NIMCET 2007 — Mathematics PYQ

Step 1: Determine the angles of the triangle

Let the angles of the triangle $ABC$ be $A$, $B$, and $C$. It is given that these angles are in an Arithmetic Progression (A.P.).

We can assume the three angles to be:

$$B-d,B,B+d$$

(where $d$ is the common difference)

By the angle sum property of a triangle, the sum of all interior angles must be $180^{\circ }$:

$$(B-d)+B+(B+d)=180^{\circ }$$

$$3B=180^{\circ }$$

$$B=60^{\circ }$$

Step 2: Apply the Sine Rule

According to the Sine Rule for any triangle $ABC$:

$$\frac{b}{\sin B}=\frac{c}{\sin C}$$

Rearranging the terms to use the given ratio $b:c=\sqrt{3}:\sqrt{2}$:

$$\frac{\sin B}{\sin C}=\frac{b}{c}=\frac{\sqrt{3}}{\sqrt{2}}$$

Substitute the value of $B=60^{\circ }$ into the equation:

$$\frac{\sin 60^{\circ }}{\sin C}=\frac{\sqrt{3}}{\sqrt{2}}$$

We know that $\sin 60^{\circ }=\frac{\sqrt{3}}{2}$:

$$\frac{\frac{\sqrt{3}}{2}}{\sin C}=\frac{\sqrt{3}}{\sqrt{2}}$$

$$\frac{1}{2\sin C}=\frac{1}{\sqrt{2}}$$

$$\sin C=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{3}\text{ or rather }}\frac{1}{\sqrt{2}}$$

This gives two possible principal values for angle $C$:

$$C=45^{\circ }\text{or}C=135^{\circ }$$

Step 3: Find the value of Angle $A$

Since the sum of all angles is $180^{\circ }$:

$$A+B+C=180^{\circ }$$

• Case 1: If $C=45^{\circ }$

  $$A+60^{\circ }+45^{\circ }=180^{\circ }$$

  $$A+105^{\circ }=180^{\circ }$$

  $$A=75^{\circ }$$

• Case 2: If $C=135^{\circ }$

  $$A+60^{\circ }+135^{\circ }=180^{\circ }$$

  $$A+195^{\circ }=180^{\circ }⟹A=-15^{\circ }$$

  (An interior angle of a triangle cannot be negative, so this case is rejected).

Thus, the exact value of angle $A$ is $75^{\circ }$.