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JEE Advanced 2010Paper 1

Exam: JEE Advanced
Year: 2010
Paper: Paper 1
Subject: Mathematics
Chapter: Properties of Triangles

JEE Advanced 2010 · Paper 1 · Mathematics · Properties of Triangles

If the angles A, B and C of a triangle are in an arithmetic...

If the angles

A, B

and

C

of a triangle are in an 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

\frac{a}{c} \sin 2 C+\frac{c}{a} \sin 2 A

is

Options

A $\frac{1}{2}$
B $\frac{\sqrt{3}}{2}$
C1
D $\sqrt{3}$

Correct answer

D. $\sqrt{3}$

Step-by-step solution

Since,

A, B, C

are in

\mathrm{AP}

\begin{aligned}& \Rightarrow \quad 2 B=A+C \text{ ie, } \angleB=60^{{}^{\circ}} \\ & \therefore \frac{a}{c}(2 \sin C \cos C)+\frac{c}{a}(2 \sin A \cos A) \\ & =2 k(a \cos C+c \cos A) \\ & \quad\left[\begin{array}{l} \text{ using, }\\ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=\frac{1}{k} \end{array}\right]\\ & =2 k(b) \\ & =2 \sin B \end{aligned}

=$\sqrt{3} \text{ [using, }b=a \cos C+c \cos A]

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