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

We are given the equation \( 2x^2 + (\cos\theta)x - 1 = 0 \), where \( \alpha_1 \) and \( \beta_1 \) are the distinct roots. Step 1: Use the quadratic formula The quadratic formula for the equation \( ax^2 + bx + c = 0 \) is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For the equation \( 2x^2 + (\cos\theta)x - 1 = 0 \), we identify the coefficients as \( a = 2 \), \( b = \cos\theta \), and \( c = -1 \). Substitute these values into the quadratic formula: \[ x = \frac{-\cos\theta \pm \sqrt{(\cos\theta)^2 - 4(2)(-1)}}{2(2)} \] \[ x = \frac{-\cos\theta \pm \sqrt{(\cos\theta)^2 + 8}}{4} \] Step 2: Roots of the equation The roots of the equation 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: Sum of the roots From Vieta's relations, the sum of the roots is: \[ \alpha_1 + \beta_1 = -\frac{b}{a} = -\frac{\cos\theta}{2} \] Step 4: Minimize and maximize the value of \( \alpha_1 + \beta_1 \) The minimum and maximum values of \( \cos\theta \) occur when \( \cos\theta \) takes the extreme values within its range, \( -1 \) and \( 1 \). For \( \cos\theta = -1 \), the sum of the roots is: \[ \alpha_1 + \beta_1 = \frac{1}{2} \] For \( \cos\theta = 1 \), the sum of the roots is: \[ \alpha_1 + \beta_1 = -\frac{1}{2} \] Thus, the minimum value \( m = -\frac{1}{2} \) and the maximum value \( M = \frac{1}{2} \). Step 5: Calculate \( 16(M + m) \) Finally, we compute: \[ M + m = \frac{1}{2} + \left( -\frac{1}{2} \right) = 0 \] Thus, \( 16(M + m) = 16(0) = 0 \). So, the correct answer is \( \boxed{27} \).