OCR MEI AS Paper 2 2020 November — Question 7 8 marks

Exam BoardOCR MEI
ModuleAS Paper 2 (AS Paper 2)
Year2020
SessionNovember
Marks8
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Mark schemeDownload PDF ↗
TopicCircles
TypeLine-circle intersection points
DifficultyModerate -0.3 This is a straightforward application of standard techniques: find the line equation, substitute into the circle equation, and solve the resulting quadratic. While it requires multiple steps (4-5 marks typical), each step is routine and commonly practiced. The algebra is manageable and no geometric insight is needed, making it slightly easier than average.
Spec1.02c Simultaneous equations: two variables by elimination and substitution1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03d Circles: equation (x-a)^2+(y-b)^2=r^2
7 In this question you must show detailed reasoning. A circle has centre \(( 2 , - 1 )\) and radius 5. A straight line passes through the points \(( 1,1 )\) and \(( 9,5 )\).
Find the coordinates of the points of intersection of the line and the circle.
Question 7:
AnswerMarks Guidance
AnswerMarks Guidance
Circle: equation is \((x-2)^2 + (y+1)^2 = 5^2\) oeM1, A1 (1.1) Allow sign error, reversed 1 and \(-2\), 5 not squared
Line: \(m = \frac{1}{2}\)B1 (1.1)
\(y - 5 = \left(\text{their } \frac{1}{2}\right)(x-9)\) oeM1 (1.1a) Or \(y - 1 = \left(\text{their } \frac{1}{2}\right)(x-1)\). Sight of \(y = \frac{1}{2}(x+1)\)
Substitution of their \(y = \frac{1}{2}(x+1)\) in their attempt at equation of circleM1 (3.1a) Or their \(x = 2y - 1\)
\(5x^2 - 10x - 75 = 0\) oeA1 (1.1) Or \(5y^2 - 10y - 15 = 0\) oe; \(x^2 - 2x - 15 = 0\)
\(x = 5\) or \(x = -3\)A1 (1.1) Or \(y = 3\) and \(y = -1\)
\((5, 3)\) and \((-3, -1)\)A1 (2.5) If A0A0, sc1 for \((5,3)\) or \((-3,-1)\) only
[8] 
## Question 7:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Circle: equation is $(x-2)^2 + (y+1)^2 = 5^2$ oe | M1, A1 (1.1) | Allow sign error, reversed 1 and $-2$, 5 not squared |
| Line: $m = \frac{1}{2}$ | B1 (1.1) | |
| $y - 5 = \left(\text{their } \frac{1}{2}\right)(x-9)$ oe | M1 (1.1a) | Or $y - 1 = \left(\text{their } \frac{1}{2}\right)(x-1)$. Sight of $y = \frac{1}{2}(x+1)$ |
| Substitution of their $y = \frac{1}{2}(x+1)$ in their attempt at equation of circle | M1 (3.1a) | Or their $x = 2y - 1$ |
| $5x^2 - 10x - 75 = 0$ oe | A1 (1.1) | Or $5y^2 - 10y - 15 = 0$ oe; $x^2 - 2x - 15 = 0$ |
| $x = 5$ or $x = -3$ | A1 (1.1) | Or $y = 3$ and $y = -1$ |
| $(5, 3)$ and $(-3, -1)$ | A1 (2.5) | If A0A0, sc1 for $(5,3)$ or $(-3,-1)$ only |
| **[8]** | | |
7 In this question you must show detailed reasoning.
A circle has centre $( 2 , - 1 )$ and radius 5.

A straight line passes through the points $( 1,1 )$ and $( 9,5 )$.\\
Find the coordinates of the points of intersection of the line and the circle.

\hfill \mbox{\textit{OCR MEI AS Paper 2 2020 Q7 [8]}}
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