2024 JPJC H1 Q4
2024 JPJC H1 Q4
10 marks
The curve \(D\) has equation \(y=2\ln(3-x)+4\).
- Sketch the graph of \(D\), stating the exact coordinates of any points of intersection with the axes and the equation of the asymptote.[3]
- Use a non-calculator method to find the equation of the tangent to \(D\) at the point where \(x=1\), giving your answer in the form \(ax+by=c\), where \(a\) and \(b\) are integers and \(c\) is exact.[4]
- The tangent in (ii) meets the \(x\)-axis at \(A\) and \(y\)-axis at \(B\). Find the length of \(AB\), giving your answer in the form \(k(\ln 4+5)\), where \(k\) is an exact constant to be determined.[3]
Solution:
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Answer:(i) Intercepts \((0,4+2\ln3)\), \((3-\mathrm{e}^{-2},0)\); asymptote \(x=3\); decreasing, concave down. (ii) \(x+y=5+2\ln2\). (iii) \(k=\sqrt2\).
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