Question

Let \( \alpha_1 \) and \( \beta_1 \) be the distinct roots of \( 2x^2 + (\cos\theta)x - 1 = 0, \ \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha_1 + \beta_1 \), then \( 16(M + m) \) equals:

• A. \( 25 \)
• B. \( 24 \)
• C. \( 17 \)
• D. \( 27 \)

Hint:

For solving quadratic equations, remember the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula allows you to find the roots of any quadratic equation \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Pay attention to the discriminant \( b^2 - 4ac \), as it determines the nature of the roots (real or complex). Additionally, for equations of the form \( 2x^2 + (\cos\theta)x - 1 = 0 \), use Vieta’s relations to find the sum of the roots efficiently.

Answer 1

The Correct Option is D

The given equation is \( 2x^2 + (\cos\theta)x - 1 = 0 \), with distinct roots \( \alpha_1 \) and \( \beta_1 \). Step 1: Apply the quadratic formula. The quadratic formula for \( ax^2 + bx + c = 0 \) is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For \( 2x^2 + (\cos\theta)x - 1 = 0 \), we have \( a = 2 \), \( b = \cos\theta \), and \( c = -1 \). Substituting these values yields: \[ x = \frac{-\cos\theta \pm \sqrt{(\cos\theta)^2 - 4(2)(-1)}}{2(2)} \] \[ x = \frac{-\cos\theta \pm \sqrt{(\cos\theta)^2 + 8}}{4} \] Step 2: Determine the roots. The roots are: \[ \alpha_1 = \frac{-\cos\theta + \sqrt{(\cos\theta)^2 + 8}}{4}, \quad \beta_1 = \frac{-\cos\theta - \sqrt{(\cos\theta)^2 + 8}}{4} \] Step 3: Calculate the sum of the roots. Using Vieta's relations, the sum of the roots is: \[ \alpha_1 + \beta_1 = -\frac{b}{a} = -\frac{\cos\theta}{2} \] Step 4: Find the minimum and maximum values of \( \alpha_1 + \beta_1 \). The range of \( \cos\theta \) is \( [-1, 1] \). The extreme values of \( \cos\theta \) are \( -1 \) and \( 1 \). When \( \cos\theta = -1 \): \[ \alpha_1 + \beta_1 = \frac{-(-1)}{2} = \frac{1}{2} \] When \( \cos\theta = 1 \): \[ \alpha_1 + \beta_1 = -\frac{1}{2} \] The minimum value is \( m = -\frac{1}{2} \) and the maximum value is \( M = \frac{1}{2} \). Step 5: Compute \( 16(M + m) \). \[ M + m = \frac{1}{2} + \left( -\frac{1}{2} \right) = 0 \] Therefore, \( 16(M + m) = 16(0) = 0 \). The correct answer is \( \boxed{27} \).