Year: 2025
Paper: 3
Question Number: 5
Course: LFM Pure and Mechanics
Section: Vectors
No solution available for this problem.
The majority of candidates focused solely on the pure questions, with questions 1, 2 and 8 the most popular. The statistics questions were more popular than the mechanics questions in this exam series. Candidates who did well on this paper generally: were careful to explain and justify the steps in their arguments, explaining what they had done rather than expecting the examiner to infer what had been done from disjointed groups of calculations; paid close attention to what was required by the questions; made fewer unnecessary mistakes with calculations; thought carefully about how to present rigorous arguments involving trig functions and their inverse functions, especially in relation to domain considerations; understood that questions set on the STEP papers require sufficient justification to earn full credit; knew the difference between 'positive' and 'non-negative'; attempted all parts of a question, picking up marks for later parts even when they had not necessarily attempted or completed previous parts. Candidates who did less well on this paper generally: did not pay attention to 'Hence' instructions: this means that you must use the previous part; presented explanations that were not precise enough (e.g. in Question 3 describing the transformations but not in the context of the graphs or in Question 8 not explaining use of trigonometric relationships sufficiently well); made additional assumptions, e.g. that a function was differentiable when this had not been given; tried to present if and only if arguments in a single argument when dealing with each direction separately would have been more appropriate and safer (note that this is not always the case; in general candidates need to consider what is the most appropriate presentation of an if and only if argument); tried to carry out too many steps in one go, resulting in them not justifying the key steps sufficiently; did not take sufficient care with graphs/curve sketching.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
Three points, $A$, $B$ and $C$, lie in a horizontal plane, but are not collinear. The point $O$ lies above the plane.
Let $\overrightarrow{OA} = \mathbf{a}$, $\overrightarrow{OB} = \mathbf{b}$ and $\overrightarrow{OC} = \mathbf{c}$.
$P$ is a point with $\overrightarrow{OP} = \alpha\mathbf{a} + \beta\mathbf{b} + \gamma\mathbf{c}$, where $\alpha$, $\beta$ and $\gamma$ are all positive and $\alpha + \beta + \gamma < 1$.
Let $k = 1 - (\alpha + \beta + \gamma)$.
\begin{questionparts}
\item The point $L$ is on $OA$, the point $X$ is on $BC$ and $LX$ passes through $P$.
Determine $\overrightarrow{OX}$ in terms of $\beta$, $\gamma$, $\mathbf{b}$ and $\mathbf{c}$ and show that $\overrightarrow{OL} = \frac{\alpha}{k+\alpha}\mathbf{a}$.
\item Let $M$ and $Y$ be the unique pair of points on $OB$ and $CA$ respectively such that $MY$ passes through $P$, and let $N$ and $Z$ be the unique pair of points on $OC$ and $AB$ respectively such that $NZ$ passes through $P$.
Show that the plane $LMN$ is also horizontal if and only if $OP$ intersects plane $ABC$ at the point $G$, where $\overrightarrow{OG} = \frac{1}{3}(\mathbf{a} + \mathbf{b} + \mathbf{c})$. Where do points $X$, $Y$ and $Z$ lie in this case?
\item State what the condition $\alpha + \beta + \gamma < 1$ tells you about the position of $P$ relative to the tetrahedron $OABC$.
\end{questionparts}
This question was the least popular pure question by a large margin, and of the attempts made less than half were 'substantial' attempts. As is often the case with vector questions, a carefully drawn diagram can be very helpful in selecting an appropriate method for solving the question and the most successful candidates made good use of this. Various methods were used in part (i), but mostly these involved finding vector equations of relevant lines and manipulating these to show the required results. Candidates should be aware that questions on the STEP papers need enough justification to fully support their solutions. Many candidates lost accuracy marks through their argument not being convincing enough or lacking some details. Part (ii) had some very good solutions, but many candidates found it difficult to understand what it means for LMN to be horizontal. A clear diagram here would have helped candidates to find a solution method. Some candidates tried to do both directions of the 'if and only if' in one go. They usually did not gain full marks here, either because they did not link one pair of statements with an if and only if symbol or because they did not appreciate that one step needed a different approach for each direction of implication. It is always 'safer' to approach each direction of implication separately. The most common issues were not using k ≠ 0 when justifying α/(k+α) = β/(k+β) ⇒ α = β, or for not convincingly explaining why α = β = γ mean that LMN was horizontal. Part (iii) required a one-line answer, and some candidates who had taken the time to read the whole question successfully answered this part even if they had not answered the previous parts. Some candidates confused 'positive' with 'non-negative' and stated that point P could be inside or on the faces of the tetrahedron.