NIMCET 2022 — Mathematics PYQ

To solve this problem, we use Napier's Analogy (also known as the Tangent Rule) and basic triangle angle properties.

1. Recall Napier's Analogy:

For any triangle $ABC$:

2. Use the Angle Sum Property:

In $△ABC$, $A+B+C=180^{\circ }$.

This means $A+B=180^{\circ }-C$.

Taking the tangent on both sides:

However, we need to relate this to $\frac{A}{2}$. Let's use $A=180^{\circ }-(B+C)$.

Then $\frac{A}{2}=90^{\circ }-\frac{B+C}{2}$, which implies $\cot \frac{A}{2}=\tan \frac{B+C}{2}$.

3. Analyze the given equation:

Given: $\tan \frac{1}{2}(B-C)=\frac{1}{3}\tan (A+B)$

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

Using Napier's Analogy:

Since $A+B+C=180^{\circ }$, then $\tan (A+B)=\tan (180^{\circ }-C)=-\tan C$.

Also, $\cot \frac{A}{2}=\tan \frac{B+C}{2}$.

The relationship simplifies significantly when we substitute the values:

4. Calculate the ratio $b:c$:

By cross-multiplication:

Conclusion:

The ratio of the sides opposite to angles $B$ and $C$ ($b:c$) is $2:1$.

Correct Option:

A) 2:1