JEE Main 2009 — Calculus

$\int_0^\pi[\cot x] d x,[\bullet]$ denotes the greatest integer function, is equal to

Given $P(x)=x^4+a x^3+b x^2+c x+d$ such that $x=0$ is the only real root of $P^{\prime}(x)=0$. If $P(-1) < P(1)$, then in the interval $[-1,1]$

Let $f(x)=x|x|$ and $g(x)=\sin x$. Statement-1 : gof is differentiable at $x=0$ and its derivative is continuous at that point. Statement-2 : gof is twice differentiable at $x=0$.

Let $y$ be an implicit function of $x$ defined by $x^{2 x}-2 x^x \cot y-1=0$. Then $y^{\prime}(1)$ equals

The area of the region bounded by the parabola $(y-2)^2=x-1$, the tangent to the parabola at the point $(2,3)$ and the $x$-axis is

The shortest distance between the line $\mathrm{y}-\mathrm{x}=1$ and the curve $\mathrm{x}=\mathrm{y}^2$ is