2023 VJC H1 Q4
2023 VJC H1 Q4
10 marks
The curve \(C\) has equation \(y = 2 - \ln(1 - x)\).
- Sketch the graph of \(C\), giving the exact coordinates of any points of intersection with the axes.[2]
- Find the equation of the tangent to \(C\) at the point where \(x = \frac{1}{2}\), giving your answer in the form of \(y = mx + a + \ln b\) where \(m\), \(a\) and \(b\) are integers to be determined.[3]
- By adding a suitable graph, solve the inequality \[3x^2 + \ln \sqrt{1 - x} \leq 1.\][3]
- Hence solve the inequality \[3e^{2x} + \ln \sqrt{1 - e^x} \leq 1.\][2]
Solution:
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Answer:(a) Intercepts \((1-e^2,0),(0,2)\); asymptote \(x=1\), domain \(x<1\). (b) \(y=2x+1+\ln2\). (c) \(-0.514\le x\le0.752\) or \(0.976\le x<1\). (d) \(x\le-0.285\) or \(-0.0248\le x<0\). Numerical boundary values are to 3 s.f.
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