| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Session | Specimen |
| Marks | 7 |
| Topic | Circles |
| Type | Area of region bounded by circle and line |
| Difficulty | Standard +0.3 Part (i) requires finding a segment area using integration or the standard formula (angle - sin), which is a routine C3/C4 technique. Part (ii) involves applying geometric transformations to a line, requiring careful angle work but following standard procedures. Both parts are straightforward applications of A-level techniques with no novel insight required, making this slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) The angle subtended at the centre by the chord is \(2 \cos^{-1}\left(\frac{4}{5}\right) = 2.2143\) | B1 | Area of minor segment is area of minor sector – area of triangle |
| (ii) After the rotation, the equation of the line has the form \(x + y = c\) | B1 | By considering a right-angled isosceles triangle, \(c = \sqrt{2}\) |
(i) The angle subtended at the centre by the chord is $2 \cos^{-1}\left(\frac{4}{5}\right) = 2.2143$ | B1 | Area of minor segment is area of minor sector – area of triangle | M1 | $\frac{1}{2} \times (\sqrt{5})^2 \times 2.2143 - \frac{1}{2} \times (\sqrt{5})^2 \times \sin 2.2143 = 3.54$ | A1, A1 | 4 marks
(ii) After the rotation, the equation of the line has the form $x + y = c$ | B1 | By considering a right-angled isosceles triangle, $c = \sqrt{2}$ | B1 | Final equation is $x - y = \sqrt{2}$ | B1 | 3 marksA circle, of radius $\sqrt{5}$ and centre the origin $O$, is divided into two segments by the line $y = 1$.
\begin{enumerate}[label=(\roman*)]
\item Determine the area of the smaller segment. [4]
\end{enumerate}
The line is rotated clockwise about $O$ through $45^{\circ}$ and then reflected in the $x$-axis.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the equation of the line in its final position. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 Q4 [7]}}