CAIE P2 2024 June — Question 5 8 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2024
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPolynomial Division & Manipulation
TypeIntegration Using Polynomial Division
DifficultyStandard +0.3 This is a straightforward two-part question combining polynomial division (routine algebraic manipulation) with integration. Part (a) is standard long division with verification. Part (b) requires recognizing that the quotient plus remainder form can be integrated term-by-term, with the remainder giving a logarithm. While it requires multiple techniques, each step follows standard procedures without requiring novel insight or problem-solving, making it slightly easier than average.
Spec1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.08j Integration using partial fractions
The polynomial \(p(x)\) is defined by \(p(x) = 9x^3 + 18x^2 + 5x + 4\).
  1. Find the quotient when \(p(x)\) is divided by \((3x + 2)\), and show that the remainder is 6. [3]
  2. Find the value of \(\int_0^2 \frac{p(x)}{3x + 2} \, dx\), giving your answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers. [5]
Question 5:
AnswerMarks Guidance
5(a)Carry out division at least as far as 3x2 k x
1M1 Or equivalent (inspection, …).
Obtain quotient 3x2 4x1A1
Confirm remainder is 6A1 Answer given – necessary detail needed.
SC B1 for use of the factor theorem to show remainder is 6 if no
other marks are awarded.
Alternative Method for Question 5(a)
Synthetic division
–2/3 9 18 5 4
–6 8 –2
AnswerMarks
9 12 –3 6(M1)
Obtain quotient 3x2 4x1(A1)
Confirm remainder is 6(A1)
3
AnswerMarks Guidance
–2/39 18
–68 –2
912 –3
QuestionAnswer Marks
AnswerMarks
5(b)6
Identify integrand as 3x2 4x1
AnswerMarks Guidance
3x2B1FT Following their quotient.
Integrate to obtain at least x3 and k ln(3x2) terms
AnswerMarks
2*M1
Obtain x3 2x2 x2ln(3x2)A1
Apply limits and appropriate logarithm propertiesDM1
Obtain 14ln16A1
5
AnswerMarks Guidance
QuestionAnswer Marks 
Question 5:
--- 5(a) ---
5(a) | Carry out division at least as far as 3x2 k x
1 | M1 | Or equivalent (inspection, …).
Obtain quotient 3x2 4x1 | A1
Confirm remainder is 6 | A1 | Answer given – necessary detail needed.
SC B1 for use of the factor theorem to show remainder is 6 if no
other marks are awarded.
Alternative Method for Question 5(a)
Synthetic division
–2/3 9 18 5 4
–6 8 –2
9 12 –3 6 | (M1)
Obtain quotient 3x2 4x1 | (A1)
Confirm remainder is 6 | (A1)
3
–2/3 | 9 | 18 | 5 | 4
–6 | 8 | –2
9 | 12 | –3 | 6
Question | Answer | Marks | Guidance
--- 5(b) ---
5(b) | 6
Identify integrand as 3x2 4x1
3x2 | B1FT | Following their quotient.
Integrate to obtain at least x3 and k ln(3x2) terms
2 | *M1
Obtain x3 2x2 x2ln(3x2) | A1
Apply limits and appropriate logarithm properties | DM1
Obtain 14ln16 | A1
5
Question | Answer | Marks | Guidance
The polynomial $p(x)$ is defined by $p(x) = 9x^3 + 18x^2 + 5x + 4$.

\begin{enumerate}[label=(\alph*)]
\item Find the quotient when $p(x)$ is divided by $(3x + 2)$, and show that the remainder is 6. [3]
\item Find the value of $\int_0^2 \frac{p(x)}{3x + 2} \, dx$, giving your answer in the form $a + \ln b$ where $a$ and $b$ are integers. [5]
\end{enumerate}

\hfill \mbox{\textit{CAIE P2 2024 Q5 [8]}}
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