NIMCET 2023 — Mathematics PYQ

$(\mathbf{a})$

Let the sides of the triangle are a, b, c.

It is given that the perimeter of a triangle ABC is 6 times the Arithmetic Mean ofthe sines of its angles

$$\therefore \mathrm{a}+\mathrm{b}+\mathrm{c}=6\left(\frac{\sin \mathrm{A}+\sin \mathrm{B}+\sin \mathrm{C}}{3}\right)$$

$$\mathbf{a}+\mathbf{b}+\mathbf{c}=2(\sin \mathbf{A}+\sin \mathbf{B}+\sin \mathbf{C})......(1)$$
From the law ofsine,

$$\frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\sin \mathrm{B}}=\frac{\mathrm{c}}{\sin \mathrm{C}}=\mathrm{k}$$

$\Rightarrow$a=k$\sin$A $\Rightarrow$b=k$\sin$B $\Rightarrow$c=k$\sin$C

$\therefore$a+b+c=k$\left(\sin \text{A}+\sin \text{B}+\sin \text{C}\right).........(2)$

Hence k$=2\Rightarrow$a$=2\sin$A$\Rightarrow 1=2\sin$A$\Rightarrow \sin$A$=\frac{1}{2}$

$$\mathrm{A}=\frac{\pi }{6}$$