Question

If the roots of quadratic equation x² + px + q = 0 are tan30° and tan15° respectively, then the value of 2 + q - p is:

Answer 1

Correct Answer - Option 3 : 3

Concept:

If the quadratic equation is Ax² + Bx + C = 0 and their roots are α and β. Then,

α + β = \(\frac{-B}{A}\)

αβ = \(\frac{C}{A}\)

We also know that,

tan(A+B) = \(\frac{tanA~+~tanB}{1~-~tanA.tanB}\)

Given:

By comparing eqaution and roots,

A = 1, B = p, C = q.

α = tan30°, β = tan15°, 

Then,

tan30° + tan15° = \(\frac{-B}{A}\) = \(\frac{-p}{1}\) = -p

tan30°.tan15° = \(\frac{C}{A}\) = q

Now,

tan(A + B) = \(\frac{tanA~+~tanB}{1~-~tanA.tanB}\)

tan(30 + 15) = \(\frac{-p}{1~-~q}\) = 1

Then,

1 - q = -p

q - p = 1

So,

2 + q - p = 2 + 1 = 3