2008 AJC P1 Q12 [Modified]

2008 AJC P1 Q12 [Modified]

11 marks
Prelims

A curve \(C\) has parametric equations

\(x=\ln (2t)\), \(y={{\tan }^{-1}}(2t)\), where \(t>0\).

  1. Describe the behaviour of the curve \(C\) as \(x\to \infty \) and \(x\to -\infty \). Sketch \(C\), showing clearly the asymptotes and the axial intercepts.[3]
  2. Show that the gradient of \(C\) at the point where \(y=p\) is \(\frac{1}{2}\sin 2p\).[3]
    Hence find the area of triangle bounded by the \(x\)-axis, the tangent and normal to \(C\) when \(p=\frac{\pi }{4}\).[5]

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Answer:\(\frac{5}{64}\pi^2\) units²

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