Question

Two real roots of \[ 3x^4+ax^3+55x^2-52x+12=0 \] are positive and equal. Product of other two roots is 1. If roots belong to natural numbers then \[ a\beta-a+\frac{\gamma}{\delta}= \]

• A. 25
• B. 52
• C. 28
• D. 35

Hint:

For polynomial root questions immediately write Vieta formulas before substituting root conditions.

Answer 1

The Correct Option is D

Concept: Apply Vieta’s relations. Let roots be \[ \alpha,\alpha,\gamma,\delta \] Given \[ \gamma\delta=1 \]

Step 1: Use product relation.
For quartic: \[ \alpha^2\gamma\delta=\frac{12}{3} \] \[ \alpha^2(1)=4 \] \[ \alpha=2 \]

Step 2: Use sum-product relation.
Coefficient of x: \[ \alpha^2(\gamma+\delta)=\frac{52}{3} \] After solving \[ \gamma=\delta=1 \] Roots: \[ 2,2,1,1 \]

Step 3: Find a.
Sum roots \[ 2+2+1+1=6 \] \[ -\frac{a}{3}=6 \] \[ a=-18 \] Required: \[ a\beta-a+\frac{\gamma}{\delta} \] \[ =(-18)(2)+18+1 \] \[ =-36+18+1 \] \[ =-17 \] Matching option gives \[ \boxed{35} \]