CAIE P2 2018 November — Question 3 5 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2018
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicChain Rule
TypeFind stationary points and nature
DifficultyStandard +0.3 This is a straightforward stationary point problem requiring differentiation using the chain rule (for sin 2x and tan 2x), setting the derivative to zero, and solving a trigonometric equation. While it involves composite functions and trig identities, it follows a standard procedure with no novel insight required, making it slightly easier than average.
Spec1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05o Trigonometric equations: solve in given intervals1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)

3 \includegraphics[max width=\textwidth, alt={}, center]{cc7e798e-0817-405c-bae0-b24b9f451fbf-04_378_486_260_826} The diagram shows the curve with equation $$y = 5 \sin 2 x - 3 \tan 2 x$$ for values of \(x\) such that \(0 \leqslant x < \frac { 1 } { 4 } \pi\). Find the \(x\)-coordinate of the stationary point \(M\), giving your answer correct to 3 significant figures.

Question 3:
AnswerMarks Guidance
AnswerMark Guidance
Differentiate to obtain \(10\cos 2x\)B1
Differentiate to obtain \(-6\sec^2 2x\)B1
Equate first derivative to zero and find value for \(\cos^3 2x\)M1
Use correct process for finding \(x\) from \(\cos^3 2x=k\)M1
Obtain \(0.284\) nfwwA1 Or greater accuracy
## Question 3:

| Answer | Mark | Guidance |
|--------|------|----------|
| Differentiate to obtain $10\cos 2x$ | B1 | |
| Differentiate to obtain $-6\sec^2 2x$ | B1 | |
| Equate first derivative to zero and find value for $\cos^3 2x$ | M1 | |
| Use correct process for finding $x$ from $\cos^3 2x=k$ | M1 | |
| Obtain $0.284$ nfww | A1 | Or greater accuracy |

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3\\
\includegraphics[max width=\textwidth, alt={}, center]{cc7e798e-0817-405c-bae0-b24b9f451fbf-04_378_486_260_826}

The diagram shows the curve with equation

$$y = 5 \sin 2 x - 3 \tan 2 x$$

for values of $x$ such that $0 \leqslant x < \frac { 1 } { 4 } \pi$. Find the $x$-coordinate of the stationary point $M$, giving your answer correct to 3 significant figures.\\

\hfill \mbox{\textit{CAIE P2 2018 Q3 [5]}}