CAIE FP1 2019 June — Question 10

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2019
SessionJune
TopicPolynomial Division & Manipulation

10 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$y = \frac { a x } { x + 5 } \quad \text { and } \quad y = \frac { x ^ { 2 } + ( a + 10 ) x + 5 a + 26 } { x + 5 }$$ respectively, where \(a\) is a constant and \(a > 2\).
  1. Find the equations of the asymptotes of \(C _ { 1 }\).
  2. Find the equation of the oblique asymptote of \(C _ { 2 }\).
  3. Show that \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
  4. Find the coordinates of the stationary points of \(C _ { 2 }\).
  5. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on a single diagram. [You do not need to calculate the coordinates of any points where \(C _ { 2 }\) crosses the axes.]