| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Geometric properties with circles |
| Difficulty | Moderate -0.8 This is a multi-part question requiring graph reading, basic circle equation manipulation, and geometric reasoning. Part (i) involves reading values from a given graph (routine skill). Part (ii) is standard substitution (x=0) into a circle equation. Part (iii) requires identifying center/radius from standard form and making a simple geometric observation about tangency, but no algebraic verification of tangency is needed. All techniques are straightforward C1-level skills with no novel problem-solving required. |
| 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 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0.2\) to \(0.3\) and \(3.7\) to \(3.8\) | \(1+1\) | [tol. 1mm or 0.05 throughout qn]; if 0, allow M1 for drawing down lines at both values |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x + \frac{1}{x} = 4 - x\) | M1 | condone one error |
| their \(y = 4 - x\) drawn | M1 | allow M2 for plotting positive branch of \(y = 2x + 1/x\) [plots at \((1,3)\) and \((2, 4.5)\)] and above other graph] or for plot of \(y = 2x^2 - 4x + 1\) |
| \(0.2\) to \(0.35\) and \(1.65\) to \(1.8\) | B2 | 1 each |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((0, \pm\sqrt{3})\) | condone \(y = \pm\sqrt{3}\) isw; 1 each or M1 for \(1 + y^2 = 4\) or \(y^2 = 3\) o.e. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| centre \((1, 0)\) radius \(2\) | \(1+1\) | allow seen in (ii) |
| touches at \((1, 2)\) [which is distance 2 from centre] | 1 | allow ft for both these marks for centre at \((-1, 0)\), rad 2; allow 2 for good sketch or compass-drawn circle of rad 2 centre \((\pm 1, 0)\) |
| all points on other branch \(> 2\) from centre | 1 |
## Question 4:
**Part i:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.2$ to $0.3$ and $3.7$ to $3.8$ | $1+1$ | [tol. 1mm or 0.05 throughout qn]; if 0, allow M1 for drawing down lines at both values |
**Part iB:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x + \frac{1}{x} = 4 - x$ | M1 | condone one error |
| their $y = 4 - x$ drawn | M1 | allow M2 for plotting positive branch of $y = 2x + 1/x$ [plots at $(1,3)$ and $(2, 4.5)$] and above other graph] or for plot of $y = 2x^2 - 4x + 1$ |
| $0.2$ to $0.35$ and $1.65$ to $1.8$ | B2 | 1 each |
**Part ii:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(0, \pm\sqrt{3})$ | | condone $y = \pm\sqrt{3}$ isw; 1 each or M1 for $1 + y^2 = 4$ or $y^2 = 3$ o.e. |
**Part iii:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| centre $(1, 0)$ radius $2$ | $1+1$ | allow seen in (ii) |
| touches at $(1, 2)$ [which is distance 2 from centre] | 1 | allow ft for both these marks for centre at $(-1, 0)$, rad 2; allow 2 for good sketch or compass-drawn circle of rad 2 centre $(\pm 1, 0)$ |
| all points on other branch $> 2$ from centre | 1 | |4 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.\\
(A) $x + \frac { 1 } { x } = 4$\\
(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 Q4 [12]}}