Q188 / 270 · NDA 2015 (Phase 2) · Mathematics

Consider a triangle ABC satisfying $2 \mathrm{a} \sin ^2\left(\frac{\mathrm{C}}{2}\right)+2 \mathrm{c} \sin ^2\left(\frac{\mathrm{~A}}{2}\right)=2 \mathrm{a}+2 \mathrm{c}-3 \mathrm{~b}$ $\sin \mathrm{A}, \sin \mathrm{B}, \sin \mathrm{C}$ are in

  • A
    G.P.
  • Correct answer
    B
    A.P.
  • C
    H.P.
  • D
    Neither in G.P. nor in A.P nor in H.P.

Solution

As we have already proven $2 b=a+c$ as- $\begin{aligned} & \mathrm{a}=\mathrm{R} \sin \mathrm{~A} \\ & \mathrm{~b}=\mathrm{R} \sin \mathrm{~B} \\ & \mathrm{c}=\mathrm{R} \sin \mathrm{C} \\ & \Rightarrow 2(\mathrm{R} \sin \mathrm{~B})=\mathrm{R} \sin \mathrm{~A}+\mathrm{R} \sin \mathrm{C} \\ & \Rightarrow 2 \mathrm{R}(\sin \mathrm{~B})=\mathrm{R}(\sin \mathrm{~A}+\sin \mathrm{C}) \\ & 2 \sin \mathrm{~B}=\sin \mathrm{A}+\sin \mathrm{C} \end{aligned}$

Hence, the correct answer is option: 'B'.

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