| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Exponential relation to line equation |
| Difficulty | Moderate -0.8 This is a straightforward logarithmic manipulation question requiring students to take logs of both sides to linearize an exponential equation, then solve a simple simultaneous equation. Part (i) is routine application of log laws (standard textbook exercise), and part (ii) involves basic algebraic substitution and calculator work. The question tests fundamental techniques without requiring problem-solving insight or multi-step reasoning beyond the prescribed method. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply \(y \ln 3 = (x+2) \ln 4\) | B1 | |
| State that this is of the form \(dy = bx + c\) and thus a straight line, or equivalent | B1 | |
| State gradient is \(\frac{\ln 4}{\ln 3}\), or equivalent (allow 1.26) | B1 | [3 marks] |
| Answer | Marks | Guidance |
|---|---|---|
| Substitute \(y = 2x\) and obtain a linear equation in \(x\) | M1* | |
| Solve for \(x\) | M1(dep*) | |
| Obtain answer 3.42 | A1 | [3 marks] |
**(i)**
State or imply $y \ln 3 = (x+2) \ln 4$ | B1 | |
State that this is of the form $dy = bx + c$ and thus a straight line, or equivalent | B1 | |
State gradient is $\frac{\ln 4}{\ln 3}$, or equivalent (allow 1.26) | B1 | [3 marks] |
**(ii)**
Substitute $y = 2x$ and obtain a linear equation in $x$ | M1* | |
Solve for $x$ | M1(dep*) | |
Obtain answer 3.42 | A1 | [3 marks] |2 The variables $x$ and $y$ satisfy the relation $3 ^ { y } = 4 ^ { x + 2 }$.\\
(i) By taking logarithms, show that the graph of $y$ against $x$ is a straight line. Find the exact value of the gradient of this line.\\
(ii) Calculate the $x$-coordinate of the point of intersection of this line with the line $y = 2 x$, giving your answer correct to 2 decimal places.
\hfill \mbox{\textit{CAIE P2 2007 Q2 [6]}}