CAIE P3 2021 March — Question 10 11 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2021
SessionMarch
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIndefinite & Definite Integrals
TypeIntegration by substitution
DifficultyStandard +0.8 Part (a) requires a non-trivial substitution with careful handling of trigonometric identities (converting cos²x and dx in terms of u) and evaluating the resulting integral. Part (b) requires differentiating using the product rule, applying double angle formulas, solving a trigonometric equation, and verifying the maximum—this involves multiple sophisticated steps beyond routine calculus. The combination of technical manipulation and multi-step reasoning places this above average difficulty.
Spec1.07n Stationary points: find maxima, minima using derivatives1.08h Integration by substitution
\includegraphics{figure_10} The diagram shows the curve \(y = \sin 2x \cos^2 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\).
  1. Using the substitution \(u = \sin x\), find the exact area of the region bounded by the curve and the \(x\)-axis. [5]
  2. Find the exact \(x\)-coordinate of \(M\). [6]
Question 10:
AnswerMarks Guidance
10(a)State or imply du = cos x dx B1
Using double angle formula for sin2x and Pythagoras, express integral in
AnswerMarks
terms of u and du.M1
( )
AnswerMarks Guidance
Obtain integral 2 u−u3 duA1 OE
Use limits u = 0 and u = 1 in an integral of the form au2 +bu4, where
AnswerMarks Guidance
ab≠0M1  1
a + b or a + b − 0  a=1 and b=− 
 2
1
Obtain answer
AnswerMarks
2A1
5
AnswerMarks Guidance
10(b)Use product rule M1
Obtain correct derivative in any formA1
Equate derivative to zero and use a double angle formula*M1
Obtain an equation in one trig variableDM1
Obtain 4 sin2x=1, 4 cos2x=3 or 3 tan2x=1A1
1
Obtain answer x= π
AnswerMarks
6A1
6
Question 10:
--- 10(a) ---
10(a) | State or imply du = cos x dx | B1
Using double angle formula for sin2x and Pythagoras, express integral in
terms of u and du. | M1
( )
Obtain integral 2 u−u3 du | A1 | OE
Use limits u = 0 and u = 1 in an integral of the form au2 +bu4, where
ab≠0 | M1 |  1
a + b or a + b − 0  a=1 and b=− 
 2
1
Obtain answer
2 | A1
5
--- 10(b) ---
10(b) | Use product rule | M1
Obtain correct derivative in any form | A1
Equate derivative to zero and use a double angle formula | *M1
Obtain an equation in one trig variable | DM1
Obtain 4 sin2x=1, 4 cos2x=3 or 3 tan2x=1 | A1
1
Obtain answer x= π
6 | A1
6
\includegraphics{figure_10}

The diagram shows the curve $y = \sin 2x \cos^2 x$ for $0 \leqslant x \leqslant \frac{1}{2}\pi$, and its maximum point $M$.

\begin{enumerate}[label=(\alph*)]
\item Using the substitution $u = \sin x$, find the exact area of the region bounded by the curve and the $x$-axis. [5]
\item Find the exact $x$-coordinate of $M$. [6]
\end{enumerate}

\hfill \mbox{\textit{CAIE P3 2021 Q10 [11]}}
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