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JEE Advance | PYQ

PARAGRAPH – 2
Let \( f : \left[0, \frac{\pi}{2}\right] \to [0,1] \) be the function defined by \( f(x) = \sin^{2} x \) and let \( g : \left[0, \frac{\pi}{2}\right] \to [0, \infty) \) be the function defined by \( g(x) = \sqrt{\frac{\pi x}{2} - x^{2}} \).
The value of \( 2 \int_{0}^{\pi/2} f(x) g(x) \, dx - \int_{0}^{\pi/2} g(x) \, dx \) is ____________.

PARAGRAPH – 2
Let \( f : \left[0, \frac{\pi}{2}\right] \to [0,1] \) be the function defined by \( f(x) = \sin^{2} x \) and let \( g : \left[0, \frac{\pi}{2}\right] \to [0, \infty) \) be the function defined by \( g(x) = \sqrt{\frac{\pi x}{2} - x^{2}} \).
The value of \( \dfrac{16}{\pi^{3}} \int_{0}^{\pi/2} f(x) g(x) \, dx \) is ____________.

Let \( S=\{(x,y)\in \mathbb{R}\times \mathbb{R}: \ge 0,\, y\ge 0,\, y^{2}\le 4x,\, y^{2}\le 12-2x \text{ and } 3y+\sqrt{8}\,x\le 5\sqrt{8}\} \). If the are of the region \( S \) is \( \alpha \sqrt{2} \), then \( \alpha \) is equal to:

Let the function \( f : [1, \infty) \to \mathbb{R} \) be defined by

Define \( g(x) = \int_{1}^{x} f(t) dt, \ x \in (1, \infty) \). Let \( \alpha \) denote the number of solutions of the equation \( g(x) = 0 \) in the interval \( (1, 8] \) and \( \beta = \displaystyle \lim_
{x \to 1^{+}} \dfrac{g(x)}{x - 1} \). Then the value of \( \alpha + \beta \) is equal to __________.