CAIE FP1 2019 November — Question 1 6 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2019
SessionNovember
Marks6
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Mark schemeDownload PDF ↗
TopicCentre of Mass 2
TypeCentre of mass of lamina by integration
DifficultyStandard +0.8 This is a standard centroid calculation by integration requiring setup of the formulas for x̄ and ȳ, evaluation of integrals involving x^a, and algebraic manipulation. While straightforward in method, it requires careful handling of the parameter 'a' throughout and is typical of Further Maths content, placing it moderately above average difficulty.
Spec4.08d Volumes of revolution: about x and y axes6.04d Integration: for centre of mass of laminas/solids
1 The curve \(C\) has equation \(y = x ^ { a }\) for \(0 \leqslant x \leqslant 1\), where \(a\) is a positive constant. Find, in terms of \(a\), the coordinates of the centroid of the region enclosed by \(C\), the line \(x = 1\) and the \(x\)-axis.
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(A = \int_0^1 x^a \, dx = \left[\frac{x^{a+1}}{a+1}\right]_0^1 = \frac{1}{a+1}\)B1 Finds area of region
\(A\bar{x} = \int_0^1 xy \, dx = \int_0^1 x^{a+1} \, dx = \left[\frac{x^{a+2}}{a+2}\right]_0^1 = \frac{1}{a+2}\)M1 A1 Finds \(\int_0^1 xy \, dx\)
\(2A\bar{y} = \int_0^1 y^2 \, dx = \int_0^1 x^{2a} \, dx = \left[\frac{x^{2a+1}}{2a+1}\right]_0^1 = \frac{1}{2a+1}\)M1 A1 Finds \(\int_0^1 y^2 \, dx\)
\((\bar{x}, \bar{y}) = \left(\frac{a+1}{a+2}, \frac{a+1}{2(2a+1)}\right)\)A1 Both coordinates correct
6 
## Question 1:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $A = \int_0^1 x^a \, dx = \left[\frac{x^{a+1}}{a+1}\right]_0^1 = \frac{1}{a+1}$ | **B1** | Finds area of region |
| $A\bar{x} = \int_0^1 xy \, dx = \int_0^1 x^{a+1} \, dx = \left[\frac{x^{a+2}}{a+2}\right]_0^1 = \frac{1}{a+2}$ | **M1 A1** | Finds $\int_0^1 xy \, dx$ |
| $2A\bar{y} = \int_0^1 y^2 \, dx = \int_0^1 x^{2a} \, dx = \left[\frac{x^{2a+1}}{2a+1}\right]_0^1 = \frac{1}{2a+1}$ | **M1 A1** | Finds $\int_0^1 y^2 \, dx$ |
| $(\bar{x}, \bar{y}) = \left(\frac{a+1}{a+2}, \frac{a+1}{2(2a+1)}\right)$ | **A1** | Both coordinates correct |
| | **6** | |

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1 The curve $C$ has equation $y = x ^ { a }$ for $0 \leqslant x \leqslant 1$, where $a$ is a positive constant. Find, in terms of $a$, the coordinates of the centroid of the region enclosed by $C$, the line $x = 1$ and the $x$-axis.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q1 [6]}}
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