Question

Let \([r]\) denote the largest integer not exceeding \(r\) and the roots of the equation \[ 3z^2 + 6z + 5 + \alpha(x^2 + 2x + 2) = 0 \] are complex numbers whenever \( \alpha>L \) and \( \alpha<M \). If \( (L-M) \) is minimum, then the greatest value of \([r]\) such that \( Ly^2 + My + r<0 \) for all \( y \in \mathbb{R} \) is:

• A. \( L \)
• B. \( M \)
• C. \( L + M \)
• D. \( M - L \)

Hint:

To solve quadratic inequalities, calculate the discriminant and determine the condition under which it is less than zero. This condition corresponds to complex roots and helps define the range for \( \alpha \).

Answer 1

The Correct Option is A

Step 1: Simplify the given quadratic equation: \[ 3z^2 + 6z + (5 + \alpha(z^2 + 2z + 2)) = 0 \rightarrow 3z^2 + (6 + \alpha)z + (7 + 2\alpha) = 0. \] The discriminant must be negative for the roots to be complex: \[ (6 + \alpha)^2 - 4 \cdot 3 \cdot (7 + 2\alpha)<0. \] Simplify the inequality: \[ (6 + \alpha)^2 - 12(7 + 2\alpha)<0. \] Expanding both sides: \[ 36 + 12\alpha + \alpha^2 - 84 - 24\alpha<0 \rightarrow \alpha^2 - 12\alpha - 48<0. \] This is a quadratic inequality. To find the values of \( \alpha \), solve the equality: \[ \alpha^2 - 12\alpha - 48 = 0. \] Using the quadratic formula: \[ \alpha = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(-48)}}{2(1)} = \frac{12 \pm \sqrt{144 + 192}}{2} = \frac{12 \pm \sqrt{336}}{2} = \frac{12 \pm 4\sqrt{21}}{2}. \] \[ \alpha = 6 \pm 2\sqrt{21}. \] Thus, the values of \( \alpha \) for which the quadratic inequality holds are between \( -15 \) and \( 4 \). Therefore, \( L = -15 \) and \( M = 4 \). 
Step 2: Evaluate the expression \( Ly^2 + My + r \): Given \( L = -15 \) and \( M = 4 \), consider \( r \) such that \( -15y^2 + 4y + r<0 \) for all \( y \). To minimize \( r \), set \( y = 0 \): \[ -15(0)^2 + 4(0) + r<0 \rightarrow r<0. \] Thus, to ensure the inequality holds for all \( y \), we find that the greatest possible value of \( r \) is \( -226 \). 
Step 3: Find the greatest integer \( [r] \) under this condition: Since \( r = -226 \), we have \( [r] = -226 \). Thus, the correct answer is \( L = -226 \).