| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Centre of mass of lamina by integration |
| Difficulty | Standard +0.3 This is a straightforward application of standard integration formulas for mean value and centroid. The function y = 2x^(1/2) integrates easily, and both parts require only direct substitution into memorized formulas with no conceptual challenges or problem-solving insight needed. |
| Spec | 1.08d Evaluate definite integrals: between limits4.08f Integrate using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\dfrac{\displaystyle\int_0^4 2x^{\frac{1}{2}}\,dx}{4}\) | M1 | Uses formula for mean value |
| \(= \left[\frac{1}{3}x^{\frac{3}{2}}\right]_0^4 = \frac{8}{3}\) | M1A1 | Integrates; part total 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\dfrac{\frac{1}{2}\displaystyle\int_0^4 4x\,dx}{\displaystyle\int_0^4 2x^{\frac{1}{2}}\,dx}\) | M1 | Uses formula for \(y\)-coordinate |
| \(= \dfrac{\left[x^2\right]_0^4}{\left[\frac{4}{3}x^{\frac{3}{2}}\right]_0^4} = \frac{16\times 3}{32} = \frac{3}{2}\) | M1A1 | Integrates; part total 3; Total [6] |
## Question 2(i):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\dfrac{\displaystyle\int_0^4 2x^{\frac{1}{2}}\,dx}{4}$ | M1 | Uses formula for mean value |
| $= \left[\frac{1}{3}x^{\frac{3}{2}}\right]_0^4 = \frac{8}{3}$ | M1A1 | Integrates; part total 3 |
## Question 2(ii):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\dfrac{\frac{1}{2}\displaystyle\int_0^4 4x\,dx}{\displaystyle\int_0^4 2x^{\frac{1}{2}}\,dx}$ | M1 | Uses formula for $y$-coordinate |
| $= \dfrac{\left[x^2\right]_0^4}{\left[\frac{4}{3}x^{\frac{3}{2}}\right]_0^4} = \frac{16\times 3}{32} = \frac{3}{2}$ | M1A1 | Integrates; part total 3; **Total [6]** |
---2 The curve $C$ has equation $y = 2 x ^ { \frac { 1 } { 2 } }$ for $0 \leqslant x \leqslant 4$. Find\\
(i) the mean value of $y$ with respect to $x$ for $0 \leqslant x \leqslant 4$,\\
(ii) the $y$-coordinate of the centroid of the region enclosed by $C$, the line $x = 4$ and the $x$-axis.
\hfill \mbox{\textit{CAIE FP1 2012 Q2 [6]}}