2025 DHS Promo Q5

2025 DHS Promo Q5

Junior College 1
8 marks

The curve \(C\) has parametric equations

  1. Explain why \(y=0\) and \(x=\ln \frac{\pi}{2}\) are asymptotes to the curve.[2]
  2. Sketch \(C\), giving the exact coordinates of any points where \(C\) crosses the \(x\)- and \(y\)-axes and the equations of any asymptotes.[3]
  3. Show that the equation of the normal to \(C\) at \(t=\frac{\pi}{3}\) is given by[3]
    \(y=-\frac{3}{4\pi}x+\frac{3}{4\pi}\ln\frac{\pi}{3}+\sqrt{3}.\)

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Answer:(a) (y=0), (x=\ln\dfrac{\pi}{2}), (x=\ln\dfrac{3\pi}{2}) (b) intercepts ((0,\tan1)), ((\ln\pi,0)) (c) (y-\sqrt3=-\dfrac{3}{4\pi}\left(x-\ln\dfrac{\pi}{3}\right))

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