2020 ACJC P2 Q4
(a)
It is given that $\text{f}\left( x \right)={{x}^{6}}-a{{x}^{4}}-{{x}^{2}}-b$, where $a$ and $b$ are real numbers.
(i) Show that $\text{f}\left( x \right)=\text{f}\left( -x \right)$.
[1]
The diagram shows the curve with equation $y=\text{f}\left( x \right)$ for $x>0$. The curve crosses the positive $x$-axis at $x=\beta $.
(ii) Determine the number of non-real roots of the equation $\text{f}\left( x \right)=0$. Justify your answer.
[2]
(iii) Given that ${{z}_{1}}=r{{\text{e}}^{\text{i}\theta }}$， where $r>0$ and $0<\theta <\frac{\pi }{2}$, is one of the roots of the equation $\text{f}\left( x \right)=0$, express all the roots of $\text{f}\left( x \right)=0$ in the modulus-argument form.
[3]
(iv) Hence, show that $\text{f}\left( x \right)$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]
(b)
Find the modulus of the complex number
$\frac{\text{i}{{\left( 2\text{i}z+{{z}^{2}} \right)}^{*}}}{z\left( 2{{z}^{*}}-4\text{i} \right)}$,
where $z$ is a complex number.
[3]
Suggested Handwritten and Video Solutions
Share with your friends!