2000 Paper 3 Q3

Year: 2000
Paper: 3
Question Number: 3

Course: UFM Pure
Section: Complex numbers 2

Difficulty: 1700.0 Banger: 1484.0

complex numbers Argand diagram equilateral triangle roots of unity Napoleon's theorem geometric proof rotation cube roots of unity

Problem

Given that \(\alpha = \e^{\mathrm{i} \pi/3}\) , prove that \(1 + \alpha^2 = \alpha\). A triangle in the Argand plane has vertices \(A\), \(B\), and \(C\) represented by the complex numbers \(p\), \(q\alpha^2\) and \(- r\alpha\) respectively, where \(p\), \(q\) and \(r\) are positive real numbers. Sketch the triangle~\(ABC\). Three equilateral triangles \(ABL\), \(BCM\) and \(CAN\) (each lettered clockwise) are erected on sides \(AB\), \(BC\) and \(CA\) respectively. Show that the complex number representing \(N\) is \mbox{\(( 1 - \alpha) p- \alpha^2 r\)} and find similar expressions for the complex numbers representing \(L\) and \(M\). Show that lines \(LC\), \(MA\) and \(NB\) all meet at the origin, and that these three line segments have the common length \(p+q+r\).

No solution available for this problem.

Rating Information

Difficulty Rating: 1700.0

Difficulty Comparisons: 0

Banger Rating: 1484.0

Banger Comparisons: 1

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Problem source

Given that $\alpha = \e^{\mathrm{i} \pi/3}$ ,  
prove that  $1 + \alpha^2 = \alpha$. 
 
A triangle in the Argand plane has vertices $A$, $B$, and $C$ 
 represented by the complex numbers  
$p$, $q\alpha^2$ and $- r\alpha$   
respectively, where  $p$, $q$ and $r$  
are positive real numbers. Sketch the triangle~$ABC$. 
 
Three equilateral triangles $ABL$, $BCM$ and $CAN$ 
(each lettered clockwise) 
 are erected on sides $AB$, $BC$ and $CA$ respectively.  
Show that the complex number representing $N$ is  
\mbox{$( 1 - \alpha) p- \alpha^2 r$}   
and find similar expressions for the  
complex numbers representing $L$ and $M$. 
 
Show that lines $LC$, $MA$ and $NB$ all meet at the origin,  
and that these three line segments have the  
common length  $p+q+r$.

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