OCR C2 2010 January — Question 5 7 marks

Exam BoardOCR
ModuleC2 (Core Mathematics 2)
Year2010
SessionJanuary
Marks7
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Mark schemeDownload PDF ↗
TopicAreas Between Curves
TypeTwo Curves Intersection Area
DifficultyStandard +0.3 This is a standard C2 area-between-curves question with intersection points given. Students must set up the integral of the difference between the curves from x=1 to x=3, integrate two straightforward functions (polynomial and x^{-2}), and evaluate. The setup is routine and the integration techniques are basic, making this slightly easier than average.
Spec1.08f Area between two curves: using integration
5 \includegraphics[max width=\textwidth, alt={}, center]{9362eb16-88c9-4279-97aa-907b4916b965-3_646_839_255_653} The diagram shows parts of the curves \(y = x ^ { 2 } + 1\) and \(y = 11 - \frac { 9 } { x ^ { 2 } }\), which intersect at \(( 1,2 )\) and \(( 3,10 )\). Use integration to find the exact area of the shaded region enclosed between the two curves.
AnswerMarks Guidance
\(\int_1^3 [(1-9x^2) - (x^2+1)]dx = [9x - \frac{1}{3}x^3 + 10x]_1^3\)M1 Attempt subtraction (correct order) at any point
\(= (3 - 9 + 30) - (9 - \frac{1}{3} + 10)\)M1 Attempt integration – inc. in power for at least one term
\(= 24 - 18\frac{1}{3} = 5\frac{1}{3}\)A1, M1 Obtain \(\pm(-\frac{1}{3}x^3 + 10x)\) or \(11x\) and \(\frac{1}{3}x^3 + x\); Obtain ± 9x⁻¹ or any unsimplified equiv
OR
AnswerMarks Guidance
\([\frac{11x + 9x - 1}{3}] - [\frac{1}{3}x^3 + x]\)M1 Use limits \(x = 1, 3\) – correct order & subtraction
\(= [(33+3) - (11+9)] - [(9+3) - (\frac{1}{3}+1)]\)A1 Obtain 5⅓, or exact equiv
\(= 16 - 10\frac{2}{3} = 5\frac{1}{3}\) 
| $\int_1^3 [(1-9x^2) - (x^2+1)]dx = [9x - \frac{1}{3}x^3 + 10x]_1^3$ | M1 | Attempt subtraction (correct order) at any point |
| $= (3 - 9 + 30) - (9 - \frac{1}{3} + 10)$ | M1 | Attempt integration – inc. in power for at least one term |
| $= 24 - 18\frac{1}{3} = 5\frac{1}{3}$ | A1, M1 | Obtain $\pm(-\frac{1}{3}x^3 + 10x)$ or $11x$ and $\frac{1}{3}x^3 + x$; Obtain ± 9x⁻¹ or any unsimplified equiv |

**OR**

| $[\frac{11x + 9x - 1}{3}] - [\frac{1}{3}x^3 + x]$ | M1 | Use limits $x = 1, 3$ – correct order & subtraction |
| $= [(33+3) - (11+9)] - [(9+3) - (\frac{1}{3}+1)]$ | A1 | Obtain 5⅓, or exact equiv |
| $= 16 - 10\frac{2}{3} = 5\frac{1}{3}$ | | |
5\\
\includegraphics[max width=\textwidth, alt={}, center]{9362eb16-88c9-4279-97aa-907b4916b965-3_646_839_255_653}

The diagram shows parts of the curves $y = x ^ { 2 } + 1$ and $y = 11 - \frac { 9 } { x ^ { 2 } }$, which intersect at $( 1,2 )$ and $( 3,10 )$. Use integration to find the exact area of the shaded region enclosed between the two curves.

\hfill \mbox{\textit{OCR C2 2010 Q5 [7]}}
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