# Question 4:

## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $V = \int_a^{3a} \pi(kx)^2\,dx$ | M1 | For $\int \ldots x^2\,dx$ |
| $= \pi\left[\frac{k^2x^3}{3}\right]_a^{3a} \left(= \frac{26\pi k^2 a^3}{3}\right)$ | A1 | For $\frac{k^2x^3}{3}$ |
| $V\bar{x} = \int \pi xy^2\,dx = \int_a^{3a}\pi x(kx)^2\,dx$ | M1 | For $\int xy^2\,dx$ |
| $= \pi\left[\frac{k^2x^4}{4}\right]_a^{3a} \left(= 20\pi k^2 a^4\right)$ | A1 | For $\frac{k^2x^4}{4}$ |
| $\bar{x} = \frac{20\pi k^2 a^4}{\frac{26}{3}\pi k^2 a^3}$ | M1 | Dependent on previous M1M1 |
| $= \frac{30a}{13}$ | A1 | Allow $2.3a$ |

**OR (large cone minus small cone method):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $mx + (M-m)z = MX$ | | M3 complete method based on large cone minus small cone |
| $x = \frac{3}{4}a$ and $X = \frac{3}{4}(3a)$ | | A1 |
| $M = 27m$ | | A1 |
| $z = 30a/13$ | | A1 |

## Part (b)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $A = \int_1^8 16(1-x^{-\frac{1}{3}})\,dx$ | M1 | For $\int 16(1-x^{-\frac{1}{3}})\,dx$ |
| $= \left[16\left(x - \frac{3}{2}x^{\frac{2}{3}}\right)\right]_1^8 \,(= 40)$ | A1 | For $16\left(x-\frac{3}{2}x^{\frac{2}{3}}\right)$ |
| $A\bar{x} = \int xy\,dx = \int_1^8 16x(1-x^{-\frac{1}{3}})\,dx$ | M1 | For $\int xy\,dx$ |
| $= \left[8x^2 - \frac{48}{5}x^{\frac{5}{3}}\right]_1^8\,(= 206.4)$ | A1 | For $8x^2 - \frac{48}{5}x^{\frac{5}{3}}$ |
| $\bar{x} = \frac{206.4}{40} = 5.16$ | A1 | |
| $A\bar{y} = \int \frac{1}{2}y^2\,dx = \int_1^8 \frac{1}{2}\{16(1-x^{-\frac{1}{3}})\}^2\,dx$ | M1 | For $\int \ldots y^2\,dx$ |
| $= \left[128\left(x - 3x^{\frac{2}{3}} + 3x^{\frac{1}{3}}\right)\right]_1^8\,(= 128)$ | A1 | For $128(x-3x^{\frac{2}{3}}+3x^{\frac{1}{3}})$ |
| $\bar{y} = \frac{128}{40} = 3.2$ | A1 | |

## Part (b)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $5\binom{x}{y} + 35\binom{5}{3} = 40\binom{5.16}{3.2}$ | M1, M1 | CM of composite body; Correct strategy including signs; (One coordinate sufficient) |
| $\binom{x}{y} = \binom{6.28}{4.6}$ | A1, A1 | FT requires $1<x<8$; FT requires $0<y<8$ |

I don't see any mark scheme content in the image you've shared. The image only shows the back cover/contact page of an OCR examination document, containing:

- OCR's postal address (1 Hills Road, Cambridge, CB1 2EU)
- Customer Contact Centre details
- Phone/fax/email information
- Company registration details
- Copyright notice (© OCR 2012)

There is **no mark scheme content** (no questions, answers, mark allocations, or guidance notes) visible in this image.

Could you please share the actual mark scheme pages? They would typically show question numbers, model answers, mark types (M1, A1, B1, etc.), and examiner guidance.