CAIE P3 2024 June — Question 8 8 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2024
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicHarmonic Form
TypeIntegration using harmonic form
DifficultyChallenging +1.2 Part (a) is a standard harmonic form conversion requiring routine application of R cos(θ+α) = R cos θ cos α - R sin θ sin α. Part (b) is more demanding, requiring recognition that the integral becomes ∫3/R² cos²(2x+α) dx = (3/R²)∫sec²(2x+α) dx, then careful handling of limits after substitution and simplification. The multi-step integration with exact values and the non-obvious sec² recognition elevates this above average difficulty, though it remains a standard Further Maths technique.
Spec1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08h Integration by substitution
  1. Express \(3 \cos 2x - \sqrt{3} \sin 2x\) in the form \(R \cos(2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). Give the exact values of \(R\) and \(\alpha\). [3]
  2. Hence find the exact value of \(\int_0^{\frac{1}{2}\pi} \frac{3}{(3 \cos 2x - \sqrt{3} \sin 2x)^2} \, dx\), simplifying your answer. [5]
Question 8:
AnswerMarks Guidance
8(a)State R = 12 or exact equivalent B1
Use trig formula to find αM1  3  3  1
Allow 30 or tan1     or cos−1      or sin−1   
 3   2   2
 3
Allow M1 if – tan1   etc.
 
3
 
NB: If cos = 3 and sin = 3 seen then M0 A0.
1
Obtain α = π
AnswerMarks Guidance
6A1  3
CWO, so A0 if from tan1  .
 
3
 
3
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
8(b)Express integral in the form Asec22x...dx or
Asec22x...dxB1FT FT α from (a).
Integrate and reach Btan2x... or Btan2x...B1FT FT α from (a).
Where B = A or 2A or 0.5A.
1
Obtain tan2x...
AnswerMarks Guidance
8B1FT OE
FT α from (a).
1 1 1
Allow as  .
8 4 2
Coefficient must be correct.
Use limits of x0and x 1 π in the correct order in expression of
12
 
form Btan2x... so Btan  ...  −Btan...
 6 
 
or Btan  ...  −Btan...
AnswerMarks Guidance
 6 M1 Allow with tan still present.
FT α from (a).
3 1  1 
SC: B1 OE after tan  2x π  with no working.
12 8  6 
1 1
Obtain answer 1 3 or or
12 4 3 48
AnswerMarks Guidance
or single term exact equivalentA1 1 1 1 31
( 3 – ) = needs simplifying.
 
8 3 8  3 
AnswerMarks
51
Note: allow all marks in (b) even if α = π found by an incorrect
6
method in (a).
AnswerMarks Guidance
QuestionAnswer Marks 
Question 8:
--- 8(a) ---
8(a) | State R = 12 or exact equivalent | B1 | ISW
Use trig formula to find α | M1 |  3  3  1
Allow 30 or tan1     or cos−1      or sin−1   
 3   2   2
 3
Allow M1 if – tan1   etc.
 
3
 
NB: If cos = 3 and sin = 3 seen then M0 A0.
1
Obtain α = π
6 | A1 |  3
CWO, so A0 if from tan1  .
 
3
 
3
Question | Answer | Marks | Guidance
--- 8(b) ---
8(b) | Express integral in the form Asec22x...dx or
Asec22x...dx | B1FT | FT α from (a).
Integrate and reach Btan2x... or Btan2x... | B1FT | FT α from (a).
Where B = A or 2A or 0.5A.
1
Obtain tan2x...
8 | B1FT | OE
FT α from (a).
1 1 1
Allow as  .
8 4 2
Coefficient must be correct.
Use limits of x0and x 1 π in the correct order in expression of
12
 
form Btan2x... so Btan  ...  −Btan...
 6 
 
or Btan  ...  −Btan...
 6  | M1 | Allow with tan still present.
FT α from (a).
3 1  1 
SC: B1 OE after tan  2x π  with no working.
12 8  6 
1 1
Obtain answer 1 3 or or
12 4 3 48
or single term exact equivalent | A1 | 1 1 1 31
( 3 – ) = needs simplifying.
 
8 3 8  3 
5 | 1
Note: allow all marks in (b) even if α = π found by an incorrect
6
method in (a).
Question | Answer | Marks | Guidance
\begin{enumerate}[label=(\alph*)]
\item Express $3 \cos 2x - \sqrt{3} \sin 2x$ in the form $R \cos(2x + \alpha)$, where $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$. Give the exact values of $R$ and $\alpha$. [3]

\item Hence find the exact value of $\int_0^{\frac{1}{2}\pi} \frac{3}{(3 \cos 2x - \sqrt{3} \sin 2x)^2} \, dx$, simplifying your answer. [5]
\end{enumerate}

\hfill \mbox{\textit{CAIE P3 2024 Q8 [8]}}
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