2001 Paper 2 Q7

Year: 2001
Paper: 2
Question Number: 7

Course: LFM Stats And Pure
Section: Complex Numbers (L8th)

Difficulty: 1600.0 Banger: 1470.8

complex numbers Argand diagram regular polygon rotation circumcentre geometric construction multiplication by cis

Problem

In an Argand diagram, $O$ is the origin and $P$ is the point $2+0\mathrm{i}$. The points $Q$, $R$ and $S$ are such that the lengths $OP$, $PQ$, $QR$ and $RS$ are all equal, and the angles $OPQ$, $PQR$ and $QRS$ are all equal to ${5{\pi}}/6$, so that the points $O$, $P$, $Q$, $R$ and $S$ are five vertices of a regular 12-sided polygon lying in the upper half of the Argand diagram. Show that $Q$ is the point $2 + \sqrt 3 + \mathrm{i}$ and find $S$. The point $C$ is the centre of the circle that passes through the points $O$, $P$ and $Q$. Show that, if the polygon is rotated anticlockwise about $O$ until $C$ first lies on the real axis, the new position of $S$ is $$ - \tfrac{1}{2} (3\sqrt 2+ \sqrt6)(\sqrt3-\mathrm{i})\;. $$

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1470.8

Banger Comparisons: 2

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Problem source

In an Argand diagram, $O$ is the origin and $P$ is the point 
$2+0\mathrm{i}$. The points  $Q$, $R$ and  $S$ are such that
the lengths  $OP$, $PQ$, $QR$ and $RS$ are all equal, and 
the angles $OPQ$, $PQR$ and $QRS$ are all equal to
${5{\pi}}/6$, so that the points $O$, $P$, $Q$, $R$ and $S$ are five vertices of 
a regular 12-sided polygon lying in the upper half of the Argand diagram. 
Show that $Q$ is the point 
$2 + \sqrt 3 + \mathrm{i}$ and find $S$.
The point  $C$ is 
 the centre of the circle that  passes through the points
$O$, $P$ and $Q$. Show that, if the polygon
is  rotated anticlockwise about $O$
until  $C$ first lies on the real axis,  the 
new position of $S$ is
$$
- \tfrac{1}{2}  (3\sqrt 2+ \sqrt6)(\sqrt3-\mathrm{i})\;. 
$$

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