Question

If the roots of the equation \[ 6x^3 - 11x^2 + 6x - 1 = 0 \] are in harmonic progression, then the roots of \[ x^3 - 6x^2 + 11x - 6 = 0 \] will be in

• A. Geometric Progression
• B. Arithmetic Progression
• C. Harmonic Progression
• D. Arithmetico-Geometric Progression

Hint:

For harmonic progression, the reciprocals of the terms are in arithmetic progression. This property can be used to determine the nature of the roots in equations involving harmonic progression.

Answer 1

The Correct Option is B

Step 1: Given that the roots of the first equation \( 6x^3 - 11x^2 + 6x - 1 = 0 \) are in harmonic progression (HP), we know that the roots \( r_1, r_2, r_3 \) of this equation satisfy the property of harmonic progression. In a harmonic progression, the reciprocals of the roots are in arithmetic progression (AP).
Step 2: The roots of the second equation \( x^3 - 6x^2 + 11x - 6 = 0 \) will follow the same pattern, and hence the roots of this equation must be in arithmetic progression (AP).
Step 3: Therefore, the correct answer is that the roots of the second equation will be in Arithmetic Progression.