| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Session | Specimen |
| Marks | 9 |
| Topic | Circles |
| Type | Circle equation from centre and radius |
| Difficulty | Standard +0.3 Part (i) is straightforward distance formula and circle equation. Part (ii)(a) requires substituting y=2x into the circle equation and solving a quadratic. Part (ii)(b) needs finding the angle subtended and using arc length formula. Multi-step but all standard techniques with no novel insight required, slightly above average due to the arc length component. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
**(i)** Radius is $\sqrt{(3-0)^2 + (1-5)^2} = 5$. B1
Equation of the circle $S$ is $(x-3)^2 + (y-1)^2 = 25$ or equivalent. B1 **[2]**
**(ii)(a)** Attempt at substitution of $x = k$ and $y = 2k$ (or $y = 2x$) into the equation for $S$: M1
$(k-3)^2 + (2k-1)^2 = 25$. A1
Expand, collect terms (and simplify):
$5k^2 - 10k - 15 = 0 \quad (k^2 - 2k - 3 = 0)$ M1
Solving: Obtain $k = -1$ and $3$
Obtain coordinates of the points $(-1, -2)$ and $(3, 6)$. A1 **[4]**
**(ii)(b)** Attempt to find the angle between the relevant radii: M1
Obtain $\frac{\pi}{2} + \tan^{-1}\left(\frac{3}{4}\right) = (2.214\ldots)$ A1
Arc length $= 5 \times 2.214 = 11.1$ A1 **[3]**5 A circle $S$ has centre at the point $( 3,1 )$ and passes through the point $( 0,5 )$.
\begin{enumerate}[label=(\roman*)]
\item Find the radius of $S$ and hence write down its cartesian equation.
\item (a) Determine the two points on $S$ where the $y$-coordinate is twice the $x$-coordinate.\\
(b) Calculate the length of the minor arc joining these two points.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 Q5 [9]}}