2021 MI P1 Q6
2021 MI P1 Q6
9 marks
Prelims
- Sketch the curve with equation \(y=\left| \frac{1}{a-x} \right|\), where \(a\) is a positive constant. State, in terms of \(a\), the equations of the asymptotes and coordinates of any intersections with the \(x\)-axis and \(y\)-axis. On the same diagram, sketch the line with equation \(y=b\left( x-a \right)\), where \(b\) is a positive constant.[4]
- Find, in terms of \(a\) and \(b\), the root of the equation \(\left| \frac{1}{a-x} \right|=b\left( x-a \right)\).[3]
- Hence solve the inequality \(\left| \frac{1}{a-x} \right|>b\left( x-a \right)\).[2]
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Answer:(i) Asymptotes \(x=a\) and \(y=0\), \(y\)-intercept \(\left(0,\dfrac1a\right)\), no \(x\)-intercept (ii) \(x=a+\dfrac1{\sqrt b}\) (iii) \(x<a+\dfrac1{\sqrt b}\), \(x\ne a\)
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