# Question 6(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dx}{d\theta}=a(1-\cos\theta)$ or $a-a\cos\theta$; $\frac{dy}{d\theta}=a\sin\theta$ | B1 | At least one correct derivative |
| $a^2(1-\cos\theta)^2+(a\sin\theta)^2 = a^2(1-2\cos\theta+\cos^2\theta+\sin^2\theta) = 2a^2(1-\cos\theta)$ | M1 | Squares and adds derivatives, uses $\cos^2\theta+\sin^2\theta=1$ to get expression in $\cos\theta$ only |
| $= 2a^2\!\left(1-\!\left(1-2\sin^2\!\tfrac{\theta}{2}\right)\right) = 4a^2\sin^2\!\tfrac{\theta}{2}$ | dM1 A1 | dM1: Replaces $\cos\theta$ with $\pm1\pm2\sin^2\!\frac{\theta}{2}$ or equivalent to get expression in $\sin^2\!\frac{\theta}{2}$ only; requires previous M; A1: Achieves $4a^2\sin^2\!\frac{\theta}{2}$ or $k=4$ from correct work |

**(4)**

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# Question 6(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{S.A.}=(2\pi)\int y\sqrt{\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2}\,d\theta = (2\pi)\int_0^{2\pi} a(1-\cos\theta)\!\left(2a\sin\tfrac{\theta}{2}\right)d\theta$ | M1 | Applies $y\sqrt{\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2}$ with $ka^2\sin^2\!\frac{\theta}{2}$ and square roots; result of square root may be incorrect but must be of form $p\sin\frac{\theta}{2}$; allow slip replacing $y$ but must not have used $x$, $\frac{dx}{d\theta}$ or $\frac{dy}{d\theta}$ for $y$ |
| $=(2\pi)2a^2\int_0^{2\pi}\!\left(\sin\tfrac{\theta}{2}-\sin\tfrac{\theta}{2}\cos\theta\right)d\theta$ → $(2\pi)2a^2\int_0^{2\pi}2\sin^3\!\tfrac{\theta}{2}\,d\theta$ | dM1 | Uses trig identity/identities (condoning sign errors) to obtain expression with arguments of $\frac{\theta}{2}$ only; dependent on previous M mark |

## Question 6(b) cont.:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $S = 8\pi a^2\left[-2\cos\frac{\theta}{2}+\frac{2}{3}\cos^3\frac{\theta}{2}\right]_{(0)}^{(2\pi)}$ or e.g. $\pi a^2\left[-16\cos\frac{\theta}{2}+\frac{16}{3}\cos^3\frac{\theta}{2}\right]_{(0)}^{(2\pi)}$ | A1ft | Correct expression ignoring limits for their numerical $k$; if piecemeal, correct expression for their $k$ when integration completed |
| $= 8\pi a^2\left[\left(-2\cos\frac{2\pi}{2}+\frac{2}{3}\cos^3\frac{2\pi}{2}\right)-\left(-2\cos 0+\frac{2}{3}\cos^3 0\right)\right]$ | ddM1 | Substitutes correct limits, subtracts either way; requires previous M marks; must have used $2\pi$; insist on limits applied on all integrations if carried out separately |
| $=\frac{64}{3}\pi a^2$ | A1 | Correct exact answer; accept equivalent fractions |
| All marks available regardless of how $k=4$ was obtained | **(5)** | **Total 9** |

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