CAIE P2 2024 November — Question 3 3 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2024
SessionNovember
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicChain Rule
TypeDifferentiation of trigonometric composites
DifficultyModerate -0.3 This is a straightforward chain rule application with a standard trigonometric function. Students need to differentiate tan²(x/2) using chain rule twice and substitute x=π/3, which is routine A-level calculus requiring no problem-solving insight. The 3 marks reflect standard working (outer derivative, inner derivative, substitution), making it slightly easier than average.
Spec1.05g Exact trigonometric values: for standard angles1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
The function \(\text{f}\) is defined by \(\text{f}(x) = \tan^2\left(\frac{1}{2}x\right)\) for \(0 \leqslant x < \pi\).
  1. Find the exact value of \(\text{f}'\left(\frac{\pi}{3}\right)\). [3]
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
The function $\text{f}$ is defined by $\text{f}(x) = \tan^2\left(\frac{1}{2}x\right)$ for $0 \leqslant x < \pi$.

\begin{enumerate}[label=(\alph*)]
\item Find the exact value of $\text{f}'\left(\frac{\pi}{3}\right)$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE P2 2024 Q3 [3]}}
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