2025 IGCSE Additional Mathematics October/November 0606/12 Q4

2025 IGCSE Additional Mathematics October/November 0606/12 Q4

10 marks
  1. Show that \(2x^2+5x+3\) can be written in the form \(2(x+a)^2+b\), where \(a\) and \(b\) are constants to be found.[2]
  2. Hence write down the coordinates of the stationary point on the curve \(y=2x^2+5x+3\).[2]
    A function \(\mathrm f\) is such that \(\mathrm f(x)=2x^2+5x+3\), for \(x\ge p\), where \(p\) is a constant.
    It is given that \(\mathrm f^{-1}\) exists.
    1. Write down the least possible value of \(p\).[1]
    2. Using your value of \(p\), sketch the graphs of \(y=\mathrm f(x)\) and \(y=\mathrm f^{-1}(x)\).
      Label each graph.
      State the intercepts of each of the graphs with the axes.[5]

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Answer:(a) \(2(x+\dfrac54)^2-\dfrac18\), so \(a=\dfrac54,b=-\dfrac18\). (b) \((-\dfrac54,-\dfrac18)\) (c)(i) \(p=-\dfrac54\) (ii) \(\mathrm f\): \((-1,0),(0,3)\); \(\mathrm f^{-1}\): \((0,-1),(3,0)\); sketches shown.

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