| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Linear relationship between log variables |
| Difficulty | Moderate -0.3 Part (a) is a straightforward application of logarithms to linearize an exponential equation (take ln of both sides, rearrange to y = mx + c form), requiring only routine manipulation. Part (b) involves calculating gradient from two points and solving simultaneous equations, which are standard techniques. The question requires multiple steps but no novel insight or challenging problem-solving. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks |
|---|---|
| 1(a) | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| 2lna | B1 | AG – necessary detail needed. |
| Answer | Marks |
|---|---|
| 1(b) | 3 |
| Answer | Marks |
|---|---|
| 2lna | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2lna 2.9 | A1 | Allow greater accuracy. |
| Substitute appropriate values to find value of k | M1 | |
| Obtain k=1.7 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| or a1.9 =e1.2+k | M1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| or a7.6 =e9.9+k | M1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain a=4.6or a=e19 | A1 | Allow greater accuracy. |
| Obtain k=1.7 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 1:
--- 1(a) ---
1(a) | 3
State or imply 2ylna=3x+k and conclude that gradient is
2lna | B1 | AG – necessary detail needed.
1
--- 1(b) ---
1(b) | 3
Equate to gradient of line
2lna | M1
3 2.85 29
Obtain = or equivalent and hence obtain a=4.6or a=e19
2lna 2.9 | A1 | Allow greater accuracy.
Substitute appropriate values to find value of k | M1
Obtain k=1.7 | A1
Alternative Method for Question 1(b)
0.95(2lna)=3(0.4)+k
Obtain
or a1.9 =e1.2+k | M1 | OE
3.80(2lna)=3(3.3)+k
Obtain
or a7.6 =e9.9+k | M1 | OE
29
Obtain a=4.6or a=e19 | A1 | Allow greater accuracy.
Obtain k=1.7 | A1
4
Question | Answer | Marks | GuidanceThe variables $x$ and $y$ satisfy the equation $a^{2y} = e^{3x+k}$, where $a$ and $k$ are constants. The graph of $y$ against $x$ is a straight line.
\begin{enumerate}[label=(\alph*)]
\item Use logarithms to show that the gradient of the straight line is $\frac{3}{2\ln a}$. [1]
\item Given that the straight line passes through the points $(0.4, 0.95)$ and $(3.3, 3.80)$, find the values of $a$ and $k$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q1 [5]}}