CAIE P2 2024 March — Question 7 10 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2024
SessionMarch
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTrig Proofs
TypeProve trigonometric identity
DifficultyStandard +0.8 Part (a) requires systematic algebraic manipulation of trigonometric identities (expanding sin2θ with cot and tan, using double angle formulas) which is moderately challenging but follows standard techniques. Part (b) applies the proven identity to evaluate a definite integral, requiring recognition and careful substitution. Part (c) involves recognizing that sin²α = ½sin2α to apply the identity with α/2, then solving a resulting trigonometric equation—this requires non-trivial insight and multiple conceptual steps beyond routine exercises.
Spec1.01a Proof: structure of mathematical proof and logical steps1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)
  1. Prove that $$\sin 2\theta (a \cot\theta + b \tan\theta) \equiv a + b + (a - b) \cos 2\theta,$$ where \(a\) and \(b\) are constants. [4]
  2. Find the exact value of \(\int_{\frac{\pi}{12}}^{\frac{\pi}{6}} \sin 2\theta (5 \cot\theta + 3 \tan\theta) \mathrm{d}\theta\). [3]
  3. Solve the equation \(\sin^2\alpha\left(2\cot\frac{1}{2}\alpha + 7\tan\frac{1}{2}\alpha\right) = 11\) for \(-\pi < \alpha < \pi\). [3]
Question 7:
AnswerMarks Guidance
7(a)Use correct identity for sin2 or for cot B1
Express left-hand side in terms of sin and cosM1
Attempt to express k cos2+k sin2 in terms of cos2 only
AnswerMarks Guidance
1 2M1
Confirm a+b+(a−b)cos2A1 Answer given – necessary detail needed.
4
AnswerMarks Guidance
7(b)Obtain integrand 8+2cos2 B1
Integrate to obtain 8+sin2B1
Apply limits correctly to obtain 2π+1 3−1
AnswerMarks Guidance
3 2 2B1 Or exact equivalent.
3
AnswerMarks Guidance
QuestionAnswer Marks
AnswerMarks Guidance
7(c)Use identity to obtain 5cos2=−2
3B1
Obtain =2.97B1 Or greater accuracy.
Obtain =−2.97B1 Or greater accuracy; and no others between −π and
π.
3
Question 7:
--- 7(a) ---
7(a) | Use correct identity for sin2 or for cot | B1
Express left-hand side in terms of sin and cos | M1
Attempt to express k cos2+k sin2 in terms of cos2 only
1 2 | M1
Confirm a+b+(a−b)cos2 | A1 | Answer given – necessary detail needed.
4
--- 7(b) ---
7(b) | Obtain integrand 8+2cos2 | B1
Integrate to obtain 8+sin2 | B1
Apply limits correctly to obtain 2π+1 3−1
3 2 2 | B1 | Or exact equivalent.
3
Question | Answer | Marks | Guidance
--- 7(c) ---
7(c) | Use identity to obtain 5cos2=−2
3 | B1
Obtain =2.97 | B1 | Or greater accuracy.
Obtain =−2.97 | B1 | Or greater accuracy; and no others between −π and
π.
3
\begin{enumerate}[label=(\alph*)]
\item Prove that
$$\sin 2\theta (a \cot\theta + b \tan\theta) \equiv a + b + (a - b) \cos 2\theta,$$
where $a$ and $b$ are constants. [4]
\item Find the exact value of $\int_{\frac{\pi}{12}}^{\frac{\pi}{6}} \sin 2\theta (5 \cot\theta + 3 \tan\theta) \mathrm{d}\theta$. [3]
\item Solve the equation $\sin^2\alpha\left(2\cot\frac{1}{2}\alpha + 7\tan\frac{1}{2}\alpha\right) = 11$ for $-\pi < \alpha < \pi$. [3]
\end{enumerate}

\hfill \mbox{\textit{CAIE P2 2024 Q7 [10]}}
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