NIMCET 2023 — Mathematics PYQ

Step 1: Write down the given mathematical conditions

Let the sides of the triangle $\Delta ABC$ be $a,b,c$ and their corresponding opposite angles be $A,B,C$.

• The perimeter of the triangle is:

  $$\text{Perimeter}=a+b+c$$

• The arithmetic mean (AM) of the sines of its angles is:

  $$\text{Arithmetic Mean}=\frac{\sin A+\sin B+\sin C}{3}$$

According to the given condition in the problem:

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

Simplifying the equation:

$$a+b+c=2(\sin A+\sin B+\sin C)$$

--- (Equation 1)

Step 2: Apply the Sine Rule

From the standard Sine Rule for any triangle, we have:

$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$$

Where $R$ is the circumradius of the triangle. From this relation, we can express the sides in terms of their angles:

$$a=2R\sin A,b=2R\sin B,c=2R\sin C$$

Substitute these values into Equation 1:

$$2R\sin A+2R\sin B+2R\sin C=2(\sin A+\sin B+\sin C)$$

$$2R(\sin A+\sin B+\sin C)=2(\sin A+\sin B+\sin C)$$

Since the sum of the sines of the angles of a valid triangle is non-zero ($\sin A+\sin B+\sin C\ne 0$), we can divide both sides by $2(\sin A+\sin B+\sin C)$:

$$R=1$$

Step 3: Solve for Angle $A$

We are given that the length of side $a=1$.

Using the Sine Rule relation for side $a$:

$$a=2R\sin A$$

Substitute $a=1$ and $R=1$ into the equation:

$$1=2(1)\sin A$$

$$\sin A=\frac{1}{2}$$

Since $A$ is an interior angle of a triangle, its value must look for primary principal angles. Checking the standard options:

$$A=\frac{\pi }{6}(30^{\circ })$$

Correct Answer:

Correct Option: A