| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| 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 | Standard +0.3 Part (i) is a standard modulus equation requiring case analysis (critical points at u=-1/3 and u=5/2), yielding two linear equations to solve. Part (ii) applies the substitution u=cot(x) and requires solving cot(x)=constant within a restricted domain, which is routine inverse trigonometry. This is slightly above average due to the two-part structure and trigonometric substitution, but both components use standard A-level techniques without requiring novel insight. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.05a Sine, cosine, tangent: definitions for all arguments1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply non-modular equation \((3u+1)^2=(2u-5)^2\) or corresponding pair of linear equations | B1 | |
| Attempt solution of 3-term quadratic equation or of 2 linear equations | M1 | |
| Obtain \(-6\) and \(\frac{4}{5}\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Evaluate \(\tan^{-1}\frac{1}{k}\) for at least one of their solutions \(k\) from part (i) | M1 | |
| Obtain 0.896 | A1 | [2] |
## Question 3:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply non-modular equation $(3u+1)^2=(2u-5)^2$ or corresponding pair of linear equations | B1 | |
| Attempt solution of 3-term quadratic equation or of 2 linear equations | M1 | |
| Obtain $-6$ and $\frac{4}{5}$ | A1 | [3] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Evaluate $\tan^{-1}\frac{1}{k}$ for at least one of their solutions $k$ from part (i) | M1 | |
| Obtain 0.896 | A1 | [2] |
---3 (i) Solve the equation $| 3 u + 1 | = | 2 u - 5 |$.\\
(ii) Hence solve the equation $| 3 \cot x + 1 | = | 2 \cot x - 5 |$ for $0 < x < \frac { 1 } { 2 } \pi$, giving your answer correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P2 2016 Q3 [5]}}