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1995 Paper 1 Q1
D: 1484.0 B: 1484.0
inequalities polynomial factorisation cubic homogeneous equation curve sketching regions in plane factoring by grouping

  1. Find the real values of \(x\) for which \[ x^{3}-4x^{2}-x+4\geqslant0. \]
  2. Find the three lines in the \((x,y)\) plane on which \[ x^{3}-4x^{2}y-xy^{2}+4y^{3}=0. \]
  3. On a sketch shade the regions of the \((x,y)\) plane for which \[ x^{3}-4x^{2}y-xy^{2}+4y^{3}\geqslant0. \]

Solution:

  1. \(\,\) \begin{align*} && 0 & \leq x^3 - 4x^2 - x + 4 \\ &&&= (x-1)(x^2-3x-4) \\ &&&= (x-1)(x-4)(x+1) \\ \Leftrightarrow && x &\in [-1, 1] \cup [4, \infty) \end{align*}
  2. \(\,\) \begin{align*} && 0 &= x^{3}-4x^{2}y-xy^{2}+4y^{3} \\ && 0 &= (x-y)(x-4y)(x+y) \end{align*} Therefore the lines are \(y = x, 4y = x, y=-x\).
  3. TikZ diagram
    (quickest way to see this is to check the \(x\) or \(y\)-axis)

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