Challenging +1.8 This question requires finding a normal line equation (needing quotient rule differentiation of a composite trigonometric function), determining intersection points, and computing a definite integral of a rational trigonometric function that likely requires substitution. The 10-mark allocation and 'show detailed reasoning' requirement indicate substantial multi-step work, but the techniques are all standard A-level methods applied systematically rather than requiring novel insight.
In this question you must show detailed reasoning.
\includegraphics{figure_9}
The diagram shows the curve \(y = \frac{4\cos 2x}{3 - \sin 2x}\) for \(x > 0\), and the normal to the curve at the point \((\frac{1}{4}\pi, 0)\).
Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac{2}{3} + \frac{1}{128}\pi^2\). [10]
In this question you must show detailed reasoning.
\includegraphics{figure_9}
The diagram shows the curve $y = \frac{4\cos 2x}{3 - \sin 2x}$ for $x > 0$, and the normal to the curve at the point $(\frac{1}{4}\pi, 0)$.
Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the $y$-axis is $\ln \frac{2}{3} + \frac{1}{128}\pi^2$. [10]
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q9 [10]}}