CAIE P2 2024 November — Question 3 4 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2024
SessionNovember
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicStandard Integrals and Reverse Chain Rule
TypeDefinite integral with trigonometric functions
DifficultyModerate -0.3 This is a straightforward definite integration question requiring standard techniques: integrating sin x (basic recall) and applying the reverse chain rule to f(x). With 4 marks and limits 0 to π/4, it involves routine substitution and evaluation at standard angles. Slightly easier than average due to the standard nature of both components and the common angle values.
Spec1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)
  1. Find the exact value of \(\int_0^{\frac{\pi}{4}} \left(\text{f}(x) + \sin x\right) dx\). [4]
Question 3:
AnswerMarks Guidance
3(a)Differentiate to obtain form ktan1xsec2 1x
2 2M1 OE. May use identities before differentiation.
Obtain correct tan1xsec2 1x
AnswerMarks Guidance
2 2A1 OE. Allow unsimplified.
Substitute 2π to obtain 4 3
AnswerMarks
3A1
3
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks
3(b)Express integrand assec2 1x−1+sinx
2B1
Integrate to obtain k tan1x−x+k cosx
AnswerMarks Guidance
1 2 2M1 Where kk 0.
1 2
Obtain correct 2tan1x−x−cosx
AnswerMarks
2A1
Apply limits correctly to obtain3−1π or exact equivalent
AnswerMarks
2A1
4
AnswerMarks Guidance
QuestionAnswer Marks 
Question 3:
--- 3(a) ---
3(a) | Differentiate to obtain form ktan1xsec2 1x
2 2 | M1 | OE. May use identities before differentiation.
Obtain correct tan1xsec2 1x
2 2 | A1 | OE. Allow unsimplified.
Substitute 2π to obtain 4 3
3 | A1
3
Question | Answer | Marks | Guidance
--- 3(b) ---
3(b) | Express integrand assec2 1x−1+sinx
2 | B1
Integrate to obtain k tan1x−x+k cosx
1 2 2 | M1 | Where kk 0.
1 2
Obtain correct 2tan1x−x−cosx
2 | A1
Apply limits correctly to obtain3−1π or exact equivalent
2 | A1
4
Question | Answer | Marks | Guidance
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the exact value of $\int_0^{\frac{\pi}{4}} \left(\text{f}(x) + \sin x\right) dx$. [4]
\end{enumerate}

\hfill \mbox{\textit{CAIE P2 2024 Q3 [4]}}
This paper (9 questions)
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Q1 5 Q2 4 Q3 3 Q3 4 Q4 5 Q4 3 Q5 17 Q6 7 Q7 11