Year: 2009
Paper: 3
Question Number: 1
Course: LFM Stats And Pure
Section: Polynomials
No solution available for this problem.
The vast majority of candidates (in excess of 95%) attempted at least five questions, and nearly a quarter attempted more than six questions, though very few doing so achieved high scores (about 2%). Most attempting more than six questions were submitting fragmentary answers, which, as the rubric informed candidates, earned little credit.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1471.4
Banger Comparisons: 2
The points $S$, $T$, $U$ and $V$ have coordinates
$(s,ms)$, $(t,mt)$, $(u,nu)$ and $(v,nv)$, respectively.
The lines $SV$ and $UT$ meet the line $y=0$ at the points
with coordinates $(p,0)$ and $(q,0)$, respectively.
Show that
\[
p = \frac{(m-n)sv}{ms-nv}\,,
\]
and write down a similar expression for $q$.
Given that $S$ and $T$ lie on the
circle $x^2 + (y-c)^2 = r^2$, find a quadratic equation
satisfied by $s$ and by $t$, and hence determine $st$ and $s+t$ in
terms of $m$, $c$ and $r$.
Given that $S$, $T$, $U$ and $V$ lie on the above circle, show that
$p+q=0$.
A popular question attempted by more than four fifths of the candidates, and scoring as well as any question, and most successfully obtained expressions for p and q. Quite a lot also obtained the quadratic equation and from it the sum and product of roots for s and t. However, a common error at this stage was to overlook the coefficient of the second degree term not being 1. For this reason, or otherwise, because they didn't know what to do many "fell at the last hurdle", although a good number completed the question successfully.