Year: 2008
Paper: 3
Question Number: 1
Course: UFM Pure
Section: Roots of polynomials
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: 1516.0
Banger Comparisons: 1
Find all values of $a$, $b$, $x$ and $y$ that satisfy the simultaneous equations
\begin{alignat*}{3}
a&+b & &=1 &\\
ax&+by & &= \tfrac13& \\
ax^2&+by^2& &=\tfrac15& \\
ax^3 &+by^3& &=\tfrac17\,.&
\end{alignat*}
\noindent{\bf [} {\bf Hint}: you may wish to start by multiplying the second equation by $x+y$. {\bf ]}This is a second order recurrence relation, so we need to find $m$ and $n$ such that;
\begin{align*}
&&\frac15 &= m\frac13 + n \\
&&\frac17 &= m \frac15 + n\frac13 \\
\Rightarrow && m,n &= \frac67, - \frac{3}{35}
\end{align*}
So we now need to solve the characteristic equation:
$\lambda^2 - \frac67 \lambda + \frac{3}{35} = 0$
So $x,y = \frac{15 \pm 2 \sqrt{30}}{35}$.
We need,
\begin{align*}
&& 1 &= a+ b \\
&& \frac13 &= a \frac{15 + 2 \sqrt{30}}{35} + b \frac{15 - 2 \sqrt{30}}{35} \\
&& \frac13 &= \frac{15}{35} + \frac{2 \sqrt{30}}{35}(a-b) \\
\Rightarrow && -\frac{\sqrt{30}}{18} &= a-b \\
\Rightarrow && a &= \frac{18-\sqrt{30}}{36} \\
&& b &= \frac{18+\sqrt{30}}{38}
\end{align*}
So our two answers are:
\[ (a,b,x,y) = \left (\frac{18\pm\sqrt{30}}{36} ,\frac{18\mp\sqrt{30}}{36},\frac{15 \pm 2 \sqrt{30}}{35},\frac{15 \mp 2 \sqrt{30}}{35}, \right)\]
This was the most popular question on the paper, and many earned good marks on it. Nearly all the candidates followed the hint, and most then applied the same trick with the third equation. Subsequent success depended on a candidate realising that they had simultaneous equations in xy and x + y, although very rarely some managed to solve directly in x and y.