One of the roots of the equation ${{x}^{3}}+p{{x}^{2}}+6x+q=0$ where $p$ and $q$ are real, is $5-\ma

Question

One of the roots of the equation ${{x}^{3}}+p{{x}^{2}}+6x+q=0$ where $p$ and $q$ are real, is $5-\mathrm{i}$. Find the other roots of the equation and the values of $p$ and $q$. [4] Do not use a calculator in answering this part. The complex number $z$ has modulus $2\sqrt{3}$ and argument $-\frac{\pi }{3}$. Find $z-2\sqrt{3}$ in the form $r{{\mathrm{e}}^{\mathrm{i}	heta }}$, where $r>0$ and $-\pi [3] Find the three smallest positive whole number values of $n$ for which $\frac{{{\left( z-2\sqrt{3} ight)}^{*}}}{{{\left( z-2\sqrt{3} ight)}^{n}}}$ is a real number. [3]

Accepted Answer

(a) $5+\mathrm{i}$, $-2$; $p=-8$, $q=52$. (b)(i) $2\sqrt{3}\,\mathrm{e}^{-\mathrm{i}2\pi/3}$. (b)(ii) $n = 2, 5, 8$

2020 NYJC P2 Q5

2020 NYJC P2 Q5

Video Solution:

Solution:

Solution locked