NIMCET 2010 — Mathematics PYQ

Solution:

1. Recall the Sine Rule:

In any triangle $ABC$, the relationship between the sides and the circumradius $R$ is given by:

From this, we can express the sides in terms of $R$ and the angles:

• $a=2R\sin A$

• $b=2R\sin B$

• $c=2R\sin C$

2. Substitute into the Given Equation:

The problem states that:

Substituting the expressions for $a,b,$ and $c$:

3. Simplify the Equation:

Divide both sides by $4R^2$ (assuming $R\ne 0$):

4. Use Trigonometric Identities:

Using the identity ${\sin }^2\theta =\frac{1-\cos 2\theta }{2}$:

Using the sum-to-product formula $\cos X+\cos Y=2\cos \frac{X+Y}{2}\cos \frac{X-Y}{2}$:

Since $A+B+C=\pi$, then $\cos (A+B)=\cos (\pi -C)=-\cos C$.

Substitute $C=\pi -(A+B)$ back into the bracket:

5. Conclusion:

For the product to be zero, at least one factor must be zero:

This implies that one of the angles ($A,B,$ or $C$) must be $90^{\circ }$ (or $\frac{\pi }{2}$).

Therefore, the triangle $ABC$ is Right angled.

Correct Option: (c)