OCR MEI C4 2006 January — Question 7 17 marks

Exam BoardOCR MEI
ModuleC4 (Core Mathematics 4)
Year2006
SessionJanuary
Marks17
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicAddition & Double Angle Formulae
TypeApplied context with trigonometry
DifficultyStandard +0.3 This is a structured multi-part question with clear signposting ('show that', 'hence') that guides students through standard techniques: applying tan subtraction formula, implicit differentiation, and finding maxima by setting derivative to zero. While it requires several steps and combines trigonometry with calculus, each part follows predictable A-level methods without requiring novel insight or particularly complex algebraic manipulation.
Spec1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.07s Parametric and implicit differentiation
7 In a game of rugby, a kick is to be taken from a point P (see Fig. 7). P is a perpendicular distance \(y\) metres from the line TOA. Other distances and angles are as shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{897205bc-2f93-4628-8f21-2ec7fd3b3699-3_509_629_513_715} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Show that \(\theta = \beta - \alpha\), and hence that \(\tan \theta = \frac { 6 y } { 160 + y ^ { 2 } }\). Calculate the angle \(\theta\) when \(y = 6\).
  2. By differentiating implicitly, show that \(\frac { \mathrm { d } \theta } { \mathrm { d } y } = \frac { 6 \left( 160 - y ^ { 2 } \right) } { \left( 160 + y ^ { 2 } \right) ^ { 2 } } \cos ^ { 2 } \theta\).
  3. Use this result to find the value of \(y\) that maximises the angle \(\theta\). Calculate this maximum value of \(\theta\). [You need not verify that this value is indeed a maximum.]
7 In a game of rugby, a kick is to be taken from a point P (see Fig. 7). P is a perpendicular distance $y$ metres from the line TOA. Other distances and angles are as shown.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{897205bc-2f93-4628-8f21-2ec7fd3b3699-3_509_629_513_715}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}

(i) Show that $\theta = \beta - \alpha$, and hence that $\tan \theta = \frac { 6 y } { 160 + y ^ { 2 } }$.

Calculate the angle $\theta$ when $y = 6$.\\
(ii) By differentiating implicitly, show that $\frac { \mathrm { d } \theta } { \mathrm { d } y } = \frac { 6 \left( 160 - y ^ { 2 } \right) } { \left( 160 + y ^ { 2 } \right) ^ { 2 } } \cos ^ { 2 } \theta$.\\
(iii) Use this result to find the value of $y$ that maximises the angle $\theta$. Calculate this maximum value of $\theta$. [You need not verify that this value is indeed a maximum.]

\hfill \mbox{\textit{OCR MEI C4 2006 Q7 [17]}}
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