Reject $-\sqrt{10} - 3$ correctly | B1 | e.g. by stating that $e^x > 0$ or $\ln(-10 - \sqrt{3})$ is impossible. Not "math error". |

**Alternative method:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| Rearrange to obtain a correct iterative formula | B1 | e.g. $x_{n+1} = -\ln(6 + e^{x_n})$ |
| Use the iterative process at least twice | M1 | |
| Obtain answer $x = -1.818$ | A1 | |
| Show sufficient iterations to at least 4 d.p. to justify $x = -1.818$ | A1 | $1, -2.165..., -1.811..., -1.819..., -1.818..., -1.818...$ |
| Clear explanation of why there is only one real root | B1 | |
| **Total** | **5** | |

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2 Find the real root of the equation $\frac { 2 \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { 2 + \mathrm { e } ^ { x } } = 3$, giving your answer correct to 3 decimal places. Your working should show clearly that the equation has only one real root.\\

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