2 questions from 2025
If \( \displaystyle \alpha = \int\limits_{\frac{1}{2}}^{2} \frac{\tan^{-1} x}{2x^2 - 3x + 2} \, dx\), then the value of \( \sqrt{7}\,\tan\!\left(\dfrac{2\alpha \sqrt{7}}{\pi}\right) \) is __________. (Here, the inverse trigonometric function \( \tan^{-1} x \) assumes values in \( \left( -\dfrac{\pi}{2} , \dfrac{\pi}{2} \right) \).)
Let \( \mathbb{R} \) denote the set of all real numbers. Then the area of the region \( \left\{ (x,y) \in \mathbb{R} \times \mathbb{R} : x > 0, \, y > \dfrac{1}{x}, \, 5x - 4y - 1 > 0, \, 4x + 4y - 17 < 0 \right\} \)
