UPSC NDA I 2024 Mathematics Paper

The slopes of the regression lines are given by the coefficients of x in the equations. So, the slopes are −1∕2 and −2∕3.
Let θ be the acute angle between the lines. Then, we have:
tan‌θ=|‌

‌ −1 2 − −2 3

1+‌ −1 2 ⋅‌ −2 3

|=‌

1

8

‌. ‌
Using the triple angle formula for tangent, we get:
tan‌3‌θ=‌

3‌tan‌θ−tan3θ

1−3tan2θ

Substituting tan‌θ=‌

1

8

, we get:
tan‌3‌θ=‌

3⋅ 1 8 −( 1 8 )3

1−3(‌ 1 8 )2

=‌

191

488

Therefore, 488‌tan‌3‌θ=191.