| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Geometric properties with circles |
| Difficulty | Moderate -0.3 This is a multi-part question combining graph reading, basic circle geometry, and geometric reasoning. Part (i) involves straightforward graphical solutions; part (ii) is routine substitution (x=0) into a circle equation; part (iii) requires identifying center/radius from standard form and making a simple geometric observation about tangency. While it spans multiple techniques, each step is standard C1 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
11 There is an insert for use in this question.
The graph of $y = x + \frac { 1 } { x }$ is shown on the insert. The lowest point on one branch is $( 1,2 )$. The highest point on the other branch is $( - 1 , - 2 )$.
\begin{enumerate}[label=(\roman*)]
\item Use the graph to solve the following equations, showing your method clearly.
$$\text { (A) } x + \frac { 1 } { x } = 4$$
$$\text { (B) } 2 x + \frac { 1 } { x } = 4$$
\item The equation $( x - 1 ) ^ { 2 } + y ^ { 2 } = 4$ represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the $y$-axis.
\item State the radius and the coordinates of the centre of this circle.
Explain how these can be used to deduce from the graph that this circle touches one branch of the curve $y = x + \frac { 1 } { x }$ but does not intersect with the other.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C1 2007 Q11 [12]}}