Prickles of Rose are

$\int_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}$ is equal to :

Let $f: R \to R$,$f(x) = \max \{ |\tan^{-1} x|, \cot^{-1} x \}$. Consider the following statements:
$I$. The function is continuous and derivable $\forall x \in R$.
$II$. The range of the function is $[\frac{\pi}{4}, \pi]$.
$III$. $f(x)$ is many-one into.
Identify the correct option.

$P$ is a point on the segment joining the feet of two vertical poles of heights $a$ and $b$. The angles of elevation of the tops of the poles from $P$ are $45^{\circ}$ each. Then,the square of the distance between the tops of the poles is

Let $A = \{x \in R, x \neq 0, -4 \leq x \leq 4\}$ and $f: A \rightarrow R$ be defined by $f(x) = \frac{|x|}{x}$ for $x \in A$. Then,the range of $f$ is

If $P_1$ and $P_2$ are two points on the ellipse $\frac{x^2}{4} + y^2 = 1$ such that the tangents at these points are parallel to the chord joining the points $(0, 1)$ and $(2, 0)$,then the distance between $P_1$ and $P_2$ is: