Year: 2010
Paper: 3
Question Number: 6
Course: LFM Pure
Section: Trigonometry 2
No solution available for this problem.
About 80% of candidates attempted at least five questions, and well less than 20% made genuine attempts at more than six. Those attempting more than six questions fell into three camps which were those weak candidates who made very little progress on any question, those with four or five fair solutions casting about for a sixth, and those strong candidates that either attempted 7th or even 8th questions as an "insurance policy" against a solution that seemed strong but wasn't, or else for entertainment!
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
The points $P$, $Q$ and $R$
lie on a sphere of unit radius centred at the origin, $O$,
which is fixed.
Initially, $P$ is at $P_0(1, 0, 0)$, $Q$ is at $Q_0(0, 1, 0)$
and $R$ is at
$R_0(0, 0, 1)$.
\begin{questionparts}
\item
The sphere is then rotated about the $z$-axis,
so that the line $OP$ turns directly
towards the positive
$y$-axis through an angle $\phi$. The position of $P$ after this
rotation is denoted by $P_1$.
Write down the coordinates of $P_1$.
\item
The sphere is now rotated about the line in the $x$-$y$ plane
perpendicular to $OP_1$, so that the line $OP$
turns directly towards the positive $z$-axis through an angle $\lambda$.
The position of $P$
after this rotation is denoted by $P_2$.
Find the coordinates of $P_2$.
Find also
the coordinates of the points $Q_2$ and $R_2$, which are
the positions of $Q$ and $R$ after
the two rotations.
\item
The sphere is now rotated for a third time,
so that $P$ returns from $P_2$ to its
original position~$P_0$. During the rotation, $P$ remains in the
plane containing $P_0$, $P_2$ and $O$.
Show that the angle of this
rotation, $\theta$, satisfies
\[
\cos\theta = \cos\phi \cos\lambda\,,
\]
and find a vector in the direction of the axis
about which this rotation takes place.
\end{questionparts}
About a tenth of the candidates attempted this, with less success than nearly all other questions on the paper. Part (i) caused few problems, but at some point in part (ii), errors were frequently made or lack of attention to which of the two angles in parts (i) and (ii) was being employed in which rotation, and so even those few that knew how to attempt part (iii) were thwarted.