CAIE P2 2011 November — Question 3 5 marks

Exam BoardCAIE
ModuleP2 (Pure Mathematics 2)
Year2011
SessionNovember
Marks5
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Mark schemeDownload PDF ↗
TopicChain Rule
TypeTangent with specified gradient
DifficultyStandard +0.3 This is a straightforward application of the chain rule to find dy/dx = sec²(2x), then solving sec²(2x) = 4 for x in the given interval. It requires standard differentiation and basic trigonometric equation solving, making it slightly easier than average but not trivial due to the multiple-step process and need to find all solutions in the range.
Spec1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07m Tangents and normals: gradient and equations
3 \includegraphics[max width=\textwidth, alt={}, center]{55794ceb-2d52-459c-8724-6a6a29ab159a-2_705_737_591_703} The diagram shows the part of the curve \(y = \frac { 1 } { 2 } \tan 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinates of the points on this part of the curve at which the gradient is 4 .
AnswerMarks Guidance
Obtain derivative of the form \(k\sec^2 2x\), where \(k = 1\) or \(k = \frac{1}{2}\)M1
Obtain correct derivative \(\sec^2 2x\)A1
Use correct method for solving \(\sec^2 2x = 4\)M1
Obtain answer \(x = \frac{1}{6}\pi\) (or 0.524 radians)A1
Obtain answer \(x = \frac{1}{3}\pi\) (or 1.05 radians) and no others in rangeA1 [5] 
Obtain derivative of the form $k\sec^2 2x$, where $k = 1$ or $k = \frac{1}{2}$ | M1 |

Obtain correct derivative $\sec^2 2x$ | A1 |

Use correct method for solving $\sec^2 2x = 4$ | M1 |

Obtain answer $x = \frac{1}{6}\pi$ (or 0.524 radians) | A1 |

Obtain answer $x = \frac{1}{3}\pi$ (or 1.05 radians) and no others in range | A1 | [5]
3\\
\includegraphics[max width=\textwidth, alt={}, center]{55794ceb-2d52-459c-8724-6a6a29ab159a-2_705_737_591_703}

The diagram shows the part of the curve $y = \frac { 1 } { 2 } \tan 2 x$ for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$. Find the $x$-coordinates of the points on this part of the curve at which the gradient is 4 .

\hfill \mbox{\textit{CAIE P2 2011 Q3 [5]}}
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Q1 3 Q2 5 Q3 5 Q4 5 Q5 7 Q6 7 Q7 8 Q8 10