| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve modulus equation then apply exponential/log substitution |
| Difficulty | Moderate -0.8 Part (i) is a straightforward modulus equation requiring splitting into two cases (3x-2=5 and 3x-2=-5). Part (ii) applies the same result with substitution x=5^y, then uses basic logarithms to solve. This is a standard two-part question testing routine modulus and logarithm techniques with minimal problem-solving demand, making it easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Either: Square both sides to obtain three-term quadratic equation | M1 | |
| Solve three-term quadratic equation to obtain two values | M1 | |
| Obtain \(-1\) and \(\frac{7}{3}\) | A1 | |
| Or: Obtain \(\frac{7}{3}\) from graphical method, inspection or linear equation | B1 | |
| Obtain \(-1\) similarly | B2 | [3] |
| (ii) Use logarithmic method to solve an equation of the form \(5^y = k\) where \(k > 0\) | M1 | |
| Obtain \(0.526\) and no others | A1 | [2] |
(i) Either: Square both sides to obtain three-term quadratic equation | M1 |
Solve three-term quadratic equation to obtain two values | M1 |
Obtain $-1$ and $\frac{7}{3}$ | A1 |
Or: Obtain $\frac{7}{3}$ from graphical method, inspection or linear equation | B1 |
Obtain $-1$ similarly | B2 | [3]
(ii) Use logarithmic method to solve an equation of the form $5^y = k$ where $k > 0$ | M1 |
Obtain $0.526$ and no others | A1 | [2]
---1 (i) Solve the equation $| 3 x - 2 | = 5$.\\
(ii) Hence, using logarithms, solve the equation $\left| 3 \times 5 ^ { y } - 2 \right| = 5$, giving the answer correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P2 2015 Q1 [5]}}