Year: 2022
Paper: 2
Question Number: 5
Course: LFM Pure
Section: Coordinate Geometry
No solution available for this problem.
Candidates appeared to be generally well prepared for most topics within the examination, but there were a few situations in questions where some did not appear to be as proficient in standard techniques as needed. In particular, the method for finding invariant lines required in question 8 and the manipulation of trigonometric functions that were needed in question 10 caused considerable difficulties for some candidates. An additional issue that occurred at numerous points in the paper relates to the direction in which a deduction is required. It is important that candidates make sure that they know which statement is the one that they should start from as they deduce the other and that it is clear in their solution that the logic has gone in the correct direction. Clarity of solution is also an issue that candidates should be aware of, especially in the situations where the result to be reached has been given. It is important to check that there are no special cases that need to be considered separately, and when dividing by a function it is necessary to confirm that the function cannot be equal to 0 (and in the case of inequalities that the function always has the same sign). When drawing diagrams and sketching graphs it is useful if significant points that need to be clear are not drawn over the lines on the page as these can be difficult to interpret during the marking process.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item Given that $a > b > c > 0$ are constants, and that $x$, $y$, $z$ are non-negative variables, show that
\[ax + by + cz \leqslant a(x + y + z).\]
\end{questionparts}
In the acute-angled triangle $ABC$, $a$, $b$ and $c$ are the lengths of sides $BC$, $CA$ and $AB$, respectively, with $a > b > c$. $P$ is a point inside, or on the sides of, the triangle, and $x$, $y$ and $z$ are the perpendicular distances from $P$ to $BC$, $CA$ and $AB$, respectively. The area of the triangle is $\Delta$.
\begin{questionparts}\setcounter{enumi}{1}
\item
\begin{enumerate}
\item Find $\Delta$ in terms of $a$, $b$, $c$, $x$, $y$ and $z$.
\item Find both the minimum value of the sum of the perpendicular distances from $P$ to the three sides of the triangle and the values of $x$, $y$ and $z$ which give this minimum sum, expressing your answers in terms of some or all of $a$, $b$, $c$ and $\Delta$.
\end{enumerate}
\item
\begin{enumerate}
\item Show that, for all real $a$, $b$, $c$, $x$, $y$ and $z$,
\[(a^2+b^2+c^2)(x^2+y^2+z^2) = (bx-ay)^2 + (cy-bz)^2 + (az-cx)^2 + (ax+by+cz)^2.\]
\item Find both the minimum value of the sum of the squares of the perpendicular distances from $P$ to the three sides of the triangle and the values of $x$, $y$ and $z$ which give this minimum sum, expressing your answers in terms of some or all of $a$, $b$, $c$ and $\Delta$.
\end{enumerate}
\item Find both the maximum value of the sum of the squares of the perpendicular distances from $P$ to the three sides of the triangle and the values of $x$, $y$ and $z$ which give this maximum sum, expressing your answers in terms of some or all of $a$, $b$, $c$ and $\Delta$.
\end{questionparts}
Many candidates appear to have spent a little time on this question before deciding to concentrate on others, meaning that the marks in general for this question were very low. In part (i) it was often unclear whether the candidate was using all of the conditions and in many cases the inequalities were written as strict when they should not have been. In part (ii) most candidates were able to write down a correct formula for the area and part (b) was also generally answered well, although in some cases the candidates appeared to guess the best point, rather than deduce it from the previous part. The incentre was a common incorrect guess for at least one of the parts. Candidates were generally most successful in part (iii)(a) with many convincing attempts seen. Fewer than half of the scripts progressed beyond this point, however. Those who did attempt part (iii)(b) managed to complete it quite well, although a common mistake was to simply square the answer from part (ii)(b) and claim that it all works out. There were very few attempts at part (iv), mostly trying to use previous parts or to guess a point. For both of the last two parts, some candidates correctly identified the maximum or minimum, but did not convincingly show where it was attained.