| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| 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. Both parts are routine applications of standard techniques with no problem-solving insight required, making it easier than average but not trivial since it involves multiple algebraic steps and logarithm laws. |
| 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 \(y \ln 3 = (2-x) \ln 4\) | B1 | |
| State that this is of the form \(ay = bx + c\) and thus a straight line, or equivalent | B1 | |
| State gradient is \(-\frac{\ln 4}{\ln 3}\), or exact equivalent | B1 | [3] |
| (ii) Substitute \(y = 2x\) and solve for \(x\), using a log law correctly at least once | M1 | |
| Obtain answer \(x = \ln 4 / \ln 6\), or exact equivalent | A1 | [2] |
**(i)** State or imply $y \ln 3 = (2-x) \ln 4$ | B1 |
State that this is of the form $ay = bx + c$ and thus a straight line, or equivalent | B1 |
State gradient is $-\frac{\ln 4}{\ln 3}$, or exact equivalent | B1 | [3]
**(ii)** Substitute $y = 2x$ and solve for $x$, using a log law correctly at least once | M1 |
Obtain answer $x = \ln 4 / \ln 6$, or exact equivalent | A1 | [2]
---2 The variables $x$ and $y$ satisfy the relation $3 ^ { y } = 4 ^ { 2 - x }$.\\
(i) By taking logarithms, show that the graph of $y$ against $x$ is a straight line. State the exact value of the gradient of this line.\\
(ii) Calculate the exact $x$-coordinate of the point of intersection of this line with the line with equation $y = 2 x$, simplifying your answer.
\hfill \mbox{\textit{CAIE P3 2016 Q2 [5]}}