OCR FP2 2009 January — Question 9

Exam BoardOCR
ModuleFP2 (Further Pure Mathematics 2)
Year2009
SessionJanuary
TopicIntegration with Partial Fractions

9 A curve has equation $$y = \frac { 4 x - 3 a } { 2 \left( x ^ { 2 } + a ^ { 2 } \right) }$$ where \(a\) is a positive constant.
  1. Explain why the curve has no asymptotes parallel to the \(y\)-axis.
  2. Find, in terms of \(a\), the set of values of \(y\) for which there are no points on the curve.
  3. Find the exact value of \(\int _ { a } ^ { 2 a } \frac { 4 x - 3 a } { 2 \left( x ^ { 2 } + a ^ { 2 } \right) } \mathrm { d } x\), showing that it is independent of \(a\).