Standard +0.3 This is a straightforward modulus inequality with an exponential. Students need to split into two cases (3^x - 8 < 0.5 and 3^x - 8 > -0.5), solve two simple exponential equations using logarithms, and combine the results. It's slightly above average due to the exponential context, but the method is standard and mechanical with no conceptual challenges.
State or imply non-modular inequality \(-0.5 < x^3 - 8 < 0.5\), or \((x^3 - 8)^2 < (0.5)^2\), or corresponding pair of linear equations or quadratic equations
B1
Use correct method for solving an equation of the form \(x^3 = a\), where \(a > 0\)
M1
Obtain critical values 1.83 and 1.95, or exact equivalents
A1
State correct answer \(1.83 < x < 1.95\)
A1
OR: Use correct method for solving an equation of the form \(x^3 = a\), where \(a > 0\)
M1
Obtain one critical value, e.g. 1.95, or exact equivalent
A1
Obtain the other critical value 1.83, or exact equivalent
A1
State correct answer \(1.83 < x < 1.95\)
A1
[Do not condone \(\leq\) for \(<\). Allow final answer given in the form \(1.83 < x\) (and) \(x < 1.95\).]
[Exact equivalents must be in terms of ln or logarithms to base 10.]
[SR: Solutions given as logarithms to base 3 can only earn M1 and B1 of the first scheme.]
4
State or imply non-modular inequality $-0.5 < x^3 - 8 < 0.5$, or $(x^3 - 8)^2 < (0.5)^2$, or corresponding pair of linear equations or quadratic equations | B1 |
Use correct method for solving an equation of the form $x^3 = a$, where $a > 0$ | M1 |
Obtain critical values 1.83 and 1.95, or exact equivalents | A1 |
State correct answer $1.83 < x < 1.95$ | A1 |
OR: Use correct method for solving an equation of the form $x^3 = a$, where $a > 0$ | M1 |
Obtain one critical value, e.g. 1.95, or exact equivalent | A1 |
Obtain the other critical value 1.83, or exact equivalent | A1 |
State correct answer $1.83 < x < 1.95$ | A1 |
[Do not condone $\leq$ for $<$. Allow final answer given in the form $1.83 < x$ (and) $x < 1.95$.] | | |
[Exact equivalents must be in terms of ln or logarithms to base 10.] | | |
[SR: Solutions given as logarithms to base 3 can only earn M1 and B1 of the first scheme.] | | |
| | 4 |
1 Find the set of values of $x$ satisfying the inequality $\left| 3 ^ { x } - 8 \right| < 0.5$, giving 3 significant figures in your answer.
\hfill \mbox{\textit{CAIE P3 2006 Q1 [4]}}