Which of the following is a fossil plant?

The differential equation of the family of parabolas with focus at the origin and the $x$-axis as axis is

The perimeter of a sector is a constant. If its area is to be maximum,the sectorial angle is:

Match the following:

$1$. Burning candle	$i$. Line spectrum
$2$. Sodium vapour	$ii$. Continuous spectrum
$3$. Bunsen flame	$iii$. Band spectrum
$4$. Dark lines in solar spectrum	$iv$. Absorption spectrum

Which of the following is concerned with eubacteria?
$(1)$ Methanogens $(2)$ Anabaena $(3)$ $N_2$ fixation $(4)$ Thermoacidophiles $(5)$ Eukaryotes $(6)$ Diatoms $(7)$ Gelatinous sheath $(8)$ Heterocyst

$\begin{aligned} & \frac{x^2+x+1}{(x-1)(x-2)(x-3)}=\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{x-3} \\ & \Rightarrow A+C= \end{aligned}$