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.

• A. A-V, B-III, C-I, D-II
• B. A-IV, B-I, C-II, D-III
• C. A-V, B-II, C-III, D-I
• D. A-IV, B-II, C-V, D-III

Hint:

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.

Answer 1

The Correct Option is D

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).