Standard +0.3 This requires understanding of absolute value equations and exponential manipulation. Students must split into two cases (3^x ≥ 1 and 3^x < 1), solve each algebraically, then verify solutions. While it involves multiple steps and case analysis, the techniques are standard for P3 level and the algebraic manipulation is straightforward once cases are identified, making it slightly easier than average.
EITHER: State or imply non-modular equation \(2^{'}(3^{x}-1)^{2}=(3^{x})^{2}\), or pair of equations \(2(3^{x}-1)=\pm 3^{x}\)
M1
Obtain \(3^{x}=2\) and \(3^{x}=\frac{2}{3}\) (or \(3^{x+1}=2\))
A1
OR: Obtain \(3^{x}=2\) by solving an equation or by inspection
B1
Obtain \(3^{x}=\frac{2}{3}\) (or \(3^{x+1}=2\)) by solving an equation or by inspection
B1
Use correct method for solving an equation of the form \(3^{x}=a\) (or \(3^{x+1}=a\)), where \(a>0\)
M1
Obtain final answers \(0.631\) and \(-0.369\)
A1
[4]
**EITHER:** State or imply non-modular equation $2^{'}(3^{x}-1)^{2}=(3^{x})^{2}$, or pair of equations $2(3^{x}-1)=\pm 3^{x}$ | M1 |
Obtain $3^{x}=2$ and $3^{x}=\frac{2}{3}$ (or $3^{x+1}=2$) | A1 |
**OR:** Obtain $3^{x}=2$ by solving an equation or by inspection | B1 |
Obtain $3^{x}=\frac{2}{3}$ (or $3^{x+1}=2$) by solving an equation or by inspection | B1 |
Use correct method for solving an equation of the form $3^{x}=a$ (or $3^{x+1}=a$), where $a>0$ | M1 |
Obtain final answers $0.631$ and $-0.369$ | A1 | [4]