Year: 2024
Paper: 3
Question Number: 4
Course: LFM Pure
Section: Coordinate Geometry
No solution available for this problem.
The total entry was an increase on that of 2023 by more than 10%. One question was attempted by more than 98% of candidates, another two by about 80%, and another five by between 50% and 70%. The remaining four questions were attempted by between 5% and 30% of candidates, these being from Section B: Mechanics, and Section C: Probability and Statistics, though the Statistics questions were in general attempted more often and more successfully. All questions were perfectly solved by some candidates. About 84% of candidates attempted no more than 7 questions.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item Show that if the acute angle between straight lines with gradients $m_1$ and $m_2$ is $45^\circ$, then
\[\frac{m_1 - m_2}{1 + m_1 m_2} = \pm 1.\]
\end{questionparts}
The curve $C$ has equation $4ay = x^2$ (where $a \neq 0$).
\begin{questionparts}
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\item If $p \neq q$, show that the tangents to the curve $C$ at the points with $x$-coordinates $p$ and $q$ meet at a point with $x$-coordinate $\frac{1}{2}(p+q)$. Find the $y$-coordinate of this point in terms of $p$ and $q$.
Show further that any two tangents to the curve $C$ which are at $45^\circ$ to each other meet on the curve $(y+3a)^2 = 8a^2 + x^2$.
\item Show that the acute angle between any two tangents to the curve $C$ which meet on the curve $(y+7a)^2 = 48a^2 + 3x^2$ is constant. Find this acute angle.
\end{questionparts}
The fourth most popular question, it was the third most successful, with a mean score of 10 marks. Part (i) needed more thoroughness than many attempts displayed. Most sensibly chose to express the gradients as tangents of angles of the lines to the x-axis, but then did not define these or consider the possible cases that could arise such as which was greater, or state that the difference between the angles is ±45° or 45°/135°. As the result was given in the question, there was an expectation that there should be complete justification. In part (ii), most attempts at the coordinates of the point of intersection were successful, though many did not use the non-equality of p and q, and a large number got the y coordinate wrong through substituting x into the equation of the parabola. Overall, many did well with the final result of this part, employing the various results from earlier in the part and that of (i). Part (iii) proved challenging for most, and there was a fair amount of guesswork based on the knowledge that 30°, 45° and 60° are angles with nice trigonometric values!