CAIE FP1 2010 November — Question 3 5 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2010
SessionNovember
Marks5
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Mark schemeDownload PDF ↗
TopicCentre of Mass 2
TypeCentre of mass of lamina by integration
DifficultyChallenging +1.2 This is a standard centroid calculation requiring integration of y and y²/2 over the given region. While it involves algebraic manipulation of surds and multiple integration steps, the method is routine for Further Maths students and the bounds are straightforward. The algebra is slightly more involved than basic examples but follows a well-practiced template.
Spec1.08f Area between two curves: using integration4.08e Mean value of function: using integral
3 A finite region \(R\) in the \(x - y\) plane is bounded by the curve with equation \(y = \sqrt { } x - \frac { 1 } { \sqrt { } x }\), the \(x\)-axis between \(x = 1\) and \(x = 4\), and the line \(x = 4\). Find the exact value of the \(y\)-coordinate of the centroid of \(R\).
AnswerMarks Guidance
\(\text{Area} = \int_1^4\left(x^2 - x^{-\frac{1}{2}}\right)dx = \left[\frac{2}{3}x^3 - 2x^{\frac{1}{2}}\right]_1^4 = \frac{8}{3}\)B1
\(\bar{y} = \frac{\frac{1}{2}\int_1^4(x - 2 + \frac{1}{x})dx}{\text{Area}} = \frac{\frac{1}{2}\left[\frac{x^2}{2} - 2x + \ln x\right]_1^4}{\text{Area}}\)M1 use of \(\frac{1}{2}\int y^2dx\) over \(A\)
M1integrate
A1correct
\(\text{Final answer: } \frac{3}{8}\left(\ln 2 + \frac{3}{4}\right)\) or \(\frac{3}{16}\left(\ln 4 + \frac{3}{2}\right)\) or \(\frac{3}{8}\ln 2 + \frac{9}{32}\) etc (ACF)A1 
$\text{Area} = \int_1^4\left(x^2 - x^{-\frac{1}{2}}\right)dx = \left[\frac{2}{3}x^3 - 2x^{\frac{1}{2}}\right]_1^4 = \frac{8}{3}$ | B1

$\bar{y} = \frac{\frac{1}{2}\int_1^4(x - 2 + \frac{1}{x})dx}{\text{Area}} = \frac{\frac{1}{2}\left[\frac{x^2}{2} - 2x + \ln x\right]_1^4}{\text{Area}}$ | M1 | use of $\frac{1}{2}\int y^2dx$ over $A$
| | M1 | integrate
| | A1 | correct

$\text{Final answer: } \frac{3}{8}\left(\ln 2 + \frac{3}{4}\right)$ or $\frac{3}{16}\left(\ln 4 + \frac{3}{2}\right)$ or $\frac{3}{8}\ln 2 + \frac{9}{32}$ etc (ACF) | A1 | | [5]

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3 A finite region $R$ in the $x - y$ plane is bounded by the curve with equation $y = \sqrt { } x - \frac { 1 } { \sqrt { } x }$, the $x$-axis between $x = 1$ and $x = 4$, and the line $x = 4$. Find the exact value of the $y$-coordinate of the centroid of $R$.

\hfill \mbox{\textit{CAIE FP1 2010 Q3 [5]}}
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