Question 3 - A-Level Maths

CAIE P3 2017 June — Question 3 4 marks

Exam Board	CAIE
Module	P3 (Pure Mathematics 3)
Year	2017
Session	June
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Exponential Equations & Modelling
Type	Quadratic in exponential form
Difficulty	Moderate -0.8 This is a straightforward substitution problem requiring students to recognize that e^(-x) = 1/u, form a quadratic equation, solve it, and take a logarithm. The substitution is explicitly given, making this easier than average with no conceptual challenges beyond standard algebraic manipulation.
Spec	1.06g Equations with exponentials: solve a^x = b

3 Using the substitution \(u = \mathrm { e } ^ { x }\), solve the equation \(4 \mathrm { e } ^ { - x } = 3 \mathrm { e } ^ { x } + 4\). Give your answer correct to 3 significant figures.

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Question 3:

Answer	Marks	Guidance
Answer	Mark	Guidance
Rearrange as \(3u^2 + 4u - 4 = 0\), or \(3e^{2x} + 4e^x - 4 = 0\), or equivalent	B1
Solve a 3-term quadratic for \(e^x\) or for \(u\)	M1
Obtain \(e^x = \frac{2}{3}\) or \(u = \frac{2}{3}\)	A1
Obtain answer \(x = -0.405\) and no other	A1
Total: 4

## Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| Rearrange as $3u^2 + 4u - 4 = 0$, or $3e^{2x} + 4e^x - 4 = 0$, or equivalent | B1 | |
| Solve a 3-term quadratic for $e^x$ or for $u$ | M1 | |
| Obtain $e^x = \frac{2}{3}$ or $u = \frac{2}{3}$ | A1 | |
| Obtain answer $x = -0.405$ and no other | A1 | |
| **Total: 4** | | |

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3 Using the substitution $u = \mathrm { e } ^ { x }$, solve the equation $4 \mathrm { e } ^ { - x } = 3 \mathrm { e } ^ { x } + 4$. Give your answer correct to 3 significant figures.\\

\hfill \mbox{\textit{CAIE P3 2017 Q3 [4]}}

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