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1996 Paper 1 Q5
D: 1484.0 B: 1500.0
complex numbers surds quadratic equation equating real and imaginary parts rationalization quadratic formula

  1. Find all rational numbers \(r\) and \(s\) which satisfy \[ (r+s\sqrt{3})^{2}=4-2\sqrt{3}. \]
  2. Find all real numbers \(p\) and \(q\) which satisfy \[ (p+q\mathrm{i})^{2}=(3-2\sqrt{3})+2(1-\sqrt{3})\mathrm{i}. \]
  3. Solve the equation \[ (1+\mathrm{i})z^{2}-2z+2\sqrt{3}-2=0, \] writing your solutions in as simple a form as possible.
{[}No credit will be given to answers involving use of calculators.{]}

Solution:

  1. Suppose \begin{align*} && 4 - 2\sqrt{3} &= (r+s\sqrt{3})^2 \\ &&&= r^2+3s^2+2sr \sqrt{3} \\ \Rightarrow && rs &= -1 \\ && r^2+3s^2 &= 4 \\ \Rightarrow && (r,s) &= (1,-1), (-1,1) \end{align*}
  2. \begin{align*} && (3-2\sqrt{3})+2(1-\sqrt{3})i &= (p+qi)^2 \\ &&&= p^2-q^2 + 2pq i \\ \Rightarrow && pq &= (1-\sqrt{3}) \\ && p^2 - q^2 &= 3-2\sqrt{3} \\ \Rightarrow &&3-2\sqrt{3} &= p^2 - \frac{(1-\sqrt{3})^2}{p^2} \\ \Rightarrow && 0 &= p^4-(3-2\sqrt{3})p^2-(4-2\sqrt{3}) \\ &&&= (p^2-(4-2\sqrt{3}))(p^2+1) \\ \Rightarrow && p &= \pm (1-\sqrt{3}) \\ && q &=\mp \frac12(1+\sqrt{3}) \end{align*}
  3. \begin{align*} && 0 &= (1+i)z^2 - 2z + 2(\sqrt{3}-1) \\ \Rightarrow && z &= \frac{2 \pm \sqrt{4-4(1+i)2(\sqrt{3}-1)}}{2(1+i)} \\ &&&= \frac{1 \pm \sqrt{1-(1+i)2(\sqrt{3}-1)}}{1+i} \\ &&&= \frac{1 \pm \sqrt{(3-2\sqrt{3})+(2-2\sqrt{3})i}}{1+i} \\ &&&= \frac{1 \pm (1 - \sqrt{3}) \mp \frac12 (1+\sqrt{3})i}{1+i} \\ &&&= \frac{5-\sqrt{3}}{4} + \frac{3-3\sqrt{3}}{4}i, \\ &&& \frac{\sqrt{3}-1}{4} + \frac{1+3\sqrt{3}}{4}i \end{align*}

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