Quadratic Equations
Trigonometric Coefficients
JEE Advanced 2014
Grade 11
Question:
Let $-\pi/6 < \theta < -\pi/12$. Suppose $\alpha_1, \beta_1$ are roots of $x^2 - 2x \sec \theta + 1 = 0$ and $\alpha_2, \beta_2$ are roots of $x^2 + 2x \tan \theta - 1 = 0$ with $\alpha_1 > \beta_1$, $\alpha_2 > \beta_2$. Then $\alpha_1 + \beta_2$ equals:
(1) $2(\sec \theta - \tan \theta)$
(2) $2 \sec \theta$
(3) $-2 \tan \theta$
(4) $0$
Step-by-Step Solution
Key Concept: Solve each equation explicitly using the quadratic formula. For the given range of $\theta$, determine the signs of $\tan \theta$ and $\sec \theta$ to correctly identify which root is larger in each pair.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)