Year: 2025
Paper: 3
Question Number: 3
Course: LFM Pure
Section: Coordinate Geometry
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
Let $f(x)$ be defined and positive for $x > 0$.
Let $a$ and $b$ be real numbers with $0 < a < b$ and define the points $A = (a, f(a))$ and $B = (b, -f(b))$.
Let $X = (m,0)$ be the point of intersection of line $AB$ with the $x$-axis.
\begin{questionparts}
\item Find an expression for $m$ in terms of $a$, $b$, $f(a)$ and $f(b)$.
\item Show that, if $f(x) = \sqrt{x}$, then $m = \sqrt{ab}$.
Find, in terms of $n$, $a$ function $f(x)$ such that $m = \frac{a^{n+1} + b^{n+1}}{a^n + b^n}$.
\item Let $g_1(x)$ and $g_2(x)$ be defined and positive for $x > 0$. Let $m = M_1$ when $f(x) = g_1(x)$ and let $m = M_2$ when $f(x) = g_2(x)$.
Show that if $\frac{g_1(x)}{g_2(x)}$ is a decreasing function then $M_1 > M_2$.
Hence show that
$$\frac{a+b}{2} > \sqrt{ab} > \frac{2ab}{a+b}$$
\item Let $p$ and $c$ be chosen so that the curve $y = p(c-x)^3$ passes through both $A$ and $B$. Show that
$$\frac{c-a}{b-c} = \left(\frac{f(a)}{f(b)}\right)^{1/3}$$
and hence determine $c$ in terms of $a$, $b$, $f(a)$ and $f(b)$.
Show that if $f$ is a decreasing function, then $c < m$.
\end{questionparts}\begin{questionparts}
\item The line $AB$ has equation:
\begin{align*}
&& \frac{y+f(b)}{x-b} &= \frac{f(a)+f(b)}{a-b} \\
\Rightarrow && \frac{f(b)}{m-b} &= \frac{f(a)+f(b)}{a-b} \\
\Rightarrow && m &= \frac{a-b}{f(a)+f(b)}f(b) + b \\
&&&= \frac{af(b)+bf(a)}{f(a)+f(b)}
\end{align*}
\item Suppose $f(x) = \sqrt{x}$ then
\begin{align*}
m &= \frac{a\sqrt{b}+b\sqrt{a}}{\sqrt{a}+\sqrt{b}} \\
&= \frac{\sqrt{ab}(\sqrt{a}+\sqrt{b})}{\sqrt{a}+\sqrt{b}} \\
&= \sqrt{ab}
\end{align*}
Suppose $f(x) = x^{-n}$ then
\begin{align*}
m &= \frac{a b^{-n}+ba^{-n}}{a^{-n}+b^{-n}} \\
&= \frac{a^{n+1}+b^{n+1}}{b^n + a^n} \\
\end{align*}
\item Without loss of generality, we can scale $g_1(x)$ and $g_2(x)$ so that $g_1(a) = g_2(a)$ and $m$ won't change for either of them. Then since $\frac{g_1(b)}{g_2(b)} < 1$ (this function is decreasing) our line connecting $(a,g_i(a))$ and $(b,-g_i(b))$ must interect the axis first for $g_2$, in particular $M_1 > M_2$.
Suppose $g_1(x) =1, g_2(x) = \sqrt{x}, g_3(x) = x^{-1}$, the notice that $\frac{g_1(x)}{g_2(x)} =\frac{g_2(x)}{g_3(x)}= x^{-1/2}$ are decreasing, therefore:
\begin{align*}
\frac{a+b}{1+1} &> \sqrt{ab} > \frac{1+1}{a^{-1}+b^{-1}} \\
\frac{a+b}{2} &> \sqrt{ab} > \frac{2ab}{a+b} \\
\end{align*}
\item We must have:
\begin{align*}
&& p(c-a)^3 &= f(a) \\
&& p(c-b)^3 &= -f(b) \\
\Rightarrow &&\left ( \frac{c-a}{c-b} \right)^3 &= -\frac{f(a)}{f(b)} \\
\Rightarrow && \frac{c-a}{b-c} &= \left (\frac{f(a)}{f(b)} \right)^{\tfrac13} \\
\Rightarrow && c-a &= \left (\frac{f(a)}{f(b)} \right)^{\tfrac13}(b-c)\\
\Rightarrow && c \left (1 + \left (\frac{f(a)}{f(b)} \right)^{\tfrac13} \right) &= \left (\frac{f(a)}{f(b)} \right)^{\tfrac13}b + a \\
\Rightarrow && c &= \frac{\left (\frac{f(a)}{f(b)} \right)^{\tfrac13}b + a}{1 + \left (\frac{f(a)}{f(b)} \right)^{\tfrac13}} \\
&&&= \frac{b[f(a)]^\tfrac13+a[f(b)]^\tfrac13}{[f(a)]^\tfrac13+[f(b)]^\tfrac13}
\end{align*}
We have that $\frac{c-a}{b-c} = \left (\frac{f(a)}{f(b)} \right)^{\tfrac13} $ and $\frac{m-a}{b-c} = \frac{f(a)}{f(b)}$. Since $f$ is decreasing, $\frac{f(a)}{f(b)} > 1$ and so $\left (\frac{f(a)}{f(b)} \right)^{\tfrac13} < \frac{f(a)}{f(b)}$, therefore $m > c$.
\end{questionparts}
Part (i) was generally well done. However, some candidates simply stated the result without showing sufficient working and full credit could not be given. Sign errors were another common pitfall and usually meant that the accuracy mark could not be awarded. Part (ii) was frequently attempted, though for many candidates it marked the end of their attempt. A common mistake was overlooking the fact that the function f(x) was defined to be positive for x > 0, leading to marks being unavailable. Another frequent issue was providing insufficient justification – some candidates simply stated a function without explaining their reasoning or showing it had the required properties, which prevented them from earning full marks. Part (iii) was less frequently attempted, especially the first subpart. Some candidates assumed that the given functions were differentiable and attempted to provide arguments involving derivatives which did not gain credit. The second subpart was generally handled better, though again, a lack of justification was common. A few candidates also attempted alternative methods not involving the previous part, thus ignoring 'hence' in the question. Part (iv) was relatively well done by those who attempted it. The first subpart was accessible to most candidates. The second subpart was more challenging and required careful attention, especially when working with inequalities and avoiding unwarranted assumptions of equality.