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# If the angles $A,B,C$ of a $\Delta$le are in arithmetic progression and if $a,b$ and $c$ denote the lengths of the sides opposite to $A,B$ and $C$ respectively,then the value of the expression $\large\frac{a}{c}$$\sin 2C+\large\frac{c}{a}$$\sin 2A$ is

$(a)\;\large\frac{1}{2}$$\qquad(b)\;\large\frac{\sqrt 3}{2}$$\qquad(c)\;1\qquad(d)\;\sqrt 3$

Since $A,B,C$ are in AP
$\Rightarrow 2B=A+C$
(i.e) $\sqrt B=60^{\large\circ}$
$\therefore \large\frac{a}{c}$$(2\sin C\cos C)+\large\frac{C}{a}$$(2\sin A\cos A)=2k(a\cos C+C\cos A)$
$\big[Using\;\large\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{C}{\sin C}=\frac{1}{k}\big]$
$\Rightarrow 2k(b)$
[Using $b=a\cos C+c\cos A$]
$\Rightarrow 2\sin B$
$\Rightarrow \sqrt 3$
Hence (d) is the correct answer.