The value of $a$ for which the equations $x^3+ax+1=0$ and $x^4+ax^2+1=0$ have a common root is

If the points $(1,2)$ and $(3,4)$ lie on the same side of the straight line $3x - 5y + a = 0$,then $a$ lies in the set

Read the following five statements ($a$ to $e$) and select the option with all correct statements:
$(a)$ Mosses and Lichens are the first organisms to colonise a bare rock.
$(b)$ $Selaginella$ is a homosporous pteridophyte.
$(c)$ Coralloid roots in $Cycas$ have $VAM$ mycorrhizae.
$(d)$ Main plant body in bryophytes is gametophytic,whereas in pteridophytes it is sporophytic.
$(e)$ In gymnosperms,male and female gametophytes are present within sporangia located on sporophyte.

$A$ long wire carries a steady current. It is bent into a circle of one turn and the magnetic field at the centre of the coil is $B$. It is then bent into a circular loop of $n$ turns. The magnetic field at the centre of the coil will be

If $u=\sin ^{-1}\left(\frac{x}{y}\right)+\tan ^{-1}\left(\frac{y}{x}\right)$,then the value of $x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}$ is

If $x = \frac{1}{2} \left( \sqrt{3} + \frac{1}{\sqrt{3}} \right)$,then the value of $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to