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2024 DHS H1 Q2

2024 DHS H1 Q2

8 marks

The diagram shows the curve \(C\) passes through the point \(A(a+1, 12)\) and has an asymptote with equation \(x = 4\). It is also given that \(C\) has a gradient function \(\frac{1}{x-a}\), where \(x > a\).

  1. Show that \(a = 4\). Hence find the equation of \(C\).[3]
  2. Use a non-calculator method to find the equation of the tangent to \(C\) at the point \(A\).[1]
  3. Hence find the area of the finite region bounded by the curve \(C\), the tangent found in part (b) and the \(x\)-axis.[4]

Solution:

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Answer:(a) \(a=4\) (shown), \(y=\ln(x-4)+12\), \(x>4\). (b) \(y=x+7\). (c) \(61.0\) square units; exactly \(61-e^{-12}\).

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