Standard +0.3 This question combines exponential and modulus inequalities, both standard P2 topics. The exponential inequality requires straightforward logarithm application, while the modulus inequality solves by squaring both sides or considering cases. The combination and intersection of solutions adds mild complexity, but each component uses routine techniques with no novel insight required.
3 It is given that the variable \(x\) is such that
$$1.3 ^ { 2 x } < 80 \quad \text { and } \quad | 3 x - 1 | > | 3 x - 10 | .$$
Find the set of possible values of \(x\), giving your answer in the form \(a < x < b\) where the constants \(a\) and \(b\) are correct to 3 significant figures.
Condone incorrect inequality signs until final answer. The first 6 marks are for obtaining the correct critical values.
Obtain \(2x < \frac{\ln 80}{\ln 1.3}\) or equivalent using \(\log_{10}\)
A1
Obtain \(x = 8.35...\)
A1
State or imply non-modulus inequality \((3x-1)^2 > (3x-10)^2\) or corresponding equation or linear equation \(3x-1 = -(3x-10)\)
B1
Attempt solution of inequality or equation (obtaining 3 terms when squaring each bracket or solving linear equation with signs of \(3x\) different)
M1
Obtain \(x = \frac{11}{6}\) or \(x = 1.83...\)
A1
Conclude \(1.83 < x < 8.35\)
A1
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Take logarithms of both sides and apply power law | M1 | Condone incorrect inequality signs until final answer. The first 6 marks are for obtaining the correct critical values. |
| Obtain $2x < \frac{\ln 80}{\ln 1.3}$ or equivalent using $\log_{10}$ | A1 | |
| Obtain $x = 8.35...$ | A1 | |
| State or imply non-modulus inequality $(3x-1)^2 > (3x-10)^2$ or corresponding equation or linear equation $3x-1 = -(3x-10)$ | B1 | |
| Attempt solution of inequality or equation (obtaining 3 terms when squaring each bracket or solving linear equation with signs of $3x$ different) | M1 | |
| Obtain $x = \frac{11}{6}$ or $x = 1.83...$ | A1 | |
| Conclude $1.83 < x < 8.35$ | A1 | |
---
3 It is given that the variable $x$ is such that
$$1.3 ^ { 2 x } < 80 \quad \text { and } \quad | 3 x - 1 | > | 3 x - 10 | .$$
Find the set of possible values of $x$, giving your answer in the form $a < x < b$ where the constants $a$ and $b$ are correct to 3 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2017 Q3 [7]}}