The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $R = \sqrt{3}$ is

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in $sq.m$) of the flowerbed is

Let $f(x) = x^3 + px + 1$ and consider the following three statements:
$(i)$ For $p \geqslant 0$,$f(x) = 0$ has one negative root and $f(x)$ is monotonic.
$(ii)$ For $-1 < p < 0$,$f(x) = 0$ has one negative root and $f(x)$ is non-monotonic.
$(iii)$ For $p < -3/\sqrt[3]{4}$,$f(x) = 0$ has three real and distinct roots.
Which of the following is correct?

For all $x \in \mathbb{R}$, the minimum value $\frac{1}{3}$ and the maximum value $3$ of $f(x) = \frac{x^2+x+1}{x^2-x+1}$ occur at $l$ and $m$ respectively. Then $l+m$ is equal to:

Let $f(x) = \alpha x^2 - 2 + \frac{1}{x}$,where $\alpha$ is a real constant. The smallest $\alpha$ for which $f(x) \geq 0$ for all $x > 0$ is

The least value of $\alpha \in R$ for which $4 \alpha x^2 + \frac{1}{x} \geq 1$,for all $x > 0$,is