Year: 2014
Paper: 2
Question Number: 1
Course: LFM Pure
Section: Introduction to trig
No solution available for this problem.
There were good solutions presented to all of the questions, although there was generally less success in those questions that required explanations of results or the use of diagrams and graphs to reach the solution. Algebraic manipulation was generally well done by many of the candidates although a range of common errors such as confusing differentiation and integration and simple arithmetic slips were evident. Candidates should also be advised to use the methods that are asked for in questions unless it is clear that other methods will be accepted (such as by the use of the phrase "or otherwise").
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
In the triangle $ABC$, the base $AB$ is of length 1 unit and the
angles at~$A$ and~$B$ are $\alpha$ and~$\beta$ respectively, where
$0<\alpha\le\beta$. The points $P$ and~$Q$ lie on the sides $AC$ and
$BC$ respectively, with $AP=PQ=QB=x$. The line $PQ$ makes an angle
of~$\theta$ with the line through~$P$ parallel to~$AB$.
\begin{questionparts}
\item Show that $x\cos\theta = 1- x\cos\alpha - x\cos\beta$, and
obtain an expression for $x\sin\theta$ in
terms of $x$, $\alpha$ and~$\beta$. Hence show that
\begin{equation}
\label{eq:2*}
\bigl(1+2\cos(\alpha+\beta)\bigr)x^2 - 2(\cos\alpha +
\cos\beta)x + 1 = 0\,. \tag{$*$}
\end{equation}
Show that $(*)$ is also satisfied if $P$ and $Q$ lie on
$AC$ produced and $BC$ produced, respectively. [By definition,
$P$ lies on $AC$ produced if $P$ lies on the line through $A$ and~$C$
and the points are in the order $A$, $C$, $P$\,.]
\item
State the condition on $\alpha$ and $\beta$ for
$(*)$
to be linear in $x$.
If this condition does not hold (but the condition
$0<\alpha \le \beta$ still holds), show that
$(*)$ has
distinct real roots.
\item Find the possible values of~$x$ in the two cases (a)
$\alpha = \beta = 45^\circ$ and (b)
$\alpha = 30^\circ$, $\beta = 90^\circ$, and
illustrate each case with a sketch.
\end{questionparts}
While the first part of the question was successfully completed by many of the candidates, there were quite a few diagrams drawn showing the point P further from the line AB than Q. Those who established the expression for cosθ were usually able to find an expression for sinθ and good justifications of the quadratic equation were given. The case where P and Q lie on the lines AC produced and BC produced caused a lot of difficulty for many of the candidates, many of whom tried unsuccessfully to create an argument based on similar triangles. The condition for (*) to be linear in ℓ did not cause much difficulty, although a number of candidates did not give the value of cosθ. Many candidates realised that the justification that the roots were distinct would involve the discriminant, although some solutions included the case where the discriminant could be equal to 0 were produced. However, very few solutions were able to give a clear justification that the discriminant must be greater than 0. In the final part some candidates sketched the graph of the quadratic rather than sketching the triangle in the two cases given. In the second case many candidates did not realise that Q was at the same point as C.