NIMCET 2017 Question Paper with solutions

Let, the altitude from A , B , C are ${\mathit{h}}_1,{\mathit{h}}_2,{\mathit{h}}_3$ respectively.
We know that,
Area of triangle Δ = $\frac{1}{2}\mathit{a}{\mathit{h}}_1$ = $\frac{1}{2}\mathit{n}{\mathit{h}}_2$ = $\frac{1}{2}\mathit{b}{\mathit{h}}_3$
∴ a = $\frac{2\mathit{\Delta }}{{\mathit{h}}_1}$ , b = $\frac{2\mathit{\Delta }}{{\mathit{h}}_2}$ , c = $\frac{2\mathit{\Delta }}{{\mathit{h}}_3}$
Since, ${\mathit{h}}_1,{\mathit{h}}_2,{\mathit{h}}_3$ are in HP
⇒ $\frac{1}{{\mathit{h}}_1},\frac{1}{{\mathit{h}}_2},\frac{1}{{\mathit{h}}_3}$ are in AP
⇒ $\frac{2\mathit{\Delta }}{{\mathit{h}}_1},\frac{2\mathit{\Delta }}{{\mathit{h}}_2},\frac{2\mathit{\Delta }}{{\mathit{h}}_3}$ are in AP
⇒ a, b, c are in AP
⇒ k sin A, k sin B, k sin C are in AP
⇒ sin A, sin B, sin C are in AP.