Question

If \(\alpha, \beta, \gamma\) are the roots of the equation \[ x^3 - 13x^2 + kx + 189 = 0 \] such that \(\beta - \gamma = 2\), then find the ratio \(\beta + \gamma : k + \alpha\).

• A. \(4 : 3\)
• B. \(2 : 1\)
• C. \(6 : 5\)
• D. \(3 : 4\)

Hint:

Use Vieta's formulas and given conditions to find relations between roots and coefficients.

Answer 1

The Correct Option is A

Step 1: Use relations between roots and coefficients
Sum of roots: \[ \alpha + \beta + \gamma = 13 \] Sum of products of roots two at a time: \[ \alpha \beta + \beta \gamma + \gamma \alpha = k \] Product of roots: \[ \alpha \beta \gamma = -189 \] Step 2: Use condition \(\beta - \gamma = 2\)
Use this to express \(\beta\) in terms of \(\gamma\) and substitute into relations. Step 3: Solve for required ratio
After substitution and simplification, ratio \(\beta + \gamma : k + \alpha = 4 : 3\).