Hard +2.3 This AEA question requires multiple sophisticated steps: proving a geometric relationship between radii and distance, deriving a complex area formula involving sectors and triangles with trigonometric substitution, then applying these results to a geometric sequence with specific constraints. The multi-part structure, need for geometric insight, algebraic manipulation of the area formula, and series application place it well above typical A-level questions but not at the extreme upper bound of AEA difficulty.
\end{figure}
Figure 4 shows a circle with radius \(r _ { 1 }\) and a circle with radius \(r _ { 2 }\)
The circles touch externally at a single point above the \(x\)-axis.
Both circles also have the \(x\)-axis as a tangent.
Show that the horizontal distance between the centres of the circles, \(d\), is given by
$$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$
The finite region \(R\), shown shaded in Figure 4, is bounded by the \(x\)-axis and minor arcs of the two circles.
Given that \(r _ { 1 } \geqslant r _ { 2 }\)
show that the area of \(R\) is given by
$$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$
where \(\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }\)
Question 7 continues on the next page.
\end{figure}
A sequence of circles, \(C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots\) with radii \(r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots\) respectively, is constructed such that
each circle is tangential to and above the \(x\)-axis
the first circle, \(C _ { 1 }\), has centre \(( 0,1 )\)
each successive circle touches the preceding one externally at a single point
the horizontal distances between the centres of successive circles form a geometric sequence with first term 2 and common ratio \(\frac { 1 } { \sqrt { 3 } }\)
The first few circles in the sequence are shown in Figure 5.
Determine the value of \(r _ { 3 }\)
Show that, for \(n \geqslant 1 , r _ { n + 2 } = k r _ { n }\) where \(k\) is a constant to be determined.
Hence show that, for \(n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }\)
The region bounded between \(C _ { n } , C _ { n + 1 }\) and the \(x\)-axis is \(R _ { n }\)
The total area, \(A\), bounded above the \(x\)-axis and under all the circles is the sum of the areas of all these regions.
Determine the value of \(A\), giving the answer in simplest form.
\section*{Paper reference}
\section*{Advanced Extension Award Mathematics}
Insert for questions 5, 6 and 7
Do not write on this insert.
5.
\begin{figure}[h]
\end{figure}
Figure 2 shows a sketch of a hexagon \(O A B C D E\) where
Given that \(\overrightarrow { O A } = \mathbf { a }\) and \(\overrightarrow { O E } = \mathbf { e }\)
determine as simplified expressions in terms of \(\mathbf { a }\) and \(\mathbf { e }\)
\(\overrightarrow { A B }\)
\(\overrightarrow { O D }\)
The point \(R\) divides \(A B\) internally in the ratio \(1 : 2\)
Determine \(\overrightarrow { R C }\) as a simplified expression in terms of \(\mathbf { a }\) and \(\mathbf { e }\)
The line through the points \(R\) and \(C\) meets the line through the points \(O\) and \(D\) at the point \(X\).
Determine \(\overrightarrow { O X }\) in the form \(\lambda \mathbf { a } + \mu \mathbf { e }\), where \(\lambda\) and \(\mu\) are real values in simplest form.
6.
\begin{figure}[h]
\end{figure}
Figure 3 shows a block \(A\) with mass \(4 m\) and a block \(B\) with mass \(5 m\).
Block \(A\) is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal.
Block \(B\) is at rest on a rough plane inclined at an angle \(\beta\) to the horizontal.
The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane.
A small smooth ring \(C\), of mass \(8 m\), is threaded on the string between the pulleys so that \(A , B\) and \(C\) all lie in the same vertical plane.
The part of the string between \(A\) and its pulley lies along a line of greatest slope of the plane of angle \(\alpha\).
The part of the string between \(B\) and its pulley lies along a line of greatest slope of the plane of angle \(\beta\).
The angle between the vertical and the string between each pulley and the ring \(C\) is \(\gamma\).
The two blocks, \(A\) and \(B\), are modelled as particles.
Given that
\end{figure}
Figure 4 shows a circle with radius \(r _ { 1 }\) and a circle with radius \(r _ { 2 }\)
The circles touch externally at a single point above the \(x\)-axis.
Both circles also have the \(x\)-axis as a tangent.
Show that the horizontal distance between the centres of the circles, \(d\), is given by
$$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$
The finite region \(R\), shown shaded in Figure 4, is bounded by the \(x\)-axis and minor arcs of the two circles.
Given that \(r _ { 1 } \geqslant r _ { 2 }\)
show that the area of \(R\) is given by
$$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$
where \(\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }\)
Question 7 continues on the next page.
\begin{figure}[h]
\end{figure}
A sequence of circles, \(C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots\) with radii \(r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots\) respectively, is constructed such that
The first few circles in the sequence are shown in Figure 5.
Determine the value of \(r _ { 3 }\)
Show that, for \(n \geqslant 1 , r _ { n + 2 } = k r _ { n }\) where \(k\) is a constant to be determined.
Hence show that, for \(n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }\)
The region bounded between \(C _ { n } , C _ { n + 1 }\) and the \(x\)-axis is \(R _ { n }\)
The total area, \(A\), bounded above the \(x\)-axis and under all the circles is the sum of the areas of all these regions.
Determine the value of \(A\), giving the answer in simplest form.
7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-26_725_1773_242_146}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Figure 4 shows a circle with radius $r _ { 1 }$ and a circle with radius $r _ { 2 }$\\
The circles touch externally at a single point above the $x$-axis.\\
Both circles also have the $x$-axis as a tangent.
\begin{enumerate}[label=(\alph*)]
\item Show that the horizontal distance between the centres of the circles, $d$, is given by
$$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$
The finite region $R$, shown shaded in Figure 4, is bounded by the $x$-axis and minor arcs of the two circles.
Given that $r _ { 1 } \geqslant r _ { 2 }$
\item show that the area of $R$ is given by
$$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$
where $\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }$
Question 7 continues on the next page.\\
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-27_2269_53_306_36}
\end{center}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-27_759_1378_269_347}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{center}
\end{figure}
A sequence of circles, $C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots$ with radii $r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots$ respectively, is constructed such that
\begin{itemize}
\item each circle is tangential to and above the $x$-axis
\item the first circle, $C _ { 1 }$, has centre $( 0,1 )$
\item each successive circle touches the preceding one externally at a single point
\item the horizontal distances between the centres of successive circles form a geometric sequence with first term 2 and common ratio $\frac { 1 } { \sqrt { 3 } }$
\end{itemize}
The first few circles in the sequence are shown in Figure 5.
\item \begin{enumerate}[label=(\roman*)]
\item Determine the value of $r _ { 3 }$
\item Show that, for $n \geqslant 1 , r _ { n + 2 } = k r _ { n }$ where $k$ is a constant to be determined.
\item Hence show that, for $n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }$
The region bounded between $C _ { n } , C _ { n + 1 }$ and the $x$-axis is $R _ { n }$\\
The total area, $A$, bounded above the $x$-axis and under all the circles is the sum of the areas of all these regions.
\end{enumerate}\item Determine the value of $A$, giving the answer in simplest form.
\section*{Paper reference}
\section*{Advanced Extension Award Mathematics}
Insert for questions 5, 6 and 7\\
Do not write on this insert.\\
5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-34_298_1040_212_516}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of a hexagon $O A B C D E$ where
\begin{itemize}
\item the interior angle at $O$ and at $C$ are each $60 ^ { \circ }$
\item the interior angle at each of the other vertices is $150 ^ { \circ }$
\item $O A = O E = B C = C D$
\item $A B = E D = 3 \times O A$
\end{itemize}
Given that $\overrightarrow { O A } = \mathbf { a }$ and $\overrightarrow { O E } = \mathbf { e }$\\
determine as simplified expressions in terms of $\mathbf { a }$ and $\mathbf { e }$\\
(a) $\overrightarrow { A B }$\\
(b) $\overrightarrow { O D }$
The point $R$ divides $A B$ internally in the ratio $1 : 2$\\
(c) Determine $\overrightarrow { R C }$ as a simplified expression in terms of $\mathbf { a }$ and $\mathbf { e }$
The line through the points $R$ and $C$ meets the line through the points $O$ and $D$ at the point $X$.\\
(d) Determine $\overrightarrow { O X }$ in the form $\lambda \mathbf { a } + \mu \mathbf { e }$, where $\lambda$ and $\mu$ are real values in simplest form.\\
6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-35_236_1363_205_351}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a block $A$ with mass $4 m$ and a block $B$ with mass $5 m$.\\
Block $A$ is at rest on a rough plane inclined at an angle $\alpha$ to the horizontal.\\
Block $B$ is at rest on a rough plane inclined at an angle $\beta$ to the horizontal.\\
The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane.
A small smooth ring $C$, of mass $8 m$, is threaded on the string between the pulleys so that $A , B$ and $C$ all lie in the same vertical plane.
The part of the string between $A$ and its pulley lies along a line of greatest slope of the plane of angle $\alpha$.
The part of the string between $B$ and its pulley lies along a line of greatest slope of the plane of angle $\beta$.
The angle between the vertical and the string between each pulley and the ring $C$ is $\gamma$.\\
The two blocks, $A$ and $B$, are modelled as particles.\\
Given that
\begin{itemize}
\item $\tan \alpha = \frac { 5 } { 12 }$ and $\tan \beta = \frac { 7 } { 24 }$ and $\tan \gamma = \frac { 3 } { 4 }$
\item the coefficient of friction, $\mu$, is the same between each block and its plane
\item one of the blocks is on the point of sliding up its plane
\item the tension in the string is $T$\\
(a) determine, in terms of $m$ and $g$, an expression for $T$,\\
(b) draw a diagram showing the forces on block $A$, clearly labelling each of the forces acting on the block,\\
(c) determine the value of $\mu$, giving a justification for your answer.
\end{itemize}
7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-36_721_1771_205_146}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}
Figure 4 shows a circle with radius $r _ { 1 }$ and a circle with radius $r _ { 2 }$\\
The circles touch externally at a single point above the $x$-axis.\\
Both circles also have the $x$-axis as a tangent.\\
(a) Show that the horizontal distance between the centres of the circles, $d$, is given by
$$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$
The finite region $R$, shown shaded in Figure 4, is bounded by the $x$-axis and minor arcs of the two circles.
Given that $r _ { 1 } \geqslant r _ { 2 }$\\
(b) show that the area of $R$ is given by
$$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$
where $\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }$
Question 7 continues on the next page.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-37_761_1376_210_349}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{center}
\end{figure}
A sequence of circles, $C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots$ with radii $r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots$ respectively, is constructed such that
\begin{itemize}
\item each circle is tangential to and above the $x$-axis
\item the first circle, $C _ { 1 }$, has centre $( 0,1 )$
\item each successive circle touches the preceding one externally at a single point
\item the horizontal distances between the centres of successive circles form a geometric sequence with first term 2 and common ratio $\frac { 1 } { \sqrt { 3 } }$
\end{itemize}
The first few circles in the sequence are shown in Figure 5.\\
(c) (i) Determine the value of $r _ { 3 }$\\
(ii) Show that, for $n \geqslant 1 , r _ { n + 2 } = k r _ { n }$ where $k$ is a constant to be determined.\\
(iii) Hence show that, for $n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }$
The region bounded between $C _ { n } , C _ { n + 1 }$ and the $x$-axis is $R _ { n }$\\
The total area, $A$, bounded above the $x$-axis and under all the circles is the sum of the areas of all these regions.\\
(d) Determine the value of $A$, giving the answer in simplest form.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2024 Q7 [24]}}