| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circles |
| Type | Point position relative to circle |
| Difficulty | Moderate -0.8 This is a straightforward C1 circle question testing basic understanding of circle equations. Part (i) requires finding the maximum y-value (centre y-coordinate plus radius), part (ii) involves substituting into the circle inequality, and part (iii) requires simultaneous equations showing no real solutions. All parts use standard techniques with clear methods and no problem-solving insight required. |
| Spec | 1.02i Represent inequalities: graphically on coordinate plane1.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| i) \(y\) coordinate of the centre is \(-5\); Radius = \(5\); Centre is five units below \(x\) axis and radius is five, so just touches the \(x\)-axis | B1, B1, B1 [3] | Correct \(y\) value. Correct radius. Correct explanation based on the above – allow clear diagram www |
| ii) \(CP^2 = (6-2)^2 + (k+5)^2\); \(CP^2 < 25 \Rightarrow 16 + k^2 + 10k + 25 < 25\); \(k^2 + 10k + 16 < 0\); \((k+2)(k+8) < 0\); \(-8 < k < -2\) | M1, A1, A1, M1, A1 [5] | Attempt CP² = r² or "CP" Puts \(x = 6\) to into equation of circle. Correct three term quadratic equation*, could be in terms of \(y\). A1. Correct method to establish quadratic has no roots e.g. considers value of \(b^2 - 4ac\), tries to find roots from quadratic formula. Must be strict inequalities for the A mark. * Or \((k+5)^2 = 9\); SC Trial and improvement. B2 if final answer correct (B1 if inequalities are not strict) |
| iii) \((2y-2)^2 + (y+5)^2 = 25\); \(5y^2 + 2y + 4 = 0\); \(b^2 - 4ac = 4 - 4 \times 5 \times 4 = -76 < 0\) so line and circle do not meet | M1*, A1, M1dep*, A1 [4] | Attempts to eliminate \(x\) or \(y\) from equation of circle. Correct three term quadratic obtained. Correct method to establish quadratic has no roots e.g. considers value of \(b^2 - 4ac\), tries to find roots from quadratic formula. Correct clear conclusion www AG. Can only get 5/5 if fully explained. If \(y\) eliminated: \(5y^2 + 4x + 16 = 0\); \(b^2 - 4ac = 16 - 4 \times 5 \times 16 = -304\). No marks for purely graphical attempts. |
**i)** $y$ coordinate of the centre is $-5$; Radius = $5$; Centre is five units below $x$ axis and radius is five, so just touches the $x$-axis | B1, B1, B1 [3] | Correct $y$ value. Correct radius. Correct explanation based on the above – allow clear diagram www
**ii)** $CP^2 = (6-2)^2 + (k+5)^2$; $CP^2 < 25 \Rightarrow 16 + k^2 + 10k + 25 < 25$; $k^2 + 10k + 16 < 0$; $(k+2)(k+8) < 0$; $-8 < k < -2$ | M1, A1, A1, M1, A1 [5] | Attempt CP² = r² or "CP" Puts $x = 6$ to into equation of circle. Correct three term quadratic equation*, could be in terms of $y$. A1. Correct method to establish quadratic has no roots e.g. considers value of $b^2 - 4ac$, tries to find roots from quadratic formula. Must be strict inequalities for the A mark. * Or $(k+5)^2 = 9$; SC Trial and improvement. B2 if final answer correct (B1 if inequalities are not strict)
**iii)** $(2y-2)^2 + (y+5)^2 = 25$; $5y^2 + 2y + 4 = 0$; $b^2 - 4ac = 4 - 4 \times 5 \times 4 = -76 < 0$ so line and circle do not meet | M1*, A1, M1dep*, A1 [4] | Attempts to eliminate $x$ or $y$ from equation of circle. Correct three term quadratic obtained. Correct method to establish quadratic has no roots e.g. considers value of $b^2 - 4ac$, tries to find roots from quadratic formula. Correct clear conclusion www AG. Can only get 5/5 if fully explained. If $y$ eliminated: $5y^2 + 4x + 16 = 0$; $b^2 - 4ac = 16 - 4 \times 5 \times 16 = -304$. No marks for purely graphical attempts.A circle with centre $C$ has equation $(x - 2)^2 + (y + 5)^2 = 25$.
\begin{enumerate}[label=(\roman*)]
\item Show that no part of the circle lies above the $x$-axis. [3]
\item The point $P$ has coordinates $(6, k)$ and lies inside the circle. Find the set of possible values of $k$. [5]
\item Prove that the line $2y = x$ does not meet the circle. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 2014 Q9 [12]}}