Year: 2008
Paper: 3
Question Number: 7
Course: LFM Stats And Pure
Section: Complex Numbers (L8th)
No solution available for this problem.
Most candidates attempted five, six or seven questions, and scored the majority of their total score on their best three or four. Those attempting seven or more tended not to do well, pursuing no single solution far enough to earn substantial marks.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
The points $A$, $B$ and $C$
in the Argand diagram are the vertices of an equilateral triangle
described
anticlockwise.
Show that the complex numbers
$a$, $b$ and $c$ representing $A$, $B$ and $C$
satisfy \[2c= (a+b) +\mathrm{i}\sqrt3(b-a).\]
Find a similar relation in the case that
$A$, $B$ and $C$
are the vertices of an equilateral triangle
described
clockwise.
\begin{questionparts}
\item The quadrilateral $DEFG$ lies in the Argand diagram. Show that
points $P$, $Q$, $R$ and $S$ can be chosen so that
$PDE$, $QEF$, $RFG$ and $SGD$ are equilateral triangles and $PQRS$ is
a parallelogram.
\item The triangle $LMN$ lies in the Argand diagram.
Show that the centroids $U$, $V$ and $W$ of the
equilateral
triangles drawn externally on the sides of $LMN$ are the vertices
of an equilateral triangle.
\noindent
[{\bf Note:} The {\em centroid} of a triangle with vertices
represented by the complex numbers $x$,~$y$ and~$z$ is the point
represented by $\frac13(x+y+z)\,$.]
\end{questionparts}
Less than a fifth attempted this and frequently with little success except for obtaining the initial result. The configuration for part (i) tripped up many, although some skipped that to do part (ii) successfully.