Integral Calculus & Area · JEE Advanced previous year questions

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

1. A.$\frac{17}{2}$
2. B.$\frac{17}{3}$
3. C.$\frac{17}{4}$
4. D.$\frac{17}{5}$

Let $f:[1,\infty )\to \mathbb{R}$ be a differentiable function such that $f(1)=\frac{1}{3}$ and $$3{\int }_1^{x}f(t)dt=xf(x)-\frac{x^3}{3},x\in [1,\infty ).$$ Let $e$ denote the base of the natural logarithm. Then the value of $f(e)$ is

1. A.$\frac{e^2+4}{3}$
2. B.$\frac{{\log }_{e}4+e}{3}$
3. C.$\frac{4e^2}{3}$
4. D.$\frac{e^2-4}{3}$

For $x\in \mathbb{R}$, let ${\tan }^{-1}(x)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the minimum value of the function $$f:\mathbb{R}\to \mathbb{R}\text{ defined by }f(x)={\int }_0^{x{\tan }^{-1}x}\frac{e^{(t-\cos t)}}{1+t^{2023}}dt$$ is ____.