CAIE P2 2021 November — Question 4 5 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2021
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
TypeFind constant from definite integral
DifficultyModerate -0.3 This is a straightforward application of logarithmic integration with a simple algebraic manipulation. Students need to integrate 1/(3x) to get (1/3)ln|x|, apply limits, use log laws to simplify ln((a+14)/a) = ln 2, then solve (a+14)/a = 2 for a = 14. While it requires multiple steps, each is routine and the question follows a standard pattern for this topic.
Spec1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits
4 Given that \(\int _ { a } ^ { a + 14 } \frac { 1 } { 3 x } \mathrm {~d} x = \ln 2\), find the value of the positive constant \(a\).
Question 4:
AnswerMarks Guidance
AnswerMark Guidance
Integrate to obtain form \(k\ln x\) or \(k\ln(3x)\)M1
Apply limits and obtain \(\frac{1}{3}\ln(a+14)-\frac{1}{3}\ln a=\ln 2\) *or* \(\frac{1}{3}\ln(3a+42)-\frac{1}{3}\ln(3a)=\ln 2\)A1 OE
Use one relevant logarithm property correctlyM1
Apply correct process to obtain equation without logarithmsM1 OE. M0 if incorrect logarithm property used
Obtain \(\frac{a+14}{a}=8\) or equivalent and hence \(a=2\)A1
Total5 
## Question 4:

| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain form $k\ln x$ or $k\ln(3x)$ | M1 | |
| Apply limits and obtain $\frac{1}{3}\ln(a+14)-\frac{1}{3}\ln a=\ln 2$ *or* $\frac{1}{3}\ln(3a+42)-\frac{1}{3}\ln(3a)=\ln 2$ | A1 | OE |
| Use one relevant logarithm property correctly | M1 | |
| Apply correct process to obtain equation without logarithms | M1 | OE. M0 if incorrect logarithm property used |
| Obtain $\frac{a+14}{a}=8$ or equivalent and hence $a=2$ | A1 | |
| **Total** | **5** | |

---
4 Given that $\int _ { a } ^ { a + 14 } \frac { 1 } { 3 x } \mathrm {~d} x = \ln 2$, find the value of the positive constant $a$.\\

\hfill \mbox{\textit{CAIE P2 2021 Q4 [5]}}
This paper (7 questions)
View full paper
Q1 8 Q2 7 Q3 6 Q4 5 Q5 6 Q6 8 Q7 10