Question:

With respect to the roots of the equation \( 3x^3 + bx^2 + bx + 3 = 0 \), match the items of List-I with those of List-II.

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For cubic equations, the sign and nature of the roots depend on the coefficient conditions. Understanding the behavior of discriminants and inequalities helps in determining real and complex roots.
Updated On: Mar 15, 2025
  •  A-V, B-III, C-I, D-II

  • A-IV, B-I, C-II, D-III

  • A-V, B-II, C-III, D-I

  • A-IV, B-II, C-V, D-III

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The Correct Option is D

Solution and Explanation


Step 1: Understanding the Given Equation 
The equation given is a cubic equation: \[ 3x^3 + bx^2 + bx + 3 = 0 \] We analyze different conditions based on the values of \( b \):
- All roots are negative: This happens when \( b = 9 \), which corresponds to IV.
- Two roots are complex: This occurs in the range \( -3<b<9 \), corresponding to II.
- Two roots are positive: This occurs when \( b = -3 \), which corresponds to V.
- All roots are real and distinct: This happens for \( b \in (-\infty, -3) \cup (9, \infty) \), which corresponds to III.
Step 2: Evaluating the Given Options 
- Option (1): Incorrect, as incorrect matches are present.
- Option (2): Incorrect, as incorrect matches are present.
- Option (3): Incorrect, as incorrect matches are present.
- Option (4): Correct, as all matches are accurate.
Thus, the correct answer is 

Option (4)

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