| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2002 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Solve by showing reduces to polynomial |
| Difficulty | Moderate -0.5 This is a straightforward logarithm manipulation question requiring standard techniques: combining logs using addition/subtraction rules, converting to exponential form, and solving a resulting quadratic. The steps are routine and well-practiced in A-level syllabi, making it slightly easier than average, though not trivial since it requires multiple connected steps. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Content | Mark | Guidance |
| (i) Use law for addition (or subtraction) of logarithms or indices | M1* | |
| Use \(\log_{10} 100 = 2\) or \(10^2 = 100\) | M1(dep*) | |
| Obtain \(x^2 + 5x = 100\), or equivalent, correctly | A1 | |
| (ii) Solve a three-term quadratic equation | M1 | |
| State answer \(7.81\) (allow \(7.80\) or \(7.8\)) or any exact form of the answer i.e. \(\frac{\sqrt{425}-5}{2}\) or better | A1 | Max 3 marks; Max 2 marks for part (ii) |
| Content | Mark | Guidance |
|---------|------|----------|
| **(i)** Use law for addition (or subtraction) of logarithms or indices | M1* | |
| Use $\log_{10} 100 = 2$ or $10^2 = 100$ | M1(dep*) | |
| Obtain $x^2 + 5x = 100$, or equivalent, correctly | A1 | |
| **(ii)** Solve a three-term quadratic equation | M1 | |
| State answer $7.81$ (allow $7.80$ or $7.8$) or any exact form of the answer i.e. $\frac{\sqrt{425}-5}{2}$ or better | A1 | Max 3 marks; Max 2 marks for part (ii) |
---3 (i) Show that the equation
$$\log _ { 10 } ( x + 5 ) = 2 - \log _ { 10 } x$$
may be written as a quadratic equation in $x$.\\
(ii) Hence find the value of $x$ satisfying the equation
$$\log _ { 10 } ( x + 5 ) = 2 - \log _ { 10 } x$$
\hfill \mbox{\textit{CAIE P3 2002 Q3 [5]}}