Challenging +1.8 This question requires finding the equation of a normal using the quotient rule for differentiation, determining intersection points, and computing a definite integral of a rational trigonometric function that likely requires substitution. While technically demanding with multiple steps and careful algebraic manipulation, the overall approach follows standard A-level techniques without requiring novel insight. The 10-mark allocation and 'show detailed reasoning' instruction indicate extended working, placing it well above average difficulty but not at the extreme end.
12 In this question you must show detailed reasoning.
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The diagram shows the curve \(y = \frac { 4 \cos 2 x } { 3 - \sin 2 x }\), for \(x \geqslant 0\), and the normal to the curve at the point \(\left( \frac { 1 } { 4 } \pi , 0 \right)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac { 9 } { 4 } + \frac { 1 } { 128 } \pi ^ { 2 }\). [0pt]
[10]
12 In this question you must show detailed reasoning.\\
\includegraphics[max width=\textwidth, alt={}, center]{1ba9fa5f-310f-4429-9bd1-4004852d5b3e-6_716_479_292_794}
The diagram shows the curve $y = \frac { 4 \cos 2 x } { 3 - \sin 2 x }$, for $x \geqslant 0$, and the normal to the curve at the point $\left( \frac { 1 } { 4 } \pi , 0 \right)$. Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the $y$-axis is $\ln \frac { 9 } { 4 } + \frac { 1 } { 128 } \pi ^ { 2 }$.\\[0pt]
[10]
\hfill \mbox{\textit{OCR H240/01 2018 Q12 [10]}}