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.
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
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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