| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Quadratic in exponential form |
| Difficulty | Moderate -0.8 This is a straightforward quadratic-in-disguise problem requiring substitution of u=5^x, solving u²+u-12=0 to get u=3, then taking logarithms. It's a standard textbook exercise with clear signposting and routine techniques, making it easier than average but not trivial since it requires recognizing the substitution and applying logarithms correctly. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply equation in the form \((5^t)^2 + 5^t - 12 = 0\) | B1 | |
| Attempt solution of quadratic equation for \(5^t\) | M1 | |
| Obtain \(5^t = 3\) only | A1 | [3] |
| (ii) Use logarithms to solve equation of the form \(5^t = k\) where \(k > 0\) | M1 | |
| Obtain \(0.683\) | A1 | [2] |
**(i)** State or imply equation in the form $(5^t)^2 + 5^t - 12 = 0$ | B1 |
Attempt solution of quadratic equation for $5^t$ | M1 |
Obtain $5^t = 3$ only | A1 | [3]
**(ii)** Use logarithms to solve equation of the form $5^t = k$ where $k > 0$ | M1 |
Obtain $0.683$ | A1 | [2]
---2 (i) Given that $5 ^ { 2 x } + 5 ^ { x } = 12$, find the value of $5 ^ { x }$.\\
(ii) Hence, using logarithms, solve the equation $5 ^ { 2 x } + 5 ^ { x } = 12$, giving the value of $x$ correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P2 2012 Q2 [5]}}