OCR MEI Paper 2 2023 June — Question 6 4 marks

Exam BoardOCR MEI
ModulePaper 2 (Paper 2)
Year2023
SessionJune
Marks4
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Mark schemeDownload PDF ↗
TopicParametric curves and Cartesian conversion
TypeConvert to Cartesian (sin/cos identities)
DifficultyEasy -1.2 This is a straightforward application of the identity cos²θ + sin²θ = 1. Students rearrange to get (x+3)/2 and (y-1)/2, square and add them to eliminate the parameter. It's a standard textbook exercise requiring only routine algebraic manipulation with no problem-solving insight needed.
Spec1.03g Parametric equations: of curves and conversion to cartesian
6 The parametric equations of a circle are \(x = 2 \cos \theta - 3\) and \(y = 2 \sin \theta + 1\).
Determine the cartesian equation of the circle in the form \(( \mathrm { x } - \mathrm { a } ) ^ { 2 } + ( \mathrm { y } - \mathrm { b } ) ^ { 2 } = \mathrm { k }\), where \(a , b\) and \(k\) are integers.
Question 6:
AnswerMarks Guidance
\(2\cos\theta = x + 3\) or \(\cos\theta = \frac{x+3}{2}\)B1 (2.1)
\(2\sin\theta = y - 1\) or \(\sin\theta = \frac{y-1}{2}\)B1 (1.1)
\(\left(\frac{x+3}{2}\right)^2 + \left(\frac{y+1}{2}\right)^2 = \cos^2\theta + \sin^2\theta\) or \((x \pm 3)^2 + (y \pm 1)^2 = 4\cos^2\theta + 4\sin^2\theta\) oeM1 (1.1) allow sign errors in their expressions for \(\sin\theta\) and \(\cos\theta\); allow if just see brackets expanded, but must be 3 terms in each case
\((x+3)^2 + (y-1)^2 = 4\)A1 (1.1) allow SC2 for \((x+3)^2 + (y-1)^2 = 4\) unsupported
*Alternatively:*
AnswerMarks Guidance
centre of circle is \((-3, 1)\)B1
radius is \(2\)B1
\((x+3)^2 + (y-1)^2 = 2^2\)M1 allow one sign error in bracket
\((x+3)^2 + (y-1)^2 = 4\)A1 
## Question 6:
$2\cos\theta = x + 3$ or $\cos\theta = \frac{x+3}{2}$ | B1 (2.1) |
$2\sin\theta = y - 1$ or $\sin\theta = \frac{y-1}{2}$ | B1 (1.1) |
$\left(\frac{x+3}{2}\right)^2 + \left(\frac{y+1}{2}\right)^2 = \cos^2\theta + \sin^2\theta$ or $(x \pm 3)^2 + (y \pm 1)^2 = 4\cos^2\theta + 4\sin^2\theta$ oe | M1 (1.1) | allow sign errors in their expressions for $\sin\theta$ and $\cos\theta$; allow if just see brackets expanded, but must be 3 terms in each case
$(x+3)^2 + (y-1)^2 = 4$ | A1 (1.1) | allow SC2 for $(x+3)^2 + (y-1)^2 = 4$ unsupported

*Alternatively:*
centre of circle is $(-3, 1)$ | B1 |
radius is $2$ | B1 |
$(x+3)^2 + (y-1)^2 = 2^2$ | M1 | allow one sign error in bracket
$(x+3)^2 + (y-1)^2 = 4$ | A1 |
6 The parametric equations of a circle are\\
$x = 2 \cos \theta - 3$ and $y = 2 \sin \theta + 1$.\\
Determine the cartesian equation of the circle in the form $( \mathrm { x } - \mathrm { a } ) ^ { 2 } + ( \mathrm { y } - \mathrm { b } ) ^ { 2 } = \mathrm { k }$, where $a , b$ and $k$ are integers.

\hfill \mbox{\textit{OCR MEI Paper 2 2023 Q6 [4]}}
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