2024 ASRJC H1 Q4
2024 ASRJC H1 Q4
10 marks
The curve \(C\) has equation \(y = x^2 + \ln(2-x)\).
- Use a non-calculator method to find the equation of the tangent to \(C\) at the point where \(x = 0\), leaving your answer in exact form.[3]
- Write down the coordinates of the point where the tangent meets \(C\) again, correct to \(3\) significant figures.[1]
- Sketch the graphs of \(C\) and the tangent found in part (a) on a single diagram, stating the equations of any asymptotes and all the coordinates of the points of intersection with the axes.[4]
- Find the area of the finite region bounded by the curve \(C\), the tangent and the line \(x = 1\). Give your answer correct to \(2\) decimal places.[2]
Solution:
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Answer:(a) \(y=\ln2-\frac x2\). (b) \((1.99,-0.300)\). (c) Sketch shown; asymptote \(x=2\); curve x-intercept approximately1.98, line x-intercept \(2\ln2\), common y-intercept \((0,\ln2)\). (d) School-intended left region 0.28 square units; other wording-permitted right region 1.40 square units.
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