If x = a(cos theta + theta sin theta), y = f( | vedclass

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(C) Given $x = a(\cos 	heta + 	heta \sin 	heta)$.First, find $\frac{dx}{d	heta}$:$\frac{dx}{d	heta} = a(-\sin 	heta + \sin 	heta + 	heta \cos

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$\frac{a}{2}$

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(C) Given $x = a(\cos 	heta + 	heta \sin 	heta)$.First, find $\frac{dx}{d	heta}$:$\frac{dx}{d	heta} = a(-\sin 	heta + \sin 	heta + 	heta \cos 	heta) = a	heta \cos 	heta$.We know that $\frac{dy}{dx} = \frac{dy/d	heta}{dx/d	heta} = \frac{	an 	heta}{	heta}$.So, $\frac{dy}{d	heta} = \frac{dx}{d	heta} \cdot \frac{	an 	heta}{	heta} = (a	heta \cos 	heta) \cdot \frac{\sin 	heta}{	heta \cos 	heta} = a \sin 	heta$.Now, integrate $\frac{dy}{d	heta}$ with respect to $	heta$ to find $f(	heta)$:$f(	heta) = \int a \sin 	heta \, d	heta = -a \cos 	heta + C$.Given $f(2\pi) = 0$, we have $-a \cos(2\pi) + C = 0 \implies -a(1) + C = 0 \implies C = a$.Thus, $f(	heta) = a - a \cos 	heta = a(1 - \cos 	heta)$.Now, calculate $f\left(\frac{\pi}{3}ight)$:$f\left(\frac{\pi}{3}ight) = a\left(1 - \cos \frac{\pi}{3}ight) = a\left(1 - \frac{1}{2}ight) = \frac{a}{2}$.

List-$I$	List-$II$
$(i)$ $x = a(\theta - \sin \theta), y = a(1 - \cos \theta)$	$(A)$ $4\sqrt{3}$
(ii) $x = 3\cos \theta - 2\cos^3 \theta, y = 3\sin \theta - 2\sin^3 \theta$	$(B)$ $-\frac{1}{3\sqrt{3}}$
(iii) $x = 3\cos \theta - \cos^3 \theta, y = 3\sin \theta - \sin^3 \theta$	$(C)$ $\sqrt{3}$
(iv) $x = a \log \sin \theta, y = a \tan \theta$	$(D)$ $\frac{1}{\sqrt{3}}$
	$(E)$ $\frac{1}{3\sqrt{3}}$

If $x = a(\cos \theta + \theta \sin \theta)$, $y = f(\theta)$, $f(2\pi) = 0$, $\frac{dy}{dx} = \frac{\tan \theta}{\theta}$, $\theta \neq 0$ and $\theta \neq (2n+1)\frac{\pi}{2}$, then $f\left(\frac{\pi}{3}\right) = $

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