| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve modulus equation then apply exponential/log substitution |
| Difficulty | Moderate -0.3 Part (i) is a standard modulus equation requiring case analysis (x≥1, 0≤x<1, x<0) with straightforward algebra. Part (ii) applies substitution u=5^x and logarithms, which is routine once part (i) is solved. This is slightly easier than average due to the guided structure and standard techniques, though the two-part nature and exponential conversion add minor complexity. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply non-modular equation \((2(x-1))^2 = (3x)^2\), or pair of linear equations \(2(x-1) = \pm 3x\) | B1 | EITHER method |
| Make reasonable solution attempt at 3-term quadratic, or solve two linear equations | M1 | |
| Obtain \(x = -2\) and \(x = \frac{2}{5}\) | A1 | |
| OR: Obtain \(x = -2\) by inspection or by solving a linear equation | (B1 | Alternative method |
| Obtain \(x = \frac{2}{5}\) similarly | B2) | |
| Total | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use correct method for solving equation of form \(5^x = a\) or \(5^{x+1} = a\), where \(a > 0\) | M1 | |
| Obtain answer \(x = -0.569\) only | A1 | |
| Total | [2] |
## Question 1:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply non-modular equation $(2(x-1))^2 = (3x)^2$, or pair of linear equations $2(x-1) = \pm 3x$ | B1 | EITHER method |
| Make reasonable solution attempt at 3-term quadratic, or solve two linear equations | M1 | |
| Obtain $x = -2$ and $x = \frac{2}{5}$ | A1 | |
| OR: Obtain $x = -2$ by inspection or by solving a linear equation | (B1 | Alternative method |
| Obtain $x = \frac{2}{5}$ similarly | B2) | |
| **Total** | **[3]** | |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use correct method for solving equation of form $5^x = a$ or $5^{x+1} = a$, where $a > 0$ | M1 | |
| Obtain answer $x = -0.569$ only | A1 | |
| **Total** | **[2]** | |
---1 (i) Solve the equation $2 | x - 1 | = 3 | x |$.\\
(ii) Hence solve the equation $2 \left| 5 ^ { x } - 1 \right| = 3 \left| 5 ^ { x } \right|$, giving your answer correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P3 2016 Q1 [5]}}