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2020 ACJC P2 Q4

2020 ACJC P2 Q4

12 marks
Prelims
  1. It is given that \(\mathrm{f}\left( x \right)={{x}^{6}}-a{{x}^{4}}-{{x}^{2}}-b\), where \(a\) and \(b\) are real numbers.
    1. Show that \(\mathrm{f}\left( x \right)=\mathrm{f}\left( -x \right)\).[1]
      The diagram shows the curve with equation \(y=\mathrm{f}\left( x \right)\) for \(x>0\). The curve crosses the positive \(x\)-axis at \(x=\beta \).
      Sub-item image
    2. Determine the number of non-real roots of the equation \(\mathrm{f}\left( x \right)=0\). Justify your answer.[2]
    3. Given that \({{z}_{1}}=r{{\mathrm{e}}^{\mathrm{i}\theta }}\)， where \(r>0\) and \(0<\theta <\frac{\pi }{2}\), is one of the roots of the equation \(\mathrm{f}\left( x \right)=0\), express all the roots of \(\mathrm{f}\left( x \right)=0\) in the modulus-argument form.[3]
    4. Hence, show that \(\mathrm{f}\left( x \right)\)can be expressed as a product of three quadratic factors of the form
      \(\left( {{x}^{2}}-A \right)\left( {{x}^{2}}-Brx\cos \theta +{{r}^{2}} \right)\left( {{x}^{2}}-Crx\cos \theta +{{r}^{2}} \right)\),
      where \(A,B\) and \(C\) are constants.[3]
  2. Find the modulus of the complex number
    \(\frac{\mathrm{i}{{\left( 2\mathrm{i}z+{{z}^{2}} \right)}^{*}}}{z\left( 2{{z}^{*}}-4\mathrm{i} \right)}\),
    where \(z\) is a complex number.[3]

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Answer:(a)(ii) Since \(\mathrm{f}(x)\)is of degree \({6}\), there is a total of \({6}\) roots. As there is only one real root for \(x\ge 0\), i.e. \(x=\beta \), \text{ and } \(\mathrm{f}(x)=\mathrm{f}(-x)\), thus \(x=-\beta \) is the other real root. With only \({2}\) real roots, the equation \(\mathrm{f}(x)=0\) has \({4}\) non-real roots. (a)(iii) The roots in modulus-argument form are: \({{z}_{1}}=r{{\mathrm{e}}^{i\theta }}\), \({{z}_{2}}=r{{\mathrm{e}}^{-i\theta }}\), \({{z}_{3}}=-r{{\mathrm{e}}^{i\theta }}\), \({{{z}_{4}}=-r{{\mathrm{e}}^{-\mathrm{i}\theta }}=r{{\mathrm{e}}^{\mathrm{i}(\pi -\theta )}}}\), \({{z}_{5}}=\beta {{\mathrm{e}}^{\mathrm{i}\pi }}\), \({{{z}_{6}}=\beta {{\mathrm{e}}^{\mathrm{i}\theta }}}\) are the remaining roots of \(\mathrm{f}(x)=0\). (a)(iv) \(A={{\beta }^{2}}\), \(B=2\), \(C=-2\) (b) \(\frac{1}{2}\)

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