| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve modulus equation then apply exponential/log substitution |
| Difficulty | Standard +0.3 Part (i) is a standard modulus equation requiring case analysis (3-4 cases), which is routine A-level work. Part (ii) adds a straightforward exponential substitution (let x = 4^y) and then solving for y using logarithms. While multi-step, both techniques are standard P3 material with no novel insight required, making it slightly easier than average overall. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Either: State or imply non-modular equation \((4x-1)^2 = (x-3)^2\) or pair of linear equations \(4x - 1 = \pm(x-3)\) | B1 | |
| Solve a three-term quadratic equation or two linear equations | M1 | |
| Obtain \(-\frac{2}{3}\) and \(\frac{4}{5}\) | A1 | |
| Or: Obtain value \(-\frac{2}{3}\) from inspection or solving linear equation | B1 | |
| Obtain value \(\frac{4}{5}\) similarly | B2 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply at least \(4^y = \frac{4}{5}\), following a positive answer from part (i) | B1\(\sqrt{}\) | |
| Apply logarithms and use \(\log a^b = b\log a\) property | M1 | |
| Obtain \(-0.161\) and no other answer | A1 | [3] |
## Question 4(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **Either:** State or imply non-modular equation $(4x-1)^2 = (x-3)^2$ or pair of linear equations $4x - 1 = \pm(x-3)$ | B1 | |
| Solve a three-term quadratic equation or two linear equations | M1 | |
| Obtain $-\frac{2}{3}$ and $\frac{4}{5}$ | A1 | |
| **Or:** Obtain value $-\frac{2}{3}$ from inspection or solving linear equation | B1 | |
| Obtain value $\frac{4}{5}$ similarly | B2 | [3] |
## Question 4(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply at least $4^y = \frac{4}{5}$, following a positive answer from part (i) | B1$\sqrt{}$ | |
| Apply logarithms and use $\log a^b = b\log a$ property | M1 | |
| Obtain $-0.161$ and no other answer | A1 | [3] |
---4 (i) Solve the equation $| 4 x - 1 | = | x - 3 |$.\\
(ii) Hence solve the equation $\left| 4 ^ { y + 1 } - 1 \right| = \left| 4 ^ { y } - 3 \right|$ correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P3 2013 Q4 [6]}}