| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2007 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve |exponential| < constant |
| Difficulty | Standard +0.3 This is a straightforward two-part question applying standard modulus inequality techniques. Part (i) is routine manipulation to get 4 < y < 5. Part (ii) requires substituting y = 3^x and solving exponential equations using logarithms, which is a standard A-level procedure with no novel insight required. The question is slightly above average difficulty only because it combines two topics (modulus and exponentials) across two parts. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02l Modulus function: notation, relations, equations and inequalities1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain critical values 4 and 6 | B1 | |
| State answer \(4 < y < 6\) | B1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Use correct method for solving an equation of the form \(3^x = a\), where \(a > 0\) | M1 | |
| Obtain one critical value, i.e. either 1.26 or 1.63 | A1 | |
| State answer \(1.26 < x < 1.63\) | A1 | [3] |
**(i)**
Obtain critical values 4 and 6 | B1 |
State answer $4 < y < 6$ | B1 | [2]
**(ii)**
Use correct method for solving an equation of the form $3^x = a$, where $a > 0$ | M1 |
Obtain one critical value, i.e. either 1.26 or 1.63 | A1 |
State answer $1.26 < x < 1.63$ | A1 | [3]3 (i) Solve the inequality $| y - 5 | < 1$.\\
(ii) Hence solve the inequality $\left| 3 ^ { x } - 5 \right| < 1$, giving 3 significant figures in your answer.
\hfill \mbox{\textit{CAIE P2 2007 Q3 [5]}}