Question

If the roots of the equation \( 4x^3 - 12x^2 + 11x + m = 0 \) are in arithmetic progression, then \( m = {?} \)

• A. \( -3 \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 3 \)

Hint:

For cubic equations with roots in arithmetic progression, use Vieta's formulas to relate the sum, product, and sum of products of the roots to the coefficients. This can help determine unknowns like \( m \).

Answer 1

The Correct Option is A

We are given the cubic equation: \[ 4x^3 - 12x^2 + 11x + m = 0, \] and the roots of this equation are in arithmetic progression. Let the roots of the equation be \( \alpha - d \), \( \alpha \), and \( \alpha + d \), where \( \alpha \) is the middle root and \( d \) is the common difference. 
Step 1:  By Vieta's formulas, we know the following relationships between the roots and the coefficients of the cubic equation \( ax^3 + bx^2 + cx + d = 0 \): - The sum of the roots is \( -\frac{b}{a} \), - The sum of the products of the roots taken two at a time is \( \frac{c}{a} \), 
- The product of the roots is \( -\frac{d}{a} \). For the equation \( 4x^3 - 12x^2 + 11x + m = 0 \), we have \( a = 4 \), \( b = -12 \), \( c = 11 \), and \( d = m \). 
Step 2: From Vieta's formulas: 1. The sum of the roots is: \[ (\alpha - d) + \alpha + (\alpha + d) = 3\alpha = -\frac{-12}{4} = 3. \] Thus, \( \alpha = 1 \). 2. The sum of the products of the roots taken two at a time is: \[ (\alpha - d)\alpha + \alpha(\alpha + d) + (\alpha - d)(\alpha + d) = \alpha^2 - \alpha d + \alpha^2 + \alpha d + \alpha^2 - d^2 = 3\alpha^2 - d^2. \] Using Vieta’s formula: \[ 3\alpha^2 - d^2 = \frac{11}{4}. \] Substitute \( \alpha = 1 \): \[ 3(1)^2 - d^2 = \frac{11}{4} \quad \Rightarrow \quad 3 - d^2 = \frac{11}{4}. \] Solving for \( d^2 \): \[ d^2 = 3 - \frac{11}{4} = \frac{12}{4} - \frac{11}{4} = \frac{1}{4}, \] \[ d = \pm \frac{1}{2}. \] 
Step 3: The product of the roots is: \[ (\alpha - d)\alpha(\alpha + d) = \alpha(\alpha^2 - d^2). \] Using Vieta’s formula: \[ \alpha(\alpha^2 - d^2) = -\frac{m}{4}. \] Substitute \( \alpha = 1 \) and \( d^2 = \frac{1}{4} \): \[ 1(1^2 - \frac{1}{4}) = -\frac{m}{4} \quad \Rightarrow \quad 1 - \frac{1}{4} = -\frac{m}{4}. \] This simplifies to: \[ \frac{3}{4} = -\frac{m}{4}. \] Multiplying both sides by 4: \[ 3 = -m \quad \Rightarrow \quad m = -3. \]
Conclusion: Thus, the value of \( m \) is \( -3 \).